Examples of asymptotically conical Ricci-flat Kähler manifolds

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VAN COEVERING C. C.

Mathematische Zeitschrift, cilt.267, sa.1, ss.465-496, 2011 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 267 Sayı: 1
  • Basım Tarihi: 2011
  • Doi Numarası: 10.1007/s00209-009-0631-7
  • Dergi Adı: Mathematische Zeitschrift
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.465-496
  • Anahtar Kelimeler: Calabi-Yau manifold, Einstein metric, Ricci-flat manifold, Sasaki manifold, Toric varieties
  • Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
  • Boğaziçi Üniversitesi Adresli: Evet

Özet

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 H2c(Y, 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. © 2009 Springer-Verlag.