The Deformation Space of Calabi - Yau n - folds with Canonical Singularities Can Be

The Bogomolov-Tian-Todorov theorem ([10] and [12]) states that a non-singular n-fold X with c 1 (X) = 0 has unobstructed deformation theory, i.e. the moduli space of X is non-singular. This theorem was reproven using algebraic methods by Ran in [7]. Since then, it has been proven for Calabi-Yau n-folds with various mild forms of isolated singularities: ordinary double points by Kawamata [5] and Tian [11], Kleinian singularities by Ran [8], and finally, in the case of threefolds, arbitrary terminal singularities by Namikawa in [6]. Now the most natural class of singularities in the context of Calabi-Yau n-folds are canonical singularities. Indeed, if X is a Calabi-Yau n-fold with terminal singularities, and f : X → Y is a birational contraction, Y normal, then Y has canonical singularities. Thus the natural question to ask is: is the deformation space of Calabi-Yau n-folds with canonical singularities unobstructed? Given the history of this problem presented above, it appears worthwhile to give a counterexample to this most general question. We give an example of a Calabi-Yau n-fold X with the simplest sort of dimension 1 canonical singularities, and show that X lies in the intersection of two distinct families of Calabi-Yau n-folds. One is a family of generically non-singular Calabi-Yaus, and the other is a family of Calabi-Yaus which generically have terminal singularities. (In the case n = 3, these are also non-singular.) In particular, the point of the moduli space corresponding to X is in the intersection of two components of moduli space, and hence has obstructed deformation theory. We do not address the issue of isolated singularities here. That issue is more of a local one, and the obstructedness of Calabi-Yaus with isolated singularities is related to the obstructedness of the singularities themselves. We will explore this in a future paper, and give applications to smoothing Calabi-Yaus with canonical singularities.