In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'. One has a satisfactory description of pure motives, whereas the mixed case is much more difficult. Pure motives are triples (X, p, m), where X is a smooth projective ...

learn more… | top users | synonyms

7
votes
1answer
173 views

Intuition for the Lefschetz motive (Tate motive)?

Yo! Maybe this question is too dumb for mathoverflow, but I believe no one will pay attention to it in math.stackexchange, so I will post it here. If this question is not suitable, just delete it. I ...
3
votes
1answer
153 views

“theta characteristics” on general motives?

Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist ...
2
votes
1answer
166 views

Reference - Generalized Hodge conjecture for triangulated motives

GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective. I would like to know some references on GHC ...
1
vote
0answers
53 views

When the tensor product of motives that are not $0$-(homotopy)-connective can be $0$-connective?

For $t$ being the homotopy $t$-structure for Voevodsky effective motivic complexes when $M\otimes N\in DM_{eff}^{t\le -1}$ ensures that either $M$ or $N$ belongs to $DM_{eff}^{t\le -1}$ (so, we use ...
5
votes
1answer
224 views

Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely Skip Garibaldi - E8 the most exceptional group , the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
5
votes
3answers
527 views

References - Voevodsky motives are the derived category of Nori motives?

First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
3
votes
0answers
101 views

Non-multiplicative Euler-Poincaré Characteristics

Are there known examples of a non-multiplicative Euler-Poincaré characteristic on varieties? Let $\mathbf{Var}/k$ be the category of varieties over a filed $k$, i.e. the category of reduced separated ...
4
votes
0answers
98 views

Polynomially countable varieties and virtual mixed Tate motives

Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$,...
3
votes
0answers
175 views

Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions). The ...
24
votes
0answers
869 views

Big list - 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 ...
5
votes
1answer
220 views

Intuition behind the definition of finite correspondences

Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...
9
votes
0answers
239 views

Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...
1
vote
0answers
92 views

On “splitting off small weights” from Chow motives

I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed). So, consider ...
2
votes
0answers
143 views

Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?

Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
8
votes
2answers
403 views

Reference for Nori motives

I would like to study Nori motives and I am a complete outsider of the subject. I do, however, have background on Chow motives, Voevodsky motives $\mathrm{DM}$ and his stable homotopy category $\...
7
votes
0answers
221 views

Quadratic twists of 1-motives

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...
17
votes
0answers
323 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 ...
4
votes
0answers
169 views

Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\...
5
votes
1answer
243 views

Generalized Euler characteristics of non-motivic origin

By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all ...
11
votes
1answer
441 views

Does the Grothendieck ring of varieties contain torsion?

Let $K_0(Var_k)$ be the abelian group generated by the isomorphism classes of varieties over the field $k$ with the relations $$[X]=[U]+[X\setminus U]$$ for every variety $X$ and open subvariety $U$. ...
8
votes
1answer
187 views

Realization Functor From $SH$ to Derived Category of $Gal$-Modules

Let $k$ be a field. I would like a reference for realization functors from Morel-Voevodksy's stable category $SH(k)$ to the derived categories of $Gal(\bar{k}/k)$-modules. Has something like this been ...
3
votes
1answer
191 views

About the decomposition of a Chow group of a variety

I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) $...
11
votes
1answer
289 views

Non-algebraic Hecke characters

Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...
2
votes
0answers
131 views

Residual Representation of a Motive

Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...
4
votes
0answers
94 views

Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
20
votes
1answer
528 views

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, ...
8
votes
1answer
291 views

Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field. According to Mazza--Voevodsky--Weibel "...
6
votes
0answers
287 views

Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
6
votes
1answer
467 views

Pure motives and compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...
1
vote
0answers
410 views

Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?
14
votes
4answers
841 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
10
votes
1answer
484 views

Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
5
votes
0answers
223 views

Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...
1
vote
1answer
160 views

Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$. A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...
2
votes
0answers
312 views

Do those manifolds atrached to L-functions give rise naturally to motives? [closed]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product $\...
5
votes
1answer
271 views

Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...
1
vote
1answer
198 views

motive of the general linear group

Let $k$ be a perfect field. Let $GL_n$ be the general linear group over $k$. Does anybody know a reference for the computation of the motive $$ M(GL_n) $$ in Voevodsky's category $DM(k)$?
5
votes
2answers
434 views

Should the Grothendieck ring of varieties be K_0 of numerical motives?

Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?
6
votes
1answer
268 views

Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details. In https://www.dpmms.cam.ac.uk/~ajs1005/...
8
votes
0answers
259 views

Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
24
votes
2answers
1k views

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 ...
9
votes
1answer
439 views

Why do we need localization by Leftschetz motive?

Definition of the Grothendieck group and Leftschetz motive. The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation: $$ [X]=[...
19
votes
1answer
521 views

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 ...
1
vote
1answer
275 views

Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning: For some $\alpha \in H^n(k,\mu_p)$ ...
5
votes
1answer
340 views

“Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
2
votes
1answer
429 views

Algebraic equivalence vs linear equivalence

Maybe the question is too general, but nevertheless: under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence? What are typical classes of ...
3
votes
1answer
289 views

Etale Realization and Gysin Sequence

Ivorra defined a tensor triangulated functor from Voevodsky's triangulated category of motives to the derived category of complexes of etale sheaves of $\mathbb{Z}/n$ modules with bounded cohomology ...
12
votes
2answers
664 views

Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.) The question As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...
4
votes
0answers
145 views

Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$ It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference ...
7
votes
0answers
425 views

Use of derivators to the theory of motives?

This is a rather imprecise question but i think this could become a interesting pool of ideas and comments. The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...