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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
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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Let (M,I,J,K) be a compact hyperkähler manifold, , 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 of the dual Kähler cone, then Hi(L)=0 for i>n. If c1(L) lies in the opposite cone , then Hi(L)=0 for i<n. Finally, if c1(L) is neither in nor in , then Hi(L)=0 for .

Keywords

hyperkählerquaternionholomorphiccohomology vanishing

MSC classification

Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 6 , November 2007 , pp. 1576 - 1592
Copyright
Copyright © Foundation Compositio Mathematica 2007