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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 $$\text{Sp}_4(\mathbb{Z})$$.
##### MSC:
 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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