Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

Details Ricci-flat Kähler metrics on crepant resolutions of Kähler cones Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

2008-06-23

arxiv.org/abs/0806.3728v3

Math. Annalen, vol. 347, no. 3 (2010), 581-611

Mathematics

Differential Geometry

26 pages. Accepted for publication in Mathematische Annalen

Scientific paper

10.1007/s00208-009-0446-1

We prove that a crepant resolution of a Ricci-flat K\"ahler cone X admits a complete Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\"ahler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat K\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.

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