Notes on absolute Hodge classes

@article{Charles2011NotesOA,
  title={Notes on absolute Hodge classes},
  author={Franccois Charles and Christian Schnell},
  journal={arXiv: Algebraic Geometry},
  year={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 field of definition of Hodge loci and the Kuga-Satake construction. 
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References

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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…
Hodge loci and absolute Hodge classes
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This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of…
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According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to…
The locus of Hodge classes in an admissible variation of mixed Hodge structure
Some aspects of the Hodge conjecture
Abstract.I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the…
The Hodge conjecture for self-products of certain K3 surfaces
Kuga-Satake varieties and the Hodge conjecture
Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge…
A counterexample to the Hodge conjecture for Kaehler varieties
Summary: The Hodge conjecture asks whether rational Hodge classes on a smooth projective manifolds are generated by the classes of algebraic subsets, or equivalently by Chern classes of coherent…
Hodge Cycles and Crystalline Cohomology
This paper is a collection of musings about several questions related to crystalline cohomology that have plagued me for the past few years. It contains many more conjectures than proofs, and my…
Mixed Motives and Algebraic K-Theory
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