The famous Bogomolov-Tian-Todorov theorem says that the moduli space of Calabi-Yau manifold is smooth, that is locally a complex manifold. Doesn't this contradicts to the fact that the moduli space of K3 surface is non-Hausdorff?
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6$\begingroup$ Sometimes people use the word manifold to mean what others call "non-Hausdorff manifold". That's probably what's going on in whatever reference you're using. As I understand it, B-T-T theorem is a local theorem, it says the local deformation space is unobstructed (so we get a formal neighborhood of every point isomorphic to power series ring). It doesn't rule out complex analogues of line with double point. And indeed there are famous examples by Atiyah showing that the moduli space is not seperated. $\endgroup$– Daniel PomerleanoSep 1, 2013 at 9:23
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