INCIDENCE DIVISOR

Abstract

Let Z be a complex manifold of dimension n+1 and (Xs)s∈S be an analytic family of n-cycles (not necessarily compact) parametrized by a reduced analytic complex space S. Denote by X the graph of this family and p1 (resp. p2) the canonical projection of S × Z on S (resp. on Z). The construction of Barlet-Magnusson assigns to each n+1-codimensionnal subspace in Z which is assumed to be a local complete intersection and to satisfy: C1[Formula: see text] is proper and finite on its image which is nowhere dense in S, an effective Cartier divisor ΣY in S. Nice functorial properties of this correspondance are proven. The purpose of this article is to generalize this result in several directions: using the relative fundamental class of the family (Xs)s∈S in Deligne cohomology, we prove the following results: (1) ΣY depends only on the cycle underlying the locally complete intersection ideal defining Y in Z (2) we generalize the construction of the incidence divisor which is Cartier and effective for any cycle Y (no nilpotent structure on Y is needed) satifying the weaker condition C2[Formula: see text] is proper and generically finite on its image which is nowhere dense in S and we extend the nice functorial properties to this case.