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).
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
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.
- 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.