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 Miles Reid (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 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 exist^[1] 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 K_Y = f*K_X.^[2]

Notes

^ T. Bridgeland, A. King, M. Reid. J. Amer. Math. Soc. 14 (2001), 535-554. Theorem 1.2.
^ C. Birkar, P. Cascini, C. Hacon, J. McKernan. J. Amer. Math. Soc. 23 (2010), 405-468. Corollary 1.4.3.

References

Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James (2010), "Existence of minimal models for varieties of log general type", Journal of the American Mathematical Society, 23 (2): 405–468, arXiv:math.AG/0610203, Bibcode:2010JAMS...23..405B, doi:10.1090/S0894-0347-09-00649-3, MR 2601039
Bridgeland, Tom; King, Alastair; Reid, Miles (2001), "The McKay correspondence as an equivalence of derived categories", Journal of the American Mathematical Society, 14 (3): 535–554, doi:10.1090/S0894-0347-01-00368-X, MR 1824990
Reid, Miles (1983), "Minimal models of canonical 3-folds", Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1, North-Holland, pp. 131–180, ISBN 978-0-444-86612-7, MR 0715649
Ruan, Yongbin (2006), "The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Providence, R.I.: American Mathematical Society, pp. 117–126, ISBN 978-0-8218-3534-0, MR 2234886

[1] T. Bridgeland, A. King, M. Reid. J. Amer. Math. Soc. 14 (2001), 535-554. Theorem 1.2.

[2] C. Birkar, P. Cascini, C. Hacon, J. McKernan. J. Amer. Math. Soc. 23 (2010), 405-468. Corollary 1.4.3.

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