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Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ also the Hodge conjecture.

Are there analogues in the general case (when cohomology is no longer pure)?

Of course there is a conjectural generalisation that says:

The $\ell$-adic (resp. Betti) realisation functors from the category of mixed motives over $K$ to the category of Galois representations (resp. mixed Hodge structures) is fully faithful.

(NB: "category of Galois representations" means that morphisms are equivariant for some open subgroup of $\mathrm{Gal}(\bar{K}/K)$.)

The problem with this generalisation is that the statement refers to the category of mixed motives, and we don't know whether it exists (i.e., the conjecture is conjectural).

Q: Is there a generalisation that does not depend on the category of mixed motives (probably involving some sort of cycle maps)?

On the one hand I expect this is possible, on the other hand maybe the same problem that pops up in defining mixed motives also appears when trying to formulate a generalisation of the Tate and Hodge conjectures. (I am still very much a novice when it comes to the mixed versions of motives and cohomology.)


Note that there is a good generalisation of the Mumford–Tate conjecture. After all, we still have Mumford–Tate groups and images of Galois; we just don't expect them to be reductive (and in general they are not).

For some reason I could not find anything about this via a search on the general internet or specifically mathscinet.

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  • $\begingroup$ Even if you assume that mixed motives exist, the conjecture as you stated it seems to be wrong over fields of trancendence degree $>1$. $\endgroup$ – Mikhail Bondarko Feb 25 '15 at 16:20
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    $\begingroup$ @MikhailBondarko — That's new to me; would you please tell me why? And if possible, how to fix it? $\endgroup$ – jmc Feb 25 '15 at 16:22
  • $\begingroup$ I don't think that one can fix it. The problem is that mixed motives are "more complicated" than Hodge structures and Galois modules. The Tate and Hodge conjectures only describe morphisms between pure motives of the same weight. $\endgroup$ – Mikhail Bondarko Feb 25 '15 at 19:32
  • $\begingroup$ @MikhailBondarko — Do you know of a reference where this is explained in more detail? You seem to suggest that if $\mathrm{tr.deg}(K) \le 1$, then such a formulation might work. How does the transcendence degree kick in? If most of this is “well-known folk lore”, then I can understand that; I'll just have to find a sage who wants to teach me in the lore… $\endgroup$ – jmc Feb 25 '15 at 19:37
  • $\begingroup$ You may start with books.google.ru/… The problem is that for "large" base fields the cohomological dimension of mixed motives should be more than $1$. $\endgroup$ – Mikhail Bondarko Feb 25 '15 at 20:30
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(Edited) The short answer is yes. There are analogues of these conjectures, starting with work of Beilinson, Jannsen and perhaps others in the 1980's. Basically, they would say that cycle maps from motivic cohomology$^1$ $\otimes \mathbb{Q}$ (resp. $\mathbb{Q}_\ell$) to $Hom_{MHS}(\mathbb{Q}(-p), H^i(X))$ (resp. $Hom_{G}(\mathbb{Q}_{\ell}(-p), H^i(X))$ should be onto. However, without further qualifications, this is known to be false$^2$.

Probably a good place to start to read about this is Jannsen's Mixed motives and algebraic K-theory.

  1. Motivic cohomology can be expressed as an $Ext$ group in Voevodsky's category of mixed motives (which exists!) or more concretely from my point of view in terms of Bloch's higher Chow groups. Since we are tensoring with $\mathbb{Q}$, this can also be defined using higher $K$-theory which is how it was done in the original papers.

  2. It is expected to hold when $p=2i$ (usual Hodge/Tate) or $p=i$ ("Milnor" case), or over $\overline{\mathbb{Q}}$ for any $p,i$.

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  • $\begingroup$ Awesome, thanks for the reference! Could you please expand a little bit on the “further qualifications”? A pointer into Jannsen's book would be great. Thanks again for your answer. $\endgroup$ – jmc Feb 25 '15 at 14:25
  • $\begingroup$ OK, I'll try to expand this a bit later when I have more time. $\endgroup$ – Donu Arapura Feb 25 '15 at 14:33
  • $\begingroup$ Ah, Springer released its grip on the electronic version of this book after I did a couple of somersaults through proxies and the like. It didn't come with a ToC, but “§5. The conjecture of Hodge and Tate for smooth varieties” (p.57) and “§7. The conjecture of Hodge and Tate for singular varieties” (p.107) look very interesting! Nevertheless, feel free to expand your answer. It is always enlightening to hear an expert about his view of the picture. $\endgroup$ – jmc Feb 25 '15 at 15:11
  • $\begingroup$ Thanks for the edit! With Voevodsky's category of mixed motives, you mean his category $DM$, which should be the derived category of mixed motives, right? Or am I missing something? $\endgroup$ – jmc Feb 25 '15 at 16:13
  • $\begingroup$ Yes. A triangulated category is all you need for the $Ext$'s. $\endgroup$ – Donu Arapura Feb 25 '15 at 16:45
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There is a version of the Hodge conjecture for open and singular varieties using Borel-Moore homology, which can be formulated without using the word "mixed motive".

Let $X$ be any complex algebraic variety. Let $H_i(X)$ be its Borel-Moore homology. It carries a mixed Hodge structure with weights in the interval $[-i,0]$. This version of the Hodge conjecture says that the image of the cycle class map $\mathrm{CH}_i(X)_{\mathbf Q} \to H_{2i}(X,{\mathbf Q})$ is exactly the subspace $W_{-2i}H_{2i}(X,\mathbf Q) \cap F^{-i}H_{2i}(X,\mathbf C)$.

It is known that this conjecture is equivalent to the usual Hodge conjecture. There is also an analogue of the generalized Hodge conjecture in this setting. If $X$ is smooth but noncompact, one can use ordinary cohomology instead of Borel-Moore homology, and consider instead $W_{2i}H^{2i}(X,\mathbf Q) \cap F^{i}H^{2i}(X,\mathbf C)$.

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  • $\begingroup$ Hi Dan, thanks for the answer! Do you have a reference for these statements of the HC? In particular the fact that it is equivalent to the usual Hodge conjecture? And do you know how this generalisation relates to the one in Jannsen's book, pointed to by Donu Arapura? $\endgroup$ – jmc Feb 26 '15 at 10:02
  • $\begingroup$ I think this can be found in a book by Lewis, "A survey of the Hodge conjecture". I don't know how it relates to the formulation in terms of motivic cohomology. $\endgroup$ – Dan Petersen Feb 26 '15 at 10:06
  • $\begingroup$ If $X$ is smooth at least, you can use duality to flip to a cohomological statement. Then it would a special case. This is actually what I meant by my somewhat cryptic comment "expected to hold for $i=2p$ (Hodge/Tate)" $\endgroup$ – Donu Arapura Feb 26 '15 at 15:06
  • $\begingroup$ Yes, I just read this in Jannsen's book. He also shows that his generalisations are equivalent to the smooth and proper case (at least in char. 0, in char p one needs the existence of “good proper covers”). $\endgroup$ – jmc Mar 3 '15 at 17:37

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