Homology and cohomology with compact supports forq-convex spaces

Abstract

In this paper we give an elementary proof of basic vanishing properties for homology and cohomology with compact supports of q-complete spaces which follow from the results of H.Hamm [16], [17] and K.-H.Fieseler-L.Kaup [13]. At the same time we obtain new finiteness results for the homology and the cohomology with compact supports in the q-convex case, which is not treated in [16], [17] and [13]. Our work extends to general q-complete spaces recent papers of M.Coltoiu-N.Mihalache [8] and M.Coltoiu [7] which treated the case of Stein spaces (q=0). A typical result is the following: if X is a q-complete space of dimension n, then Hi (X, Z)=0 for i>n+q and Hn+q (X, Z) is free, if X is also purely dimensional and locally a set-theoretic complete intersection, then Hci(X, Z)=0 for i<n −q and Hcn−q(X, Z) is free. The vanishing of the cohomology with compact supports for q-complete spaces has as consequence Lefschetz-type theorems for singular spaces (the homology statements) proved by C.Okonek [24] using Goresky-MacPherson stratified Morse theory.