Deformations of fibered Calabi--Yau varieties

TL;DR

This paper extends Kollár's results on the stability of elliptic fibrations under small deformations to a broader class of fibered Calabi--Yau varieties, using Hodge theory and deformation criteria.

Contribution

It generalizes deformation stability results for fibered Calabi--Yau varieties beyond elliptic cases, employing Hodge theoretic methods and the $T^1$-lifting criterion.

Findings

Small deformations preserve elliptic fibration structure under broader conditions.

The approach applies to semiample line bundles without cohomological restrictions.

Deformations maintain semiample line bundles up to homological equivalence.

Abstract

Koll\'{a}r showed that small deformations of elliptically fibered smooth $K$-torsion varieties with $H^2(X,\mathcal{O}_X)=0$ remain elliptically fibered. We extend this result to any fibered smooth $K$-torsion variety $X$ with $H^2(X,\mathcal{O}_X)=0$, using Hodge theoretic techniques and the $T^1$-lifting criterion of Kawamata--Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth $K$-torsion variety remain semiample up to homological equivalence.