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.

Filter by
Sorted by
Tagged with
4 votes
1 answer
243 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 ...
user avatar
  • 19.3k
18 votes
1 answer
887 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 ...
user avatar
  • 19.3k
4 votes
1 answer
159 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 ...
user avatar
  • 2,541
3 votes
1 answer
319 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 ...
user avatar
  • 1,317
1 vote
0 answers
57 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^{\...
user avatar
8 votes
1 answer
375 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}(...
user avatar
2 votes
0 answers
160 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 $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
user avatar
4 votes
0 answers
155 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 ...
user avatar
11 votes
1 answer
501 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 $k$, ...
user avatar
  • 133
4 votes
0 answers
130 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 ...
user avatar
3 votes
0 answers
206 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 $\...
user avatar
  • 213
1 vote
0 answers
172 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 ...
user avatar
  • 159
1 vote
0 answers
116 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 ...
user avatar
1 vote
0 answers
199 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 ...
user avatar
  • 19
4 votes
0 answers
355 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 ...
user avatar
5 votes
0 answers
348 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 ...
user avatar
  • 2,041
5 votes
0 answers
165 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 ...
user avatar
2 votes
0 answers
172 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 ...
user avatar
5 votes
1 answer
244 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 ...
user avatar
  • 101
1 vote
1 answer
297 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 ...
user avatar
  • 3,067
8 votes
0 answers
398 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 ...
user avatar
  • 4,926
9 votes
0 answers
274 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) ...
user avatar
  • 10k
7 votes
0 answers
162 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 $X(\mathbb C)$ ...
user avatar
  • 6,782
3 votes
1 answer
485 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 ...
user avatar
  • 1,287
1 vote
0 answers
247 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 ...
user avatar
  • 485
1 vote
0 answers
111 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 ...
user avatar
  • 3,665
4 votes
0 answers
196 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.
user avatar
  • 485
7 votes
0 answers
365 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 ...
user avatar
  • 13.9k
1 vote
0 answers
68 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 $X$ ...
user avatar
  • 1,287
20 votes
1 answer
927 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 ...
user avatar
  • 794
11 votes
1 answer
554 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 ...
user avatar
4 votes
1 answer
192 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 ...
user avatar
  • 1,243
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-...
user avatar
  • 2,371
7 votes
1 answer
430 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 ...
user avatar
9 votes
0 answers
397 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 ...
user avatar
2 votes
0 answers
184 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}...
user avatar
4 votes
0 answers
170 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{...
user avatar
9 votes
0 answers
279 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 ...
user avatar
4 votes
0 answers
100 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 ...
user avatar
4 votes
0 answers
257 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. ...
user avatar
8 votes
0 answers
195 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? ...
user avatar
3 votes
1 answer
154 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 $\...
user avatar
2 votes
0 answers
105 views

Analytic properties of motivic L-functions twisted by Dirichlet characters

Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic ...
user avatar
  • 83
2 votes
0 answers
225 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 $A$,...
user avatar
  • 3,067
5 votes
0 answers
343 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 ...
user avatar
  • 4,004
2 votes
0 answers
109 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 ...
user avatar
2 votes
1 answer
287 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 ...
user avatar
  • 479
5 votes
3 answers
786 views

Motivation for Karoubi envelope/ idempotent completion

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...
user avatar
  • 479
3 votes
0 answers
175 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 ...
user avatar
1 vote
1 answer
194 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 ...
user avatar
  • 1,015

1
2 3 4 5
9