[Paper Review] Examples of asymptotically conical Ricci-flat K\

Craig van Coevering|arXiv (Cornell University)|Dec 30, 2008

Geometry and complex manifolds34 references3 citations

TL;DR

This paper constructs infinitely many new examples of asymptotically conical Ricci-flat Kähler manifolds by proving that every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution supporting a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class. It further provides two families of hypersurface singularities with distinct topological invariants, yielding examples with b₃(Y) = 0 or b₃(Y) > 0.

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.

Motivation & Objective

• To extend the construction of Ricci-flat Kähler metrics beyond known ALE spaces to asymptotically conical manifolds.
• To identify broad classes of singularities whose crepant resolutions admit complete Ricci-flat Kähler metrics.
• To provide explicit, infinite families of such manifolds with controlled topology, particularly distinguishing cases by b₃(Y).
• To generalize prior results on Ricci-flat metrics on crepant resolutions to the asymptotically conical setting.

Proposed method

• Prove that every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution supporting a complete Ricci-flat Kähler metric in every Kähler class.
• Apply the general existence theorem for Ricci-flat Kähler metrics on crepant resolutions of Ricci-flat Kähler cones.
• Construct explicit examples using toric geometry and the classification of Gorenstein toric singularities.
• Analyze the topology of the resolutions, particularly focusing on the third Betti number b₃(Y), to distinguish two families.
• Use known results on hypersurface singularities admitting Ricci-flat Kähler cone metrics, as established by Boyer, Galicki, Kollár, and others.
• Distinguish examples by the value of b₃(Y), yielding two distinct families: one with b₃(Y) = 0 and one with b₃(Y) > 0.

Experimental results

Research questions

• RQ1 Which classes of singularities admit crepant resolutions that support complete Ricci-flat Kähler metrics asymptotic to a cone metric?
• RQ2 Can the existence of such metrics be guaranteed for all 3-dimensional Gorenstein toric Kähler cones?
• RQ3 What topological invariants, such as b₃(Y), distinguish different families of these asymptotically conical Ricci-flat manifolds?
• RQ4 How do the topological properties of the resolution relate to the geometry of the original cone?
• RQ5 Can explicit constructions be given for infinite families of such manifolds with controlled Betti numbers?

Key findings

• Every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution supporting a complete Ricci-flat Kähler metric in every Kähler class.
• Infinitely many new examples of asymptotically conical Ricci-flat Kähler manifolds are constructed via this method.
• Two distinct families of hypersurface singularities yield crepant resolutions with b₃(Y) = 0 and b₃(Y) > 0, respectively.
• The construction generalizes known results on ALE spaces to the asymptotically conical setting.
• The topology of the resolution, particularly b₃(Y), serves as a key invariant to classify the resulting Ricci-flat manifolds.
• The results confirm the existence of Ricci-flat metrics on crepant resolutions of singularities known to admit Ricci-flat Kähler cone metrics.