On the positivity of direct image bundles. II

Abstract

We show that if an upper semi-continuous function $\varphi$ on $\Omega\in\mathbb C^n$ satisfies some natural conditions, and for any non-negative psh function $\psi$ on $\mathbb D\times]Omega$, the fiberwise Bergman kernel associated to $p_2^*\varphi+\psi$ is log-psh, then $\varphi$ must be psh. This offers a version of converse to Berndtsson’s theorem concerning log-plurisubharmonic variation of Bergman kernels and generalizes a previous result of Li-Zhou. Additionally, we obtain a parallel result for the positivity of Hermitian holomorphic line bundles on Kähler manifolds.