Fibered stable varieties

TL;DR

This paper proves that fibrations of stable varieties with stable fibers and base deform uniquely under small deformations, and establishes a related vanishing theorem for certain differential forms.

Contribution

It demonstrates the deformation invariance of fibrations in stable varieties and introduces a Bogomolov-Sommese type vanishing result for vector bundles.

Findings

Fibration structures deform uniquely under small deformations.

Established a Bogomolov-Sommese type vanishing theorem.

Results apply to stable varieties in the sense of Kollár and Shepherd-Barron.

Abstract

We show that if a stable variety (in the sense of Koll\'ar and Shepherd-Barron) admits a fibration with stable fibers and base, then this fibration structure deforms (uniquely) for all small deformations. During our proof we obtain also a Bogomolov-Sommese type vanishing for vector bundles and reflexive differential $n-1$-forms.