Introduction

The Gauß-Manin connection $\mathcal{G}$ is a regular $\mathcal{D}$-module associated to an isolated hypersurface singularity [32]. The V-filtration on $\mathcal{G}$ is defined by the $\mathcal{D}$-module structure. One can describe $\mathcal{G}$ in terms of integrals of holomorphic differential forms over vanishing cycles [1]. Classes of these differential forms in the Brieskorn lattice $\mathcal{H}''$ can be considered as elements of $\mathcal{G}$. The V-filtration on $\mathcal{H}''$ reflects the embedding of $\mathcal{H}''$ in $\mathcal{G}$ and determines the singularity spectrum which is an important invariant of the singularity.

E. Brieskorn [1] gave an algorithm to compute the complex monodromy based on the $\mathcal{D}$-module structure which is implemented in the computer algebra system SINGULAR [17] in the library mondromy.lib [35]. In many respects, the microlocal structure of $\mathcal{G}$ and $\mathcal{H}''$ [32] seems to be more natural.

After a brief introduction to the theory of the Gauß-Manin connection, we describe how to use this structure for computing in $\mathcal{G}$ and give an explicit algorithm to compute the V-filtration on $\mathcal{H}''$. This also leads to a much more efficient algorithm to compute the complex monodromy and the singularity spectrum of an arbitrary isolated hypersurface singularity. All algorithms are implemented in the SINGULAR library gaussman.lib [36] and are distributed with version 2.0.

For more theoretical background on this section see [37].