Harmonic forms and $\overline{\partial}$ -cohomology groups

8.6 Harmonic forms and $\overline{\partial }$ -cohomology groups

The $(r,s)$ th $\overline{\partial }$ -cohomology group is defined by

$$H_{\partial }^{r,s}(M)\equiv Z_{\partial }^{r,s}(M)/B_{\partial }^{r,s}(M).\text{\hspace{1em}(8.102)}$$

An element $[\omega ]\in H_{\overline{\partial }}^{r,s}(M)$ is an equivalence class of $\overline{\partial }$ -closed forms of bidegree $(r,s)$ which differ from $\omega$ by a $\overline{\partial }$ -exact form,

$$[\omega ]=\left\{\eta \in {\Omega }^{r,s}(M)|\bar{\partial }\eta =0,\omega -\eta =\bar{\partial }\psi ,\psi \in {\Omega }^{r,s-1}(M)\right\}.\text{\hspace{1em}(8.103)}$$

Clearly $H_{\overline{\partial }}^{r,s}(M)$ is a complex vector space. Similarly to the de Rham cohomology groups, the $\overline{\partial }$ -cohomology groups of ${\mathbb{C}}^m$ are trivial, that is, all the closed $(r,s)$ -forms are exact. The $\overline{\partial }$ -cohomology groups measure the topological non-triviality of a complex manifold $M$ .

8.6.1 The adjoint operators ${\partial }^{†}$ and ${\overline{\partial }}^{†}$

Let $M$ be a Hermitian manifold with ${\dim }_{\mathbb{C}}M=m$ . Define the inner product between $\alpha ,\beta \in {\Omega }^{r,s}(M)$ $(0\le r,s\le m)$ by

$$(\alpha ,\beta )\equiv {\int }_M\alpha \wedge \bar{\ast }\beta \text{\hspace{1em}(8.104)}$$

where $\overline{\ast }:{\Omega }^{r,s}(M)\to {\Omega }^{m-r,m-s}(M)$ is the Hodge $\ast$ defined by

$$\overline{\ast }\beta \equiv \overline{\ast \beta }=\ast \overline{\beta }\text{\hspace{1em}(8.105)}$$

where $\ast \beta$ is computed according to (7.173) extended to ${\Omega }^{r+s}(M{)}^{\mathbb{C}}$ . [Remark: * maps an $(r,s)$ -form to an $(m-s,m-r)$ -form since it acts on a basis of ${\Omega }^{r,s}(M)$ , up to an irrelevant factor, as

$$\begin{array}{rl}& \ast \mathrm{d}{z}^{{\mu }_1}\wedge \cdots \wedge \mathrm{d}{z}^{{\mu }_r}\wedge \mathrm{d}{\bar{z}}^{{\nu }_1}\wedge \cdots \wedge \mathrm{d}{\bar{z}}^{{\nu }_s}\sim {\epsilon }_{{\overline{\mu }}_{r+1}\dots {\overline{\mu }}_m}^{{\mu }_1\dots {\mu }_r}{\epsilon }_{{\nu }_{s+1}\dots {\nu }_m}^{{\overline{\nu }}_1\dots {\overline{\nu }}_s}\\ & \times \mathrm{d}{\bar{z}}^{{\mu }_{r+1}}\wedge \cdots \wedge \mathrm{d}{\bar{z}}^{{\mu }_m}\wedge \mathrm{d}{z}^{{\nu }_{s+1}}\wedge \cdots \wedge \mathrm{d}{z}^{{\nu }_m}.\end{array}$$

Note that the above $\epsilon$ -symbols are the only non-vanishing components in a Hermitian manifold. Now it follows that $\overline{\ast }:{\Omega }^{r,s}(M)\to {\Omega }^{m-r,m-s}(M).]$

We define the adjoint operators ${\partial }^{†}$ and ${\overline{\partial }}^{†}$ of $\partial$ and $\overline{\partial }$ by

$$(\alpha ,\partial \beta )=({\partial }^{†}\alpha ,\beta )(\alpha ,\bar{\partial }\beta )=({\bar{\partial }}^{†}\alpha ,\beta ).\text{\hspace{1em}(8.106)}$$

The operators ${\partial }^{†}$ and ${\overline{\partial }}^{†}$ change the bidegrees as ${\partial }^{†}:{\Omega }^{r,s}(M)\to {\Omega }^{r-1,s}(M)$ and ${\overline{\partial }}^{†}:{\Omega }^{r,s}(M)\to {\Omega }^{r,s-1}(M)$ . Clearly ${\mathrm{d}}^{†}={\partial }^{†}+{\overline{\partial }}^{†}$ . Noting that a complex manifold $M$ is even dimensional as a differentiable manifold, we have (see (7.184a))

$${\mathrm{d}}^{†}=-\ast \mathrm{d}\ast .\text{\hspace{1em}(8.107)}$$

Proposition 8.3

$${\partial }^{†}=-\ast \bar{\partial }\ast ,{\bar{\partial }}^{†}=-\ast \partial \ast .\text{\hspace{1em}(8.108)}$$

Proof

Let $\omega \in {\Omega }^{r-1,s}(M)$ and $\psi \in {\Omega }^{r,s}(M)$ . If we note that $\omega \wedge \overline{\ast }\psi \in {\Omega }^{m-1,m}(M)$ and hence $\overline{\partial }(\omega \wedge \overline{\ast }\psi )=0$ , we find that

$$\begin{array}{rl}d(\omega \wedge \bar{\ast }\psi )& =\partial (\omega \wedge \bar{\ast }\psi )=\partial \omega \wedge \bar{\ast }\psi +(-1{)}^{r+s-1}\omega \wedge \partial (\bar{\ast }\psi )\\ & =\partial \omega \wedge \bar{\ast }\psi +(-1{)}^{r+s-1}\omega \wedge (-1{)}^{r+s+1}\bar{\ast }\bar{\ast }\partial (\bar{\ast }\psi )\\ & =\partial \omega \wedge \bar{\ast }\psi +\omega \wedge \bar{\ast }\bar{\ast }\partial \bar{\ast }\psi \end{array}\text{\hspace{1em}(8.109)}$$

where use has been made of the facts $\partial \overline{\pi }\psi \in {\Omega }^{2m-r-s-1}(M)$ , $\overline{\ast \ast }\beta =\ast \beta$ and (7.176a). If (8.109) is integrated over a compact complex manifold $M$ with no boundary, we have

$$0=(\partial \omega ,\psi )+(\omega ,\overline{\ast }\partial \overline{\ast }\psi ).$$

The second term is

$$(\omega ,\overline{\ast }\partial \overline{\ast }\psi )=(\omega ,\ast \overline{\partial \ast \overline{\psi }})=(\omega ,\ast \overline{\partial }\ast \psi ).$$

We finally find $0=(\partial \omega ,\psi )+(\omega ,\ast \overline{\partial }\ast \psi )$ , namely ${\partial }^{†}=-\ast \overline{\partial }\ast$ . The other formula ${\overline{\partial }}^{†}=-\ast \partial \ast \mathrm{follows}\mathrm{similarly}$ .

As a corollary of proposition 8.3, we have

$${\left({\partial }^{†}\right)}^2={\left({\bar{\partial }}^{†}\right)}^2=0.\text{\hspace{1em}(8.110)}$$

8.6.2 Laplacians and the Hodge theorem

Besides the usual Laplacian $\Delta =(\mathrm{d}{\mathrm{d}}^{†}+{\mathrm{d}}^{†}\mathrm{d})$ , we define other Laplacians ${\Delta }_{\partial }$ and ${\Delta }_{\overline{\partial }}$ on a Hermitian manifold,

$${\Delta }_{\partial }\equiv (\partial +{\partial }^{†}{)}^2=\partial {\partial }^{†}+{\partial }^{†}\partial \text{\hspace{1em}(8.111a)}$$

$${\Delta }_{\bar{d}}\equiv (\bar{\partial }+{\bar{\partial }}^{†}{)}^2=\bar{\partial }{\bar{\partial }}^{†}+{\bar{\partial }}^{†}\bar{\partial }.\text{\hspace{1em}(8.111b)}$$

An $(r,s)$ -form $\omega$ which satisfies ${\Delta }_{\partial }\omega =0$ ( ${\Delta }_{\overline{\partial }}\omega =0$ ) is said to be $\partial$ -harmonic ( $\overline{\partial }$ -harmonic). If ${\Delta }_{\partial }\omega =0$ ( ${\Delta }_{\overline{\partial }}\omega =0$ ), $\omega$ satisfies $\partial \omega ={\partial }^{†}\omega =0$ ( $\overline{\partial }\omega ={\overline{\partial }}^{†}\omega =0$ ).

We have the complex version of the Hodge decomposition. Let ${\mathrm{Harm}}_{\partial }^{r,s}(M)$ be the set of $\overline{\partial }$ -harmonic $(r,s)$ -forms,

$${\mathrm{Harm}}_{\bar{\partial }}^{r,s}(M)\equiv \left\{\omega \in {\Omega }^{r,s}(M)\mid {\Delta }_{\bar{\partial }}\omega =0\right\}.\text{\hspace{1em}(8.112)}$$

Theorem 8.8 · Hodge's theorem

${\Omega }^{r,s}(M)$ has a unique orthogonal decomposition:

$${\Omega }^{r,s}(M)=\bar{\partial }{\Omega }^{r,s-1}(M)\oplus {\bar{\partial }}^{†}{\Omega }^{r,s+1}(M)\oplus {\mathrm{Harm}}_{\frac{\partial }{\partial }}^{r,s}(M)\text{\hspace{1em}(8.113a)}$$

namely an $(r,s)$ -form $\omega$ is uniquely expressed as

$$\omega =\bar{\partial }\alpha +{\bar{\partial }}^{†}\beta +\gamma \text{\hspace{1em}(8.113b)}$$

where $\alpha \in {\Omega }^{r,s-1}(M)$ , $\beta \in {\Omega }^{r,s+1}(M)$ and $\gamma \in {\mathrm{Harm}}_{\partial }^{r,s}(M)$ .

The proof is found in lecture 22, Schwartz (1986), for example. If $\omega$ is $\overline{\partial }$ -closed, we have $\overline{\partial }\omega =\overline{\partial }{\overline{\partial }}^{†}\beta =0$ . Then $0=⟨\beta ,\overline{\partial }{\overline{\partial }}^{†}\beta ⟩=⟨{\overline{\partial }}^{†}\beta ,{\overline{\partial }}^{†}\beta ⟩\ge 0$ implies ${\overline{\partial }}^{†}\beta =0$ . Thus, any closed $(r,s)$ -form $\omega$ is written as $\omega =\gamma +\overline{\partial }\alpha$ , $\alpha \in {\Omega }^{r,s-1}(M)$ . This shows that $H_{\overline{\partial }}^{r,s}(M)\subset {\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)$ . Note also that ${\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)\subset Z_{\overline{\partial }}^{r,s}(M)$ since $\overline{\partial }\gamma =0$ for $\gamma \in {\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)$ . Moreover, ${\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)\cap B_{\overline{\partial }}^{r,s}(M)=\varnothing$ since $B_{\overline{\partial }}^{r,s}(M)=\overline{\partial }{\Omega }^{r,s-1}(M)$ is orthogonal to ${\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)$ . Then it follows that ${\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)\cong H_{\overline{\partial }}^{r,s}(M)$ . If $P:{\Omega }^{r,s}(M)\to {\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)$ denotes the projection operator to a harmonic $(r,s)$ -form, $[\omega ]\in H_{\overline{\partial }}^{r,s}(M)$ has a unique harmonic representative $P\omega \in {\mathrm{Harm}}_{\overline{\partial }}^{r,s}(M)$ .

8.6.3 Laplacians on a Kähler manifold

In a general Hermitian manifold, there exist no particular relationships among the Laplacians $\Delta$ , ${\Delta }_{\partial }$ and ${\Delta }_{\overline{\partial }}$ . However, if $M$ is a Kähler manifold, they are essentially the same. [Note that the Levi-Civita connection is compatible with the Hermitian connection in a Kähler manifold.]

Theorem 8.9

Let $M$ be a Kähler manifold. Then

$$\Delta =2{\Delta }_{\partial }=2{\Delta }_{\overline{\partial }}.\text{\hspace{1em}(8.114)}$$

The proof requires some technicalities and we simply refer to Schwartz (1986) and Goldberg (1962). This theorem puts constraints on the cohomology groups of a Kähler manifold $M$ . A form $\omega$ which satisfies $\overline{\partial }\omega ={\overline{\partial }}^{†}\omega =0$ also satisfies $\partial \omega ={\partial }^{†}\omega =0$ . Let $\omega$ be a holomorphic $p$ -form; $\overline{\partial }\omega =0$ . Since $\omega$ contains no $\mathrm{d}{\overline{z}}^{\mu }$ in its expansion, we have ${\overline{\partial }}^{†}\omega =0$ , hence ${\Delta }_{\overline{\partial }}\omega =(\overline{\partial }{\overline{\partial }}^{†}+{\overline{\partial }}^{†}\overline{\partial })\omega =0$ . According to theorem 8.9, we then have $\Delta \omega =0$ , that is any holomorphic form is automatically harmonic with respect to the Kähler metric. Conversely $\Delta \omega =0$ implies $\overline{\partial }\omega =0$ , hence every harmonic form of bidegree $(p,0)$ is holomorphic.

8.6.4 The Hodge numbers of Kähler manifolds

The complex dimension of $H_{\frac{\partial }{\partial }}^{r,s}(M)$ is called the Hodge number $b^{r,s}$ . The cohomology groups of a complex manifold are summarized by the Hodge

diamond,

$$\left(\begin{array}{ccccccc}& & & b^{m,m}& & & \\ & & b^{m,m-1}& & b^{m-1,m}& & \\ b^{m,0}& b^{m-1,1}& & \dots & & & \\ & & & \dots & & b^{1,m-1}& b^{0,m}\\ & & & \dots & & & \\ & & b^{1,0}& & b^{0,1}& & \\ & & & b^{0,0}& & & \end{array}\right).\text{\hspace{1em}(8.115)}$$

These $(m+1{)}^2$ Hodge numbers are far from independent as we shall see later.

Theorem 8.10

Let $M$ be a Kähler manifold with ${\dim }_{\mathbb{C}}M=m$ . Then the Hodge numbers satisfy

(a) $b^{r,s}=b^{s,r}$ (8.116)
(b) $b^{r,s}=b^{m-r,m-s}$ (8.117)

Proof

(a) If $\omega \in {\Omega }^{r,s}(M)$ is harmonic, it satisfies ${\Delta }_{\overline{\partial }}\omega ={\Delta }_{\partial }\omega =0$ . Then the $(s,r)$ -form $\overline{\omega }$ is also harmonic, ${\Delta }_{\overline{\partial }}\overline{\omega }=0$ since ${\Delta }_{\overline{\partial }}\overline{\omega }=\overline{{\Delta }_{\partial }\omega }=\overline{{\Delta }_{\overline{\partial }}\omega }=0$ (note that ${\Delta }_{\partial }={\Delta }_{\overline{\partial }}$ ). Thus, for any harmonic form of bidegree $(r,s)$ , there exists a harmonic form of bidegree $(s,r)$ and vice versa. Thus, it follows that $b^{r,s}=b^{s,r}$ . (b) Let $\omega \in {\Omega }^{r,s}(M)$ and $\psi \in H_{\overline{\partial }}^{m-r,m-s}(M)$ . Then $\omega \wedge \psi$ is a volume element and it can be shown (Schwartz 1986) that ${\int }_M\omega \wedge \psi$ defines a non-singular map $H_{\overline{\partial }}^{r,s}(M)\times H_{\overline{\partial }}^{m-r,m-s}(M)\to \mathbb{C}$ , hence the duality between $H_{\overline{\partial }}^{r,s}(M)$ and $H_{\overline{\partial }}^{m-r,m-s}(M)$ . This shows that $H_{\overline{\partial }}^{r,s}(M)$ is isomorphic to $H_{\overline{\partial }}^{m-r,m-s}(M)$ as a vector space and it follows that ${\dim }_{\mathbb{C}}{H}_{\overline{\partial }}^{r,s}(M)={\dim }_{\mathbb{C}}{H}_{\overline{\partial }}^{m-r,m-s}(M)$ hence $b^{r,s}=b^{m-r,m-s}$ .

Accordingly, the Hodge diamond of a Kähler manifold is symmetric about the vertical and horizontal lines. These symmetries reduce the number of independent Hodge numbers to ${\left(\frac{1}{2}m+1\right)}^2$ if $m$ is even and $\frac{1}{4}(m+1)(m+3)$ if $m$ is odd.

In a general Hermitian manifold, there are no direct relations between the Betti numbers and the Hodge numbers. If $M$ is a Kähler manifold, however, theorem 8.11 establishes close relationships between them.

Theorem 8.11

Let $M$ be a Kähler manifold with ${\dim }_{\mathbb{C}}M=m$ and $\partial M=\varnothing$ . Then the Betti numbers $b^p$ ( $1\le p\le 2m$ ) satisfy the following conditions;

(a) $b^p={\sum }_{r+s=p}{b}^{r,s}$ (8.118)
(b) $b^{2p-1}$ is even $(1\le p\le m)$ (8.119)
(c) $b^{2p}\ge 1$ $(1\le p\le m)$ (8.120)

Proof

(a) $H_{\overline{\partial }}^{r,s}(M)$ is a complex vector space spanned by ${\Delta }_{\overline{\partial }}$ -harmonic $(r,s)$ -forms, $H_{\overline{\partial }}^{r,s}(M)=\{[\omega ]|\omega \in {\Omega }^{r,s}(M),{\Delta }_{\overline{\partial }}\omega =0\}$ . Note also that, $H^p(M)$ is a real vector space spanned by $\Delta$ -harmonic $p$ -forms, $H^p(M)=\{[\omega ]|\omega \in {\Omega }^p(M),\Delta \omega =0\}$ . Then the complexification of $H^p(M)$ is $H^p(M{)}^{\mathbb{C}}=\{[\omega ]|\omega \in {\Omega }^p(M{)}^{\mathbb{C}},\Delta \omega =0\}$ . Since $M$ is Kähler, any form $\omega$ which satisfies ${\Delta }_{\overline{\partial }}\omega =0$ also satisfies $\Delta \omega =0$ and vice versa. Since

$${\Omega }^p(M{)}^{\mathbb{C}}={\oplus }_{r+s=p}{\Omega }^{r,s}(M)$$

we find that

$$H^p(M{)}^{\mathbb{C}}={\oplus }_{r+s=p}{H}^{r,s}(M).$$

Noting that ${\dim }_{\mathbb{R}}{H}^p(M)={\dim }_{\mathbb{C}}{H}^p(M{)}^{\mathbb{C}}$ , we obtain $b^p={\sum }_{r+s=p}{b}^{r,s}$ .

(b) From (a) and (8.116), it follows that

$$b^{2p-1}=\sum _{r+s=2p-1}{b}^{r,s}=2\sum _{\begin{array}{c}r+s=2p-1\\ r>s\end{array}}{b}^{r,s}.$$

Thus, $b^{2p-1}$ must be even.

(c) The crucial observation is that the Kähler form $\Omega$ is a closed real two-form, $\mathrm{d}\Omega =0$ , and the real $2p$ -form

$${\Omega }^p=\underset{p}{\underbrace{\Omega \wedge ...\wedge \Omega }}$$

is also closed, $\mathrm{d}{\Omega }^p=0$ . We show that ${\Omega }^p$ is not exact. Suppose ${\Omega }^p=\mathrm{d}\eta$ for some $\eta \in {\Omega }^{2p-1}(M)$ . Then ${\Omega }^m={\Omega }^{m-p}\wedge {\Omega }^p=\mathrm{d}({\Omega }^{m-p}\wedge \eta )$ . It follows from Stokes’ theorem that

$${\int }_M{\Omega }^m={\int }_M\mathrm{d}({\Omega }^{m-p}\wedge \eta )={\int }_{\partial M}{\Omega }^{m-p}\wedge \eta =0.$$

Since the LHS is the volume of $M$ , this is in contradiction. Thus, there is at least one non-trivial element of $H^{2p}(M)$ and we have proved that $b^{2p}\ge 1$ .

If a Kähler manifold is Ricci flat, there exists an extra relationship among the Hodge numbers, which further reduces the independent Hodge numbers, see Horowitz (1986) and Candelas (1988).

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