Kodaira vanishing theorem

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In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch-Riemann-Roch theorem.

[edit] The complex analytic case

The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold, L any holomorphic line bundle on M that is a positive line bundle, and K is the canonical line bundle, then

H^q(M, KL) = {0}

for q > 0. Here KL stands for the tensor product of line bundles. By means of Serre duality, K can be removed. There is a generalisation, the Kodaira-Nakano vanishing theorem, in which K, the nth exterior power of the holomorphic cotangent bundle where n is the complex dimension of M, is replaced by the rth exterior power. Then the cohomology group vanishes whenever

q + r > n.

[edit] 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 field of characteristic zero, X is a smooth and projective k-scheme of dimension d, and L is an ample invertible sheaf on X, then

$H^q(X,L\otimes\Omega^p_{X/k}) = 0$ for

p + q > d

, and

$H^q(X,L^{\otimes-1}\otimes\Omega^p_{X/k}) = 0$ for

p + q < d

where the Ώ^p denote the sheaves of relative (algebraic) differential forms (see Kähler differential.

This result does not always hold over fields of characteristic p > 0; counterexamples are known.

Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the GAGA comparison theorems. However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in (Deligne & Illusie 1987). Their proof is based on showing that Hodge-de Rham spectral sequence for algebraic de Rham cohomology degenerates in degree 1. It is remarkable that 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.

[edit] References

Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Chapter 1.2
Deligne, Pierre & Illusie, Luc (1987), “Relèvements modulo p² et décomposition du complexe de de Rham”, Inventiones Mathematicae 89: 247 - 270