Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric

TL;DR

This paper derives formulas for the curvature of the $L^2$ metric on direct images of twisted relative canonical bundles with singular metrics, extending Berndtsson's results to singular settings and providing explicit curvature bounds.

Contribution

It generalizes Berndtsson's curvature formulas to cases with singular metrics and offers explicit lower bounds when the twist is relatively big.

Findings

Derived curvature formulas for singular metrics

Extended Berndtsson's results to singular settings

Provided explicit lower bounds for curvature

Abstract

In this note, we obtain various formulas for the curvature of the $L^2$ metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positively curved metric with analytic singularities, generalizing some of Berndtsson's seminal results in the smooth case. When the twist is assumed to be relatively big, we further provide a very explicit lower bound for the curvature of the $L^2$ metric.