Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection image on M? We consider the maximal Cohen–Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections. Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a image-Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ωA to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ωA admits an integrable connection if and only if A is Gorenstein.