Unobstructed deformations for singular Calabi-Yau varieties

TL;DR

This paper extends deformation theory results for singular Calabi-Yau varieties to broader classes with Du Bois singularities, showing unobstructed deformations under new conditions involving the bla-lemma.

Contribution

It generalizes Imagi's deformation results to varieties with Du Bois singularities without weighted homogeneous assumptions, replacing the Ke4hler condition with a bla-7-lemma-based resolution.

Findings

Deformations are unobstructed for varieties with Du Bois and local complete intersection singularities.

Results apply to log Calabi-Yau and Fano varieties.

Generalizes deformation theory beyond weighted homogeneous singularities.

Abstract

Let $Y$ be a compact Gorenstein analytic space with only isolated singularities and trivial dualizing sheaf. A recent paper of Imagi studies the deformation theory of $Y$ in case the singularities of $Y$ are weighted homogeneous and rational and $Y$ is K\"ahler. In this note, assuming that $H^1(Y;\mathcal{O}_Y) =0$, we generalize Imagi's results to the case where the singularities of $Y$ are Du Bois, with no assumption that they be weighted homogeneous, and where the K\"ahler assumption is replaced by the hypothesis that there is a resolution of singularities of $Y$ satisfying the $\partial\bar\partial$-lemma. As a consequence, if the singularities of $Y$ are additionally local complete intersections, then the deformations of $Y$ are unobstructed. The log Calabi-Yau and Fano cases are also discussed.