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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?

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    $\begingroup$ Yes, Calabi-Yau manifolds have unobstructd 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. $\endgroup$ – abx Apr 23 '15 at 12:49
  • $\begingroup$ @abx Great! That answers my question. The smoothness of the moduli stack is equivalent to "unobstructed deformations" by general abstract nonsense, right? Could you put your comment as an answer, so that I can accept it? Many thanks! $\endgroup$ – El Nino Apr 23 '15 at 12:58
  • $\begingroup$ As additional comment, moduli space of polarized Calabi-Yau is an orbifold since there is no nonzero holomorphic vector field on a Calabi-Yau manifold $\endgroup$ – user21574 Jun 2 '17 at 13:46
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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.

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