Journal of the Mathematical Society of Japan

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

Ryushi GOTO

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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

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

First available in Project Euclid: 24 July 2012

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Zentralblatt MATH identifier

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

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


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.

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