Singular Hermitian metrics and positivity of direct images of pluricanonical bundles

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.

Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics

“…-67 -Next, we review the definitions of singular Hermitian metrics. For more details, we refer the reader to [18,Section 2]. Let H r be the set of r × r semipositive definite Hermitian matrices and H r be the space of semipositive, possibly unbounded Hermitian forms on C r .…”

“…Recently Cao and Păun [4] used this result to prove Iitaka's conjecture when the base space is an Abelian variety. For more details, we refer the reader to [18].…”

A decomposition theorem for projective manifolds with nef anticanonical bundle

“…Now we recall two basic properties of possibly singular hermitian metrics proved in [PT14,Pău16] which are extremely useful.…”

Kodaira dimension of algebraic fiber spaces over abelian varieties

“…We recall first the definition of singular Hermitian metrics on vector bundles cf. [2], [36] and [34]. Let E → X be a holomorphic vector bundle of rank r on a complex manifold X.…”