nLab Hodge conjecture

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.)

Related concepts

• Hodge theory, Hodge theorem
• integral Hodge conjecture
• Tate conjecture

References

• Pierre Deligne, The Hodge conjecture (pdf)

• Claire Voisin, Hodge theory and Complex algebraic geometry.

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).

• Mathoverflow, Equivalent descriptions of Hodge conjecture, Why is the hodge conjecture so important, Refinement of Hodge conjecture, Generalized Hodge conjecture for triangulated motives, Equivalence between statements of Hodge conjecture, Interesting implications on the theory of Motives if the hodge conjecture holds, Heuristics for the hodge conjecture

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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.