# Kodaira theorem

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Kodaira's vanishing theorem

A theorem on the vanishing of the cohomology groups , , where is the sheaf of holomorphic sections of the negative vector bundle of rank on a compact complex manifold . An equivalent statement of Kodaira's vanishing theorem is that

for any positive vector bundle of rank 1 (here denotes the canonical line bundle on ). In terms of divisors (cf. Divisor) Kodaira's vanishing theorem is stated as the equation for and any divisor such that for some , is a hyperplane section in some projective imbedding of .

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:

where is a weakly-positive vector bundle of rank on the compact complex manifold , and is the sheaf of holomorphic forms (cf. Holomorphic form) of degree with values in . For weakly-negative vector bundles , vanishing takes place when . Analogues of these theorems have been obtained for weakly-complete manifolds , that is, manifolds admitting a smooth pluriharmonic function such that the set is relatively compact in for all , and for compact complex spaces having 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 [2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) [3] D. Mumford, "Pathologies III" Amer. J. Math. , 89 : 1 (1967) pp. 94–104 [4] O. Zariski, "Algebraic surfaces" , Springer (1971) [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