Global smoothing of singular Fano and Calabi-Yau varieties

TL;DR

This paper investigates smoothing techniques for Fano and Calabi-Yau varieties with specific singularities, providing new criteria and extending previous results to higher dimensions and broader singularity classes.

Contribution

It introduces new smoothing criteria for singular Fano and Calabi-Yau varieties, generalizing prior results and extending to higher dimensions with Du Bois lci singularities.

Findings

Fano varieties with isolated Du Bois lci singularities can be deformed to those with only 1-rational singularities.

Calabi-Yau varieties can be deformed to those with only 1-Du Bois singularities.

Global criteria involving Hodge-Du Bois numbers ensure smoothability when allowing 1-liminal singularities.

Abstract

We study the problem of smoothing Fano and Calabi-Yau varieties with isolated Du Bois lci singularities. For Fano varieties, we show that any such $Y$ admits a deformation to a Fano variety with only $1$-rational singularities, and if none of the singularities of $Y$ are $1$-rational, then $Y$ is smoothable. For Calabi-Yau varieties, we show first that any such $Y$ deforms to a Calabi-Yau with only $1$-Du Bois singularities. Moreover, if none of the singularities of $Y$ are $1$-Du Bois then $Y$ is smoothable. When we allow $1$-liminal singularities, we give a global criterion in terms of the Hodge-Du Bois numbers of $Y$ which ensures that $Y$ is smoothable. These theorems recover and generalize results for threefolds of Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza. In higher dimensions, our results provide alternative smoothing conditions and also extend the work of Friedman-Laza from the case of rational hypersurface singularities to Du Bois lci singularities.