nLab
Hodge conjecture

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Complex geometry

Contents

Statement

On a smooth projective complex manifold, every rational Hodge cycle is a linear combination of classes cl(Z)cl(Z) of algebraic cycles.

A motivic reformulation is as follows. Let SmProj C corSmProj^{cor}_\mathbf{C} denote the category of algebraic correspondences of smooth projective algebraic varieties over the complex numbers. The canonical functor

SmProj corHS pure SmProj^{cor} \to HS^{pure}

to the category of rational pure Hodge structures, given by taking rational Betti cohomology, is fully faithful.

(The Tate conjecture is the analogue where C\mathbf{C} is replaced by a finite field or number field, and the target category of pure Hodge structures is replaced by the l-adic representations? of the absolute Galois group.)

References

Lecture notes from a summer school on Hodge theory:


Add info also on Hodge conjecture: Lewis: A Survey of the Hodge conjecture. Deligne’s Clay formulation.

<http://burttotaro.wordpress.com/2012/03/18/why-believe-the-hodge-conjecture/>

Can add Jossen’s work as an answer here, when it appears: <http://mathoverflow.net/questions/17020/why-do-people-think-that-abelian-varieties-are-the-hardest-case-for-the-hodge-con>

Van Geemen: An introduction to the Hodge conjecture for abelian varieties

arXiv:0907.2503 The Hodge conjecture for self-products of certain K3 surfaces from arXiv Front: math.AG by Ulrich Schlickewei We use a result of van Geemen to determine the endomorphism algebra of the Kuga–Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of PP 2\PP^2 which are ramified along six lines.

Last revised on January 11, 2017 at 13:30:40. See the history of this page for a list of all contributions to it.