Contents

cohomology

complex geometry

# Contents

## Statement

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

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

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

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

(The Tate conjecture is the analogue where $\mathbf{C}$ 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:

• Claire Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 69, 2 (2011), pp. 149-198, pdf.

• Annette Huber, Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub (pdf).

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 $\PP^2$ 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.