Algebraic connections on ellipsoid surfaces

Abstract

This paper is part of a series of papers where the aim is to give explicit formulas for algebraic differential operators on a finitely generated projective module E on a commutative unital ring A. In previous papers on the subject the Kodaira-Spencer map and Kodaira-Spencer class was used to give explicit formulas for flat algebraic connections on a class of maximal Cohen-Macaulay modules on isolated hypersurface singularities. In this paper we give explicit formulas for algebraic connections on a class of finitely generated projective modules on ellipsoid surfaces. The connections we construct are non-flat with trace of curvature equal to zero. We construct these formulas using the fundamental matrix M of the module E. This matrix may in the case when A is a finitely generated algebra over a field be calculated using Groebner bases. We also discuss a possible relationship to algebraic cycles and a problem on existence of holomorphic connections in complex analysis.Comment: Example 2.21 added. A minor correction to Example 3.1