Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

TL;DR

This paper explores a new positivity concept for Bismut curvature on non-Kähler Hermitian manifolds and examines its evolution under Hermitian curvature flows, highlighting cases where positivity is not preserved.

Contribution

Introduces Bismut-Griffiths-positivity for Hermitian manifolds and analyzes its behavior under curvature flows, with specific examples showing non-preservation of positivity.

Findings

Bismut-Griffiths-positivity is a meaningful curvature notion for non-Kähler manifolds.

Certain Hermitian curvature flows do not preserve Bismut-Griffiths-non-negativity.

Examples include linear Hopf manifolds and Calabi-Yau solvmanifolds.

Abstract

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-K\"ahler manifolds. Since the K\"ahler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these examples we identify some HCFs which do not preserve Bismut-Griffiths-non-negativity.