Vanishing Theorems on Compact Hyperk\"ahler Manifolds

TL;DR

This paper proves new vanishing theorems for cohomology groups of line bundles on compact hyperk"ahler manifolds, extending previous results and providing conditions under which certain cohomology groups vanish.

Contribution

It establishes vanishing theorems for cohomology groups involving positive line bundles on hyperk"ahler manifolds, generalizing earlier results by Verbitsky.

Findings

Vanishing of H^p(M, Ω^q ⊗ B) for p > n + [k/2] when B is k-positive.

Special case k=0 and q=0 recovers and slightly strengthens Verbitsky's vanishing theorem.

Provides conditions linking positivity of line bundles to cohomology vanishing on hyperk"ahler manifolds.

Abstract

We prove that if $B$ is a $k$-positive holomorphic line bundle on a compact hyperk\"ahler manifold $M,$ then $H^p (M,\Omega^q\otimes B)=0$ for $p>n+[\frac{k}{2}]$ and any nonnegative integer $q.$ In a special case $k=0$ and $q=0$ we recover a vanishing theorem of Verbitsky's with a little stronger assumption.