First order extensions of holomorphic foliations

Let S be a subvariety of a complex manifold M. Let F be a holomorphic foliation on S and [ a coherent sheaf on S. We give a definition of first order tangency extension of F to M with respect to ɛ and prove that, under some suitable hypotheses, the existence of extensions give rise to localization of certain characteristic classes on S. This point of view includes both the classical Camacho-Sad index theorem, variation and the newer indices theorems for holomorphic self-maps along fixed points sets.