  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.
