Talk:Kodaira vanishing theorem

Untitled

Also called the Kodaira-Nakano Vanishing Theorem:

Let M be a compact Kähler manifold. If L → M is a positive holomorphic line bundle, then

$H^{q}(M,\Omega ^{p}(L))=0$ for p+q > n

A holomorphic line bundle is positive if it admits a hermitian metric such that the associated connection has curvature tensor $\Theta$ satisfying:

$-i<\Theta (x);v,v^{*}>\in Hom(L_{x},L_{x})=\mathbb {C}$ is positive for all v in the holomorphic tangent bundle of M at x.