All Questions
31 questions
2
votes
0
answers
483
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Absolute Hodge cycles over $\mathbf{Q}$
In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
- 63.9k
6
votes
0
answers
265
views
Rank $2$ motivic local systems on a curve
This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...
- 10.5k
31
votes
2
answers
3k
views
On Grothendieck's idea on his Standard Conjecture B
Let me recall the Standard Conjecture B (see [1,2] below):
The $\Lambda$-operation of Hodge theory is algebraic.
It more or less says that the partial inverse to “cupping with the class of a ...
CommunityBot
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8
votes
2
answers
2k
views
Hodge standard conjecture in positive characteristic
In the Wikipedia article on the Hodge Standard Conjecture it is written (note [Oct. 2015]: it has since been fixed):
In characteristic zero the Hodge standard conjecture holds, being a consequence ...
- 4,713
14
votes
2
answers
1k
views
Is Deligne cohomology the motivic cohomology of analytic spaces?
Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...
CommunityBot
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3
votes
0
answers
114
views
Multiplicative structure on Deligne cohomology
Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...
CommunityBot
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5
votes
1
answer
348
views
Spectral sequence in Betti cohomology
Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X_{\rm an}\to X_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $...
- 16.9k
1
vote
0
answers
118
views
Torsion homologically trivial cycles
Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?
CommunityBot
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1
vote
0
answers
117
views
Filtrations and the Betti cycle map
Let $X$ be a smooth projective complex variety.
Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...
CommunityBot
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0
votes
0
answers
88
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Cycles modulo homological equivalence
Let $\text{CH}^p(X)_{\rm hom}$ be the abelian group of codimension $p$ algebraic cycles on a smooth projective variety over a field $k$, modulo homological equivalence. Is $\text{CH}^p(X)_{\rm hom}$ ...
CommunityBot
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3
votes
1
answer
568
views
Absolute Hodge cycles
Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...
- 35.2k
4
votes
0
answers
92
views
Locus of Hodge classes
Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...
CommunityBot
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11
votes
1
answer
967
views
How to think about infinite generatedness of motivic cohomology
In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
- 16.5k
8
votes
1
answer
989
views
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
A refresher (you may skip to the question at the bottom)
One defines
(1) $z_n(X,d) :=$...
CommunityBot
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9
votes
0
answers
463
views
Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
- 13.6k
12
votes
3
answers
2k
views
Motivic vs Deligne cohomology
Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch ...
- 35.2k
16
votes
1
answer
3k
views
Tate twists and cohomology of $\mathbf{P}^1$
I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to ...
- 13.6k
2
votes
1
answer
164
views
Full lattice images and Hodge decomposition
Let $X$ be a smooth and proper complex analytic space, or a Kahler complex manifold. Is the image $\Lambda$ of the finitely generated abelian group $H^{2i}(X,\mathbf{Z})$ into $J := H^{2i}(X,\mathbf{C}...
- 38k
8
votes
1
answer
432
views
Finiteness aspects of Deligne cohomology
Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.
For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{...
CommunityBot
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6
votes
1
answer
1k
views
Intuition for polarized Hodge structures
A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The ...
- 35.2k
10
votes
2
answers
1k
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What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?
I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I don'...
CommunityBot
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7
votes
1
answer
718
views
Generalised Hodge Conjecture
Further to my question,
A Naive Question on Mixed Motives and Mixed Hodge Structures
that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on ...
- 35.2k
8
votes
1
answer
846
views
A Naive Question on Mixed Motives and Mixed Hodge Structures
As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated!
The category of mixed motives ...
- 16.9k
35
votes
2
answers
3k
views
Equivalent descriptions of Hodge conjecture?
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 ...
CommunityBot
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17
votes
2
answers
1k
views
Hodge standard conjecture for étale cohomology
It is known that Hodge standard conjecture is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\...
- 28.7k
2
votes
1
answer
334
views
Reference - Generalized Hodge conjecture for triangulated motives
GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
I would like to know some references on GHC ...
- 1,720
6
votes
1
answer
899
views
Interesting implications on the theory of motives if the Hodge conjecture holds
For example,
Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as ...
- 1,720
13
votes
1
answer
591
views
Is there a yoga of effectivity for motives and their realizations?
Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
- 1,720
15
votes
2
answers
2k
views
Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...
- 10.5k
8
votes
1
answer
958
views
Motives over the complex numbers versus mixed Hodge structures
Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...
- 13.6k
8
votes
1
answer
1k
views
If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?
Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...
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