Examples of asymptotically conical Ricci-flat K\"{a}hler manifolds

TL;DR

This paper constructs numerous examples of asymptotically conical Ricci-flat Kähler manifolds by demonstrating the existence of crepant resolutions for certain Ricci-flat Kähler cones, expanding known classes of such manifolds.

Contribution

It proves that crepant resolutions of Ricci-flat Kähler cones admit complete Ricci-flat Kähler metrics and provides explicit examples, including Gorenstein toric and hypersurface singularities.

Findings

Every 3D Gorenstein toric Kähler cone admits a crepant resolution with Ricci-flat metric.

Constructs infinitely many asymptotically conical Ricci-flat Kähler manifolds.

Provides examples distinguished by the third Betti number b_3(Y).

Abstract

The author has proved that a crepant resolution Y of a Ricci-flat K\"{a}hler cone X admits a complete Ricci-flat K\"{a}hler metric asymptotic to the cone metric in every K\"{a}hler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE K\"{a}hler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric K\"{a}hler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat K\"{a}hler cone metrics by the work of C. Boyer, K. Galicki, J. Koll\'{a}r, and others. Two families of hypersurface examples are given which are distinguished by the condition b_3(Y)=0 or b_3(Y)>0.