Crepant resolution
Encyclopedia
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 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
is isomorphic to a semiclassical limit of the quantum cohomology of a crepant resolution.
In 2 dimensions, crepant resolutions always exist and are unique, in 3 dimensions they exist but need not be unique as they can be related by flops, and in dimensions greater than 3 they need not exist.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a crepant resolution of a singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
is a resolution
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...
that does not affect the canonical class of the manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. 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
Daniel Gorenstein
Daniel E. Gorenstein was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings...
orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
is isomorphic to a semiclassical limit of the quantum cohomology of a crepant resolution.
In 2 dimensions, crepant resolutions always exist and are unique, in 3 dimensions they exist but need not be unique as they can be related by flops, and in dimensions greater than 3 they need not exist.