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Abelian varieties and the general Hodge conjecture. (English) Zbl 0891.14003

An abelian variety \(A\) over \(\mathbb{C}\) is said to be of PEL-type if the family of abelian varieties constructed by D. Mumford from \(A\) [in: Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, 347-351 (1966; Zbl 0199.24601)] is a PEL-family in Shimura’s sense [G. Shimura, Ann. Math., II. Ser. 83, 294-338 (1966; Zbl 0141.37503)]. The main theorem of this article is the following: Let \(A\) be an abelian variety of PEL-type with semisimple Hodge group. Suppose that for every simple factor \(B\) of \(A\), if \(B\) is of type III, then \(H^1(B,\mathbb{Q})\) has odd dimension as a vector space over the endomorphism algebra of \(B\). Then the usual Hodge conjecture for all powers of \(A\) implies the Grothendieck’s general Hodge conjecture for \(A\). – Note that the general Hodge conjecture for the products of abelian varieties of “\(sp_n\)-type” is proved by S. G. Tankeev [Russ. Acad. Sci., Izv., Math. 43, No. 1, 179-191 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 4, 192-206 (1993; Zbl 0871.14009)] and by F. Hazama [Compos. Math. 93, No. 2, 129-137 (1994; Zbl 0848.14003)] independently.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K05 Algebraic theory of abelian varieties
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