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Concept# Kodaira vanishing theorem

Summary

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.
The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on M that is positive, and KM is the canonical line bundle, then
for q > 0. Here stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of for q < n. There is a generalisation, the Kodaira–Nakano vanishing theorem, in which , where Ωn(L) denotes the sheaf of holomorphic (n,0)-forms on M with values on L, is replaced by Ωr(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group Hq(M, Ωr(L)) vanishes whenever q + r > n.
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
where the Ωp denote the sheaves of relative (algebraic) differential forms (see Kähler differential).
showed that this result does not always hold over fields of characteristic p > 0, and in particular fails for Raynaud surfaces. Later give a counterexample for singular varieties with non-log canonical singularities, and also, gave elementary counterexamples inspired by proper homogeneous spaces with non-reduced stabilizers.

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