Geodesic
On the locus of Hodge classes
Journal of the American Mathematical Society, Tome 08 (1995) no. 2, pp. 483-506

Voir la notice de l'article provenant de la source American Mathematical Society

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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     title = {On the locus of {Hodge} classes},
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Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo. On the locus of Hodge classes. Journal of the American Mathematical Society, Tome 08 (1995) no. 2, pp. 483-506. doi: 10.1090/S0894-0347-1995-1273413-2

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[2] Cattani, Eduardo, Kaplan, Aroldo Degenerating variations of Hodge structure Astérisque 1989

[3] Cattani, Eduardo, Kaplan, Aroldo, Schmid, Wilfried Degeneration of Hodge structures Ann. of Math. (2) 1986 457 535

[4] Deligne, Pierre Équations différentielles à points singuliers réguliers 1970

[5] Topics in transcendental algebraic geometry 1984

[6] Schmid, Wilfried Variation of Hodge structure: the singularities of the period mapping Invent. Math. 1973 211 319

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