Singular Hermitian metrics and positivity of direct images of pluricanonical bundles

TL;DR

This paper reviews the theory of singular Hermitian metrics on sheaves and explores their application in understanding the positivity properties of direct images of twisted pluricanonical bundles.

Contribution

It provides an exposition connecting singular Hermitian metrics with the positivity of direct images of pluricanonical bundles, highlighting recent developments.

Findings

Clarifies the role of singular Hermitian metrics in algebraic geometry

Summarizes key results on positivity of direct image sheaves

Provides a comprehensive survey of recent advances

Abstract

This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.