Deformations of singular Fano and Calabi-Yau varieties

TL;DR

This paper extends deformation theory results for singular Fano and Calabi-Yau varieties, especially in higher dimensions, highlighting the roles of higher Du Bois and rational singularities in smoothability.

Contribution

It generalizes deformation results from threefolds to higher dimensions, replacing terminal with canonical singularities and identifying new classes of singularities where smoothing applies.

Findings

Smoothability results for generalized Fano varieties with non-1-rational singularities

Smoothability of Calabi-Yau varieties with certain non-1-rational but 1-Du Bois singularities

Extension of deformation theory to higher dimensions with new singularity classes

Abstract

The goal of this paper is to generalize results concerning the deformation theory of Calabi-Yau and Fano threefolds with isolated hypersurface singularites, due to the first author, Namikawa and Steenbrink. In particular, under the assumption of terminal singularities, Namikawa proved smoothability in the Fano case and also for generalized Calabi-Yau threefolds assuming that a certain topological first order condition is satisfied. In the case of dimension $3$, we extend their results by, among other things, replacing terminal with canonical. In higher dimensions, we identify a class of singularities to which our method applies. A surprising aspect of our study is the role played by the higher Du Bois and higher rational singularities. Among other deformation theoretic results in higher dimensions, we obtain smoothing results for generalized Fano varieties whose singularities are not $1$-rational, and for generalized Calabi-Yau varieties whose singularities are not $1$-rational but are $1$-Du Bois under a topological condition on the links which is similar to the first order obstruction in dimension $3$.