Harmonic Theory on Compact Manifolds

Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217

2. Harmonic Theory on Compact Manifolds

where M^′=N −(A∪B∪M), as before. On the other hand, we see that, using Lemma 1.2 and the definition of Λ,

(1.5) Λ(z_A∧ ¯z_B∧w_M)= 2

iz_A∧ ¯z_B∧3

µ∈M

w_M−|µ|4 .

Using these formulas, one obtains easily, assuming that z_A∧ ¯z_B∧w_M has total degree p,

ΛL−LΛ=(n−p)z_A∧ ¯z_B∧w_M, and part (c) of Proposition 1.1 follows immediately.

Q.E.D.

IfXis an oriented Riemannian manifold, thenXcarries avolume element dV, which is nothing but a d-form ϕ ∈ E^d(X), with the property that in any oriented system of local coordinates U ⊂X

ϕ(x)=f (x)dx₁∧ · · · ∧dx_d,

wheref (x) >0 for allx∈U. By means of the∗-operator it is easy to see that ϕ= ∗(1), (1∈C⊂E⁰(X))

is indeed a volume element on X.

Remark: Denote in local coordinates the Riemannian metric on X by ds²=gijdxⁱ⊗dx^j,

using the summation convention, where g_ij is a symmetric positive definite matrix of functions. If we let g^ij be defined by

g^ijgj k=δⁱ_k (Kronecker delta), and if we raise indices by setting

a^i1···ip=g^i1j1•g^i2j2•· · ·•g^ipjpai₁···ip,

then we can express the ∗-operator given by the metric ds² explicitly in terms of these quantities (cf., deRham [1], pp. 119–122). Namely, we have, if

α=

i₁<···<ip

α_i1···ipdxⁱ¹ ∧ · · · ∧dx^ip, then

(∗α)=

j₁<···<jd−p

(∗α)j₁···j_d−pdx^j1∧ · · · ∧dx^jd−p, where

(∗α)j₁···j_d−p= ±

5det(gij)α^i1···ip,

where {i₁, . . . , i_p, j₁, . . . , j_d−p} = {1, . . . , d}, and we have the positive sign if the permutation is even and negative sign in the other case (just as in the case of an orthonormal basis). Thus in particular

∗(1)=5

det(gij) dx¹∧ · · · ∧dx^d is the volume element in this case.

Define

(ϕ, ψ )=

$

X

ϕ∧ ∗ ¯ψ , ϕ, ψ ∈E^p(X)

(ϕ, ψ )=0, ϕ∈E^p(X), ψ∈E^p(X), p=q (2.1)

and the integral is well defined since ϕ∧ ∗ ¯ψ is a d-form on X. We can extend this definition to noncompact manifolds by considering only forms with compact support. We then have the following proposition.

Proposition 2.1: The form ( , ) defined by (2.1) defines a positive defi- nite, Hermitian symmetric, sesquilinear form on the complex vector space E∗(X)= ⊕^dp=0E^p(X).

Proof: The Riemannian metric onXinduces an Hermitian inner product , on ∧^pT∗x(X) for eachx ∈X, given by, for ϕ, ψ p-forms onX,

ϕ∧ ∗ ¯ψ= ϕ, ψvol as we saw in (1.3). It is then clear that

(ϕ, ψ )=

$

X

ϕ∧ ∗ ¯ψ=

$

X

ϕ, ψvol

is a positive semidefinite, sesquilinear Hermitian form on E^p(X). To see that ( , ) is positive definite, suppose that ϕ ∈E^p(X) is not equal to zero at x0 ∈ X, then near x0, we can express ϕ in terms of a local oriented orthonormal frame for T∗(X)⊗C,{e1, . . . , e_d},

ϕ =

|I|=p

′ϕ_Ie_I, and

ϕ∧ ∗ ¯ϕ =

|I|=p

′|ϕ_I|² vol near x₀, and_′

|J|=p|ϕ_I|²>0 nearx₀. Then the contribution to the integral (ϕ, ϕ)=

$

X

ϕ∧ ∗ ¯ϕ will be nonzero, and thus (ϕ, ϕ) >0.

Q.E.D.

Thus the elliptic complex (E^∗(X), d) is equipped with a canonical inner product depending only on the orientation and Riemannian metric of the base spaceX(in Sec. 5 of Chap. IV we had allowed arbitrary metrics on each of the vector bundles appearing in the complex). We would have arrived at the same inner product had we merely used the metric on∧^pT^∗(X)naturally induced by that of T (X)and for our strictly positive measuredλ used the volume element ∗(1). However, the representation we have given here for the inner product on E^∗(X) will prove to be very useful, as we shall see.

For convenience, we shall call the inner product (2.1) on E^∗(X)the Hodge inner product on E^∗(X).

Suppose thatXis a Hermitian complex manifold. Then we can define the Hodge inner product on E^∗(X)with respect to the underlying Riemannian metric and a fixed orientation given by the complex structure (all complex manifolds are orientable).

Proposition 2.2: The direct sum decompositionE^r(X)=

p+q=rE^p,q(X)is an orthogonal direct sum decomposition with respect to the Hodge inner product.

Proof: Suppose thatϕ ∈E^p,q(X) and that ψ∈E^r,s(X), where p+q = r+s. Then we see that ϕ∧ ∗ ¯ψ is of type(n−r+p, n−s+q), since ψ¯ is of type (s, r) and ∗ ¯ψ is then of type (n−r, n−s) by Proposition 1.1.

Therefore ϕ∧ ∗ ¯ψ is a 2n-form if and only if r =p and s=q. Otherwise, ϕ∧ ∗ ¯ψ is identically zero. This proves the proposition.

Q.E.D.

Using the Hodge inner product, it will be very easy to compute the adjoints of various linear operators acting on E^∗(X) (cf. the computation of L∗in Sec. 1). First we want to modify the ∗-operator in a manner which will be convenient for this purpose. On an oriented Riemannian manifold we define

¯∗: E^∗(X)−→E^∗(X)

by setting ¯∗(ϕ)= ∗ ¯ϕ. Thus ¯∗ is a conjugate-linear isomorphism of vector bundles,

¯∗: ∧^pT^∗(X)_c −→ ∧^m−pT^∗(X)_c,

where m=dim_RX. Suppose that X is now a Hermitian complex manifold and that E−→X is a Hermitian vector bundle. Let

τ: E−→E^∗

be a conjugate-linear bundle isomorphism ofEonto its dual bundleE∗. The mapping τ depends on the Hermitian metric of E and is defined fibrewise in a standard manner. We then define

¯∗E: ∧^pT^∗(X)_c⊗E−→ ∧^2n−pT^∗(X)_c⊗E^∗ by setting

¯∗E(ϕ⊗e)= ¯∗(ϕ)⊗τ (e)

for ϕ ∈ ∧^pT_x^∗(X)_c and e∈E_x. Thus ¯∗E is a conjugate-linear isomorphism of Hermitian vector bundles. We recall that we defined E^r(X, E) to be the sections of∧^rT^∗(X)_c⊗E and that, moreover, there is a decomposition into bidegrees

E^r(X, E)=

p+q=r

E^p,q(X, E).

Thus we note that first the Hodge inner product onE^∗(X)can be written as (ϕ, ψ )=

$

X

ϕ∧ ¯∗ψ,

and we extend this to a Hodge inner product on E^∗(X, E) by setting

(2.2) (ϕ, ψ )=

$

X

ϕ∧ ¯∗^Eψ

if ϕ, ψ ∈E^r(X, E). It is easy to see that ϕ∧ ¯∗Eψ does make sense and is a scalar 2n-form which can be integrated over X (where n=dimCX). In fact, if we let, represent the bilinear duality pairing betweenE andE^∗, then we set, for ϕ∈ ∧^pT_x^∗(X)_c, e∈E_x, ψ∈ ∧^2n−pT_x^∗(X)_c, f ∈E_x^∗,

(ϕ⊗e)∧(ψ⊗f )=ϕ∧ψ•e, f ∈ ∧²ⁿT_x^∗(X)_c.

By using a basis for Eand a dual basis for E^∗, we can extend this exterior product to vector-bundle-valued differential forms, and it is easily checked that the resulting exterior product is independent of the choice of basis.

Thus (2.4) defines what we shall call a Hodge inner product on E^∗(X, E).

Then it is easy to see that ¯∗E preserves the bigrading on E^∗(X, E), and that, in fact,

¯∗E: E^p,q(X, E)−→^∼ E^n−p,n−q(X, E^∗)

is a conjugate-linear isomorphism. It is then clear that Proposition 2.2 extends to this case.

We are now in a position to compute the adjoints of various operators with respect to the Hodge inner product. Moreover, all adjoints in this and later sections of the book will be with respect to the Hodge inner product.

Proposition 2.3: Let X be an oriented compact Riemannian manifold of real dimension m and let =dd^∗+d^∗d, where the adjoint d^∗ is defined with respect to the Hodge inner product on E^∗(X). Then

(a) d^∗=(−1)^m+mp+1¯∗d¯∗ =(−1)^m+mp+1∗d∗on E^p(X).

(b) ∗=∗, ¯∗=¯∗.

Proof: The basic fact we need is that ∗∗ = w, as defined in Sec. 1.

Suppose that ϕ∈E^p−1(X) and that ψ∈E^p(X). Then we consider (dϕ, ψ )=

$

X

dϕ∧ ¯∗ψ

=

$

X

d(ϕ∧ ¯∗ψ )−(−1)^p−1

$

X

ϕ∧d¯∗ψ,

by the rule for differentiating a product of forms. Moreover, by Stokes’

theorem, we see that the first term vanishes, and hence we obtain (noting that ∗∗ = ¯∗¯∗ =w, since ∗is real)

(dϕ, ψ )=(−1)^p

$

X

ϕ∧ ¯∗(¯∗⁻¹d¯∗)ψ

=(−1)^p

$

X

ϕ∧ ¯∗(¯∗wd¯∗)ψ

=(−1)^m+mp+1(ϕ,¯∗d¯∗ψ ), and thus we have

d^∗=(−1)^m+mp+1¯∗d¯∗, and since d is real, we also obtain

d^∗=(−1)^m+mp+1∗d∗. To prove (b), we compute, for ϕ ∈E^p(X),

∗ϕ=(−1)^m+mp+1(∗d∗d∗ +(−1)^m∗∗d∗d)ϕ ∗ϕ=(−1)^m+m(m−p)+1(d∗d∗∗ +(−1)^m∗d∗d∗)ϕ,

and so it suffices to show that (recall that w= ∗∗) wd∗dϕ=d∗dwϕ.

But this is simple, since w=

(−1)^p+mp_p, and thus the right-hand side is d∗d(−1)^p+mpϕ, whereas the left-hand side has degree m−p, and so

wd∗dϕ=(−1)^{m−p+m(m−p)}d∗dϕ=(−1)^p+mpd∗dϕ.

Q.E.D.

We have a similar result for the Hermitian case. Note that ¯∗E∗ is defined in the same way as ¯∗E by usingτ⁻¹: E^∗−→E.

Proposition 2.4: LetXbe a Hermitian complex manifold and letE−→X be a Hermitian holomorphic vector bundle. Then

(a) ∂¯:E^p,q(X, E)−→E^p,q+1(X, E) has an adjoint∂¯^∗ with respect to the Hodge inner product on E^∗∗(X, E) given by

¯

∂^∗= −¯∗E^∗∂¯¯∗E.

(b) If= ¯∂∂¯^∗+ ¯∂^∗∂¯ is the complex Laplacian acting onE^∗∗(X, E), then

¯∗E = ¯∗E.

Proof: In this case we also have ¯∗E¯∗E^∗ =w =

(−1)^p_p, a simpler expression since the real dimension of X is even. The proof of (a) then follows as before, with minor modification. Suppose that ϕ∈E^p,q−1(X, E) and that ψ∈E^p,q(X, E). Then we have thatϕ∧ ¯∗^Eψ is a scalar differential form of type (n, n−1), and hence ∂(ϕ¯ ∧ ¯∗^Eψ )=d(ϕ∧ ¯∗^Eψ ). Moreover,

¯

∂(ϕ∧ ¯∗^Eψ )= ¯∂ϕ∧ ¯∗^Eψ+(−1)^p+q−1ϕ∧ ¯∂¯∗^Eψ.

Substituting into the inner product, we obtain, using Stokes’ theorem as in the proof of Proposition 2.3,

(¯∂ϕ, ψ )=(−1)^p+q

$

X

ϕ∧ ¯∂¯∗Eψ

=(−1)^p+q

$

ϕ∧ ¯∗E(w¯∗E^∗∂¯¯∗Eψ )

= −

$

ϕ∧ ¯∗^E(¯∗^E∗∂¯¯∗^Eψ )

=(ϕ,−¯∗E^∗∂¯¯∗Eψ ),

and hence (a) is proved. The proof of (b) is exactly the same as in Propo- sition 2.3 (Note that acting on E^∗∗(X, E) and E^∗∗(X, E^∗) denotes two different operators).

Q.E.D.

Remark: We note that only∂¯acts naturally onE^p,q(X, E)for a nontrivial holomorphic vector bundle E, whereas ∂ and hence d do not, since they

do not annihilate the transition functions definingE. However, in the scalar case, we have∂:E^p,q(X)−→E^p+1,q(X), and by the same calculation as above we obtain that∂^∗= −¯∗∂¯∗and that=∂∂^∗+∂^∗∂commutes with ¯∗, exactly the same as the ∂-operator case.¯

Using the above results we can derive two well-known duality theo- rems. We first remark that a finite dimensional complex vector space E is conjugate-linearly isomorphic to a complex vector space F if and only if F is complex-linearly isomorphic to E^∗, the dual of E (and the bilinear pairing ofE toF can be obtained from a Hermitian inner product onE).

Theorem 2.5 (Poincaré duality): Let X be a compact m-dimensional ori- entable differentiable manifold. Then there is a conjugate linear isomorphism

σ: H^r(X,C)−→H^m−r(X,C),

and hence H^m−r(X,C) is isomorphic to the dual ofH^r(X,C).

Proof: Introduce a Riemannian metric and an orientation onXand let

∗ be the associated ∗-operator. Then we have the commutative diagram E^r(X) ^¯∗ //

H

E^m−r(X) H

H^r(X) ^¯∗ // H^m−r(X)

≀ ≀

H^r(X,C) ^σ // H^m−r(X,C),

where H is the projection onto the harmonic forms given by Theo- rem IV.4.12, and the mapping ¯∗ maps harmonic forms to harmonic forms since ¯∗ = ¯∗, as we saw in Proposition 2.3. Moreover, the de Rham groups H^r(X,C) are isomorphic to H^r(X) (Example IV.5.4), and σ is the induced conjugate linear isomorphism.

Q.E.D.

Remark: We could have restricted ourselves to real-valued differential forms and obtained the same result. Also, the more general Poincaré duality theorem of algebraic topology is true with coefficients inZand is indepen- dent of any differentiable structure on X, but one needs a different type of proof for that (see, e.g., Greenberg [1]).

Corollary 2.6: Let X be as in Theorem 2.5. Then b_r(X)=b_m−r(X), r=0, . . . , m.

Our next result is more analytical in nature and depends very much on the complex structures involved, in contrast to the Poincaré duality above.

Theorem 2.7 (Serre duality): Let X be a compact complex manifold of complex dimension n and let E −→ X be a holomorphic vector bundle over X. Then there is a conjugate linear isomorphism

σ: H^r(X,^p(E))−→H^n−r(X,^n−p(E^∗)), and hence these spaces are dual to one another.

Proof: By introducing Hermitian metrics on X and E, we can define the ¯∗E operator. Then we obtain the following commutative diagram,

E^p,q(X, E)

¯∗E

//

H

E^n−p,n−q(X, E^∗) H

H^p,q(X, E)

¯∗E

// H^n−p,n−q(X, E^∗)

≀ ≀

H^p,q(X, E)

τ // H^n−p,n−q(X, E^∗)

≀ ≀

H^p(X,^p(E))

σ // H^n−q(X,^n−p(E^∗)),

which proves the result immediately. Once again, ¯∗^E maps harmonic forms to harmonic forms by Proposition 2.4, and the {H^p,q(X, E)} are the Dol- beault groups [the cohomology of the complex (E^p,∗(X, E),∂)], which are¯ isomorphic to H^q(X,^p(E)), as we saw in Theorem II.3.20.

Q.E.D.

Remark: Serre proved this also in the case of noncompact manifolds, under certain closed range hypotheses on ∂¯ and by using cohomology with compact supports, i.e., H_∗^q(X,^p(E)) is the topological dual of H^n−q(X,^n−q(E^∗)), where H_∗^q( ) denotes cohomology with compact sup- ports. In our case we have finite dimensional vector spaces (due to the harmonic theory), in which case Serre’s hypothesis is fulfilled and the com- pact support is automatic. Serre’s proof (in Serre [1]) used resolutions of ^p(E) by both C^∞ forms and by distribution forms, and he was able to utilize the natural duality of these spaces to obtain his results. The proof above is due to Kodaira [1].

Corollary 2.8: LetXbe a compact complex manifold of complex dimension n. Then

(a) b_r(X)=b2n−r(X), r =0, . . . ,2n.

(b) h^p,q(X)=h^n−p,n−q(X), p, q =0, . . . , n.