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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
DOI: https://doi.org/10.1090/S0894-0347-1995-1273413-2
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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Article copyright: © Copyright 1995 American Mathematical Society