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 Gröbner basis computations in the Weyl algebra.

25 pages, amsart, uses verbatim, amsmath, latexsym, amssymb, xypic, fixed typos