Classes de Chern Des Ensembles Analytiques

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 H_2n(V;). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*)(V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] 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 N_V and T_vir(V). However we shall associate, to each desingularisation of V, Chern classes c_n-*(N_V, ) and in the homology H_2(n-*)(V), which coincide respectively with the Poincaré duals and of the cohomological Chern classes c^(*)(N_V) 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.Mathematics Subject Classification: 57R20, 57R25, 19E20.