2 results
ON A BOGOMOLOV TYPE VANISHING THEOREM
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- Nagoya Mathematical Journal , First View
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- 13 January 2025, pp. 1-13
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Let X be a compact Kähler manifold, and let
$L \rightarrow X$ be a holomorphic line bundle equipped with a singular metric h such that the curvature 
$\mathrm {i}\Theta _{L,h}\geqslant 0$ in the sense of currents. The main result of this paper is the vanishing of 
$H^n(X,\mathcal {O}(\Omega ^p_X\otimes L)\otimes \mathcal {I}(h))$ for 
$p\geqslant n-\operatorname {nd}(L,h)+1$, which generalizes Bogomolov’s vanishing theorem and Watanabe’s result.
Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds
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- Journal:
- Compositio Mathematica / Volume 143 / Issue 6 / November 2007
- Published online by Cambridge University Press:
- 01 November 2007, pp. 1576-1592
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- 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 
.
, and 
of the dual Kähler cone, then 
, then 
.