Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

Abstract

Abstract. We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree n of compact congruence p-dimensional hyperbolic manifolds "of simple type" as long as n is strictly smaller than p 3 . We also prove that for connected Shimura varieties associated to O(p, 2) the Hodge conjecture is true for classes of degree < p+13 . The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [5]. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection 1.19 below.

The Modularity of Special Cycles on Orthogonal Shimura Varieties over Totally Real Fields under the Beilinson–Bloch Conjecture

“…We follow the strategy of Yuan-Zhang-Zhang [22]. First, we shall prove Theorem 1.5 (2). To prove Theorem 1.5 (2), we calculate the cohomology of the Shimura variety M K f .…”

Tautological classes on moduli spaces of hyper-Kähler manifolds

“…The following theorem is proved in [6,Theorem 7.7] when E is the trivial representation. It was proved in [7] for general E but under the hypothesis that π is cuspidal. To be able to deal with residual representations as well was the main input of [6].…”

“…The inclusion (6.5) follows from (the proof of) [7,Proposition 11.3]. 2 The last assertion follows from Theorem 6.5, the decomposition (6.4) and the fact that if π R is an irreducible unitary representation of G(R) that satisfies H r,r (g, K R ; π ∞ R ⊗ E) = 0 for some r < b then π ∞ R is isomorphic to A r,r (E).…”

“…If x ∈ V r we then have ϕ(x) ∈ A 2r (D, T ℓ (V )). We shall rather work with the polynomial Fock model for the dual pair O(b, 2)×Sp 2r (R), see [7,Section 7]. We denote by ι the intertwining operator from the Schödinger model to the Fock model, see [29,Section 6].…”

Periods and (χ, b)-factors of cuspidal automorphic forms of symplectic groups

“…It is worth mentioning that the periods considered in [GJS09] have been used in a recent work of Bergeron, Millson and Moeglin on Hodge type theorems for arithmetic manifolds associated to orthogonal groups ( [BMM16b]). It is expected that similar applications will be found for the periods studied in [JW16] and [JW14] for unitary groups and those investigated in this paper for symplectic groups ( [BMM16a] and [HH12]).…”