Suppose that X is a smooth complex algebraic variety. Riemann–Hilbert correspondence : 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 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 : 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 then there is a more general Riemann–Hilbert correspondence for algebraic connections with no regularity assumption described in Malgrange, the Riemann–Hilbert–Birkhoff correspondence.
Examples
An example where the theorem applies is the differential equation on the punctured affine lineA1 − . Here a is a fixed complex number. This equation has regular singularities at 0 and ∞ in the projective lineP1. The local solutions of the equation are of the form cza for constants c. If a is not an integer, then the function za cannot be made well-defined on all of C −. That means that the equation has nontrivial monodromy. Explicitly, the monodromy of this equation is the 1-dimensional representation of the fundamental group 1 = Z in which the generator acts by multiplication by e2ia. To see the need for the hypothesis of regular singularities, consider the differential equation on the affine line A1. This equation corresponds to a flat connection on the trivial algebraic line bundle over A1. The solutions of the equation are of the form cez for constants c. Since these solutions do not have polynomial growth on some sectors around the point ∞ in the projective line P1, the equation does not have regular singularities at ∞. Since the functions cez are defined on the whole affine line A1, the monodromy of this flat connection is trivial. But this flat connection is not isomorphic to the obvious flat connection on the trivial line bundle over A1, because its solutions do not have moderate growth at ∞. This shows the need to restrict to flat connections with regular singularities in the Riemann–Hilbert correspondence. On the other hand, if we work with holomorphic vector bundles with flat connection on a noncompact complex manifold such as A1 = C, then the notion of regular singularities is not defined. A much more elementary theorem than the Riemann–Hilbert correspondence states that flat connections on holomorphic vector bundles are determined up to isomorphism by their monodromy.
