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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Known norm varieties and the Bloch-Kato conjecture
The Bloch-Kato conjecture states that
$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to ...
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motives of elliptic curves, modular forms, Hecke characters
Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives:
(a) The Chow motive $h^1(E)$ ...
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why Borel's computation implies Beilinson-Soulé?
Let $k$ be a field of characteristic zero and $DM(k)_{\mathrm{Q}}$ Voevodsky's category of motives over $k$ with rational coefficients. The Beilinson-Soulé conjecture says
$$
\mathrm{Hom}_{DM(k)_{\...
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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 ...
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What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?
Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...
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$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?
I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...
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Cohomology of relative motives
Notation
Let $S$ be a scheme, proper over a field $k$. Let $\mathrm{SmPr}_{S}$ denote the category of smooth projective $S$-schemes. Let $\mathcal{M}_{S}$ denote the category of relative Chow motives ...
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Two questions on motivic homotopy theory
I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions:
Why is the question of $\...
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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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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 ...
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The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?
Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...
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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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Finer motivic decomposition in a bigger motivic category
In his Ph.D. thesis, Semenov shows that the motivic decomposition of a variety in general is not unique. He works in the category of Chow-Motives and not in a bigger category of motives.
Is there an ...
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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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Nice references for injective model structures and Quillen functors between motivic homotopy categories
It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...
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Moduli space of motives vs moduli space of varieties
A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...
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Why is the Category of Correspondences not pseudo abelian?
i've just gotten into the theory of motives.I understand the construction of the Karoubian envelope (pseudo-abelian completion) to ensure that morphisms have kernels and images in order to get certain ...
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"Motivic structure on higher homotopy of non-nilpotent spaces" ?
Has anyone an idea where one can read more about Deepam Patel's talk "Motivic structure on higher homotopy of non-nilpotent spaces" http://www.ihes.fr/~abbes/SGA/patel.html ?
Edit/Answer: The video ...
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The historical development of automorphic geometry
Background:
Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
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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 ...
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On triangulated categories of pro-objects
Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
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Is there a motive attached to half weight modular forms?
The question is kind of contained in the title but let me add a few words.
If $f$ is a cusp form of weight $k$ for $SL(2, \mathbb{Z})$ then Scholl constructed a Grothendieck motive $M(f)$. In this ...
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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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Motives over the complex numbers versus mixed Hodge structures
Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...
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On Deligne's determinant of motives
This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...
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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 ...
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"Modular forms from Feynman integrals "?
I would like to learn more about the background of this talk, but found no text on that theme. Do you know more? Edit: An interesting talk by Miranda Cheng (slides).
Edit: A talk today on the theme, ...
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Decomposition of Motives of cellular varieties
Dear community,
in his 2005 Inventiones Paper "On motivic decompositions arising from the method
of Białynicki-Birula" P. Brosnan deduced from the classical (?) theorem of Bialynicki-Birula on ...
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Status of Beilinson conjectures?
(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...
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Chow-Künneth decomposition for hypersurfaces
Short version: is the Chow-Künneth motivic decomposition known for $X \hookrightarrow \mathbb{P}^n_k$ a hypersurface over a field $k$?
Long version: let $M(X)$ be the Chow motive of $X$ with ...
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critical values of motives
Hi friends,
I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic ...
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Realisation functor
Let k be a field. Is there a realization functor
$DM_{gm}(k,\mathbb{Z}/n)^{op} \to D^b_c(k, \mathbb{Z}/n)$
from category of motives to category of complexes of étale sheaves of $\mathbb{Z}/n$ ...
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Pull-back of algebraic cycles under holomorphic maps
Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...
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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 ...
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Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?
$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$
Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...
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Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$
In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. ...
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Ring structure for the motivic spectrum/complex that represents singular cohomology?
As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex?
shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex (...
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Corresponding notion of unramified for motives (or de Rham cohomology)
The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if $X$...
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chow kunneth motivic decomposition for dummies
Hi everybody,
I have just realized that, if I had to explain Chow-Kunneth decomposition for Chow motives to someone who is a newbie of the subject, I would hardly find a way which is far from the ...
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Motivic generalisation of Neron-Ogg-Shaferevich criterion
Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...
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Voevodsky's 'split standard triple' argument: an explanation; does it work with $Z/nZ$-coefficients?
For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see http://books.google.ru/books?id=TzUmk87bN9cC&pg=PA85&...
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Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?
It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...
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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 ...
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(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 ...
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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 ...
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Motivic cohomology vs. K-theory for singular varieties
As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\...
- 1,553
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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 ...
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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\...
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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 ...
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Followup questions about the relationship between modular forms and motives
It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here ...
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