Riemann–Hilbert correspondence
Encyclopedia
In mathematics, the Riemann–Hilbert correspondence is a generalization of Hilbert's twenty-first problem
to higher dimensions. The original setting was for Riemann surface
s, where it was about the existence of regular differential equations with prescribed monodromy
groups. In higher dimensions, Riemann surfaces are replaced by complex manifold
s of dimension > 1, and there is a correspondence between certain systems of partial differential equation
s (linear and having very special properties for their solutions) and possible monodromies of their solutions.
Such a result was proved independently by Masaki Kashiwara
(1980) and Zoghman Mebkhout
(1980).
Riemann–Hilbert correspondence (general form): there is a functor DR called the de Rham functor, that is an equivalence from the category of holonomic
D-module
s on X with regular singularities to the category of perverse sheaves on X.
By considering the irreducible elements of each category, this gives a 1:1 correspondence between isomorphism classes of
and
A D-module
is something like a system of differential equations on X, and a local system on a subvariety is something like a description of possible monodromies, so this correspondence can be thought of as describing certain systems of differential equations in terms of the monodromies of their solutions.
Hilbert's twenty-first problem
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this ....
to higher dimensions. The original setting was for Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s, where it was about the existence of regular differential equations with prescribed monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
groups. In higher dimensions, Riemann surfaces are replaced by complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s of dimension > 1, and there is a correspondence between certain systems of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (linear and having very special properties for their solutions) and possible monodromies of their solutions.
Such a result was proved independently by Masaki Kashiwara
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Sato and Kashiwara have collaborated on algebraic analysis and D-module theory.He is a member of the French Academy of Sciences.- Concepts named after Kashiwara :...
(1980) and Zoghman Mebkhout
Zoghman Mebkhout
Zoghman Mebkhout is an Algerian mathematician known for his work in algebraic analysis, geometry, and representation theory, more precisely on the theory of D-modules ....
(1980).
Statement
Suppose that X is a complex variety.Riemann–Hilbert correspondence (general form): there is a functor DR called the de Rham functor, that is an equivalence from the category of holonomic
Holonomic
In mathematics and physics, the term holonomic may occur with several different meanings.-Holonomic basis:A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:[e_j,e_k]=0 \,...
D-module
D-module
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
s on X with regular singularities to the category of perverse sheaves on X.
By considering the irreducible elements of each category, this gives a 1:1 correspondence between isomorphism classes of
- irreducible holonomic D-modules on X with regular singularities,
and
- intersection cohomologyIntersection cohomologyIn topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
complexes of irreducible closed subvarieties of X with coefficients in irreducible local systems.
A D-module
D-module
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
is something like a system of differential equations on X, and a local system on a subvariety is something like a description of possible monodromies, so this correspondence can be thought of as describing certain systems of differential equations in terms of the monodromies of their solutions.