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Deformations of Calabi-Yau Kleinfolds. (English) Zbl 0827.32021
Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 451-457 (1992).
Summary: Recall that a Calabi-Yau manifold is a compact Kähler manifold $$X$$ whose canonical bundle $$K_X$$ is trivial (we do not assume any extra conditions such as on the Hodge numbers of $$X)$$. A well-known theorem, due to Bogomolov, Tian and Todorov, asserts that such $$X$$ always has unobstructed deformations, i.e. its local moduli (Kuranishi) space is smooth. By a Calabi-Yau Kleinfold we shall mean a compact analytic variety $$X_0$$ which admits a resolution of singularities by a compact Kähler manifold, and whose singularities are Kleinian i.e. isolated simple hypersurface singularities (type $$A - D - E)$$, and satisfy an additional local condition of “goodness” (see Definition below; this condition is automatically satisfied at least for all 3-fold Kleinian, singularities and $$A_1$$ singularities in all dimensions, and may well hold for all Kleinian singularities). By a theorem of Grauert, $$X_0$$ admits a local moduli space or semiuniversal deformation. Our purpose here is to prove the following.
Theorem 1. Any Calabi-Yau Kleinfold of dimension $$\geq 3$$ has unobstructed deformations.
For the entire collection see [Zbl 0816.00010].

##### MSC:
 32G10 Deformations of submanifolds and subspaces 32J27 Compact Kähler manifolds: generalizations, classification 32S25 Complex surface and hypersurface singularities 32S45 Modifications; resolution of singularities (complex-analytic aspects)