Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

2008/06/30 by Craig van Coevering
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.DG #msc:14M25 #msc:53C25 #msc:53C55

paper · pdf · doi:10.1007/s00208-009-0446-1

published as Math. Annalen, vol. 347, no. 3 (2010), 581-611 · 26 pages. Accepted for publication in Mathematische Annalen

arxiv created 2009/10/29 · openalex publication_date 2009/11/17 · arxiv updated 2010/07/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29

Abstract

We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H²_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=Cⁿ /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 Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.

Citations

• Symmetry-breaking vacua and baryon condensates in AdS/CFT correspondence
• Examples of asymptotically conical Ricci-flat Kähler manifolds
• On the existence of a complete Kähler metric on non‐compact complex manifolds and the regularity of fefferman's equation
• On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth