Holonomic D-modules of Fuchsian and logarithmic types

Abstract

The article deals with holonomic systems of differential equations with regular singularities. The main result is formulated as follows: any integrable Fuchsian D-module is meromorphically isomorphic to integrable logarithmic D-module. As a consequence we obtain a classification of integrable Fuchsian D-modules at non-singular points of their singular loci. A complete description of regular singular integrable systems with one unknown is another important result. In particular, it delivers an explicit representation of closed logarithmic forms in terms of eigenvalues of the monodromy associated with the corresponding logarithmic holonomic D-module. A method of computation of eigenvalues of the monodromy, the most important invariants of a regular singular holonomic integrable system, is also described. In conclusion, several examples for a clear interpretation of the obtained results are constructed and discussed in detail.