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1
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1answer
723 views

Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...
3
votes
1answer
508 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 ...
9
votes
1answer
716 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 ...
1
vote
2answers
303 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 ...
7
votes
1answer
915 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 ...
1
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0answers
149 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 ...
3
votes
1answer
254 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} ...
2
votes
0answers
342 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 ...
3
votes
1answer
268 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 ...
4
votes
1answer
515 views

Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on ...
2
votes
1answer
419 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 ...
3
votes
1answer
300 views

Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...
10
votes
1answer
534 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 ...
6
votes
0answers
278 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
5
votes
1answer
439 views

The conjectural relation between mixed motivic sheaves and the perverse t-structure.

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. ...
5
votes
2answers
602 views

Automorphic form encoding the orders of $N$ modulo $p$.

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...
1
vote
1answer
340 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 $d$. ...
2
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0answers
173 views

A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...
2
votes
0answers
132 views

On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...
5
votes
0answers
198 views

Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
7
votes
0answers
474 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, ...
8
votes
1answer
384 views

Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...
6
votes
0answers
381 views

Is singular cohomology representable by a (Voevodsky's) motivic complex?

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of ...
9
votes
1answer
705 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 ...
15
votes
2answers
3k views

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 ...
5
votes
0answers
313 views

Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...
38
votes
2answers
2k 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. ...
6
votes
1answer
711 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 ...
2
votes
0answers
158 views

Transformation rule for motivic integration: divisors or divisors classes?

In the Kontsevich's 'change of variables for motivic integral' rule does one consider any possible choices for the corresponding canonical divisors, or is it necessary to fix certain representatives ...
3
votes
1answer
374 views

Morava's “Motives and cell bundles”?

Hello, do you know more about, or some exposition of Morava's talk?
6
votes
2answers
690 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 ...
9
votes
3answers
2k 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 ...
22
votes
3answers
1k 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 ...
16
votes
1answer
3k 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 ...
20
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8answers
4k 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 ...
6
votes
1answer
732 views

Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed): In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge ...
16
votes
2answers
1k 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. ...
3
votes
0answers
619 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 ...
7
votes
0answers
1k 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 ...
11
votes
0answers
976 views

Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
17
votes
3answers
1k 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 ...
14
votes
4answers
1k 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, ...
27
votes
4answers
2k views

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 ...
13
votes
2answers
2k views

Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
16
votes
1answer
923 views

constants in Gamma factors in functional equation for zeta functions.

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a ...
8
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2answers
1k views

Artin motives, References for.

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
25
votes
4answers
3k views

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 ...
4
votes
1answer
421 views

Is the scalar extension functor for Chow motives conservative?

Denote $CHM(F)$ to be the category of Chow motives over a field $F$. Let's consider an algebraic exension $E/F$, then there is a natural extension of scalars functor $CHM(F) \to CHM(E)$. I was ...
9
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6answers
1k views

Kunneth formula for motivic cohomology

I was wondering when the Kunneth formula holds for motivic cohomology: $$ H^p(X,A(\alpha)) = \bigoplus_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma)) $$ where ...
6
votes
2answers
563 views

Néron theory for motives of arbitrary weight

SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...