Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

Abstract

Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given by T.~Oaku and N.~Takayama (\cite{O-T2}), who considered the case where $Y$ is a hypersurface, and our methods from \cite{W-1} for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with support $H^i_{dR,Z}(U,\C)$ where again $U$ is an arbitrary Zariski-open subset of $X$ and $Z$ is an arbitrary Zariski-closed subset of $U$. Our main tool is the generalization of the restriction process from \cite{O-T1} to complexes of modules over the Weyl algebra. All presented algorithms are based on Gr\"obner basis computations in the Weyl algebra.