Positivity preserving along a flow over projective bundle

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.

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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