Kodaira vanishing theorem

In

holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem

The complex analytic case

The statement of

holomorphic line bundle on M that is positive, and K_M is the canonical line bundle

H^{q}(M,K_{M}\otimes L)=0

for q > 0. Here $K_{M}\otimes L$ stands for the

tensor product of line bundles. By means of Serre duality

, one also obtains the vanishing of

H^{q}(M,L^{\otimes -1})

for q < n. There is a generalisation, the Kodaira–Nakano vanishing theorem, in which

K_{M}\otimes L\cong \Omega ^{n}(L)

, where Ωⁿ(L) denotes the sheaf of

holomorphic (n,0)-forms

on M with values on L, is replaced by Ω^r(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group H^q(M, Ω^r(L)) vanishes whenever q + r > n.

The algebraic case

The Kodaira vanishing theorem can be formulated within the language of algebraic geometry without any reference to transcendental methods such as Kähler metrics. Positivity of the line bundle L translates into the corresponding invertible sheaf being ample (i.e., some tensor power gives a projective embedding). The algebraic Kodaira–Akizuki–Nakano vanishing theorem is the following statement:

If k is a

projective k-scheme

of dimension d, and L is an ample invertible sheaf on X, then

H^{q}(X,L\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q>d,{\text{ and}}

H^{q}(X,L^{\otimes -1}\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q<d,

where the Ω^p denote the

differential forms (see Kähler differential

).

Raynaud (1978) showed that this result does not always hold over fields of characteristic p > 0, and in particular fails for Raynaud surfaces. Later Sommese (1986) give a counterexample for singular varieties with non-log canonical singularities,^[1] and also,Lauritzen & Rao (1997) gave elementary counterexamples inspired by proper homogeneous spaces with non-reduced stabilizers.

Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the

algebraic de Rham cohomology

degenerates in degree 1. This is shown by lifting a corresponding more specific result from characteristic p > 0 — the positive-characteristic result does not hold without limitations but can be lifted to provide the full result.

Consequences and applications

Historically, the Kodaira embedding theorem was derived with the help of the vanishing theorem. With application of Serre duality, the vanishing of various sheaf cohomology groups (usually related to the canonical line bundle) of curves and surfaces help with the classification of complex manifolds, e.g. Enriques–Kodaira classification.