Vanishing theorems on Hermitian manifolds

TL;DR

This paper establishes vanishing theorems for Dolbeault cohomology on certain Hermitian manifolds, linking geometric conditions to topological properties and providing explicit computations for specific Lie group structures.

Contribution

It proves new vanishing theorems for Dolbeault cohomology on Hermitian manifolds with special Ricci form conditions, and computes cohomology for invariant structures on Lie groups.

Findings

Vanishing of Dolbeault cohomology on Hermitian manifolds with positive Ricci form.

Complex surfaces with such metrics are rational with positive first Chern class.

Explicit cohomology calculations for invariant structures on compact Lie groups.

Abstract

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with $dd^c$-harmonic K\"ahler 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.