Tame topology of arithmetic quotients and algebraicity of Hodge loci

Abstract

In this paper we prove the following results: 1 ) 1) We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. 2 ) 2) We prove that the period map associated to any pure polarized variation of integral Hodge structures V \mathbb {V} on a smooth complex quasi-projective variety S S is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. 3 ) 3) As a corollary of 2 ) 2) and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of ( S , V ) (S, \mathbb {V}) is a countable union of algebraic subvarieties of S S , a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable S L 2 SL_2 -orbit theorem of Cattani-Kaplan-Schmid.

On the closure of the Hodge locus of positive period dimension

“…In the geometric case Weil [28] asked whether HL(S, V) is a countable union of closed algebraic subvarieties of S (he noticed that a positive answer follows easily from the rational Hodge conjecture). In [4] Cattani, Deligne and Kaplan proved the following unconditional celebrated result (see [5], we also refer to [3] for an alternative proof): Theorem 1.1 (Cattani-Deligne-Kaplan) Let S be a smooth connected complex quasi-projective algebraic variety and V be a polarizable ZVHS over S. Then HL(S, V) (thus also HL(S, V ⊗ )) is a countable union of closed irreducible algebraic subvarieties of S.…”

“…The very definition of the Hodge locus HL(S, V ⊗ ) implies that special subvarieties of S for V in the sense of Definition 4.5 coincide with the ones defined in Definition 1.2. In particular, in view of Theorem 1.1, any special subvariety of S (hence any special intersection in S) is a closed irreducible algebraic subvariety of S. An alternative proof of Theorem 1.1 using o-minimal geometry was provided in [3,Theor. 1.6].…”

“…1.6]. The approach of [3] gives immediately the following more general algebraicity result, which is implicit in the discussion of [15,Section 7]: Proposition 4.8 Any weakly special subvariety Z for V (hence also any weakly special intersection for V) is an algebraic subvariety of S.…”

“…1.1(1)] the Hodge variety Hod 0 (S, V) = \D + is an arithmetic quotient endowed with a natural structure of R alg -definable manifold. By [3,Theor. 1.3] the period map S is R an,exp -definable with respect to the natural R alg -structures on S and Hod 0 (S, V).…”

“…Case 2: the period map S has constant relative dimension d. The proof is the same as in the first case, replacing (V i ) 0 ≥1 and "of positive period dimension" by (V i ) 0 ≥d and "of period dimension at least (d + 1)". General case: As the period map S is definable in the o-minimal structure R an,exp (see [3]), it follows from the trivialization theorem [25,Theor.…”