Topological and algebraic cycles in Kuga-Shimura varieties
Let X denote a nonsingular connected complex projective variety, \(F^ rH^ i(X):=\oplus _{p\geq r}H^{p,i-p}(X)\quad its\) Hodge filtration, and \(F'{}^ rH^ i(X,{\mathbb{Q}})\) the arithmetic filtration, defined as the set of classes \(\xi \in H^ i(X,{\mathbb{Q}})\) which vanish when restricted to the complement of some Zariski-closed subset of X of codimension at least r. Then the generalized Hodge conjecture asserts that \(F'{}^ rH^ i(X,{\mathbb{Q}})\) is the largest sub-Hodge structure contained in \(F^ rH^ i(X)\cap H^ i(X,{\mathbb{Q}})\). In this paper a slightly stronger statement, namely that \(F'{}^ rH^ i(X,{\mathbb{Q}})=F^ rH^ i(X)\cap H^ i(X,{\mathbb{Q}})\) is proved when X is either a generic fiber or the total space of a family of abelian varieties parametrized by a Shimura curve C associated to an indefinite division quaternion algebra over \({\mathbb{Q}}.\) The proofs of these results rely heavily on the methods by which \textit{M. Kuga} [Proc. U.S.-Japan Semin. differential geometry, Kyoto 1965, 72-81 (1966; Zbl 0173.489)] and \textit{R. Hall} and \textit{M. Kuga} [Sci. Papers College General Educ., Univ. Tokyo 25, 1-6 (1975; Zbl 0315.14001)] verified the usual Hodge conjecture, that \(H^{p,p}(X)\cap H^{2p}(X,{\mathbb{Q}})\) is generated by algebraic cycles (or equivalently, that \(F^ pH^{2p}(X)\cap H^{2p}(X,{\mathbb{Q}})=F'{}^ pH^{2p}(X,{\mathbb{Q}}))\), for these same varieties. Here the proofs proceed by decomposing \(H^ i(X,{\mathbb{Q}})\) into subspaces isotypic under the monodromy action of \(\pi _ 1(C)\), and then analyzing the Hodge structure of these subspaces, using Kuga's and Hall's results to characterize the algebraic cycles.
scientific article; zbMATH DE number 3981325
scientific article