Journal of the Mathematical Society of Japan

Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities

Ryushi GOTO

Full-text: Open access

Abstract

Let X0 be an affine variety with only normal isolated singularity at p. We assume that the complement X0 \ {p} is biholomorphic to the cone C(S) of an Einstein-Sasakian manifold S of real dimension 2n − 1. If there is a resolution of singularity π: XX0 with trivial canonical line bundle KX, then there is a Ricci-flat complete Kähler metric for every Kähler class of X. We also obtain a uniqueness theorem of Ricci-flat conical Kähler metrics in each Kähler class with a certain boundary condition. We show there are many examples of Ricci-flat complete Kähler manifolds arising as crepant resolutions.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 3 (2012), 1005-1052.

Dates
First available in Project Euclid: 24 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1343133753

Digital Object Identifier
doi:10.2969/jmsj/06431005

Mathematical Reviews number (MathSciNet)
MR2965437

Zentralblatt MATH identifier
1262.53041

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Keywords
Ricci-flat Kähler metric Einstein-Sasakian metric Monge-Ampère equation Calabi-Yau structures

Citation

GOTO, Ryushi. Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities. J. Math. Soc. Japan 64 (2012), no. 3, 1005--1052. doi:10.2969/jmsj/06431005. https://projecteuclid.org/euclid.jmsj/1343133753


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References

  • T. Aubin, Nonlinear Analysis on Manifolds, \MA equations, Grundlehren Math. Wiss., 252, A Series of Comprehensive Studies on Mathematics, Springer-Verlag, 1982.
  • S. Bando and R. Kobayashi, Ricci-flat Kähler metrics on affine algebraic manifolds, Geometry and Analysis on Manifolds, In: (ed. T. Sunada), Lecture Notes in Math., 1339, Springer-Verlag, 1988, pp.,20–31.
  • S. Bando and R. Kobayashi, Ricci-flat Kähler metrics on affine algebraic manifolds, II, Math. Ann., 287 (1990), 175–180.
  • C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monogr., Oxford University Press, Oxford, 2008.
  • C. P. Boyer, K. Galicki and M. Nakamaye, On positive Sasakian geometry, Geom. Dedicata, 101 (2003), 93–102.
  • E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. (4), 12 (1979), 269–294.
  • P. Candelas and Xenia C. de la Ossa, Comments on conifolds, Nuclear Phys. B, 342 (1990), 246–268.
  • R. J. Conlon, A vanishing theorem for positive Sasaki manifolds, preprint, 2011.
  • A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, [Transversely elliptic operators on a Riemannian foliation, and applications], Compositio Math., 73 (1990), 57–106.
  • L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333–363.
  • A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom., 83 (2009), 585–635.
  • S. Gallot, A Sobolev inequality and some geometric applications, Spectra of Riemannian manifolds, Kaigai Publications, Tokyo, 1983, pp.,45–55.
  • R. Goto, On hyper-Kähler manifolds of type $A_\infty$ and $D_\infty$, Comm. Math. Phys., 198 (1998), 469–491.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., 224, A Series of Comprehensive Studies in Mathematics, Springer-Verlag, 1983.
  • A. Grigor'yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble), 55 (2005), 825–890.
  • H.-J. Hein, Weighted Sobolev inequalities under lower Ricci curvature bounds, Proc. Amer. Math. Soc., 139 (2011), 2943–2955.
  • N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys., 108 (1987), 535–589.
  • S. Ishii, Introduction to singularities (Tokuiten nyumon), Springer-Verlag, Tokyo, 1997.
  • D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford Math. Monogr., Oxford University Press, Oxford, 2000.
  • P. B. Kronheimer, The construction of ALE spaces as Hyper-Kähler quotients, J. Differential Geom., 29 (1989), 665–683.
  • N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 487–523 [Russian], English translation in Math. USSR-Izv., 20 (1983), 459–492.
  • F. W. Kamber and P. Tondeur, de Rham-Hodge theory for Riemannian foliations, Math. Ann., 277 (1987), 415–431.
  • R. B. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12 (1985), 409–447.
  • R. B. Melrose, Atiyah-Patodi-Singer Index Theorem, Res. Notes in Math., 4, A K Peters Ltd., Wellesley, MA, 1993.
  • D. Martelli and J. Sparks, Symmetry-breaking vacua and baryon condensates in AdS/CFT correspondence, Phys. Rev D, 79 (2009), 065009.
  • D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys., 262 (2006), 51–89.
  • D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Math. Phys., 280 (2008), 611–673.
  • V. Minerbe, Weighted Sobolev inequality and Ricci flat manifolds, Geom. Funct. Anal., 18 (2009), 1696–1749.
  • H. Nakajima, Nonlinear Analysis and Complex Geometry, Gendai suugaku no tenkai (in Japanese), Iwanami Shoten, 1999.
  • T. Oota and Y. Yasui, Explicit toric metric on resolved Calabi-Yau cone, Phys. Lett. B, 639 (2006), 54–56.
  • B. Santoro, Existence of complete Kähler Ricci-flat metrics on crepant resolutions, math.DG/0902.0595
  • Y. T. Siu, Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics, DMV Seminar, 8, Birkhäuser, 1987.
  • G. Tian, Canonical Metrics in Kähler Geometry, Lectures in Math. ETH Zürich, Birkhäuser, 2000.
  • G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math., 101 (1990), 101–172.
  • G. Tian and S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., 1, World Scientific Publishing, 1987, pp.,574–628.
  • G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature, I, J. Amer. Math. Soc., 3 (1990), 579–609.
  • G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature, II, Invent. Math., 106 (1991), 27–60.
  • G. Tian and S.-T. Yau, Kähler-Einstein metrics on complex surfaces with $C_1>0$, Comm. Math. Phys., 112 (1987), 175–203.
  • N. S. Trudinger, Lectures on nonlinear elliptic equations of second order, Lectures in Mathematical Sciences, The University of Tokyo, 9, 1995.
  • C. van Coevering, Ricci-flat Kähler metrics on crepant resolutions of Kähler cones, Math. Ann., 347 (2010), 581–611.
  • C. van Coevering, Regularity of asymptotically conical Ricci-flat Kähler metrics, arXiv:0912.3946