Let Ω be a bounded hyperconvex domain in C n , 0 ∈ Ω, and S ε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each S ε we associate its vanishing ideal I ε and pluricomplex Green function G ε = G Iε . Suppose that, as ε tends to 0, (I ε ) ε converges to I (local uniform convergence), and that (G ε ) ε converges to G, locally uniformly away from 0; then G ≥ G I . If the Hilbert-Samuel multiplicity of I is strictly larger than its length (codimension, equal to N here), then (G ε ) ε cannot converge to G I . Conversely, if I is a complete intersection ideal, then (G ε ) ε converges to G I . We work out the case of three poles.
We introduce a class of normal complex spaces having only mild singularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients.AMS classification. 32 C 20-32 C 25-32 C 36.
The main purpose of this article is to increase the efficiency of the tools introduced in [B. Mg. 98] and [B.Mg. 99], namely integration of meromorphic cohomology classes, and to generalize the results of [B.Mg. 99]. They describe how positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n + 1 in a complex manifold Z can be transformed into positivity conditions for a Cartier divisor in a space parametrizing n−cycles in Z .As an application of our results we prove that the following problem has a positive answer in many cases :Let Z be a compact connected complex manifold of dimension n+p. Let Y ⊂ Z a submanifold of Z of dimension p − 1 whose normal bundle N Y |Z is (Griffiths) positive. We assume that there exists a covering analytic family (Xs) s∈S of compact n−cycles in Z parametrized by a compact normal complex space S.Is the algebraic dimension of Z ≥ p ? *
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