On the logarithmic Riemann-Hilbert correspondence

Abstract

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 $\mathbb{C}$. 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 equivalence between the relevant category and a suitable combinatorial or topological category. In the equivariant case, the objects of the target category are graded modules endowed with a group action. We then show that every germ of a holomorphic connection has a canonical equivariant model. Global connections are classified by locally constant sheaves of modules over a (varying) sheaf of graded rings on the topological space $X_{\log }$. Each of these equivalences is compatible with tensor product and cohomology.