Global smoothing of singular Fano and Calabi-Yau varieties

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.