Examples of asymptotically conical Ricci-flat Kähler manifolds

By Craig van Coevering. 2008

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Previously the author has proved 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 $${H^2_c(Y,\mathbb R)}$$. These manifolds can be considered to be generalizations of the Ricci-flat ALE Kähler 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 Kähler 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 Kähler cone metrics by the work of C. Boyer, K. Galicki, J. Kollár, and others. We concentrate on 3-dimensional examples. Two families of hypersurface examples are given which are distinguished by the condition b3(Y) = 0 or b3(Y) ≠ 0.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/511042ecb34fcb801d230228658322ae6fe1e0ee

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018, methodology): “In order to show that X is homeomorphic to a normal affine variety, we can use the method in Donaldson-Sun [11], which in turn is based on an idea of Van Coevering [19] in the setting of Kähler cones with smooth links.”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, methodology): “We will make use of an idea due to Van Coevering [34], who proved essentially the same result for a Kähler cone with smooth cross section.”