Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

Abstract

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, R). A Kähler cone (X,ḡ) is a metric cone over a Sasaki manifold (S, g), i.e. X = C(S) := S × R >0 withḡ = dr 2 + r 2 g, and (X,ḡ) 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 π : Y → X = C n / , with ⊂ SL(n, 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,ḡ) in every Kähler class in H 2 c (Y, 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.

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

“…As mentioned before, due to their importance in the AdS/CFT correspondence, Sasaki-Einstein manifolds have been widely studied by mathematicians and physicists. Inspired by some of these applications, and by following [19], [65], [30], our forth result provides a constructive method to describe the resolution of Calabi-Yau cones, with certain homogeneous Sasaki-Einstein manifolds realized as links of isolated singularities, by means of the Cartan-Remmert reduction [31] and the Calabi Ansatz technique [17]. The result is precisely the following.…”

Asymptotically conical Calabi–Yau manifolds, I

“…We thank MPIM and HIM Bonn for excellent working conditions during our stay in Bonn in Fall 2011, the members of the Imperial geometry group for many helpful conversations, and Jeff Cheeger and Gang Tian for their willingness to discuss [17]. Finally, we must acknowledge an intellectual debt of gratitude to Craig van Coevering, whose articles [54,55,56, 57] inspired much of our research.…”

“…Van Coevering [54] pointed out that all AC Kähler manifolds can be made Stein by contracting compact analytic sets. We then have the following rough general picture, which we will flesh out in Sections 4-5 by looking at some (extreme) special cases.…”

Examples of asymptotically conical Ricci-flat Kähler manifolds

“…Already in [59] many examples of crepant resolutions of Gorenstein toric Kähler cones which satisfy Corollary 1.2 are given. But for n = 3 a much more exhaustive existence result can be given.…”

“…The proof of (iii) follows from the following commutative diagram with exact rows (59) where K is the group generated by divisors with support in E = π −1 (o). It is easy to see that ι is an inclusion because the codimension of o ∈ X is at least 2.…”