## On the locus of Hodge classes

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- by Eduardo Cattani, Pierre Deligne and Aroldo Kaplan
- J. Amer. Math. Soc.
**8**(1995), 483-506 - DOI: https://doi.org/10.1090/S0894-0347-1995-1273413-2
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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.## References

- Eduardo Cattani and Aroldo Kaplan,
*Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure*, Invent. Math.**67**(1982), no. 1, 101–115. MR**664326**, DOI 10.1007/BF01393374 - Eduardo Cattani and Aroldo Kaplan,
*Degenerating variations of Hodge structure*, Astérisque**179-180**(1989), 9, 67–96. Actes du Colloque de Théorie de Hodge (Luminy, 1987). MR**1042802** - Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid,
*Degeneration of Hodge structures*, Ann. of Math. (2)**123**(1986), no. 3, 457–535. MR**840721**, DOI 10.2307/1971333 - Pierre Deligne,
*Équations différentielles à points singuliers réguliers*, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR**0417174** - Phillip Griffiths (ed.),
*Topics in transcendental algebraic geometry*, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR**756842**, DOI 10.1515/9781400881659 - Wilfried Schmid,
*Variation of Hodge structure: the singularities of the period mapping*, Invent. Math.**22**(1973), 211–319. MR**382272**, DOI 10.1007/BF01389674
A. Weil,

*Abelian varieties and the Hodge ring*, André Weil: Collected Papers III, Springer-Verlag, Berlin and New York, 1979, pp. 421-429.

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- 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