# Hodge Cycles on Abelian Varieties

@inproceedings{Delign1982HodgeCO, title={Hodge Cycles on Abelian Varieties}, author={Pierre Delign{\'e}}, year={1982} }

The main result proved in these notes is that any Hodge cycle on an abelian variety (in characteristic zero) is an absolute Hodge cycle — see §2 for definitions and (2.11) for a precise statement of the result.

## 255 Citations

Notes on absolute Hodge classes

- Mathematics
- 2011

We survey the theory of absolute Hodge classes. The notes include a full proof of Deligne's theorem on absolute Hodge classes on abelian varieties as well as a discussion of other topics, such as the…

A Survey of the Hodge Conjecture for Abelian Varieties

- Mathematics
- 1997

We review what is known about the Hodge conjecture for abelian varieties, with some emphasis on how Mumford-Tate groups have been applied to this problem.

Chow motives of universal families over some Shimura surfaces

- Mathematics
- 2007

We prove an absolute Chow-Kuenneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surface. We furthermore prove the Hodge conjecture for general…

On the Mumford–Tate conjecture for 1-motives

- Mathematics
- 2011

We show that the statement analogous to the Mumford–Tate conjecture for Abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge…

The symmetric Square of the Theta Divisor in Genus 4

- Mathematics
- 2011

We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4

Addendum to: Hodge Classes on Self-Products of a Variety with an Automorphism

- MathematicsCompositio Mathematica
- 1998

There are infinitely many fundamentally distinct families of polarized Abelian fourfolds of Weil type with multiplication from the cyclotomic field of cube roots of unity. The Hodge conjecture is…

An Introduction to the Hodge Conjecture for Abelian Varieties

- Mathematics
- 1994

In this lecture we give a brief introduction to the Hodge conjecture for abelian varieties. We describe in some detail the abelian varieties of Weil-type. These are examples due to A. Weil of abelian…

Abelian varieties and the general Hodge conjecture

- MathematicsCompositio Mathematica
- 1997

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A…

Hodge structures on abelian varieties of type IV

- Mathematics
- 2004

Abstract.Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k.…

A CHARACTERIZATION OF CERTAIN SHIMURA CURVES IN THE MODULI STACK OF ABELIAN VARIETIES

- Mathematics
- 2002

Let f : X → Y be a semistable family of complex abelian varieties over a curve Y of genus g(Y ), and smooth over the com

## References

SHOWING 1-10 OF 101 REFERENCES

Conjugates of Shimura Varieties

- Mathematics
- 1982

In the first three sections we review the definition of a Shimura variety of abelian type, describe how certain Shimura varieties are moduli varieties for abelian varieties with Hodge cycles and…

On the periods of abelian integrals and a formula of Chowla and Selberg

- Mathematics
- 1978

Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a…

NUMERICAL AND HOMOLOGICAL EQUIVALENCE OF ALGEBRAIC CYCLES ON HODGE MANIFOLDS.

- Mathematics
- 1968

with rational coefficients). We denote by 3h (M), (resp. (3(M)) the cycles which are homologous, (resp. numerically equivalent) to zero. In general g n D3-hand 3m, = m. Matsusaka's work [5] implies…

On the Periods of Certain Rational Integrals: II

- Mathematics
- 1969

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how…

Resolution Of Singularities Of Embedded Algebraic Surfaces

- Mathematics
- 1966

0 Introduction.- 1. Local Theory.- 1 Terminology and preliminaries.- 2 Resolvers and principalizers.- 3 Dominant character of a normal sequence.- 4 Unramified local extensions.- 5 Main results.- 2.…

On Fermat varieties

- Mathematics
- 1979

(0.1) x? + x? + + x?+ί = 0 . Throughout this paper, we denote it by Xrm, or by X r m(p), when we need to specify the characteristic p of the base field k; we always assume that m ^ 0 (mod p). The…

Applications de la formule des traces aux sommes trigonométrigues

- Philosophy
- 1977

Dans cet expose, j’explique comment la formule des traces permet de calculer ou d’etudier diverses sommes trigonometriques et comment, jointe a la conjecture de Weil, elle peut permettre de les…

Introduction to Affine Group Schemes

- Mathematics
- 1979

I The Basic Subject Matter.- 1 Affine Group Schemes.- 1.1 What We Are Talking About.- 1.2 Representable Functors.- 1.3 Natural Maps and Yoneda's Lemma.- 1.4 Hopf Algebras.- 1.5 Translating from…

Nilpotent connections and the monodromy theorem: Applications of a result of turrittin

- Mathematics
- 1970

© Publications mathématiques de l’I.H.É.S., 1970, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www.…

La conjecture de Weil pour les surfacesK3

- Mathematics
- 1971

1. Enonc6 du th~or~me Soient Fq un corps ~t q ~l~ments, Fq une cl6ture alg6brique de Fq, r la substitution de Frobenius xv-~x q et F= tp -1 le << Frobenius g6om6trique >>. Soit X un sch6ma (s6par6 de…