algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a crepant resolution of a singularity is a resolution that does not affect the

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

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 N ...

. The term "crepant" was coined by 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 states that the orbifold cohomology of a 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 that 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 In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with mi ...

) always exist and are unique, in 3 dimensions they exist but need not be unique as they can be related by

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

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 exampl ...

singularities (for example,

rational Rationality is the quality of being guided by or based on reason. 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 ...

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

terminal Terminal may refer to: Computing Hardware * Computer terminal, a set of primary input and output devices for a computer * Terminal (electronics), a device for joining electrical circuits together ** Battery terminal, electrical contact used to ...

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 rational f ...

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