##
**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.

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 |