Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

By Craig van Coevering. 2008

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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 Tn-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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0e9dc0cfce8d34de219313e98426a171b3a07ab5

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In Applied Mathematics

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In Geometry and Topology

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  2. M. Gualtieri

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