Deformations of Calabi-Yau varieties with isolated log canonical singularities

TL;DR

This paper extends deformation theory results of Calabi-Yau varieties to include those with isolated Gorenstein log canonical singularities, demonstrating existence of smoothings and unobstructed deformations in new singularity cases.

Contribution

It generalizes deformation theory to Calabi-Yau varieties with isolated 0-liminal singularities, including existence of smoothings and extending unobstructedness results.

Findings

Existence of first order smoothings for isolated 0-liminal hypersurface singularities.

Extension of Namikawa's unobstructedness theorem to certain non-canonical singularities.

Presentation of examples illustrating the extended deformation theory.

Abstract

Recent progress in the deformation theory of Calabi-Yau varieties $Y$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $k$-liminal singularities for $k\ge 1$. The goal of this paper is to show that certain aspects of this study extend naturally to the $0$-liminal case as well, i.e. to Calabi-Yau varieties $Y$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $Y$ in the case of isolated $0$-liminal hypersurface singularities, and extend Namikawa's unobstructedness theorem for deformations of singular Calabi-Yau threefolds $Y$ with canonical singularities to the case where $Y$ has an isolated $0$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples.