Connections on modules over quasi-homogeneous plane curves

Abstract

Let k be an algebraically closed field of characteristic 0, and let $A=k[x,y]/(f)$ be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e. a graded A-linear homomorphism $\nabla :{\mathrm{Der}}_k(A)\to {\mathrm{End}}_k(M)$ that satisfy the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring $B=\hat{A}$ admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module.

AMS-LaTeX, 12 pages, minor changes. To appear in Comm. Algebra