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Vanishing theorems on Hermitian manifolds

Bogdan Alexandrov·Stefan Ivanov
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Abstract

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with $dd^c$-harmonic Kähler form and positive (1,1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with $c_1^2 >0$. As an application, the pth Dolbeault cohomology groups of a left-invariant complex structure compatible with a bi-invariant metric on a compact even dimensional Lie group are computed.

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Bogdan Alexandrov, Stefan Ivanov. 2003-01-09. Vanishing theorems on Hermitian manifolds. https://arxiv.org/abs/math/9901090

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