# 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 [1] (see also [2]) 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 [4].

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 [5].

#### References

 [1] K. Kodaira, "On a differential geometric method in the theory of analytic stacks" Proc. Nat. Acad. Sci. USA , 39 (1953) pp. 1268–1273 MR0066693 Zbl 0053.11701 [2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004 [3] D. Mumford, "Pathologies III" Amer. J. Math. , 89 : 1 (1967) pp. 94–104 MR0217091 Zbl 0146.42403 [4] O. Zariski, "Algebraic surfaces" , Springer (1971) MR0469915 Zbl 0219.14020 [5] A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 4 (1980) pp. 1363–1407 Itogi Nauk. Algebra Topol. Geom. , 15 (1977) pp. 93–171 Zbl 0449.32020