A new positivity condition for the curvature of Hermitian manifolds

TL;DR

This paper introduces a new positivity condition for Hermitian manifold curvature, extending previous notions to non-Kähler cases, and explores its geometric implications and examples.

Contribution

It proposes a novel curvature positivity condition for Hermitian manifolds that generalizes existing concepts and analyzes its geometric consequences and specific examples.

Findings

The positivity condition leads to parallel representatives in Bott-Chern cohomology.

It holds on certain generalized Hopf and Vaisman manifolds.

The condition naturally arises from a derived Bochner formula.

Abstract

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-K\"ahler case. We derive a Bochner formula for closed $(1, 1)$-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfy our positivity condition, then any class $\alpha \in H^{1, 1}_{BC}(X)$ can be represented by a closed $(1, 1)$-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.