Connections on modules over singularities of finite CM representation type

Let AA be a commutative kk-algebra, where kk is an algebraically closed field of characteristic 00, and let MM be an AA-module. We consider the following question: Under what conditions is it possible to find a connection ∇:Derk(A)→Endk(M) on MM?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≤2d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3d≥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 AA be a singularity not necessarily of finite CM representation type, and consider the condition that AA is a Gorenstein curve or a Q-Gorenstein singularity of dimension d≥2d≥2. We show that this condition is sufficient for the canonical module ωAωA to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if AA is a monomial curve singularity, then the canonical module ωAωA admits an integrable connection if and only if AA is Gorenstein.