Riemann–Hilbert correspondence
- Riemann–Hilbert correspondence
In mathematics, the Riemann-Hilbert correspondence is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups. In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1, and there is a correspondence between certain systems of partial differential equations (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).
tatement
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 D-modules on "X" with regular singularities to the bounded derived category of the 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 cohomology complexes of irreducible closed subvarieties of "X" with coefficients in irreducible local systems.
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.
References
* A. Borel "Algebraic D-modules" ISBN 0-12-117740-8
*P. Deligne, "Equations differentials a points singuliers reguliers", Springer Lecture notes in mathematics 163 (1970).
*M. Kashiwara, "Faiseaux constructibles et systems holonomes d'equations aux derivees partielles lineaires a points singuliers reguliers", Se. Goulaouic-Schwartz, 1979-80, Exp. 19.
*Z. Mebkhout, "Sur le probleme de Hilbert-Riemann", Lecture notes in physics 129 (1980) 99-110.
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