Abundance for varieties with many differential forms

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.

References in corpus (2)

• Nef line bundles on Calabi-Yau threefolds, I
• Versions of injectivity and extension theorems

Cited by in corpus (11)

• On Generalised Abundance, I
• Injectivity theorem for pseudo-effective line bundles and its applications
• On the existence of minimal models for log canonical pairs
• Nef line bundles on Calabi-Yau threefolds, I
• On Generalised Abundance, II
• The Nonvanishing problem for varieties with nef anticanonical bundle
• Special MMP for log canonical generalised pairs
• Rationally connected varieties - on a conjecture of Mumford
• Asymptotic estimate of cohomology groups valued in pseudo-effective line bundles
• Non-vanishing theorem for lc pairs admitting a Calabi--Yau pair
• On Nonvanishing for uniruled log canonical pairs