Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds

Details Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds

2006-12-22

arxiv.org/abs/math/0612715v1

Proceedings of G\"okova Geometry-Topology Conference, pages 137-153, International Press, 2006

Mathematics

Differential Geometry

16 pages

Scientific paper

We study a class of asymptotically cylindrical Ricci-flat K\"ahler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any `small' asymptotically cylindrical Ricci-flat deformations of asymptotically cylindrical Ricci-flat K\"ahler metrics are again K\"ahler, possibly with respect to a perturbed complex structure. We also find the dimension of the moduli space for these small deformations. In the class of asymptotically cylindrical Ricci-flat metrics on $2n$-manifolds, the holonomy reduction to SU(n) is an open condition.

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