Dual Nakano positivity and singular Nakano positivity of direct image sheaves

TL;DR

This paper investigates the positivity properties of direct image sheaves under a surjective projective map, establishing dual Nakano semi-positivity in smooth cases and local Nakano semi-positivity in singular cases.

Contribution

It introduces new positivity results for direct image sheaves with singular Hermitian metrics, expanding understanding of their geometric and analytic properties.

Findings

Dual Nakano semi-positivity for smooth Hermitian metrics

Local Nakano semi-positivity in the singular setting

Analysis of canonical metrics on direct image sheaves

Abstract

Let $f:X\to Y$ be a surjective projective map and $L$ be a holomorphic line bundle on $X$ equipped with a (singular) semi-positive Hermitian metric $h$. In this article, by studying the canonical metric on the direct image sheaf of the twisted relative canonical bundles $K_{X/Y}\otimes L\otimes\mathscr{I}(h)$, we obtain that this metric has dual Nakano semi-positivity when $h$ is smooth and there is no deformation by $f$ and that this metric has locally Nakano semi-positivity in the singular sense when $h$ is singular.