In "The work of Tate" Milne says:

"The relation between the two conjectures has been greatly clarified by the work of Deligne. He defines the notion of an absolute Hodge class on a (complete smooth) variety over a field of characteristic zero, and conjectures that every Hodge class on a variety over C is absolutely Hodge. The Tate conjecture for a variety implies that all absolute Hodge classes on the variety are algebraic. Therefore, in the presence of Deligne’s conjecture, the Tate conjecture implies the Hodge conjecture. As Deligne has proved his conjecture for abelian varieties, this gives another proof of Piatetski- Shapiro’s theorem."

In which work (articles) Deligne shows this conjecture?

Is there some book (or expository article) about this conjecture and its connections with Tate and Hodge conjectures?


1 Answer 1


Deligne, Hodge Cycles on Abelian Varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, 1981, pp. 9-100. Maybe you could use Google to find these references by yourself?


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