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Hodge cycles on abelian varieties. (Notes by J. S. Milne). (English) Zbl 0537.14006
Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 9-100 (1982).
[For the entire collection see Zbl 0465.00010.]
Let $$k$$ be an algebraically closed field of finite transcendence degree over $${\mathbb{Q}}$$ and let $$X$$ be a nonsingular complete algebraic variety over k. One disposes of the algebraic de Rham cohomology groups $$H^ i_{DR}(X)$$ and the etale cohomology groups $$H^ i_{et}(X)=\lim H^ i_{et}(X,{\mathbb{Z}}/n)\times {\mathbb{Q}}.$$ An embedding $$\sigma:k\subset {\mathbb{C}}$$ gives rise to a complex manifold $$\sigma$$ X which has singular cohomology groups $$H^ i_ B(\sigma X)=H^ i(\sigma X,{\mathbb{Q}}).$$ One has comparison isomorphisms $$\sigma^*_{DR}:H^ i_{DR}(X)\otimes_{k,\sigma}{\mathbb{C}}\to H^ i_{DR}(\sigma X)$$ and $$\sigma^*_{et}:H^ i_{et}(X)\to H^ i_{et}(\sigma X)$$ and embeddings of $$H^ i_ B(\sigma X)$$ in $$H^ i_{DR}(\sigma X)$$ and $$H^ i_{et}(\sigma X)$$. An element $$(t_ 1,t_ 2)\in H^{2p}_{DR}(X)\times H^{2p}_{et}(X)(p)$$ is called a Hodge cycle relative to $$\sigma$$ if $$(\sigma^*_{DR}(t_ 1),\sigma^*_{et}(t_ 2))$$ lies in the image of $$H_ B^{2p}(\sigma X)(p)$$ (here $$(p)$$ denotes the suitable Tate twists), and $$t_ 1\in F^ pH^{2p}_{DR}(X)$$. An absolute Hodge cycle on $$X$$ is a pair $$(t_ 1,t_ 2)$$ which is a Hodge cycle relative to all embeddings of $$k$$ in $${\mathbb{C}}$$. The main result of this paper states that, if $$X$$ is an abelian variety over $$k$$ and $$t$$ is a Hodge cycle relative to one embedding of $$k$$ in $${\mathbb{C}}$$, then $$t$$ is an absolute Hodge cycle. This result is proven first for abelian varieties of CM-type. It relies on the following principle. Let $$(X_ j)$$ be a family of complex algebraic varieties and let $$t_ 1,...,t_ n$$ be absolute Hodge classes, where each $$t_ i$$ lies in a space obtained by tensoring spaces of the form $$H_ B^{n_ j}(X_ j)$$, $$H_ B^{n_ j}(X_ j)$$ and $${\mathbb{Q}}(1)$$. Let G be the subgroup of $$\prod GL(H_ B^{n_ j}(X_ j))\times {\mathbb{G}}_ m$$ fixing the $$t_ i$$. If t lies in a similar space and is fixed by G, then t is an absolute Hodge cycle. To derive the result for general abelian varieties one exploits the families of abelian varieties parametrized by Shimura varieties and a second principle: let $$f:X\to S$$ be smooth and proper and let $$(t_ 1,t_ 2)$$ be a global horizontal section of $${\mathcal H}^{2p}_{DR}(X/S)\times {\mathcal H}^{2p}_{et}(X/S)(p)$$ such that $$t_ 1(s)\in F^ pH^{2p}_{DR}(X_ s)$$ for all $$s\in S$$. If t(s) is an absolute Hodge cycle for one $$s$$ and $$S$$ is connected, then $$t(s)$$ is an absolute Hodge cycle for all $$s$$. The paper ends with a consideration of the periods of Fermat varieties with an application to the algebraicity of certain products of special values of the gamma function.
Reviewer: J.H.M.Steenbrink

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F40 de Rham cohomology and algebraic geometry 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 33B15 Gamma, beta and polygamma functions