Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?


In many cases I know, the moduli stack is smooth (e.g., complete intersections, rigid Calabi-Yau threefolds, Calabi-Yau's with $\tau=1$). Is this the case in general?



Singularities of the moduli stack of Calabi-Yau threefolds
Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.


