The Chern-Ricci flow and holomorphic bisectional curvature

TL;DR

This paper demonstrates that on Hopf manifolds, the non-negativity of holomorphic bisectional curvature is not maintained during the evolution of the Chern-Ricci flow, highlighting limitations of curvature preservation in complex geometry.

Contribution

The paper provides a counterexample showing the non-preservation of non-negative holomorphic bisectional curvature under the Chern-Ricci flow on Hopf manifolds.

Findings

Non-negativity of holomorphic bisectional curvature is not preserved

Hopf manifold serves as a counterexample

Curvature properties can change under Chern-Ricci flow

Abstract

In this note, we show that on Hopf manifold $\mathbb S^{2n-1}\times \mathbb S^1$, the non-negativity of the holomorphic bisectional curvature is not preserved along the Chern-Ricci flow.