In algebraic geometry, a crepant resolution of a singularity is a

resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual ma ...

that does not affect the

canonical class In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...

of the

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

. The term "crepant" was coined by by removing the prefix "dis" from the word "discrepant", to indicate that the resolutions have no

discrepancy Discrepancy may refer to: Mathematics * Discrepancy of a sequence * Discrepancy theory in structural modelling * Discrepancy of hypergraphs, an area of discrepancy theory * Discrepancy (algebraic geometry) Statistics * Discrepancy function in th ...

in the canonical class. The crepant resolution conjecture of states that the orbifold cohomology of a

Gorenstein Gorenstein may refer to: * Daniel Gorenstein (1923–1992), American mathematician, known for **Alperin–Brauer–Gorenstein theorem **Gorenstein–Harada theorem **Gorenstein ring **Gorenstein scheme **Gorenstein–Walter theorem * Eli Gorenstein ...

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

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 but need not be unique as they can be related by

flop In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate meas ...

s, 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 The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...

singularities (for example,

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...

Gorenstein singularities), there is a variety ''Y'' with Q-factorial

terminal Terminal may refer to: Computing Hardware * Terminal (electronics), a device for joining electrical circuits together * Terminal (telecommunication), a device communicating over a line * Computer terminal, a set of primary input and output devic ...

singularities and a

birational In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rationa ...

projective morphism ''f'': ''Y'' → ''X'' which is crepant in the sense that ''K''_''Y'' = ''f''*''K''_''X''.C. Birkar, P. Cascini, C. Hacon, J. McKernan. ''J. Amer. Math. Soc.'' 23 (2010), 405-468. Corollary 1.4.3.

Notes

References

* * * *{{Citation , last1=Ruan , first1=Yongbin , title=Gromov-Witten theory of spin curves and orbifolds , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

, location=Providence, R.I. , series=Contemp. Math. , mr=2234886 , year=2006 , volume=403 , chapter=The cohomology ring of crepant resolutions of orbifolds , pages=117–126 , isbn=978-0-8218-3534-0 Algebraic geometry Singularity theory