3 Incidence Divisor

The main purpose of this paper is to prove an important generalization of the construction of the Incidence Divisor given in [BMg1] in the case of an ambient manifold. Let us first recall briefly the setting : let Z be a complex manifold and (X s) s∈S an analytic family of (closed) n−cycles in Z parametrized by a reduced complex space S. To a (n+1)−codimensional subspace Y in Z, which is assumed to be a locally complete intersection and to satisfy the following condition : (C1) the analytic set (S × |Y |) ∩ |X| in S × Z(**) is S-proper and finite on its image |Σ Y | which is nowhere dense in S, an effective Cartier divisor Σ Y in S, called the " incidence divisor of Y in S " , is defined with support |Σ Y |, and nice functorial properties of this construction are proven. Of course, no assumption is made on the singularities of S. A relative version is also given : when Y varies in a flat family over T in such a way that (C1) remains true, Σ Y moves in a flat family over T. As a consequence, Σ Y depends only on the underlying cycle of Y for a connected flat deformation of a locally complete intersection (nilpotent) structure inducing a fixed cycle. But the general invariance question was not solved in [BMg1]. [Q1] Does the Cartier divisor Σ Y only depend on the cycle underlying the locally complete intersection ideal of O Z defining Y ? (*) Senior chair of Complex Analysis and Geometry in Institut Universitaire de France (**) X denotes the graph of the family (X s) S∈s , which is a cycle in S × Z and |X| the support of this cycle.