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Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds
Part of
- Global differential geometry
Misha Verbitsky
Journal:

Compositio Mathematica / Volume 143 / Issue 6 / November 2007

Published online by Cambridge University Press:

01 November 2007, pp. 1576-1592

Print publication:

November 2007
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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 .