All Questions
Tagged with motives ag.algebraic-geometry
359 questions
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
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
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
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
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
9
votes
0
answers
276
views
Which field extensions do not affect Chow groups?
Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...
- 16.9k
3
votes
1
answer
174
views
Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant?
Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...
- 12.9k
3
votes
0
answers
300
views
Why the scissor relations in Grothendieck rings?
Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. One type of relation which defines $K_0(V_k)$ is the following: if $A$ is a $k$-variety and $C$ a closed subset of $A$,...
- 4,547
5
votes
0
answers
383
views
What is the motive of $\operatorname{Bun}_G(X)$?
$\DeclareMathOperator\Bun{Bun}$Let $X$ be a scheme over algebraically closed field $k$, $G$ a reductive group and $\Bun_G(X)$ the stack of $G$ bundles on $X$. Write $[\Bun_G(X)]\in K_\text{st}$ for ...
- 5,701
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
2
votes
0
answers
118
views
Adjoining data about singularities to "correct" the category of pure motives?
There are a few well known constructions of potential categories of pure motives for smooth projective varieties over a field. My understanding is that modulo the standard conjectures these should be ...
- 1,635
15
votes
2
answers
853
views
Galois group for 0-dimensional motives
It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.
One can ...
- 171
3
votes
1
answer
477
views
Arc space & formal loops in motivic integration
One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...
- 45.7k
3
votes
0
answers
178
views
Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy
In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...
- 4,418
1
vote
1
answer
245
views
Pseudo-Abelian Completion in the constrution of Motifs (by Y. Manin)
I reading Yuri Manin's famous paper on "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS" and struggle with his definition for so called pseudo-abelian completion given on page 453 by a reason I ...
- 333
1
vote
0
answers
107
views
Why is $\Delta - p_0 - p_{2}$ a projector?
I apologize in advance, since I am probably doing a very naive mistake in my computation. I am learning about pure (Chow / Grothendieck) motives. One of the first steps is to consider the category ...
- 533
4
votes
1
answer
273
views
Definition field of weight homomorphism and moduli interpretation of Shimura varieties
In "Canonical models of Shimura curves" by J.S. Milne (avaliable at https://www.jmilne.org/math/articles/2003a.pdf), he explains the definition of quaternion Shimura curve, and explains the modern ...
- 184
8
votes
1
answer
603
views
How much of the category of motives can be recovered from automorphisms of the Betti functor
Say we are working with schemes over a field $k\subset \mathbb{C}.$ A motive in the sense of Voevodsky is a functor $Sch\to D^bVect$ from (an appropriate category of) schemes to the DG category of ...
- 16.5k
4
votes
1
answer
1k
views
Gauss--Manin connection for de Rham realisation
Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a ...
CommunityBot
- 1
1
vote
0
answers
228
views
Motivic integration of an Abelian variety and its dual are same?
Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are ...
- 519
21
votes
1
answer
1k
views
When simple cohomological computations predict ingenious algebro-geometric constructions?
Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...
2
votes
0
answers
263
views
Brauer groups and del Pezzo surfaces
Let $k$ be a field of characteristic $0$ und let $X$ be a del Pezzo surface over $k$. Note that $X$ may not have points.
Let us consider $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$.
...
- 1,479
3
votes
1
answer
249
views
Motivic class of mixed Tate motive
Let $k$ be a field (of characteristic zero), $R$ be a ring and let $X\in DM(k;R)$ be a Tate motive. By definition, this means that $X$ is a summand of an object of the smallest strictly full ...
- 16.9k
7
votes
1
answer
552
views
Motivation for Suslin’s Rigidity Conjecture
Suslin Rigidity conjecture states that motivic cohomology
$$
H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n))
$$
of the field $F$ coincides with motivic cohomology for the subfield of ...
- 17.4k
12
votes
1
answer
596
views
An inverse problem for Grothendieck rings of varieties
Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...
- 4,547
21
votes
1
answer
757
views
What should motives for $L(E,n)$ look like?
Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...
- 4,316
42
votes
0
answers
2k
views
Are we better in computing integrals than mathematicians of 19th century?
When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
- 4,316
6
votes
2
answers
600
views
Why is the triangulated category of motives easier than the abelian one?
There are several expository articles with the title "You could have invented [insert something mysterious here]" (a notable one being about spectral sequences, possibly it even started this genre). ...
- 4,316
1
vote
0
answers
305
views
Does the pure motive determine the Voevodsky motive?
I do not quite understand the construction of Voevodsky motives yet. Let $k$ be a field (possibly not algebraically closed), $X$ be a connected smooth projective $k$-scheme. Does the motive of $X$ in ...
CommunityBot
- 1
6
votes
3
answers
1k
views
Motives and homotopy theories of algebraic varieties
The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a ...
- 20.8k
4
votes
2
answers
569
views
Current status of independence of Betti numbers for different Weil cohomology theories
Previous problem: Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Let $X$ be an smooth projective variety over a field $k$. For any Weil cohomology theory for ...
- 208