Positivity preserving along a flow over projective bundle

Abstract

In this paper, we introduce a flow over the projective bundle $p:P(E^{\ast })\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle ${\mathcal{O}}_{P(E^{\ast })}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under Kähler-Ricci flow.

arXiv commentFinal version, to appear in Communications in Analysis and Geometry