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].

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) |