Deformations of Compact Calabi--Yau Conifolds

Abstract

Let $X$ be a compact normal Kähler space whose canonical sheaf is a rank-one free ${\mathcal{O}}_X$ module and whose singularities are isolated, rational and quasi-homogeneous. We prove then that under a topological hypothesis the obstruction to deforming $X$ concentrates upon its singularities, generalizing partially the results of Namikawa--Gross. We prove also that under a certain hypothesis the locally trivial deformations of $X$ are unobstructed.

arXiv comment76 pages. I have corrected the errors in the version 1. Theorem 3.4 and Corollary 3.5 were not true as they are. There is a counterexample to Corollary 3.5 with $p=q=0,$ given by the harmonic function $r^{2-2n}.$ Also Lemma 6.5 was not really proved