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Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

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We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in \({H^2_c(Y,\mathbb{R})}\). A Kähler cone \({(X,\bar{g})}\) is a metric cone over a Sasaki manifold (S, g), i.e. \({X=C(S):=S\times\mathbb{R}_{ >0 }}\) with \({\bar{g}=dr^2 +r^2 g}\), and \({(X,\bar{g})}\) is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat Kähler metrics on crepant resolutions \({\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}\), with \({\Gamma\subset SL(n,\mathbb{C})}\), due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric Kähler cone admits a Ricci-flat Kähler cone metric. It follows that if a toric Kähler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T n-invariant Ricci-flat Kähler metric asymptotic to the cone metric \({(X,\bar{g})}\) in every Kähler class in \({H^2_c(Y,\mathbb{R})}\). A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.

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  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Craig van Coevering

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van Coevering, C. Ricci-flat Kähler metrics on crepant resolutions of Kähler cones. Math. Ann. 347, 581–611 (2010). https://doi.org/10.1007/s00208-009-0446-1

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