nLab Serre duality

Contents

Idea

Serre duality in complex analytic geometry is the duality induced by the Hodge star operator on the Dolbeault complex. This generalizes to suitable non-singular projective algebraic varieties over other base rings.

Statement

In complex analytic geometry

Let $X$ be a Hermitian manifold of complex dimension $dim_{\mathbb{C}}(\Sigma) = n$. Its Riemannian metric induces a Hodge star operator which acts on the pieces in the Dolbeault complex as

$$\star \colon \Omega^{p,q}(X)\to \Omega^{n-q, n-p}(X)$$

(see at Hodge star operator – On a Kähler manifold).

Moreover, complex conjugation gives $\mathbb{C}$-antilinear functions

$$\overline{(-)} \colon \Omega^{p,q}(X)\to \Omega^{q,p}(X) \,.$$

Definition

Write

$$\bar \star \;\coloneqq\; \overline{(-)} \circ \star = \star \circ \overline{(-)} \;\colon\; \Omega^{p,q}(X)\to \Omega^{n-p, n-q}(X)$$

for the composite antilinear function.

e.g. (Huybrechts 04, def. 4.1.6)

Remark

By the basic properties of the Hodge star it follows that restricted to $\Omega^{p,q}(X)$

$$\bar \star \circ \bar \star = (-1)^{p+q} \,.$$

Definition

For $X$ a compact Hermitian manifold, define a bilinear form

$$\int_X (-) \wedge \bar \star (-) \;\colon\; \Omega^{p,q}(X) \otimes \Omega^{n-p,n-q}(X) \longrightarrow \mathbb{C}$$

as the integration of differential forms

$$(\alpha,\beta) \mapsto \int_X \alpha \wedge \bar \star \beta$$

of the wedge product of $\alpha$ with the image of $\beta$ under the complex conjugated Hodge star operator of def. .

Proposition

(Serre duality)

The pairing of def. induces a non-degenerate sesquilinear (i.e. hermitian) form on Dolbeault cohomology

$$\int_X (-) \wedge \bar \star (-) \;\colon\; \H^{p,q}(X) \otimes H^{n-p,n-q}(X) \longrightarrow \mathbb{C}$$

e.g. (Huybrechts 04, prop 4.1.15)

Properties

Relation to Poincaré duality

Remark

For $\Sigma$ a compact Kähler manifold the Hodge theorem gives an isomorphism

$$H^k(X, \mathbb{C}) \simeq \underset{p+q = k}{\oplus} H^{p,q}(X)$$

between the ordinary cohomology of the underlying topological space with coefficients in the complex numbers, and the direct sum of all the Dolbeault cohomology groups in the same total degree.

Therefore for $\Sigma$ of complex dimension $dim_{\mathbb{C}}(\Sigma)= n$ then Serre duality in the form of prop. induces an isomorphism in ordinary cohomology of the form

$$H^k(X, \mathbb{C}) \stackrel{\simeq}{\longrightarrow} H^{2n-k}(X, \mathbb{C}) \,.$$

Proposition

The isomorphism in remark coincides with Poincaré duality.

Related concepts

• complex conjugation, Hodge star operator

References

• Wikipedia, Serre duality

Discussion in complex analytic geometry (Hermitian manifolds) includes

• Daniel Huybrechts around prop. 4.1.15 of Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)

• R.O. Wells, Differential Analysis on Compact Manifolds, Second Edition, Springer, 1980. 14

and review with emphasis on the case of Kähler manifolds includes

• Julien Meyer, Hodge theory on Kähler manifolds (pdf)