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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the locus of Hodge classes
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by Eduardo Cattani, Pierre Deligne and Aroldo Kaplan PDF
J. Amer. Math. Soc. 8 (1995), 483-506 Request permission


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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 8 (1995), 483-506
  • MSC: Primary 14D07; Secondary 14C30, 32G20, 32J25
  • DOI:
  • MathSciNet review: 1273413