# Kodaira theorem

Kodaira's vanishing theorem

A theorem on the vanishing of the cohomology groups $H ^ {i} ( X, {\mathcal O} ( L) )$, $i < \mathop{\rm dim} X$, where ${\mathcal O} ( L)$ is the sheaf of holomorphic sections of the negative vector bundle $L$ of rank $1$ on a compact complex manifold $X$. An equivalent statement of Kodaira's vanishing theorem is that

$$H ^ {i} ( X, {\mathcal O} ( L \otimes K _ {X} ) ) = 0 ,\ \ i > 0 ,$$

for any positive vector bundle of rank 1 (here $K _ {X}$ denotes the canonical line bundle on $X$). In terms of divisors (cf. Divisor) Kodaira's vanishing theorem is stated as the equation $H ^ {i} ( X , {\mathcal O} _ {X} ( - D ) ) = 0$ for $i < \mathop{\rm dim} X$ and any divisor $D$ such that for some $n \geq 1$, $n D$ is a hyperplane section in some projective imbedding of $X$.

The theorem was proved by transcendental methods by K. Kodaira  (see also ) as a generalization to arbitrary dimension of the classical theorem on the regularity of an adjoint system on an algebraic surface. There exists an example of a normal algebraic surface over a field of positive characteristic for which Kodaira's vanishing theorem is false .

Kodaira's theorem also holds for holomorphic vector bundles of arbitrary rank that are negative in the sense of J. Nakano. The following result is also a generalization of Kodaira's theorem:

$$H ^ {i} ( X , \Omega ^ {p} ( L) ) = 0 \ \ \textrm{ for } \ p + i \geq \mathop{\rm dim} X + r ,$$

where $L$ is a weakly-positive vector bundle of rank $r$ on the compact complex manifold $X$, and $\Omega ^ {p} ( L) = \Omega ^ {p} \otimes {\mathcal O} ( L)$ is the sheaf of holomorphic forms (cf. Holomorphic form) of degree $p$ with values in $L$. For weakly-negative vector bundles $L$, vanishing takes place when $p + i \leq \mathop{\rm dim} X - r$. Analogues of these theorems have been obtained for weakly-complete manifolds $X$, that is, manifolds admitting a smooth pluriharmonic function $\psi$ such that the set $\{ {x \in X } : {\psi ( x) < c } \}$ is relatively compact in $X$ for all $c \in \mathbf R$, and for compact complex spaces $X$ having $n = \mathop{\rm dim} X$ algebraically-independent meromorphic functions .

How to Cite This Entry:
Kodaira theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kodaira_theorem&oldid=47510
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article