Abundance for varieties with many differential forms

TL;DR

This paper proves the abundance conjecture for varieties with many differential forms, showing that certain positivity conditions lead to semiample canonical divisors, with some results unconditional in specific cases.

Contribution

It establishes the abundance conjecture under conditions of many differential forms and extends results to uniruled varieties and Calabi-Yau cases, assuming the MMP in lower dimensions.

Findings

Abundance conjecture holds for varieties with many differential forms.

Hermitian semipositive canonical divisors are often semiample.

Uniruled klt pairs have good models under these conditions.

Abstract

We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This implies, for instance, that under this condition, hermitian semipositive canonical divisors are almost always semiample, and that klt pairs whose underlying variety is uniruled have good models in many circumstances. When the numerical dimension of $K_X$ is $1$, our results hold unconditionally in every dimension. We also treat a related problem on the semiampleness of nef line bundles on Calabi-Yau varieties.