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  (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:
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 .
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Counterexamples to Kodaira's theorem for non-singular varieties over a field of positive characteristic were given by M. Raynaud [a1]. There exists a much stronger version of Kodaira's theorem, due to E. Viehweg and Y. Kawamata [a2].
Recently, many generalizations of Kodaira vanishing have been found, see [a3].
|[a1]||M. Raynaud, "Contre-example du "vanishing theorem" en caractéristique " K.G. Ramanathan (ed.) , C.P. Ramanujam, a tribute , Springer (1978) pp. 273–278|
|[a2]||E. Viehweg, "Vanishing theorems and positivity in algebraic fibre spaces" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 682–688 MR0934270 Zbl 0685.14013|
|[a3]||J. Kollar, "Vanishing theorems for cohomology groups" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 : 2 , Amer. Math. Soc. (1987) pp. 233–243 MR0927959 Zbl 0658.14012|
Kodaira theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kodaira_theorem&oldid=23879