Riemann–Hilbert correspondence

Remarks

Masaki Kashiwara and Zoghman Mebkhout proved the theorem independently around 1980. Holonomic means that the characteristic variety of the module, a subvariety of the cotangent bundle, has dimension $d$. Regular is a growth condition on solutions near singularities. It extends the notion of a regular singular point of an ordinary differential equation.

The simplest case: a holomorphic vector bundle $E$ with a flat connection $\nabla$ is a regular holonomic $\mathcal{D}_X$-module. Its de Rham complex is quasi-isomorphic to the local system of flat sections $\ker \nabla$, shifted by $d$. On this part, the correspondence is the classical equivalence between flat bundles and local systems, that is, representations of the fundamental group when $X$ is connected.

There is also an algebraic version, for a smooth complex algebraic variety $X$ and algebraic regular holonomic modules, where regularity includes a condition at infinity. There regularity is needed. On the affine line, the algebraic connections $d$ and $d - dz$ on the trivial bundle are not isomorphic, because an isomorphism would be multiplication by $e^{z}$, which is not a polynomial. Yet their analytic flat sections, the constants and the functions $c \, e^{z}$, form isomorphic local systems. The connection $d - dz$ has an irregular singular point at infinity.

Going deeper

For a student

The theorem gives an answer to a modern form of Hilbert's twenty-first problem. Its classical forerunner, due to Deligne, treats algebraic vector bundles with regular singular connections on smooth varieties, and matches them with local systems. In the Kashiwara–Mebkhout form, the category of perverse sheaves belongs to topology and the category of D-modules to analysis. Many results move across. For example, the intersection cohomology complex corresponds to a minimal extension of a D-module. It is a basic tool of geometric representation theory, for example in the proof of the Kazhdan–Lusztig conjecture.