Microlocal Riemann-Hilbert correspondence

The Riemann-hilbert Correspondence for Algebraic Stacks

Using the theory ∞-categories we construct derived (dg-)categories of regular, holonomic D-modules and algebraically constructible sheaves on a complex smooth algebraic stack. We construct a natural ∞-categorical equivalence between these two categories generalising the classical Riemann-Hilbert correspondence.

Some examples of the Riemann-Hilbert correspondence

1.1 Fix a variety X over C. The Riemann-Hilbert correspondence identifies the category of perverse sheaves on X(C) with the (abelian) category of regular holonomic D-modules on X. This is a remarkable and deep theorem in the theory of linear partial differential equations. In this note we will investigate this correspondence in simple examples, exploring the topological and algebraic sides as w...

On the Logarithmic Riemann-Hilbert Correspondence

We construct a classification of coherent sheaves with an integrable log connection, or, more precisely, sheaves with an integrable connection on a smooth log analytic space X over C. We do this in three contexts: sheaves and connections which are equivariant with respect to a torus action, germs of holomorphic connections, and finally global log analytic spaces. In each case, we construct an e...

Renormalization and the Riemann-Hilbert correspondence

Classical Motivation for the Riemann–hilbert Correspondence

These notes explain the equivalence between certain topological and coherent data on complex-analytic manifolds, and also discuss the phenomenon of “regular singularities” connections in the 1-dimensional case. It is assumed that the reader is familiar with the equivalence of categories between the category of locally constant sheaves of sets on a topological space X and the category of coverin...