A converse of Berndtsson's theorem on the positivity of direct images

TL;DR

This paper proves a converse to Berndtsson's theorem, showing that if certain direct image bundles are Griffiths semi-positive for all semi-positive line bundles, then the original line bundle must have semi-positive curvature.

Contribution

It establishes a new converse result for Berndtsson's theorem in the projective case, linking the positivity of direct image bundles to the semi-positivity of the original line bundle.

Findings

If $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for all semi-positive $E$, then $L$ has semi-positive curvature.

The result applies specifically to projective fibrations.

It extends the understanding of the relationship between direct image bundle positivity and line bundle curvature.

Abstract

Berndtsson's famous theorem asserts that, for a compact K\"ahler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive.