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I would like to know equivalent descriptions of the Hodge conjecture (with references).

  • Dan Freed's Version:

Consider a topological cycle (boundary less chains that are free to deform) on a projective manifold. The topological cycle is homologous to a rational combination of algebraic cycles, if and only if the topological cycle has rotation number zero.

  • Deligne's version (Clay's official description):

On a projective non-singular algebraic variety over $\mathbb{C}$ , any Hodge class is a rational combination of classes $\rm{Cl(Z)}$ of algebraic cycles.

  • nLab ((Pure)Motivic description):

Let $SmProj^{cor}_\mathbf{C}$ denote the category of algebraic correspondences of smooth projective algebraic varieties over the complex numbers. Then the canonical functor

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

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

Equivalence between statements of Hodge conjecture

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    $\begingroup$ The equivalence between the classical version (by Deligne above) and the fullness (=nlab above) version is explained in the book of André "Introduction aux motifs". It is just a reformulation of the definitions, though. What is nice about the fullness version is that you can use another cohomology theory. For example, replacing the Hodge realization by the l-adic one, you get Tate's conjecture. Grothendieck's period conjecture is equivalent to the fullness of the de-Rham-Betti realization. $\endgroup$ – Jakob Aug 20 '16 at 19:37
  • $\begingroup$ math.ru.nl/~bmoonen/Lecturenotes/MTGps.pdf (pp 5). $\endgroup$ – tttbase Jan 19 '17 at 3:34
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  • [...] "Hodge conjecture can be reformulated by saying that the Hodge realization of the algebraically defined $\mathbb{Q}$-vector space of codimension p algebraic cycles modulo numerical (or homological) equivalence is the 1-motive part of $H^{2p}(X, \mathbb{Q}(p))$."[...] On algebraic 1-motives related to hodge cycles, Luca Barbieri-Viale.

  • "In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles." Clay's Informal description.

  • "The Hodge conjecture holds and a pure motive M ∈M is effective if and only if H(M) is effective."

  • "(for triangulated motives) The Hodge conjecture holds and an object M ∈ DMgm is effective if and only if its Hodge realization is effective. Slice filtration on motives and the Hodge conjecture, with an appendix by J. Ayoub, A. Huber.

  • $∀X, GMot_{k,Hσ}(X) = MT(X) ⊆ GL(H_σ(X))$. Motivic galois groups, Kahn

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A possibly less known reformulation of the Hodge conjecture was given by Richard Thomas. He relates to a problem about finding nodal hypersurfaces with sufficient homology.

https://arxiv.org/abs/math/0212216

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