A new positivity condition for the curvature of Hermitian manifolds

Freid Tong

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arXiv · 2005.13411

A new positivity condition for the curvature of Hermitian manifolds

Freid Tong

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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.

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Freid Tong. 2020-05-27. A new positivity condition for the curvature of Hermitian manifolds. https://arxiv.org/abs/2005.13411

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