All Questions
Tagged with motives ag.algebraic-geometry
359 questions
6
votes
1
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389
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Virtual mixed Tate motives
Let $\mathbf{Sch}_k$ be the category of $k$-schemes of finite type, and let $K_0(\mathbf{Sch}_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the ...
- 7,377
3
votes
0
answers
154
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Connecting Quillen functors between motivic homotopy categories (of different "types"): references?
For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it:
(a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
- 16.9k
5
votes
0
answers
530
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What is the algebro-geometric or measure-theoretic "content" of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?
1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$...
- 954
16
votes
3
answers
3k
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Motives versus Motifs
I was in Paris recently for a meeting about motives or motifs, and since I'm too jet lagged
for real work let me ask the following somewhat frivolous question. The word "motif" is
usually translated ...
- 43
4
votes
0
answers
477
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Why is the Hodge conjecture equivalent to the assertion that $ \mathcal{R}_{ \mathrm{Hodge} } $ is fully faithfull?
On pages 17 and 18 of the following document: https://www.math.tifr.res.in/~sujatha/ihes.pdf, we find the following paragraph:
Let $ \mathbb{Q} \mathrm{HS}$ be the category of pure Hodge structures ...
- 35.5k
7
votes
1
answer
787
views
Special cases of the Kimura-O’Sullivan conjecture, i.e., examples of finite dimensional motives?
In this 2005 paper, Kimura introduces a notion of finite dimensionality for Chow motives, defined in terms of vanishing of high symmetric and wedge powers. Toward the end of his paper, he conjectures ...
- 954
35
votes
4
answers
8k
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What would a "moral" proof of the Weil Conjectures require?
At the very end of this 2006 interview (rm), Kontsevich says
"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
- 43
2
votes
1
answer
367
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Dualizability and motivic cohomology
Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is ...
- 17.6k
35
votes
2
answers
3k
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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
- 1
8
votes
2
answers
793
views
Some questions about the map $K_0(\text{Var})\to K_0(\text{Mot})$
Let $k$ be a field. The naive Grothendieck ring of varieties $K_0(\text{Var})$ is generated by isomorphism classes of varieties over $k$ with the scissors relation $[X]=[X-Y]+[Y]$ for $Y$ a closed ...
- 693
17
votes
2
answers
1k
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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
18
votes
3
answers
1k
views
Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply ...
- 16.9k
20
votes
3
answers
2k
views
Voevodsky's Triangulated Categories of Motives and their Relationships
As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...
- 251
17
votes
1
answer
1k
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Why presheaves with transfer?
Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH ...
- 16.5k
6
votes
4
answers
2k
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References - Voevodsky motives are the derived category of Nori motives?
First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
CommunityBot
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3
votes
1
answer
465
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Relations between Motivic Galois groups and Motivic t-structure?
What are some relations between the existence of Motivic t-structures and Motivic galois groups?
I heard that indeed the existence of the Motivic t-structure implies the isomorphism between Ayoub's ...
- 1,720
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
8
votes
1
answer
567
views
What are Motivic homotopy types?
There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, ...
- 1,720
6
votes
1
answer
899
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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
4
votes
0
answers
161
views
Quadrics contained in the (complex) Cayley plane
In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...
- 17.4k
12
votes
0
answers
726
views
What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
- 2,227
2
votes
0
answers
304
views
Should all cohomology theories have a smooth proper base change
Let $H$ be a cohomology theory with respect to some Grothendieck topology (e.g. Zariski, analytic, etale, fppf, Nisnevich, etc.)
Does H satisfy smooth proper base?
If yes, does this mean that "...
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10
votes
1
answer
947
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Derived version of equivalence between motives and representations of Motivic galois groups?
A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $...
- 1,720
3
votes
0
answers
128
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On Abhyankar's results cited in a paper of Manin titled "Correspondences, Motifs and Monoidal Transformations"
Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.
Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over ...
CommunityBot
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6
votes
1
answer
1k
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Relationship between motivic Galois groups and Langlands program [duplicate]
I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.
- 17.6k
20
votes
2
answers
5k
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What does the Jacobian of a curve tell us about the curve?
A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell ...
- 1,720
8
votes
0
answers
603
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A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures
Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite.
A weakening of this conjecture states that the $\ell$-...
- 15.4k
7
votes
0
answers
811
views
Roadmap to study (Deligne) Algebraic geometry over Tannakian categories
I would like to know the way to proceed in the first lecture of Deligne's Le groupe fondamental de la droite projective moins trois points.
General advices for reading Deligne's paper.
What should I ...
CommunityBot
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2
votes
1
answer
474
views
What is explicitly, $ \mathcal{P} \mathrm{er} ( X ) / \mathbb{Q} $?
In the following link$^{[1]}$, page $2$, we find the following question :
Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ ...
- 17.6k
8
votes
1
answer
814
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Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?
First of all, I need to declare my extreme ignorance on the topic of modular forms, so, please, does not assume that I know Deligne's construction in details.
In Motives for modular forms, Scholl ...
- 17.6k
7
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1
answer
1k
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Intuition for the Lefschetz motive (Tate motive)?
Yo! Maybe this question is too dumb for mathoverflow, but I believe no one will pay attention to it in math.stackexchange, so I will post it here. If this question is not suitable, just delete it.
I ...
CommunityBot
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35
votes
1
answer
4k
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Grothendieck's letter to Faltings: On the yoga of motives and the degeneration of Leray spectral sequence
In Grothendieck's letter to Faltings, he writes
There exists at this time a kind of “yoga des motifs”, which is familiar to a handful of initiates, and in some situations provides a firm support for ...
- 16.9k
3
votes
1
answer
251
views
"theta characteristics" on general motives?
Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist ...
- 2,327
1
vote
0
answers
81
views
When the tensor product of motives that are not $0$-(homotopy)-connective can be $0$-connective?
For $t$ being the homotopy $t$-structure for Voevodsky effective motivic complexes
when $M\otimes N\in DM_{eff}^{t\le -1}$ ensures that either $M$ or $N$ belongs to $DM_{eff}^{t\le -1}$ (so, we use ...
- 16.9k
5
votes
1
answer
450
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Constructing groups of Type E7 with certain Tits Index
In a new survey on $E_8$, namely
Skip Garibaldi - E8 the most exceptional group
, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
- 1,479
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
3
votes
0
answers
172
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Non-multiplicative Euler-Poincaré Characteristics
Are there known examples of a non-multiplicative Euler-Poincaré characteristic on varieties?
Let $\mathbf{Var}/k$ be the category of varieties over a filed $k$, i.e. the category of reduced separated ...
- 349
5
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0
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172
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Polynomially countable varieties and virtual mixed Tate motives
Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$,...
- 4,547
18
votes
4
answers
2k
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Applications of homotopy purity theorem of Morel-Voevodsky
One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...
- 2,871
4
votes
0
answers
220
views
Passing motivic decompositions from rational to algebraic equivalence
It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The ...
- 1,479
21
votes
1
answer
734
views
Why is the CM closure of $\mathbb{Q}$ the "ultimate" coefficient field for motives?
In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the ...
- 6,920
11
votes
2
answers
1k
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The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?
I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory $...
- 16.9k
7
votes
1
answer
919
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Intuition behind the definition of finite correspondences
Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...
- 35.2k
6
votes
0
answers
351
views
Correspondences of schemes induced by a finite Galois extension
I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes.
Recall that if $...
- 596
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
1
vote
0
answers
116
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On "splitting off small weights" from Chow motives
I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed).
So, consider ...
- 16.9k
8
votes
0
answers
184
views
Can the failure of the multiplicativity of archimedean L-factors be corrected?
My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-...
CommunityBot
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7
votes
0
answers
279
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Quadratic twists of 1-motives
Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...
- 16.9k
39
votes
4
answers
10k
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difference between equivalence relations on algebraic cycles
For the definitions of the equivalence relations on algebraic cycles see http://en.wikipedia.org/wiki/Adequate_equivalence_relation.
I want to know how far away from each other the equivalence ...
- 105
4
votes
0
answers
248
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Derived equivalent varieties with differing integral Mukai-Hodge structures?
For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\...
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