HOLONOMIC D-MODULES // FLAT CONNECTIONS // DE RHAM RH FUNCTOR // REGULAR SINGULARITIES
The Riemann-Hilbert correspondence is a categorical equivalence between two seemingly different worlds: algebraic D-modules on a smooth variety X and constructible sheaves on X^an. For regular holonomic D-modules, this is an exact equivalence of abelian categories.
The de Rham functor DR(M) = Omega_X tensor_{D_X} M takes a D-module M to a complex of sheaves. When M is regular holonomic, DR(M) is a perverse sheaf. The inverse is the solution functor Sol(M) = RHom_{D_X}(M, O_X).
A D-module M on X has regular singularities if, roughly, solutions grow at most polynomially near the singular locus. This is the algebraic incarnation of Fuchsian differential equations — every singularity is regular singular.
For a connection nabla on a vector bundle E, regularity means the connection has only logarithmic poles: nabla: E -> E tensor Omega^1_X(log D) where D is the divisor at infinity. Deligne proved the 1D case; Mebkhout and Kashiwara proved the general theorem.
The de Rham functor is the bridge. Given a left D-module M on smooth X of dimension n:
This is a complex of sheaves on X^an, shifted by dimension n (the perverse shift). For M = O_X (the structure sheaf as a D-module), DR(O_X) = C_X[n], the constant sheaf shifted — the simplest perverse sheaf. The key theorem: DR is t-exact with respect to the standard t-structure on D-modules and the perverse t-structure on constructible sheaves.
On the constructible side, flat connections correspond to local systems — locally constant sheaves, equivalently representations of pi_1(X \ D). The monodromy representation rho: pi_1 -> GL(V) determines the flat connection up to isomorphism. RH recovers this classical picture as a special case.
RH intertwines the six operations on D-modules (direct image f_+, inverse image f^+, tensor, dual, f_!, f^!) with the six operations on constructible sheaves (Rf_*, f^{-1}, tensor, RHom, Rf_!, f^!). This is the deepest structural consequence — the entire formalism transports.