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A geometric description of Hazama’s exceptional classes. (English) Zbl 1004.14002
The paper gives an explicit description of the exceptional class found by F. Hazama [J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 487-520 (1984; Zbl 0591.14006)] in \(X\times X\), where \(X\) is a variety of Mumford-type [belonging to a family of four dimensional polarized Abelian varieties constructed by D. Mumford, Math. Ann. 181, 345-351 (1969; Zbl 0169.23301)]. Let \(Y\) be a complex Abelian variety and let \(B^p(Y)\) denote the Hodge classes \(H^{2p}(Y, \mathbb{Q})\cap H^{p, p}(Y)\). These can be described as the classes in \(H^{2p}(Y, \mathbb{Q})\) that are invariant under the Mumford-Tate group. The exceptional classes discussed in this paper are classes in \(B^2(X\times X)\) that are not in \(\bigwedge^2B^1(X\times X)\). The study of them is made possible by the explicit description of \(X\) and the Mumford-Tate group of \(X\) and is ultimately reduced to a problem in representation theory of the groups \(\text{Sl}_2(\mathbb{Z})^3\) and \(\text{Sp}_4(\mathbb{Z})\).
14C25 Algebraic cycles
14K12 Subvarieties of abelian varieties
14F25 Classical real and complex (co)homology in algebraic geometry
20G05 Representation theory for linear algebraic groups
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