Global division of cohomology classes via injectivity

TL;DR

This paper uses algebraic vanishing and injectivity theorems to strengthen and extend division theorems for cohomology classes, connecting algebraic and analytic methods in complex geometry.

Contribution

It provides a new algebraic proof of a strengthened Skoda-type division theorem and extends it to higher cohomology classes, incorporating multiplier ideals and injectivity results.

Findings

Strengthened division theorem for global sections

Extension to higher cohomology classes

Injectivity statement for multiplier ideals

Abstract

We note that the vanishing and injectivity theorems of Koll\'ar and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves, and to extend it to higher cohomology classes as well. For global sections, this is also a slightly more general statement of the algebraic translation of an analytic result of Siu. Along the way we write down an injectivity statement for multiplier ideals, and its standard consequences.