Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | 陈家明 博士，Université de Lorraine
| | Inviter: | | Title:
| Atypical Intersections For Variations of Hodge Structures(III) | Time & Venue:
| 2021.04.20 15:00-16:00 腾讯会议室 ID：223 767 119 | Abstract:
| Given a polarizable integral variation of Hodge structure (VHS) on a smooth complex quasi-projective variety S, the associated Hodge locus is the subset of points of S for which the corresponding fiber admits more Hodge tensors than the very general fiber. A classical result of Cattani, Deligne and Kaplan states that this Hodge locus is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for the VHS. In this series of talks, I will first introduce a set of conjectures formulated by Bruno Klingler concerning the structure of this Hodge locus in terms of atypical intersections, which generalize the Zilber-Pink conjectures in the Shimura case. Then I will focus on a geometric part of these conjectures concerning the distributions of positive dimensional special subvarieties of S. More precisely, I will show that (1) there are only finitely many maximal "Hodge-theoretic rigid" special subvarities of S of Shimura type with dominant period maps, which generalizes previous results of Clozel-Ullmo and Ullmo in the Shimura case. (2) (joint work with Richard Rodolphe and Emmanuel Ullmo) if the union of horizontal (weakly) special subvarities of S is dense in S, then the period imagewill be essentially a product (of a locally symmetric variety with another period image). The techniques involved will be a combination of o-minimality and homogeneous dynamics.
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