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Rigidité des fibres des réductions algébriques. (Rigidity of the fibers of algebraic reductions). (French) Zbl 0636.32015

Let S, X be irreducible compact complex spaces, let \(S\subset C(S)\), the space of cycles of X, and let \(Z_ s\) be the cycle of X parametrized by \(s\in S\). Assume that S covers X (i.e.: \(\forall x\in X\), \(\exists s\in S\) s.t.: \(x\in Z_ s)\). The so-called Kodaira map \(K_ s: X\to C(S)\) was constructed by the author in Invent. Math. 63, 187-223 (1981; Zbl 0436.32024), p. 202 in this situation, generalizing the classical map \(\Phi_ L: X\to {\mathbb{P}}(H^ 0(X,L)')\) associated to a linear system of divisors.
In this paper, a Kodaira map is more generally constructed when S, not necessarily compact, contains a compact subspace R which covers X. The natural factorization \(\alpha\) : \(K_ S(X)\to K_ R(X)\) of \(K_ R\) through \(K_ S\) is then algebraic. From this, the rigidity of the algebraic reduction \(e: X\to A\) follows (i.e.: its fibers form an irreducible component of C(S)).
Reviewer: F.Campana

MSC:

32J10 Algebraic dependence theorems
32J99 Compact analytic spaces
14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
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