All Questions
Tagged with ag.algebraic-geometry motives
359 questions
14
votes
1
answer
746
views
Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?
Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
- 16.9k
7
votes
1
answer
2k
views
(Mixed) Tate motives
Hi there,
in recent times I was reading texts about motives, and I want to ask
something about Tate motives which is not clear to me (as I came across
different definitions in different texts).
Let ...
- 4,547
2
votes
1
answer
555
views
Chow group of a (particular) motive [+ reference request]
I have two (not unrelated) questions. Let me first give a short introduction.
Introduction
For a general overview of the setup I refer to the introduction (§1) of [Zhang].
Let $k$ be a number field ...
- 5,504
18
votes
2
answers
1k
views
Grothendieck ring of "varieties carrying a function"
Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme
of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not
constructible!) function on $X$.
I want to consider a ...
- 27.8k
8
votes
1
answer
1k
views
motivic t-structure and realisations
Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $
\mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...
- 81
71
votes
1
answer
8k
views
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 ...
- 5,929
4
votes
1
answer
767
views
Realization of Voevodsky Motives over a perfect field in mixed categories.
Let $k$ be a finite field and $l$ be different from characteristic of $k$. Is there a realization functor from the Voevodsky's category with $\mathbb Q$ coefficients to the constructible mixed étale ...
- 113
4
votes
1
answer
603
views
A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,213
3
votes
0
answers
638
views
Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?
I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of ...
- 16.9k
26
votes
1
answer
4k
views
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 "...
- 982
15
votes
1
answer
769
views
Crystalline realization of mixed Tate motives
Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...
- 313
5
votes
2
answers
848
views
Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?
This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics.
Let $...
- 5,929
10
votes
1
answer
2k
views
How does the conjectural Langlands group fit into the Tannakian point of view?
I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...
14
votes
2
answers
2k
views
How would a motivic proof of the Riemann hypothesis over finite fields go?
It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...
- 6,348
3
votes
1
answer
777
views
Is the "L-function of the complex cohomology" of a motive equal to the L-function of its l-adic realization?
Let's say I have a motive in $\mathcal{M}_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}_l$. ...
- 6,348
1
vote
0
answers
299
views
(why) Are the following two constructions of zeta functions equal?
Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function.
Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...
- 6,348
23
votes
2
answers
2k
views
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
19
votes
2
answers
3k
views
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/...
- 6,348
6
votes
0
answers
242
views
Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?
Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
- 16.9k
141
votes
0
answers
13k
views
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 $...
- 7,527
8
votes
1
answer
985
views
commutativity constraint in Grothendieck's motives
This is a basic question about Grothendieck's conjectural category $M_k$ of pure motives (over a field $k$). This construction first produces a category (the "false category of motives") which need ...
- 3,867
3
votes
1
answer
767
views
Is there a 'classical' definition for the support of a perverse sheaves.
I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter?
I suspect that ...
- 16.9k
14
votes
1
answer
1k
views
Are two conjectural descriptions of the motivic t-structure (via cohomology and via affine varieties) known to be equivalent?
I know of two conjectural descriptions of the motivic $t$-structure for Voevodsky's motives.
The first one is based on the conjecture that Weil cohomology theories should yield exact and ...
- 16.9k
1
vote
2
answers
392
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 16.9k
8
votes
1
answer
1k
views
If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?
Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...
- 16.9k
1
vote
0
answers
178
views
$G_m$-cohomology of a motif (that corresponds to a stack?)
As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-...
- 16.9k
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
2
votes
0
answers
515
views
A motivic complex
By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...
- 16.9k
3
votes
1
answer
284
views
For a G-variety, what could one say about the motif of the corresponding simplicial variety
Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex ...
- 16.9k
2
votes
1
answer
589
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,871
10
votes
1
answer
743
views
Stable motivic cohomology with finite coefficients?
In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
- 15.6k
1
vote
1
answer
661
views
A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?
Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...
- 16.9k
7
votes
0
answers
729
views
Integral decomposition of the diagonal (Chow motives)
Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...
- 4,015
9
votes
1
answer
804
views
Motivic characterization of affine spaces
Let $k$ be a field and let $X$ be a smooth irreducible variety over $k$.
Suppose that I know that the image of $X$ in the Grothendieck group
of varieties over $k$ is equal to that of
a) ${\mathbb A}^...
- 7,037
45
votes
2
answers
3k
views
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. ...
- 1,704
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: ...
- 2,734
6
votes
1
answer
1k
views
When is the K-theory presheaf a sheaf?
Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale ...
- 562
7
votes
2
answers
882
views
Rankin-Selberg convolutions of motivic L-series
Background:
Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms
$f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively.
The Rankin-Selberg convolution ...
- 1,704
24
votes
3
answers
4k
views
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 ...
- 7,442
30
votes
3
answers
4k
views
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 ...
- 23.5k
18
votes
1
answer
6k
views
Deligne's proof of Ramanujan's conjecture
I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I ...
- 1,783
34
votes
2
answers
6k
views
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
40
votes
8
answers
12k
views
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 ...
- 1,539
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 ...
- 1,479
20
votes
2
answers
2k
views
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. ...
- 2,976
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 ...
- 4,015
11
votes
0
answers
2k
views
What are "fractional motives"?
Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?
Edit: Further cases of "fractional motives" as ...
- 10.8k
16
votes
3
answers
3k
views
Motives versus Motifs
I was in Paris recently for a meeting about motives or motifs, and since I'm too jet lagged
for real work let me ask the following somewhat frivolous question. The word "motif" is
usually translated ...
28
votes
3
answers
2k
views
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
18
votes
4
answers
2k
views
Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a category of schemes, ...
- 12.3k