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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. ==Statement== 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''. 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. In the case ''X'' has dimension one (a complex algebraic curve) then there is a more general Riemann–Hilbert correspondence for algebraic connections with no regularity assumption (or for holonomic D-modules with no regularity assumption) described in Malgrange (1991), the Riemann–Hilbert–Birkhoff correspondence. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann–Hilbert correspondence」の詳細全文を読む スポンサード リンク
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