The Hodge conjecture and arithmetic quotients of complex balls
type: Item
instance of: scholarly article
title: The Hodge conjecture and arithmetic quotients of complex balls
zbMATH Open document ID: 1349.14031
author: Nicolas Bergeron
author: Colette Moeglin
author: John Millson
published in: Acta Mathematica
publication date: 2016-07-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.1515
review text: This paper studies the cohomology of a Shimura variety associated to a unitary group, with results related to the Hodge and Tate conjectures for these varieties. The Shimura variety is assumed to be smooth, connected and compact, and associated to an algebraic group \(G\) such that \(G(\mathbb R) \cong U(p,q) \times G^c\), where \(p, q > 0\), and \(G^c\) is compact. For the sake of simplicity we describe the results of the paper under the assumption that \(q=1\), though a more general situation is considered in the paper. In this situation, the authors show that for ``small'' \(n\), the cohomology groups \(H^n (S(\mathbb C), \mathbb C)\) are generated by cup products of classes of type \((1,1)\), holomorphic forms, anti-holomorphic forms, and Kudla-Millson cycle classes. This is used to obtain the general Hodge conjecture (even without Grothendieck's modification) for \(N^cH^n(S(\mathbb C), \mathbb Q)\), where \(N^{\bullet}\) is the coniveau filtration, and either \(2n-c \leq p\) or \(2n+c \geq 3p\). As a corollary they obtain the usual Hodge conjecture for \(H^{2n}(S(\mathbb C), \mathbb Q)\) when either \(0 \leq n \leq \frac{1}{3} p\), or, \(\frac{2}{3} p \leq n \leq p\). A similar result holds for the Tate conjecture.
reviewed by: Salman Abdulali
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6607754
zbMATH Keywords: Shimura variety
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: Tate conjecture
zbMATH Keywords: unitary group
MaRDI profile type: Publication
OpenAlex ID: W1699191428
Wikidata QID: Q101504524
arXiv ID: 1306.1515
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The Hodge conjecture and arithmetic quotients of complex balls (DOI)
DOI: 10.1007/S11511-016-0136-2