A new positivity condition for the curvature of Hermitian manifolds

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-Kähler 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 $\alpha \in H_{BC}^{1,1}(X)$ 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.

13 pages

References in corpus (1)

• Geometric Partial Differential Equations from Unified String Theories