Positivity of twisted relative pluricanonical bundles and their direct images

TL;DR

This paper establishes quantitative positivity properties of twisted relative pluricanonical bundles and their direct images using singular Hermitian metrics and Ohsawa-Takegoshi extension, advancing understanding in complex geometry.

Contribution

It provides a new quantitative framework for positivity of twisted relative pluricanonical bundles via singular Hermitian metrics and extension theorems.

Findings

Quantitative positivity results for twisted relative pluricanonical bundles

Development of a framework using singular Hermitian metrics

Extension of previous joint work with Berndtsson

Abstract

Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.