Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds

Misha Verbitsky

doi:10.1112/S0010437X07002746

Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds

Part of: Global differential geometry

Published online by Cambridge University Press: 01 November 2007

Misha Verbitsky

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Misha Verbitsky*: Affiliation:
Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, 117259, Russia (email: verbit@maths.gla.ac.uk, verbit@mccme.ru)

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Abstract

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Abstract

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Let (M,I,J,K) be a compact hyperkähler manifold, $\dim _{\mathbb {H}}M=n$ , and L a non-trivial holomorphic line bundle on (M,I). Using the quaternionic Dolbeault complex, we prove the following vanishing theorem for holomorphic cohomology of L. If c1(L) lies in the closure $\hat K$ of the dual Kähler cone, then Hi(L)=0 for i>n. If c1(L) lies in the opposite cone $-\hat K$ , then Hi(L)=0 for i<n. Finally, if c1(L) is neither in $\hat K$ nor in $-\hat K$ , then Hi(L)=0 for $i\neq n$ .