Decomposition of direct images of logarithmic differentials

TL;DR

This paper provides a concise proof of Kollár's theorem on the splitting of derived direct images of the canonical sheaf, utilizing a stronger decomposition theorem for logarithmic differentials and weak semistable reduction.

Contribution

It introduces a new decomposition theorem for direct images of logarithmic differentials and offers a simplified proof of Kollár's splitting theorem.

Findings

Derived direct image of the canonical sheaf splits into a sum of cohomology sheaves.

Decomposition theorem for direct images of logarithmic differentials established.

Utilizes weak semistable reduction to achieve the results.

Abstract

The purpose of this note is to give a short proof of a theorem of Koll\'ar that the derived direct image of the canonical sheaf splits into a sum of its cohomology sheaves. This is deduced from a stronger decomposition theorem for direct images of sheaves of logarithmic differentials, together with the weak semistable reduction theorem.