Questions tagged [motives]
for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.
459 questions
5
votes
1
answer
234
views
Motive associated to a cuspidal representation of $GSp_{4}$
In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)
Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...
- 37k
5
votes
0
answers
138
views
Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$
My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.
It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...
- 2,871
6
votes
1
answer
383
views
Nisnevich topology inspired by Adeles
I'm quite a newbe in the field of motives & A1 homotopy theory,
so please forgive me if the question is too elementary:
In the intro from wikipedia on Nisnevish topology
is remarked that it's ...
CommunityBot
- 1
3
votes
0
answers
206
views
Generalization of conjectures involving Beilinson regulators
I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
- 9,501
32
votes
4
answers
3k
views
Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
- 4,713
4
votes
1
answer
544
views
What is motivic sheaf intuitively?
I am not very familiar with motif theory, but I do know a little about Hodge theory.
I view (mixed) motif theory as an enhancement of (mixed) Hodge structures.
Q1. Is (mixed) motivic sheaf theory an ...
- 16.9k
11
votes
3
answers
1k
views
Motive of CM elliptic curve and modular forms
I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especially when there is complex ...
- 37k
5
votes
1
answer
397
views
Which varieties are sums of tensor powers of the Lefschetz motive?
Any smooth projective variety $X$ gives an object $h(X)$ in the category of pure Chow motives. If $X$ is a generalized flag variety, i.e. a quotient $G/P$ where $G$ is semisimple linear algebraic ...
- 35.2k
23
votes
1
answer
1k
views
Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?
For algebraic curves we can state the Riemann hypothesis part of the Weil conjectures directly as a formula for the number of points on the curves, sidestepping the zeta function. Namely, given a ...
- 894
4
votes
1
answer
238
views
Functoriality conjectures on the slice filtration
Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that
Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...
- 2,806
3
votes
1
answer
533
views
Homotopy invariance of $\ell$-adic cohomology
In the end of the Voevodsky’s lectures on cross functors, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we ...
- 148k
9
votes
1
answer
472
views
Why is the category of motives generated by varieties?
I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...
- 16.9k
1
vote
0
answers
79
views
Localization with or without transfers
Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\...
- 309
3
votes
0
answers
433
views
Stable $\infty$-category of motives
In nLab motive, it defines the derived category of motives as the full sub-$\infty$-category of the $\infty$-category of functors $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
- 988
11
votes
1
answer
1k
views
How should I think about 1-motives?
By definition, a 1-motive over an algebraically closed field $k$ is the data
$$
M = [X\stackrel{u}{\to}G]
$$
where $X$ is a free abelian group of finite type, $G$ is a semi-abelian variety over $k$, ...
- 148k
4
votes
0
answers
251
views
Motives based on Hodge cycles vs algebraic cycles
I am not a specialist of motives. I am afraid my questions are rather naive.
We have the category of (pure) motives based on Hodge cycles by Deligne. In his articles with Milne, morphisms between ...
- 153
4
votes
0
answers
219
views
Generate periods only by smooth varieties
Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\...
- 275
1
vote
0
answers
213
views
Algebraic correspondence as morphisms in Betti cohomology
$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
- 63.9k
1
vote
0
answers
123
views
Motivic homotopy categories closed under subobjects and quotients
It is well known that the category $\mathbf{HI}_{\rm Nis}^{\rm tr}(k)$ of $\mathbb{A}^1$-local Nisnevich sheaves with transfers is closed under subobjects and quotients, from the highly nontrivial ...
- 16.9k
1
vote
0
answers
260
views
Non-examples of mixed Tate motives
I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written ...
- 63.9k
4
votes
0
answers
488
views
Deligne's letter to Soulé from 1985
There is a famous letter of Deligne to C. Soulé in which, apparently, Deligne first formulated the conjecture on the existence of an abelian category of mixed motives, extending Grothendieck's pure ...
- 12.9k
6
votes
0
answers
439
views
Cohomology theories for algebraic varieties over number fields
There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
- 2,751
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
- 1
5
votes
0
answers
176
views
Can a Chow motif be isomorphic to its own direct summand?
Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus ...
- 16.9k
3
votes
0
answers
331
views
Motives (and examples) of projective bundles over projective spaces
If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in ...
- 16.9k
5
votes
1
answer
300
views
A question on motivic zeta-function
It's well-known that over $\mathbb F_q$ every smooth projective conic $C$ is isomorphic to a projective line. So the formula for the motivic zeta-function $Z_{mot}(C)$ is evident since $S^n\mathbb P^1 ...
- 148k
7
votes
1
answer
474
views
Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
CommunityBot
- 1
1
vote
1
answer
383
views
Grothendieck rings and the Tannakian formalism
I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives ...
- 63.9k
8
votes
0
answers
587
views
Values of cohomology theory on a point
$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar ...
- 5,901
10
votes
0
answers
481
views
What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?
The question is as in the title, but here is some background:
Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
- 10.5k
3
votes
1
answer
758
views
Motives and topological data analysis
Here is some meta mathematics question.
During the last decade there has been some progress in the field of applied maths, called topological data analysis.
The setup starts with some set of points in ...
- 3,065
7
votes
0
answers
181
views
In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
- 7,746
11
votes
1
answer
893
views
K-equivalence ⇒ isomorphism of Chow motives?
An old conjecture of Bondal–Orlov–Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper D-equivalence and K-equivalence for definitions. In particular this applies to ...
- 2,300
26
votes
1
answer
1k
views
What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.
I ...
- 13.6k
1
vote
0
answers
270
views
Eilenberg-Steenrod cohomological theory versus Weil cohomological theory [closed]
Can someone enlighten me what is the difference between an Eilenberg-Steenrod cohomological theory ( See here, https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms ), and a Weil ...
- 595
2
votes
0
answers
137
views
Thickness of category of abelian motives
A motive over a field $k$ is of abelian type if it belongs to the thick and rigid
subcategory of Chow motives spanned by the motives of abelian varieties over $k$. I understand that this is the ...
- 3,779
4
votes
0
answers
238
views
Motivic Galois correspondence
Is there a Galois correspondence in motivic Galois theory ? If so, is there a mathematical work on this correspondence that i can find on the net ?
Thanks in advance for your help.
- 16.9k
7
votes
0
answers
444
views
Status of the conjectured vanishing of Bloch-Kato H^2
There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
- 35.5k
1
vote
0
answers
90
views
The splitting pattern of the Killing form of an algebraic group and the Tits index
Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...
- 12.9k
37
votes
1
answer
3k
views
Morava on Shafarevich conjecture
$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: ...
- 63.9k
4
votes
1
answer
237
views
What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$?
Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed ...
- 20.8k
6
votes
1
answer
652
views
$l$-adic periods?
For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
- 148k
35
votes
1
answer
2k
views
The modularity theorem as a special case of the Bloch-Kato conjecture
In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
- 37k
9
votes
1
answer
623
views
When does a triangulated category have a heart?
Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that ...
- 16.9k
9
votes
0
answers
439
views
Uncountably many non-isomorphic Tate modules
Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?
If we took $l$-adic Tate modules there would be ...
CommunityBot
- 1
2
votes
0
answers
245
views
What unramified Galois representations come from geometry?
I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
CommunityBot
- 1
4
votes
0
answers
293
views
Galois representation with infinite image but finite image everywhere locally
Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
CommunityBot
- 1
9
votes
0
answers
361
views
Would full resolution of singularities have cohomological implications beyond the alteration theory?
De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of ...
CommunityBot
- 1
4
votes
0
answers
108
views
Algorithmically recover the $l'$-adic Galois representation from the $l$-adic one (assuming the Tate conjecture)
Let $E$ be a number field. For any finite Galois extension $E\subset F$ there is a continuous homomorphism $\pi_F:\mathrm{Gal}(\overline{E}/E)\to \mathrm{Gal}(F/E)$.
Let $X$ be a smooth projective ...
CommunityBot
- 1
4
votes
0
answers
265
views
Explicit linear object underlying $l$-adic cohomology for almost all $l$
If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...
CommunityBot
- 1