Griffiths positivity for Bismut curvature and its behaviour along Hermitian curvature flows

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.