Author:
Klingler B.,Otwinowska A.
Abstract
AbstractGiven $${{\mathbb {V}}}$$
V
a polarizable variation of $${{\mathbb {Z}}}$$
Z
-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$
V
⊗
is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$
V
s
has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$
V
⊗
is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$
V
. Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$
V
is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$
V
whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$
A
g
of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$
A
g
is either a closed algebraic subvariety of S or is Zariski-dense in S.
Funder
Humboldt-Universität zu Berlin
Publisher
Springer Science and Business Media LLC
Reference29 articles.
1. André, Y.: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82, 1–24 (1992)
2. Bakker, B., Tsimerman, J.: The Ax-Schanuel conjecture for variations of Hodge structures. Inventiones Math. Invent. Math. 217(1), 77–94 (2019)
3. Bakker, B., Klingler, B., Tsimerman, J.: Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Am. Math. Soc. 33(4), 917–939 (2020)
4. Cattani, E., Deligne, P., Kaplan, A.: On the locus of Hodge classes. J. Am. Math. Soc. 8, 483–506 (1995)
5. Cattani, E., Kaplan, A.: Algebraicity of Hodge loci for variations of Hodge structure. In: Hodge Theory, Complex Geometry and Representation Theory, pp. 59–83, Contemporary Mathematics, vol. 608. Amer. Math. Soc., Providence (2014)
Cited by
5 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献