Communications in Analysis and Geometry

Volume 30 (2022)

Number 7

Positivity preserving along a flow over projective bundle

Pages: 1541 – 1574



Xueyuan Wan (Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China)


In this paper, we introduce a flow over the projective bundle $p : P(E^\ast) \to M$, a natural generalization of both Hermitian–Yang–Mills flow and Kähler–Ricci flow. We prove that the semi-positivity 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 semi-positivity is preserved along the flow if the base manifold $M$ is a curve, which implies that the Griffiths semi-positivity 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.

Xueyuan Wan is partially supported by the National Natural Science Foundation of China (Grant No. 12101093) and the Natural Science Foundation of Chongqing (Grant No. CSTB2022NSCQ-JQX0008), the Scientific Research Foundation of the Chongqing University of Technology.

Received 1 February 2018

Accepted 2 March 2020

Published 25 May 2023