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 the Riemann sphere, where it was about the existence of regular differential equations with prescribed monodromy groups. First the Riemann sphere may be replaced by an arbitrary Riemann surface and then, in higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. 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 for algebraic connections with regular singularities by Pierre Deligne (1970) and more generally for regular holonomic D-modules by Masaki Kashiwara (1980, 1984) and Zoghman Mebkhout (1980, 1984) independently.

Suppose that X is a smooth complex algebraic variety.

Riemann–Hilbert correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X. For X connected, the category of local systems is also equivalent to the category of complex representations of the fundamental group of X.

The condition of regular singularities means that locally constant sections of the bundle (with respect to the flat connection) have moderate growth at points of Y − X, where Y is an algebraic compactification of X. In particular, when X is compact, the condition of regular singularities is vacuous.

More generally there is the

Riemann–Hilbert correspondence (for regular holonomic D-modules): 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.

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