Reflexive modules on normal surface singularities and representations of the local fundamental group

Abstract

We consider the Riemann–Hilbert correspondence on the complement of a normal surface singularity ( X , x ) . Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group π 1 loc ( X , x ) and the category of left D X , x -modules that are reflexive as O X , x -modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left D X , x -modules. We prove that the fundamental module is an invariant module if and only if ( X , x ) is a quotient singularity. Finally we investigate some algebraisation aspects.