The Hodge conjecture and arithmetic quotients of complex balls. (English) Zbl 1349.14031
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.
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.
Reviewer: Salman Abdulali (Greenville)
MSC:
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14G35 | Modular and Shimura varieties |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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