General Hodge conjecture for abelian varieties of CM-type. (English) Zbl 1087.14503

Hazama, Fumio

Proc. Japan Acad., Ser. A 78, No. 6, 72-75 (2002).

From the introduction: The purpose of this article is to announce that the validity of the Hodge conjecture in codimension two implies that of the whole general Hodge conjecture (GHC for short) for any abelian varieties of CM-type. The main ingredient is the theory of abelian varieties associated to hyperplane arrangements as is developed in our previous paper [J. Algebr. Geom. 9, No. 4, 711–753 (2000; Zbl 1004.14003)]. In particular, the notion of “\(N\)-dominatedness” introduced in [loc. cit.] plays an essential role for us to understand what kind of exceptional Hodge cycles should be proved to be algebraic. Our strategy for the proof goes roughly as follows: Given a Galois CM-field \(K\) with \(\text{Gal}(K/\mathbb Q)\cong G\), we associate an abelian variety \(A_{{\mathcal A}(2^n)}(G; K)\) to a hyperplane arrangement \({\mathcal A}(2^n)\) in \(\mathbb R^n\). Thereafter, we show an arbitrary abelian variety \(A\) of CM-type split by \(K\) can be embedded into an appropriate self-product \(A_{{\mathcal A}(2^n)}(G; K)^m\). Thus GHC for \(A\) is reduced to GHC for \(A_{{\mathcal A}(2^n)}(G; K)^m\). Furthermore, we reduce GHC for \(A_{{\mathcal A}(2^n)}(G; K)^m\) to the usual Hodge conjecture for \(A_{{\mathcal A}(2^n)}(G; K)\) by translating the properties of various rational sub-Hodge structures of its cohomology spaces into some combinatorial properties of the arrangement \({\mathcal A}(2^n)\). Thus the fact that \(A_{{\mathcal A}(2^n)}(G; K)\) is 2-dominated implies the aforementioned result.

MSC:

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

Citations:

Zbl 1004.14003

References:

[1]	Dodson, B.: The structure of Galois groups of CM-fields. Trans. Amer. Math. Soc., 283 , 1-32 (1984). · Zbl 0535.12008
[2]	Hazama, F.: Hodge cycles on Abelian varieties of CM type. Res. Act. Fac. Sci. Engrg. Tokyo Denki Univ., 5 , 31-33 (1983).
[3]	Hazama, F.: Hodge cycles on abelian varieties of \(S_n\)-type. J. Alg. Geom., 9 , 711-753 (2000). · Zbl 1004.14003