Extendable Cohomologies for Complex Analytic Varieties

We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern-Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As an application of our theory, we prove a Camacho-Sad type index theorem for holomorphic foliations of singular complex varieties. Introduction. One of the more important contributions to the study of complex vector bundles over differentiable manifolds has been given by the Chern-Weil theory. Thanks to such a theory it is possible to describe the topological Chern classes of a complex vector bundle on a manifold (which lie in the topological cohomology groups of the manifold) by means of the differentiable Chern classes of the bundle (which belong to the de Rham cohomology groups of the manifold). By their very definition, the differentiable Chern classes of a complex vector bundle are built starting from suitable differentiable differential forms on the manifold. This is the reason why, until now, it was impossible to achieve a generalization of the Chern-Weil theory allowing to study complex vector bundles over singular varieties. In fact, the hurdles for having such a theory are tied to the difficulties of giving an appropriate definition of differential forms on singular spaces. In this paper we solve the problem of extending the Chern-Weil theory to the case of abstract complex analytic varieties. Namely, we introduce a suitable notion of differential forms, the extendable differentiable differential forms, we develop a cohomology theory based on such forms, we define the extendable Chern classes for differentiable complex vector bundles over complex analytic varieties and we prove that these classes represent the topological Chern classes of the bundle. The starting point is the following. In the case of complex analytic varieties, it can be given several natural definitions of holomorphic differential forms. Nevertheless, although remarkable results have been obtained, the development of the theories based on such holomorphic forms did not carry on, because of the failure of the Poincaré lemma. Namely, the cohomologies associated with these holomorphic forms are not, in general, locally trivial (cp. [Fe 1], [Fe 2], [He 1], [He 2], [Bl-He]). Anyway, all these definitions of holomorphic differential form are such that the Date: October 16, 2008. 2000 Mathematics Subject Classification. 32C99, 20G10, 14C17, 32S65, 37F75.