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On the Locus of Hodge Classes

Eduardo Cattani, Pierre Deligne and Aroldo Kaplan
Journal of the American Mathematical Society
Vol. 8, No. 2 (Apr., 1995), pp. 483-506
DOI: 10.2307/2152824
Stable URL: http://www.jstor.org/stable/2152824
Page Count: 24
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Locus of Hodge Classes
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Abstract

Let S be a nonsingular complex algebraic variety and V a polarized variation of Hodge structure of weight 2p with polarization form Q. Given an integer K, let S(K) be the space of pairs (s, u) with s ∈ S, u ∈ Vs integral of type (p, p), and Q(u, u) ≤ K. We show in Theorem 1.1 that S(K) is an algebraic variety, finite over S. When V is the local system H2p (Xs, Z)/torsion associated with a family of nonsingular projective varieties parametrized by S, the result implies that the locus where a given integral class of type (p, p) remains of type (p, p) is algebraic.

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