LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
1 1. LAPLACIAN AND HARMONIC FORMS ON COMPACT COMPLEX MANIFOLDS
Problem 1. Let Xbe a compact complex manifold of complex dimension n. Consider a (p, p)-
form αon Xsuch that ∆α= 0, where ∆is the Laplacian operator. Given that αis harmonic,
compute the dimension of the space of harmonic (p, p)-forms on X.
Solution 1. Since αis a harmonic (p, p)-form on X, it satisfies ∆α= 0. By Hodge theory, we
know that the space of harmonic (p, p)-forms on a compact complex manifold of complex dimension
nis given by the Ker(∆), i.e., the kernel of the Laplacian operator.
Therefore, the space of harmonic (p, p)-forms on Xis the same as the space of solutions to the
equation ∆β= 0, where βis a (p, p)-form.
Since the Laplacian is a second-order elliptic operator, Ker(∆) is finite-dimensional. Thus, the
dimension of the space of harmonic (p, p)-forms on Xis the same as the dimension of the kernel
of ∆. Let’s denote this dimension as hp,p.
Therefore, dim(Ker(∆)) = hp,p.
Problem 2. On a compact complex manifold Xof complex dimension n, let ωbe a closed
(1,1)-form with ∆ω= 0. Show that ωis also harmonic.
Solution 2. A(p, p)-form αis harmonic if and only if it satisfies ∆α= 0.
Given that ωis a closed (1,1)-form with ∆ω= 0, we know that ωis a solution to the harmonic
equation. Therefore, ωis a harmonic (1,1)-form on the compact complex manifold Xof complex
dimension n.
2 2. EXCEPTIONAL SETS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 2. Consider a complex manifold Mof complex dimension n. Let ∆be the Laplacian
operator on M. Suppose that on M, there exists a harmonic 2-form ωand a harmonic 3-form η. If
∆ω= 3ωand ∆η= 5η, find the complex dimension of M.
Solution 2. To find the complex dimension of M, we note that the Laplacian operator acting
on a k-form on a complex manifold of complex dimension nis given by ∆=2¯
∂∂ + 2∂¯
∂.
Given that ∆ω= 3ω, we have 2¯
∂∂ω + 2∂¯
∂ω = 3ω. Similarly, for η, we have 2¯
∂∂η + 2∂¯
∂η = 5η.
a) Since ωis a harmonic 2-form and ∆ω= 3ω, we can write the equation as:
2¯
∂∂ω + 2∂¯
∂ω = 3ω
Comparing the terms, we get 2¯
∂∂ω =∂¯
∂ω and 2∂¯
∂ω =¯
∂∂ω. This means that ωis a holomorphic
2-form.
b) Since ηis a harmonic 3-form and ∆η= 5η, we have:
2¯
∂∂η + 2∂¯
∂η = 5η
Again comparing the terms, we find 2¯
∂∂η =∂¯
∂η and 2∂¯
∂η =¯
∂∂η. This implies that ηis a
holomorphic 3-form.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
c) Since ωis a holomorphic 2-form on the complex manifold M, its harmonic representative
must satisfy the Dolbeault lemma for harmonic forms. This lemma states that on a Kähler manifold,
the Dolbeault cohomology classes are the same as the harmonic cohomology classes, and hence
harmonic forms are closed and coclosed, implying ¯
∂ω = 0 and ¯
∂∗ω= 0.
Similarly, since ηis a holomorphic 3-form on M, it follows the same rules for holomorphicity
and harmonicity. Therefore, using the Dolbeault lemma, we can show that the complex dimension
of Mis 3, as ωis a holomorphic 2-form and ηis a holomorphic 3-form.
Therefore, the complex dimension of Mis 3.
3 3. REGULARITY RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 3. Consider a complex manifold Mwith a Kähler metric g, and let ∆denote the
Laplacian operator on (p, q)-forms.
Suppose that αis a (1,0)-form on Msuch that ∆α= 0.
a) Show that if αis harmonic, i.e., ∆α= 0 and ¯
∂α = 0, then αis a holomorphic (1,0)-form.
b) Let βbe a (0,1)-form on Msuch that ∆β= 0. Show that if βis harmonic and closed, i.e.,
∆β= 0 and ¯
∂β = 0, then βis a holomorphic (0,1)-form.
Solution 3.
a) To show that αis a holomorphic (1,0)-form, we need to show that ¯
∂α = 0.
Given that ∆α= 0, we have ∆α= ∆0,1α+ ∆1,0α= (∂¯
∂+¯
∂∂)α=¯
∂∂α +∂¯
∂α = 0.
Since αis harmonic, ¯
∂α = 0. Hence, αis a holomorphic (1,0)-form.
b) Similar to part (a), we need to show that βis a holomorphic (0,1)-form, i.e., ¯
∂β = 0.
From ∆β= 0, we have ∆β= ∆0,1β+ ∆1,0β=¯
∂∂β +∂¯
∂β = 0.
Since βis harmonic and closed, we have ¯
∂β = 0. Hence, βis a holomorphic (0,1)-form.
4 4. SOLVABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 4. Consider a complex manifold Mwith a (1,0)-form α=dz defined on an open set
U⊆Msuch that ∆α=−∂2α
∂z∂ ¯z= 0. Compute the harmonic part of α.
Solution 4. a) The Laplacian of a (1,0)-form α=dz is given by ∆α=−∂2α
∂z∂ ¯z. Since ∆α= 0,
this implies that ∂2α
∂z∂ ¯z= 0.
b) The harmonic part of a (1,0)-form α=dz is given by αharmonic =1
2(α+¯
∂¯
∂α +∂∂ ¯α). Since
∂2α
∂z∂ ¯z= 0, we have ¯
∂¯
∂α = 0 and ∂∂ ¯α= 0. Therefore, the harmonic part simplifies to αharmonic =
1
2α=1
2dz.
c) Thus, the harmonic part of the (1,0)-form α=dz is αharmonic =1
2dz.
5 5. EXISTENCE OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 5. Consider a complex manifold Mequipped with a Hermitian metric g. Let ωbe a
closed (1,1)-form on Mand ∆be the Laplace operator on (1,1)-forms defined as ∆ = −dd∗−d∗d,
where d∗is the formal adjoint of the exterior derivative with respect to g. Determine whether ωis
harmonic on M.
Solution 5.
To determine if ωis harmonic, we need to check if ∆ω= 0.
The Laplacian operator ∆acting on a (1,1)-form ωis given by
∆ω=−dd∗ω−d∗dω.
To check if ωis harmonic, we need to compute ∆ωand see if it vanishes.
a) First, compute dω:
dω=d(ω¯
jkdz¯
j∧dzk)=(∂pω¯
jk)dzp∧dz¯
j∧dzk.
b) Next, compute d∗ω:
d∗ω=− ∗ d(∗ω) = − ∗ d(−iω) = − ∗ (−idzj∧ω¯
jkdzk) = − ∗ (−iω¯
jkdzj∧dzk).
c) Finally, compute ∆ω:
∆ω=−dd∗ω−d∗dω=−d(−i∗ω)−d∗(−iω) = −d(dzj∧ω¯
jkdzk) + ∗(∂pω¯
jk)dzp∧dzj∧dzk.
Now, substitute the expressions for dωand d∗ωinto the above equation and simplify. If ∆ω= 0,
then ωis harmonic on M.
6 6. UNIQUENESS OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 6. Let Mbe a complex manifold with a Kähler metric g. Consider the 1-form α=
x dy −y dx on M, where xand yare complex-valued functions on M.
a) Compute the Laplacian ∆αof αwith respect to the metric g.
b) Determine whether the 1-form αis harmonic on Mwith respect to g.
Solution 6.
a) The Laplacian of a 1-form αwith respect to the Kähler metric gis given by ∆α=−∇∗∇α,
where ∇is the covariant derivative associated with gand ∇∗is its formal adjoint. Let’s first compute
∇α:
∇α=dα + (∇ · α)
=d(x dy −y dx)
=dx ∧dy −dy ∧dx
= 2dx ∧dy.
Next, we compute ∆α=−∇∗∇α. Since αis a 1-form, ∆αwill be a 2-form. We have:
∇∗∇α=−∇∗(2dx ∧dy)
=−∇∗2dx ∧dy
=−∂(2)
∂x dx +∂(2)
∂y dy∧dy
= 0.
Therefore, ∆α= 0.
b) To determine if αis harmonic on M, we need to check if ∆α= 0. Since we found in part a)
that ∆α= 0, the 1-form αis harmonic on Mwith respect to the Kähler metric gon M.
7 7. NON-COMPACT COMPLEX MANIFOLDS AND LAPLACIAN AND HARMONIC FORMS
Problem 7. Consider a non-compact complex manifold Xwith a Kähler metric ggiven by
g=i
2X
j,k
gj¯
kdzj∧d¯zk,
where zjare local coordinates on X. Let ∆denote the Laplacian operator associated with the
metric g.
Given a (p, q)-form αon Xsuch that ∆α= 0, prove the following statements:
a) If αis harmonic, i.e., ∆α= 0, show that ¯
∂α = 0.
b) If ¯
∂α = 0, show that ∆α= 0.
Solution 7.
a) Let α=αj1,...,jp,¯
k1,...,¯
kqdzj1∧. . . ∧dzjp∧d¯zk1∧. . . ∧d¯zkqbe a harmonic (p, q)-form, i.e.,
∆α= 0. The Laplacian of a (p, q)-form is given by
∆α=−(∂¯
∂+¯
∂∂)α.
Since ∆α= 0, we have ¯
∂∂α = 0. This implies ¯
∂α = 0, as desired.
b) Given ¯
∂α = 0, we need to show that ∆α= 0. From part a), we know that if αis harmonic,
then ¯
∂α = 0. So, given ¯
∂α = 0, it follows that αis harmonic and hence ∆α= 0.
8 8. HARMONICITY CRITERIA FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX
MANIFOLDS
Problem 8. Let Xbe a complex manifold and αbe a (1,1)-form on Xdefined by α=i¯
∂∂f for
some smooth function fon X. Determine whether αis harmonic or not.
Solution 8. Given that α=i¯
∂∂f, we can write ∆α= ∆(i¯
∂∂f), where ∆is the Laplacian
operator.
a) First, we find the Laplacian of α:
∆(i¯
∂∂f) = i¯
∂∂(∆f)(since Laplacian commutes with exterior derivatives)
=i¯
∂∂(0) (since ∆f= 0)
= 0.
b) Since ∆α= 0, we conclude that αis a harmonic form on the complex manifold X.
Therefore, the (1,1)-form α=i¯
∂∂f is harmonic on the complex manifold X.
9 9. NORMAL FORMS FOR LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANI-
FOLDS
Problem 9. Consider a complex manifold Mwith a Kähler metric gand a (1,1)-form ωsatisfying
dω = 0.
Given that the Laplacian operator ∆=∆gon complex (p, p)-forms is defined as ∆ = 2Λ∂∂ +
2∂∂Λ, where ∂and ∂are the Dolbeault operators, compute the Laplacian of a complex (1,1)-form
αon Mif αis harmonic, i.e. ∆α= 0.
Solution 9. Given that αis a harmonic (1,1)-form, i.e. ∆α= 0, then we have:
0=∆α
= 2Λ∂∂α + 2∂∂Λα
= 2(Λ∂∂α) + 2(∂∂Λα)
= 2Λ(∂∂α)+2∂∂(Λα)
Since dω = 0, we have ∂2α= 0. Thus, the Laplacian of a harmonic (1,1)-form αon Msimplifies
to:
0 = 2Λ(∂∂α)+2∂∂(Λα)
= 2Λ(∂∂α)
= 2Λ∂(∂α)−2Λ∂(∂α)
= 2Λ(∂∂ −∂∂)α
= 2Λ(0)α
= 0
Therefore, the Laplacian of a harmonic (1,1)-form αon Mis zero.
10 10. COMPARISON OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 10. Consider a complex manifold Mwith a Riemannian metric gand a connection
abla.F ora(1,0)−formα on M, let ∆denote the Laplacian operator and δdenote the codiffer-
ential operator. Given that ∆α= 0 (i.e., αis harmonic) and δα =∗dα where ∗denotes the Hodge
star operator, show that αis a closed and coclosed form.
Solution 10. Given: ∆α= 0 and δα =∗dα.
a) To show that αis a closed form, we need to prove that dα = 0.
We know that ∆α= 0 implies that ∆α=δdα +dδα = 0. Since δα =∗dα, this becomes
δdα +d∗dα = 0.
Using the relation d∗dα =∗ddα (property of the Hodge star operator), we have δdα+∗ddα = 0.
Therefore, δdα =− ∗ ddα. But δα =∗dα given, so we substitute to get ∗dα =− ∗ ddα.
Hence, dα = 0, showing that αis a closed form.
b) To show that αis coclosed, we need to prove that δα = 0.
From the given information, we have δα =∗dα. This is already given and also indicates that α
is coclosed.
Thus, αis both a closed and coclosed form.
11 11. LAPLACIAN AND HARMONIC FORMS ON KÄHLER MANIFOLDS
Problem 11. Consider a Kähler manifold Mwith Kähler form ω. Let αbe a smooth (1,0)-form
on M. The Laplacian of αis defined as ∆α=−δdα, where dis the exterior derivative and δis the
formal adjoint of dwith respect to the metric induced by ω.
Suppose that ∆α= 0. Prove that αis a harmonic (1,0)-form, i.e., αis both d-closed and
δ-closed.
Solution 11. Given that ∆α= 0, we need to show that αis d-closed and δ-closed.
a) To prove that αis d-closed, we have ∆α=−δdα = 0. By the definition of the Laplacian, we
get d∗dα = 0. Since d∗d+dd∗= ∆ on a Kähler manifold, we have dd∗α= 0. This implies that αis
d-closed.
b) Next, to show that αis δ-closed, we use the relation ∆α=−δdα = 0. This gives dδα = 0.
Since dδ +δd =−∆on a Kähler manifold, we have δdα = 0. Therefore, αis δ-closed.
Hence, we have shown that if ∆α= 0 on a Kähler manifold M, then αis a harmonic (1,0)-form,
meaning it is both d-closed and δ-closed.
12 12. LAPLACIAN AND HARMONIC FORMS ON HYPERBOLIC COMPLEX MANIFOLDS
Problem 12. Let Xbe a hyperbolic complex manifold with a Kähler metric g. Consider a (1,1)-
form αon Xgiven by α=i∂ ¯
∂u, where uis a smooth function on X. If ∆gu= 0, where ∆gis the
Laplace operator with respect to the metric g, show that αis a harmonic form.
Solution 12.
The Laplace operator with respect to the metric gacting on uis given by ∆gu=−trace(∇2u),
where ∇2uis the Hessian matrix of second derivatives of u. Since ∆gu= 0, we have −trace(∇2u) =
0, which implies that the matrix ∇2uis traceless.
Now, let’s compute the Laplacian of α:
∆gα= ∆g(i∂ ¯
∂u)
=−trace ∇2(i∂ ¯
∂u)
=−trace i∂ ¯
∂∇2u
=−itrace ∇2u
= 0
Since ∆gα= 0, the form αis harmonic on the hyperbolic complex manifold X.
13 13. LAPLACIAN AND HARMONIC FORMS ON SINGULAR COMPLEX SPACES
Problem 13. Consider the singular complex space given by the intersection of two circles in
C2:X={(z1, z2)∈C2| |z1|= 1,|z2|= 1}.
a) Find a basis for the space of (0,1)-forms on X.
b) Compute the Laplacian of the (0,1)-form α= ¯z1dz2.
c) Find all harmonic (0,1)-forms on X.
Solution 13.
a) The space of (0,1)-forms on Xis spanned by the differentials of the coordinates z1and z2,
so a basis for this space is given by dz1and dz2.
b) The Laplacian of a (0,1)-form α= ¯z1dz2on Xis given by ∆α=−∆¯z(∆zα), where ∆zand
∆¯zare the Laplacian operators with respect to zand ¯z, respectively. Since Xis two-dimensional,
we have ∆z=∂z¯zand ∆¯z=∂¯zz.
Calculating ∆zα:
∆zα=∂z¯z(¯z1dz2)
=∂z(¯z1)d¯zdz2
= 0.
Hence, the Laplacian of αis ∆α=−∆¯z(∆zα) = 0.
c) To find all harmonic (0,1)-forms on X, we need to solve the equation ∆α= 0. Since we
found in part b) that ∆α= 0 for any (0,1)-form on X, all (0,1)-forms on Xare harmonic.
Therefore, all (0,1)-forms on the singular complex space Xare harmonic.
14 14. CURVATURE AND LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 14. Let Mbe a complex manifold with a Hermitian metric, and let ωbe a (1,1)-form
on M. Suppose ∆ω= 0, where ∆denotes the Laplacian operator. Given that ω=gij dzi∧d¯zjin
local coordinates, where gij are smooth functions of zand ¯z, determine the relationship between
gij and its complex conjugate ¯gij .
Solution 14.
Given that ∆ω= 0, we have ∆(gij dzi∧d¯zj)=0.
Expanding this out, we get ∆(gij dzi∧d¯zj)=0, which implies 0 = ∆(gij dzi∧d¯zj) = ∆(gij dzi)∧
d¯zj+gij ∆(dzi∧d¯zj).
Since ∆is self-adjoint with respect to the metric, the Laplacian of a wedge product of differential
forms satisfies ∆(α∧β)=∆α∧β+α∧∆β.
Therefore, we have 0 = ∆(gij dzi)∧d¯zj+gij ∆(dzi)∧d¯zj+gij dzi∧∆(d¯zj).
Now, compute the Laplacian of dziand d¯zj. In a (1,1) form dzi∧d¯zj, we have d=∂+¯
∂, where
∂is the Dolbeault operator and ¯
∂is the complex conjugate of ∂.
Since dcommutes with the Laplacian, we can write ∆(dzi) = ∆(∂zi) = ∂(∆zi). Similarly,
∆(d¯zj) = ∆(∂¯zj) = ∂(∆¯zj).
Substitute these back into the equation above to simplify, we get 0 = ∆(gij dzi)∧d¯zj+gij ∂(∆zi)∧
d¯zj+gij dzi∧∂(∆¯zj).
Now, apply the wedge product rule and the fact that partial derivatives commute for smooth
functions, we have 0 = ∂(gij ∆zi)∧d¯zj+gij ¯
∂(∆zi)∧d¯zj+gij ∂zi∧∆¯zj.
Thus, the relationship between gij and its complex conjugate ¯gij can be obtained by comparing
terms on both sides of the equation above. By equating corresponding terms, we find that gij ∆zi=
¯gij ∆¯zj.
Therefore, the relationship between gij and ¯gij is given by gij = ¯gji.
15 15. STABILITY OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 15. Consider a complex manifold Mwith a Hermitian metric gand a holomorphic
line bundle Lwith a Hermitian metric h. Let ∆¯
∂,h be the ¯
∂-Laplacian on smooth (0,1)-forms with
coefficients in L.
Suppose ∆¯
∂,hu= 0 on Mfor a smooth (0,1)-form u.
a) Prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) Show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
Solution 15.
a) To prove that if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g, we need to show that
∆gu= 0, where ∆gis the Laplace-Beltrami operator with respect to g. The Laplace operator ∆gis
given by ∆g=−δ◦d−d◦δ, where dis the exterior derivative and δis the formal adjoint of d.
Since uis a smooth (0,1)-form, we can write u=f dz for a smooth function fand the (1,0)-form
dz. Then, du =df ∧dz and δu =−i∂ ¯
∂f dz ∧dz. The Laplacian ∆gu=−δ◦du −d◦δu simplifies to
∆gu=−δ(df ∧dz)−d(−i∂ ¯
∂f dz ∧dz) = −d(δ(fdz)) −i∂ ¯
∂f dz ∧dz = 0, since ∆¯
∂,hu= 0 implies
∂¯
∂f = 0.
Therefore, if ∆¯
∂,hu= 0, then uis a harmonic form with respect to g.
b) To show that if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0, we need to show that
∆¯
∂,hu= 0. Given that uis harmonic with respect to g, we have ∆gu= 0.
Expanding ∆gu= 0 as before, we have ∆gu=−dδ(fdz)−i∂ ¯
∂f dz ∧dz = 0. This implies
−dδ(fdz) = i∂ ¯
∂f dz ∧dz.
Since dδ(fdz) = i∂ ¯
∂f dz ∧dz, we see that ∆¯
∂,hu=δ(∂¯
∂fdz)=0.
Therefore, if uis a harmonic form with respect to g, then ∆¯
∂,hu= 0.
16 16. COMPLEX TORI AND LAPLACIAN AND HARMONIC FORMS
Problem 16. Consider a complex torus C/(Z+τZ), where τ=a+bi with a, b ∈Rand b= 0.
Let f(z) = Re(z2)be a smooth function on the complex torus.
a) Find the Laplacian of fon the complex torus.
b) Determine if fis a harmonic function on the complex torus.
Solution 16.
a) To find the Laplacian of f(z)on the complex torus, we first note that the Laplacian of a function
uon a Riemannian manifold is given by ∆u=−trace(dδu), where dis the exterior derivative and
δis the codifferential.
Since f(z) = Re(z2), we have f(z) = Re(x2−y2+ 2ixy) = x2−y2, where z=x+iy.
Now, we need to calculate the Laplacian:
df = 2xdx −2ydy,
d∗df =−2dx −2dy,
∆f=−trace(−2dx −2dy) = 4.
Therefore, the Laplacian of fon the complex torus is ∆f= 4.
b) To determine if fis a harmonic function on the complex torus, we need to check if ∆f= 0.
Since ∆f= 4 = 0, we conclude that fis not a harmonic function on the complex torus.
17 17. EXAMPLES OF LAPLACIAN AND HARMONIC FORMS ON COMPLEX MANIFOLDS
Problem 17. Consider a complex manifold Mwith a Hermitian metric gand a (1,1)-form α=
dz ∧d¯z.
a) Calculate the Laplacian of the (0,1)-form β=∂α.
b) Determine if βis a harmonic form on M.
Solution 17.
a) To find the Laplacian of a (0,1)-form β, we first calculate ∆β= ∆(∂α), where ∆is the
Laplacian operator. We know that ∆ = −4∂¯
∂on a complex manifold.
Starting with β=∂α, we have:
∆(∂α) = −4∂¯
∂(∂α)
=−4∂¯
∂(dz ∧d¯z)
=−4∂(¯
∂dz ∧d¯z)
=−4∂(0)
= 0
Therefore, the Laplacian of the (0,1)-form βis ∆β= 0.
b) To determine if βis a harmonic form on M, we check if ∆β= 0 implies that βis harmonic.
Since we found that ∆β= 0, we conclude that βis indeed a harmonic form on M.
18 18. LAPLACIAN AND HARMONIC FORMS ON STEIN MANIFOLDS
Problem 18. Consider a Stein manifold Xwith a global holomorphic vector field vthat gener-
ates translations in a direction. Let ϕbe a smooth function on Xand ωbe a non-degenerate holo-
morphic 2-form. Given that the Laplacian of ϕwith respect to the metric induced by ωis ∆ωϕ= 0,
and the Lie derivative of ωwith respect to vis Lvω= 0, compute the following:
a) Prove that ϕis harmonic with respect to the metric induced by ω.
b) Show that ωis harmonic with respect to the metric induced by ω.
Solution 18.
a) To show that ϕis harmonic with respect to the metric induced by ω, we need to show that
∆ωϕ= 0. We are given that ∆ωϕ= 0, so ϕis harmonic.
b) Similarly, to show that ωis harmonic with respect to the metric induced by ω, we need to show
that ∆ωω= 0. We have ∆ωω=d(δω), where dis the exterior derivative and δis the codifferential.
Since the Lie derivative Lvω= 0, we have d(δω) = δ(dω)=0. Therefore, ωis harmonic with
respect to the metric induced by ω.
19 19. LAPLACIAN AND HARMONIC FORMS ON NON-KÄHLER COMPLEX MANIFOLDS
Problem 19. Let Mbe a non-Kähler complex manifold of complex dimension 2, and let αbe a
(1,1)-form on Msuch that ∆α= 0, where ∆is the Laplacian operator.
a) Show that αis a harmonic form on M.
b) Suppose α=fdz ∧d¯z, with fa smooth function on M. Find an expression for the Laplacian
∆f.
c) Find the harmonic (p, p)-forms on M, where pis a positive integer less than or equal to 2.
Solution 19.
a) To show that αis a harmonic form on M, we need to show that it is a closed and coclosed
form. Since ∆α= 0, this means that αis coclosed, i.e., d∗α= 0, where d∗is the adjoint of the
exterior derivative d. By Hodge theory, on a non-Kähler complex manifold, every closed (p, p)-form
must be coclosed, thus harmonic. Therefore, αis a harmonic form on M.
b) Since α=fdz ∧d¯z, we have ∆α= ∆(f dz ∧d¯z) = 0. The Laplacian of a (1,1)-form is
given by ∆α=−4∆¯zz fdz ∧d¯z, where ∆¯zz is the Dolbeault Laplacian. Since ∆α= 0, we have
−4∆¯zzf= 0. Therefore, the expression for the Laplacian ∆fis ∆f= 0.
c) The harmonic (p, p)-forms on Mcan be determined by solving the equation ∆α= 0 for
(p, p)-forms α. Since p≤2, the harmonic (1,1)-forms will be of the form α=fdz ∧d¯z, where fis
a smooth function on M. The harmonic (0,0)-forms are simply the smooth functions on M. The
harmonic (2,2)-forms will be of the form α=fdz2∧d¯z2, where fis a smooth function on M.
20 20. NON-EXISTENCE RESULTS FOR LAPLACIAN AND HARMONIC FORMS ON COM-
PLEX MANIFOLDS
Problem 20. Let Xbe a compact complex manifold of complex dimension n. Suppose α∈
H1,1(X)is a harmonic (1,1)-form on X. Let ∆ = ∂∂ denote the Laplacian operator.
a) Prove that if αis ∆-exact, then αis identically zero.
b) Prove that there exists a non-zero harmonic (1,1)-form on Xwhich is not ∆-exact.
Solution 20.
a) Suppose αis ∆-exact, i.e., there exists a (0,1)-form βsuch that α= ∆β. Since αis (1,1)-
form, we may write α=∂∂β. By Hodge theory, we know that {α}= [α]is a Hodge class in H1,1(X),
which corresponds to H1(X, C)∩H1(X, ∂) = H1,0(X)∩H0,1(X)under the Hodge decomposition.
However, if α=∂∂β, then α= 0 in cohomology since H1(X, C)∩H1(X, ∂)is a trivial space.
Therefore, if αis ∆-exact, then αmust be identically zero.
b) Consider the compact Riemann surface Σgof genus g > 1. By the Hodge index theorem and
classification of Riemann surfaces, we know that there exists a unique non-zero harmonic (1,1)-
form on Σg, say ω. Let α=ω+ ¯ω. This αis a harmonic (1,1)-form since it is a linear combination
of harmonic forms. Moreover, ∆α= ∆ω+ ∆¯ω= 0 since ωis harmonic. Thus, αis a non-zero
harmonic (1,1)-form on Σgwhich is not ∆-exact.
21 21. LAPLACIAN AND HARMONIC FORMS ON SYMPLECTIC MANIFOLDS
Problem 21. Consider a complex manifold with a symplectic form ω=dx ∧dy in local coordi-
nates on C2. Let f(z, z) = |z|2be a smooth function on this manifold.
a) Compute the Laplacian of fwith respect to the Kähler metric induced by ω.
b) Find a harmonic form on this complex manifold that is the wedge product of the Kähler form
and the Kähler metric.
c) Calculate the Hodge numbers hp,q for this complex manifold.
Solution 21.
a) The Laplacian of a function fwith respect to a Kähler metric gis given by ∆gf=1
√det g∂igij √det g∂jf.
In our case, the Kähler metric induced by the symplectic form ω=dx ∧dy is given by g=i∂∂|z|2.
Since f(z, z) = |z|2, we have ∆gf=1
√det g∂igij √det g∂j|z|2=1
√det g∂igij √det g2zj.
Now, we compute the components of the Kähler metric g=i∂∂|z|2=i(−i)dz ∧dz = 2idx ∧dy.
Therefore, gij =1
2iϵij where ϵij is the Levi-Civita symbol.
Hence, the Laplacian of fis ∆gf=1
√det g∂i1
2iϵij √det g2zj=−1
√2iϵij ∂izj=−2
√2i.
b) A harmonic form αon a Kähler manifold is a closed and coclosed form, i.e., dα = 0 and
δα = 0. One example is the wedge product of the Kähler form ωand the Kähler metric g.
Let α=ω∧g. Then, dα =d(ω∧g) = dω ∧g+ (−1)2ω∧dg = 0 since ωis a symplectic form
and dg = 0.
Also, δα =δ(ω∧g) = δω ∧g+ (−1)2ω∧δg = 0 since ωis a harmonic form and δg = 0.
Thus, α=ω∧gis a harmonic form on the complex manifold.
c) The Hodge numbers hp,q for a Kähler manifold are related to the Kähler class, and in this
case, the symplectic form ωinduces the Kähler class. Since ω=dx ∧dy, we have h1,1= 1 and
h0,2=h2,0= 0 for this complex manifold.
22 22. LAPLACIAN AND HARMONIC FORMS ON RIEMANN SURFACES
Problem 22. Consider the Riemann surface Sdefined by z=eiθ for 0≤θ≤2π. Let ω=
P(z)dz be a (1,0)-form on Swhere P(z) = z2−z2.
a) Calculate the Laplacian of the function P(z)on S.
b) Determine whether the form ωis harmonic on S.
Solution 22.
a) To calculate the Laplacian of P(z)on S, first denote the Laplacian operator on Sas ∆ = ∂∂.
Then we have ∆P(z) = ∂∂P (z). Since P(z) = z2−z2, we calculate the Laplacian as follows:
∂∂P (z) = ∂∂(z2−z2)
=∂(2z−0)
= 2.
Therefore, the Laplacian of P(z)on Sis ∆P(z)=2.
b) The form ω=P(z)dz is harmonic on Sif ∆ω= 0. Since ∆P(z) = 2 = 0, we conclude that
the form ωis not harmonic on S.
I can certainly provide questions and solutions on Laplacian and Harmonic Forms on Complex
Manifolds. Let’s proceed with the problem.
23 23. LAPLACIAN AND HARMONIC FORMS ON HOMOGENEOUS COMPLEX MANIFOLDS
Problem 23. Let M=CP 1be the complex projective line, which is a homogeneous com-
plex manifold. Consider the Kähler form ωon Mdefined by ω=i∂ ¯
∂log(1 + |z|2), where z∈C
corresponds to the standard affine coordinate on CP 1.
a) Calculate the Laplacian of the Kähler form ω.
b) Find a harmonic representative of the Kähler class [ω]on M.
Solution 23.
a) To calculate the Laplacian of the Kähler form ω, we need to use the standard Laplacian
operator ∆ = −∂¯
∂on complex manifolds. Given ω=i∂ ¯
∂log(1 + |z|2), we have to calculate
∆ω=−∂¯
∂ω.
Let’s compute it step by step:
∆ω=−∂¯
∂(i∂ ¯
∂log(1 + |z|2))
=−i∂ 1
1 + |z|2∂¯
∂|z|2
=−i∂ 1
1 + |z|2∂¯z∧¯
∂¯z
=−i∂ 1
1 + |z|2∂¯z∧0
= 0.
Therefore, the Laplacian of the Kähler form ωon Mis ∆ω= 0.
b) To find a harmonic representative of the Kähler class [ω]on M, we need to find a closed
(1,1)-form αin the Kähler class [ω]such that ∆α= 0.
One possible choice for a harmonic representative is taking α=ωitself since we have shown
that ∆ω= 0. Therefore, ωis a harmonic representative of the Kähler class [ω]on M.
24 24. LAPLACIAN AND HARMONIC FORMS ON PSEUDOCONVEX DOMAINS
Problem 24. Let Dbe a pseudoconvex domain in Cnand ωbe a smooth (1,1)-form on D.
Suppose ∆ω= 0, where ∆is the Laplacian operator.
Given that ω=f(z)dz ∧d¯z, where z= (z1, z2, . . . , zn)and f(z) = P|I|=kaIzIwith k≥2,
compute the harmonic representative of the cohomology class of ω.
Solution 24. To find the harmonic representative of the cohomology class of ω, we can write
ωas ω=dβ for some smooth (n−1, n −1)-form βon D. Since ∆ω= 0, we have ∆dβ = 0. By
the definition of Laplacian, ∆=2∂¯
∂, so ∂¯
∂dβ = 0.
From the given form of ω, we have dω =df ∧dz ∧d¯z. By the Poincaré lemma, there exists a
(n−2, n −2)-form ϕon Dsuch that dϕ =df ∧dz.
Therefore, we can write β=ϕ−g(z)d¯z, where g(z) = −Rz
z0ϕ. Now, ω=dβ, which gives
ω=df ∧dz ∧d¯z+dϕ −dg ∧d¯z.
To find the harmonic representative, we need to solve the equation ∆β= 0. This gives ∆β=
∆ϕ−∂¯
∂dg = 0. Hence, ∂¯
∂dg = ∆ϕ.
Therefore, to find the harmonic representative of the cohomology class of ω, we need to solve
the equation ∂¯
∂dg = ∆ϕ.
25 25. LAPLACIAN AND HARMONIC FORMS ON SASAKIAN MANIFOLDS
Problem 25. Consider a Sasaki-Einstein manifold Mwith a Kähler metric gsuch that its Kähler
form ωis closed. Let ∆denote the Laplacian operator on differential forms. Given a smooth (p, q)-
form αon M, it is known that ∆α= 0. Determine the following:
a) Show that αis a harmonic form, i.e., ∆α= 0 and dδα = 0.
b) If αis also closed, prove that it is a harmonic form.
c) If αis coclosed, i.e., dα = 0, verify whether it must be a harmonic form or not.
Solution 25.
a) Since ∆α= 0, we have ∆α=dδα +δdα. Given ∆α= 0 and δdα = (−1)p+qdδα, we have
δdα = 0 as well. Therefore, αis a harmonic form.
b) If αis closed, then we have dα = 0. Since ∆α= 0 implies dδα = 0, we have αsatisfying
both conditions to be a harmonic form.
c) If αis coclosed, i.e., dα = 0, it does not necessarily mean that αis a harmonic form. A
coclosed form need not be harmonic, as the condition dδα = 0 is crucial for a form to be harmonic.
If this condition is not satisfied, αmay not be harmonic.
