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Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric. (English) Zbl 0696.53040

Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 629-646 (1987).
[For the entire collection see Zbl 0651.00012.]
It is proved that a Kähler manifold X with trivial canonical bundle has a smooth Kuranishi space, or, in other words, the deformation of X is unobstructed. According to the author, this result was first proved in Bogomolov’s preprint, but with a much more complicated proof. The present proof is by constructing a vector (0,1)-form \(\phi\) (t) with the integrability condition for the deformation based on an isomorphism \(T_ x \overset \sim \rightarrow \Omega^{n-1}.\) As an application the author starts studying the period mapping for such X. It is proved that the Peterson-Weil metric on the universal polarized deformation space is the pull-back of an invariant metric on the period space D.
Reviewer: E.Horikawa

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32G05 Deformations of complex structures
14J30 \(3\)-folds

Citations:

Zbl 0651.00012