Talk:Kodaira vanishing theorem

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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.