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.

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. The condition on L is, for an algebraic geometer, reducible to ample line bundle (i.e. some tensor power gives a projective embedding).

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