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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


On the locus of Hodge classes

Authors: Eduardo Cattani, Pierre Deligne and Aroldo Kaplan
Journal: J. Amer. Math. Soc. 8 (1995), 483-506
MSC: Primary 14D07; Secondary 14C30, 32G20, 32J25
MathSciNet review: 1273413
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Abstract: Let $ S$ be a nonsingular complex algebraic variety and $ \mathcal{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 \in S$, $ u \in {\mathcal{V}_s}$ integral of type $ (p,p)$, and $ Q(u,u) \leq K$. We show in Theorem 1.1 that $ {S^{(K)}}$ is an algebraic variety, finite over $ S$. When $ \mathcal{V}$ is the local system $ {H^{2p}}({X_s},\mathbb{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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PII: S 0894-0347(1995)1273413-2
Article copyright: © Copyright 1995 American Mathematical Society

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