In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of Bismut-Griffiths-positivity for complex Hermitian non-Kähler manifolds. Since the Kähler-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 Hermitian curvature flows which do not preserve Bismut-Griffiths-non-negativity.

在本笔记中，我们研究了 Bismut 连接曲率的正性概念；更准确地说，我们研究了复杂 Hermitian 非 Kähler 流形的Bismut-Griffiths-positive概念。由于 Kähler-Ricci 流保留并规范了通常的 Griffiths 正性，我们研究了在 Hermitian 曲率流作用下 Bismut-Griffiths 正性的行为。特别地，我们研究了两类具体的例子，即线性 Hopf 流形和具有全纯平凡规范丛的六维 Calabi-Yau 求解流形。从这些例子中，我们确定了一些不保持 Bismut-Griffiths 非负性的 Hermitian 曲率流。