crepant resolution
In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant" was coined by {{harvs |txt |last=Reid |first=Miles |authorlink=Miles Reid |year=1983}} by removing the prefix "dis" from the word "discrepant", to indicate that the resolutions have no discrepancy in the canonical class.
The crepant resolution conjecture of {{harvtxt|Ruan|2006}} states that the orbifold cohomology of a Gorenstein orbifold is isomorphic to a semiclassical limit of the quantum cohomology of a crepant resolution.
In 2 dimensions, crepant resolutions of complex Gorenstein quotient singularities (du Val singularities) always exist and are unique, in 3 dimensions they existT. Bridgeland, A. King, M. Reid. J. Amer. Math. Soc. 14 (2001), 535-554. Theorem 1.2. but need not be unique as they can be related by flops, and in dimensions greater than 3 they need not exist.
A substitute for crepant resolutions which always exists is a terminal model. Namely, for every variety X over a field of characteristic zero such that X has canonical singularities (for example, rational Gorenstein singularities), there is a variety Y with Q-factorial terminal singularities and a birational projective morphism f: Y → X which is crepant in the sense that KY = f*KX.C. Birkar, P. Cascini, C. Hacon, J. McKernan. J. Amer. Math. Soc. 23 (2010), 405-468. Corollary 1.4.3.
Notes
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References
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