Deformations of singular Fano and Calabi-Yau varieties

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, recently introduced by Mustat ¸ȃ, Popa, Saito and their coauthors. While the results in this paper only require the notion of 1rational and 1-Du Bois singularities, we also consider the case of k-rational and k-Du Bois isolated hypersurface singularities for k ≥ 1, which is of independent interest. In particular, we propose a general definition for k-rational singularities and show that it agrees with the previous definition in the case of isolated hypersurface 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.

Deformations of some local Calabi-Yau manifolds

“…This is the second paper in our study of deformations of Calabi-Yau varieties with canonical singularities. In [FL22], strengthening previous results of [Fri86], [NS95], [Nam94,Nam02], [Kaw92] among others, we obtained some general unobstructedness results ([FL22, Thm. 0.15]) and some general smoothing results (e.g.…”

“…0.14]). A key aspect of [FL22] is the fact that we work modulo the deformations induced from a good resolution of the singularity, both in the local and global setting. The purpose of this follow-up paper is to study the complementary question of the relationship between deformations of the resolutions and the deformations of the original variety, somewhat in the spirit of Wahl [Wah76] for the case of surface singularities.…”

“…In the case of a small resolution in dimension three, this is closely related to the Clemens conjecture for Calabi-Yau threefolds. Thus, the strategy of [FL22] is to work modulo this image, i.e. with K = Coker H 1 ( X;…”

Higher Du Bois and higher rational singularities

“…A brief description of the contents of this paper is as follows. After a quick review of the definition and the basic facts about higher Du Bois singularities (following [MOPW21], [JKSY21], and [FL22,§3]) and examples illustrating some of the issues occurring in the singular case, we prove Theorem 1.5 regarding the flatness of the relative Kähler differentials. After these preliminaries, we establish Theorem 1.2 and Corollary 1.4.…”

“…(2) There is a companion definition to that of k-Du Bois singularities, namely the k-rational singularities defined in [KL20, §4], [FL22,§3]. Is there a result analogous to Theorem 1.2 for k-rational singularities?…”