Several complex variables and analytic spaces Analytic spaces Theorem

Grauert's semicontinuity theorem

Builds on Complex analytic space Coherent analytic sheaf Proper map Sheaf cohomology Semicontinuity Euler characteristic Complex analytic variety Cartan–Serre finiteness theorem Grauert's direct image theorem

Statement

Let $f \colon X \to Y$ be a proper holomorphic map of complex analytic spaces and let $\mathcal{F}$ be a coherent analytic sheaf on $X$ . Assume that $\mathcal{F}$ is flat over $Y$ : for each $x \in X$ the stalk $\mathcal{F}_x$ is a flat module over $\mathcal{O}_{Y, f(x)}$ . For $y \in Y$ let $\mathfrak{m}_y \subseteq \mathcal{O}_{Y,y}$ be the maximal ideal. Let $X_y = f^{-1}(y)$ be the fibre with the structure sheaf $\mathcal{O}_X / \mathfrak{m}_y \mathcal{O}_X$ . Let $\mathcal{F}(y) = \mathcal{F} / \mathfrak{m}_y \mathcal{F}$ , a coherent sheaf on $X_y$ . For an integer $q \ge 0$ let $h^q(y) = \dim_\C H^q(X_y, \mathcal{F}(y))$ . Then:

for every $q \ge 0$ and every integer $k$ , the set $\set{y \in Y : h^q(y) \ge k}$ is an analytic subset of $Y$ ; in particular $h^q$ is upper semicontinuous;
the Euler characteristic $y \mapsto \sum_{q \ge 0} (-1)^q h^q(y)$ is locally constant on $Y$ .

Remarks

Each $h^q(y)$ is finite by the Cartan–Serre finiteness theorem, since $X_y$ is compact. The proof uses Grauert's direct image theorem in a refined form: locally on $Y$ , the cohomology of all fibres is computed by one bounded complex of finite free $\mathcal{O}_Y$ -modules. Ranks of matrices of holomorphic functions can only drop on analytic subsets, which gives part 1, and the alternating sum of the ranks of the free modules gives part 2.

Example: let $E$ be a compact Riemann surface of genus $1$ and let $\mathcal{P}$ be the Poincaré line bundle on $E \times E$ . For $y \in E$ , its restriction to $E \times \set{y}$ is a line bundle $L_y$ of degree $0$ on $E$ , trivial exactly when $y = 0$ . For the projection $f \colon E \times E \to E$ onto the second factor, the fibre over $y$ is $E \times \set{y}$ , and one gets $h^0(y) = h^1(y) = 1$ for $y = 0$ and $h^0(y) = h^1(y) = 0$ otherwise. The dimensions jump up on the analytic set $\set{0}$ , and the Euler characteristic is constantly $0$ .

Going deeper

For a student

The theorem is a basic tool in deformation theory. If $h^q$ is constant and $Y$ is reduced, then $R^q f_* \mathcal{F}$ is locally free and its fibre at $y$ is $H^q(X_y, \mathcal{F}(y))$ . This base change property is used to build vector bundles over moduli spaces. It is the analytic counterpart of the semicontinuity theorem for flat proper morphisms of schemes.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Coherent analytic sheaf concept
Compact space concept
Complex analytic space concept
Complex analytic variety concept
Dimension of an analytic set concept
Euler characteristic concept
Germ of a holomorphic function concept
Proper map concept
Ring of convergent power series concept
Semicontinuity concept
Sheaf cohomology concept
Stein space concept
Remmert's proper mapping theorem statement
Cartan's theorem B statement
Weierstrass division theorem statement
Cartan–Serre finiteness theorem statement
Grauert's direct image theorem statement
Grauert's semicontinuity theorem statement

Tag	`MREZVW7JTW`
URI	`m2://MREZVW7JTW`
Kind	statement, theorem
Writing	W0 stub
MSC	32C35, 32G13
Status	This copy of the site has no build data. Run `m2 status MREZVW7JTW`.
About	Coherent analytic sheaf, Semicontinuity
Uses	Complex analytic space, Coherent analytic sheaf, Proper map, Sheaf cohomology, Semicontinuity, Euler characteristic, Complex analytic variety, Cartan–Serre finiteness theorem, Grauert's direct image theorem