Classes de Chern Des Ensembles Analytiques

摘要

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V0 its regular part, and by [V] its fundamental class in H2n(V; ). If V is a locally complete intersection (LCI), it is known that the normal bundle NV_0 in M to V0 in M has a natural extension NV to all of V, so that we can define its Chern classes c(*)(NV) in cohomology, as well as the Chern classes cvir(*). If V is a locally complete intersection (LCI), it is known that the normal bundle NV_0 in M to V0 in M has a natural extension NV to all of V, so that we can define its Chern classes c(*)(NV) in cohomology, as well as the Chern classes cvir(*) (V) of the virtual tangent bundle Tvir(V):=[TM|V - NV] in the K-theory K0(V). This has applications – on one hand to the definition of various indices associated to a singular foliation on M with respect to which V is invariant (cf. [23–25]), and– on the other hand to the definition of the Milnor numbers and classes of the singular part of V (cf. [7,8]).In the general case, we can no more define NV and Tvir(V). However we shall associate, to each desingularisation of V, Chern classes cn-*(NV, ) and in the homology H2(n-*)(V), which coincide respectively with the Poincaré duals and of the cohomological Chern classes c(*)(NV) and c vir(*)(V) when V is LCI. Our classes do not coincide with the inverse Segre classes and the Fulton–Johnson classes respectively, except for LCIs. Moreover, it turns out that this is sufficient for being able to generalize to compact pure dimensional complex analytic subsets of a holomorphic manifold the two kinds of applications mentioned above. These constructions depend on in general. However, in the case of curves, there is only one desingularisation, so that all these constructions become intrinsic.