All Questions
Tagged with ag.algebraic-geometry motives
359 questions
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 ...
- 1,479
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 $X$ ...
- 1,479
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 ...
- 1,514
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
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 ...
- 1,479
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 ...
user158636
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}...
user158636
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 ...
user158636
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 ...
user158636
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.
...
user158636
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
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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 $\...
- 16.9k
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
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
3
votes
1
answer
476
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 ...
- 6,018
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 ...
- 4,408
1
vote
1
answer
245
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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 ...
- 6,018
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 ...
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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 ...
- 7,952
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
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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 ...
- 113
7
votes
1
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551
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 ...
- 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 ...
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). ...
user141010
1
vote
0
answers
304
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 ...
user138661
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 ...
- 6,622
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 ...
- 61
2
votes
0
answers
209
views
Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?
Let $X$ be a smooth projective variety over $\mathbb{Q}$. The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $GL(H^k_{\bullet}(X)...
- 9,061
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 ...
user138661
8
votes
1
answer
492
views
Motives of complex-analytic spaces
In any setting where we have a notion of space and a notion of cohomology theory, we in principle could ask "what are motives in this setting?". In some settings the question can be interesting (i.e. ...
user138661
3
votes
1
answer
217
views
How to cook up an Artin motive from a positive-dimensional variety
I am trying to make sense of the paper "Eigenvalues of Frobenius and Hodge Numbers" (Kisin--Lehrer). I have not succeeded after some hours of intent staring at the screen.
In the proof of Corollary ...
- 31
3
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0
answers
193
views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
user138661
4
votes
1
answer
372
views
$p$-adic realisation of Kummer motive and Frobenius matrix
Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$
\begin{equation}
0 \rightarrow \mathbb{Q}(1) \...
- 2,971
9
votes
0
answers
291
views
Searching for hypergeometric motives that split
Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...
- 3,337
7
votes
1
answer
433
views
Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.
Is it ...
- 6,622
3
votes
0
answers
182
views
Where is smoothness used in Voevodsky's homotopy theory of schemes? [duplicate]
Let $S$ be a smooth noetherian scheme, and let $Sm/S$ be the category of smooth schemes over $S$. Voevodsky constructs the homotopy category of motives (resp. the stable homotopy category of motives) $...
- 3,019
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
6
votes
1
answer
525
views
Functoriality for $\ell$-adic cohomology - a question
This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...
user134132
4
votes
0
answers
306
views
Etale cohomology of projective spaces in the rigid analytic setting
Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...
4
votes
0
answers
232
views
holomorphic continuation of motivic $L$-functions
The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
- 41
8
votes
1
answer
417
views
Motive of a conic without points
Let $C\subset \mathbb{P}^2$ be a smooth conic without $k$-points.
Call the Chow $k$-motive in zero-dimensional if it is a sum of $M\mathbb{L}^n$ where $M$ is an Artin motive, i.e. a part of a motive ...
- 399
10
votes
2
answers
534
views
A question about the vanishing of motivic cohomology in negative Tate twist
Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...
- 761
2
votes
0
answers
187
views
Artin-Tate chow motives and graded Galois representations
Consider the category $\mathsf{Chow}_{\mathbb{Q}}$ of rational pure effective chow $k$-motives. The full subcategory of Artin motives (generated by (X,p,0), X smooth projective zero-dimensional, let $...
- 399