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).

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 D-modules 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 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

• Borel, Armand (1987), Algebraic D-Modules, Perspectives in Mathematics, 2, Boston, MA: Academic Press, ISBN 978-0-12-117740-9 
• Deligne, Pierre (1970), Équations différentielles à points singuliers réguliers, Springer Lecture notes in Mathematics, 163, OCLC 169357 
• M. Kashiwara, Faisceaux constructibles et systemes holonomes d'équations aux derivées partielles linéaires à points singuliers réguliers, Se. Goulaouic-Schwartz, 1979–80, Exp. 19.
• Z. Mebkhout, Sur le probleme de Hilbert-Riemann, Lecture notes in physics 129 (1980) 99–110.