Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

TL;DR

This paper proves a decomposition theorem for torsion-free sheaves with singular hermitian metrics, showing they split into flat and ample parts, extending previous results on direct images of pluricanonical bundles in algebraic geometry.

Contribution

It establishes a new decomposition theorem for sheaves with singular hermitian metrics, generalizing earlier work by Fujita, Catanese--Kawamata, and Iwai.

Findings

Sheaves with semi-positive curvature metrics decompose into flat and ample parts.

The theorem applies to direct images of relative pluricanonical bundles.

It extends classical decomposition results in algebraic geometry.

Abstract

We prove that a torsion-free sheaf $\mathcal F$ endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition $\mathcal F \simeq \mathcal U \oplus \mathcal A$ where $\mathcal U$ is a hermitian flat bundle and $\mathcal A$ is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles $f_* \omega_{X/Y}^{\otimes m}$ under a surjective morphism $f\colon X \to Y$ of smooth projective varieties with $m\geq 2$. This extends previous results of Fujita, Catanese--Kawamata, and Iwai.