Nowhere vanishing 1-forms on varieties admitting a good minimal model

TL;DR

This paper proves conjectures linking nonvanishing 1-forms on varieties with good minimal models, revealing structural properties and decompositions involving Calabi-Yau and uniruled varieties over abelian schemes.

Contribution

It establishes new connections between nonvanishing 1-forms and smooth morphisms, and provides a structure theorem for certain varieties assuming minimal model existence.

Findings

Proved conjectures relating 1-forms to morphisms over abelian varieties.

Decomposition result for families of Calabi-Yau varieties with surjective maps.

Structure theorem for uniruled varieties with nowhere vanishing 1-forms.

Abstract

We prove several conjectures relating the existence of nonvanishing 1- forms to smooth morphisms over abelian varieties, assuming the existence of good minimal models. The proof involves a decomposition result for a family of Calabi-Yau varieties equipped with a surjective map to an abelian scheme. In the uniruled case, supposing the MRC base admits a good minimal model, we also achieve a structure theorem for those varieties admitting nowhere vanishing 1-forms.