Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

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VAN COEVERING C. C.

Mathematische Annalen, cilt.347, sa.3, ss.581-611, 2010 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 347 Sayı: 3
  • Basım Tarihi: 2010
  • Doi Numarası: 10.1007/s00208-009-0446-1
  • Dergi Adı: Mathematische Annalen
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.581-611
  • 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

We prove 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). A Kähler cone (x, g)is a metric cone over a Sasaki manifold (S, g), i. e. X = C(S):= S × R>0 with g = dr2 + r2g, and (X, g)is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat Kähler metrics on crepant resolutions, with π: Y → X = Cn/ Γ, with Γ ⊂ SL(n, C), due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric Kähler cone admits a Ricci-flat Kähler cone metric. It follows that if a toric Kähler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a Tn-invariant Ricci-flat Kähler metric asymptotic to the cone metric (X, g) in every Kähler class in {H2c(Y, R). A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry. © 2009 Springer-Verlag.