Kodaira vanishing theorem
= Kodaira vanishing theorem
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On a compact <Kähler manifold> $X$, if $P$ is a <positive holomorphic line bundle>, then $H^i(X,K_X\otimes P)=0$ for $i>0$. The canonical-bundle twist is part of the hypothesis and conclusion; positivity alone does not remove it on a general manifold. For a <complex torus>, the <canonical bundle> is trivial, so higher cohomology of a positive line bundle vanishes without an additional twist.