Irregularity of an analogue of the Gauss-Manin systems

Abstract

In the D-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf O by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex of a D-module twisted by the exponential of a polynomial g by another polynomial f, where f and g are two polynomials in two variables. The analogue of the Gauss-Manin systems can have irregular singularities (at finite distance and at infinity). We express an invariant associated with the irregularity of these systems by the geometry of the map (f,g).