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. 
On the Mumford–Tate conjecture for 1-motives
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
Families of Motives and the Mumford–Tate Conjecture
We give an overview of some results and techniques related to the Mumford–Tate conjecture for motives over finitely generated fields of characteristic 0. In particular, we explain how working in
On the fields of definition of Hodge loci.
A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the
Some remarks concerning the Grothendieck Period Conjecture
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the
The Tate conjecture for K3 surfaces over finite fields
Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s
Height bounds for certain exceptional points in some variations of Hodge structures
We consider smooth projective morphisms f : X → S of Kvarieties with S an open curve and K a number field. We establish upper bounds of the Weil height h(s) by [K(s) : K] at certain points s ∈ S(K̄)
Lectures on K3 Surfaces
TLDR
Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular and each chapter ends with questions and open problems.
Some non-special cubic fourfolds
In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz
A characterization of complex quasi-projective manifolds uniformized by unit balls
In 1988 Simpson extended the Donaldson–Uhlenbeck–Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds
MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES
Abstract For any subfield $K\subseteq \mathbb{C}$ , not contained in an imaginary quadratic extension of $\mathbb{Q}$ , we construct conjugate varieties whose algebras of $K$ -rational ( $p,p$
...
...

References

SHOWING 1-10 OF 34 REFERENCES
Hodge Cycles on Abelian Varieties
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
  • C. Voisin
  • Mathematics
    Compositio Mathematica
  • 2007
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
Extensions of mixed Hodge structures
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
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
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
Mixed motives for absolute hodge cycles.- Algebraic cycles, K-theory, and extension classes.- K-theory and ?-adic cohomology.
...
...