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Hodge structures of CM-type. (English) Zbl 1094.14004

The author proves that any effective polarized Hodge structure of CM-type comes from the cohomology of some CM abelian variety over complex numbers. As an application he proves that the usual Hodge conjecture for CM abelian varieties implies the general Hodge conjecture for the same class. This result has been already obtained in [F. Hazama, Proc. Japan Acad., Ser. A 78, No. 6, 72–75 (2002; Zbl 1087.14503); Publ. Res. Inst. Math. Sci. 39, No. 4, 625–655 (2003; Zbl 1067.14010)] using different methods. The proof uses an observation of A. Grothendieck that if a smooth complex projective variety \(X\) is dominated by a family \(\mathcal Y\) of smooth complex projective varieties, then the usual Hodge conjecture for all \(X \times Y\) with \(Y \in \mathcal Y\) implies the general Hodge conjecture for \(X\). At the end the author gives known examples of CM abelian varieties for which the general Hodge conjecture is well-known. This includes S. G. Tankeev’s example [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)], powers of a CM elliptic due to T. Shioda [in: Algebraic varieties and analytic varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 55–68 (1983; Zbl 0527.14010)] and powers of CM abelian surfaces.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K22 Complex multiplication and abelian varieties
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