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
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Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
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votes
7
answers
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What is the field with one element?
I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-...
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votes
4
answers
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What's the "Yoga of Motives"?
There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-...
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What is the relationship between motivic cohomology and the theory of motives?
I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...
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What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
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answers
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Langlands in dimension 2: the Yoshida conjecture
Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. ...
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42
votes
1
answer
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Progress on the standard conjectures on algebraic cycles
What's the current state of these conjectures?
Who is working on them?
In "Standard conjectures on algebraic cycles" Grothendieck says:
"They would form the basis of the so-called "theory of ...
user55909
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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 ...
41
votes
4
answers
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Understanding the definition of the Lefschetz (pure effective) motive
For all those who are unlikely to have answers to my questions, I provide some
Background:
In some sense, pure motives are generalisations of smooth projective varieties. Every Weil cohomology ...
- 1,926
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votes
8
answers
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What is the proper initiation to the theory of motives for a new student of algebraic geometry?
A preliminary apology is in order: I realize that most of my contributions to this site are in the form of reference requests. I understand that this makes it seem as though I do nothing more than sit ...
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4
answers
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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 ...
user19475
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1
answer
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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: ...
- 2,734
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votes
4
answers
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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......
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answer
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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-...
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35
votes
2
answers
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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 ...
35
votes
1
answer
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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 ...
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votes
2
answers
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Derived Algebraic Geometry and Chow Rings/Chow Motives
I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all ...
- 12.1k
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votes
4
answers
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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 ...
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answer
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For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
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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 ...
- 5,504
30
votes
3
answers
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t-structures and higher categories?
I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one ...
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Reasons for the use of Nisnevich topology in motivic homotopy theory
The objects of interest in motivic homotopy theory are "spaces"-which are simplicial sheaves of sets on the big Nisnevich site $Sm/k$ of smooth schemes of finite type over a field $k$. I understand ...
- 805
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votes
4
answers
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What is the status of the theory of motives?
It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.
But what is the ...
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1
answer
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Is there a higher Grothendieck ring of varieties?
Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor ...
- 293
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votes
3
answers
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$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
- 291
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votes
3
answers
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What do you lose when passing to the motive?
I contemplated about what information about a scheme we lose when passing to its motive. I came up with the following examples:
The projective bundle of a vector bundle does only depend on the rank ...
user19475
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votes
1
answer
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Voevodsky's counterexample to the existence of a motivic t-structure
I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts".
Voevodsky in "...
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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
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votes
2
answers
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Most significant results in motivic integration theory?
I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, ...
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votes
3
answers
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How are motives related to anabelian geometry and Galois-Teichmuller theory?
In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...
- 3,309
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votes
3
answers
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Are there "motivic" proofs of Weil conjectures in special cases?
This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...
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2
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Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself
Preface: I am not an expert in the work of Scholze, or anything for that matter.
Question
Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the ...
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votes
1
answer
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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 ...
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Why would the category of Motives be Tannakian?
After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify?
I decided to read up on Tannakian formalism.
Given the ...
- 6,348
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votes
2
answers
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Kontsevich's conjectures on the Grothendieck-Teichmüller group?
Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...
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answer
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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 ...
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answer
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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-...
21
votes
1
answer
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Spectral sequences in $K$-theory
There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.
For a field $k$, let $X$ be smooth variety $X$ ...
user87684
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votes
1
answer
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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 ...
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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 ...
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votes
2
answers
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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 ...
20
votes
2
answers
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Unipotency in realisations of the motivic fundamental group
Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. ...
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1
answer
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Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?
I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...
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votes
1
answer
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views
Is there any publication of Bombieri about the standard conjectures on algebraic cycles?
In "Standard conjectures of algebraic cycles" Grothendieck says:
"... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and ...
user51486
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votes
1
answer
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Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology
In
[S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70
Serre presents nine conjectures *C*$_1$...
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2
answers
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What are the different theories that the motivic fundamental group attempts to unify?
I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In http://www.math.ias.edu/files/deligne/...
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3
answers
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$p$-adic periods
For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism
$$
H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}.
$$
If we pick $\mathbb{Q}$-bases for ...
- 9,061
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2
answers
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What is the relationship between these two notions of "period"?
The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...
- 9,061
19
votes
1
answer
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Analogue of Tate or Hodge conjecture for varieties over $\mathbb Q_p$
I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly background;...
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1
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Connes-Kreimer Hopf algebra and cosmic Galois group
Hi,
I'm interested in the relation between the two following constructions motivated by renormalization:
Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...
- 8,524