Deformations of Compact Calabi--Yau Conifolds

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.