# 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.

389
questions

**1**

vote

**0**answers

233 views

### Eilenberg-Steenrod cohomological theory versus Weil cohomological theory [closed]

Can someone enlighten me what is the difference between an Eilenberg-Steenrod cohomological theory ( See here, https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms ), and a Weil ...

**1**

vote

**0**answers

92 views

### Thickness of category of abelian motives

A motive over a field $k$ is of abelian type if it belongs to the thick and rigid
subcategory of Chow motives spanned by the motives of abelian varieties over $k$. I understand that this is the ...

**4**

votes

**0**answers

170 views

### Motivic Galois correspondence

Is there a Galois correspondence in motivic Galois theory ? If so, is there a mathematical work on this correspondence that i can find on the net ?
Thanks in advance for your help.

**7**

votes

**0**answers

309 views

### Status of the conjectured vanishing of Bloch-Kato H^2

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...

**1**

vote

**0**answers

60 views

### The splitting pattern of the Killing form of an algebraic group and the Tits index

Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...

**20**

votes

**1**answer

703 views

### 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 ...

**8**

votes

**0**answers

309 views

### K-equivalence => isomorphism of Chow motives?

An old conjecture of Bondal-Orlov-Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper for definitions. In particular this applies to birational Calabi-Yau varieties. ...

**4**

votes

**1**answer

167 views

### What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$?

Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed ...

**33**

votes

**1**answer

1k views

### 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-...

**7**

votes

**1**answer

353 views

### When does a triangulated category have a heart?

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that ...

**9**

votes

**0**answers

384 views

### Uncountably many non-isomorphic Tate modules

Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?
If we took $l$-adic Tate modules there would be ...

**2**

votes

**0**answers

167 views

### What unramified Galois representations come from geometry?

I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...

**4**

votes

**0**answers

155 views

### Galois representation with infinite image but finite image everywhere locally

Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...

**9**

votes

**0**answers

209 views

### Would full resolution of singularities have cohomological implications beyond the alteration theory?

De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of ...

**4**

votes

**0**answers

95 views

### Algorithmically recover the $l'$-adic Galois representation from the $l$-adic one (assuming the Tate conjecture)

Let $E$ be a number field. For any finite Galois extension $E\subset F$ there is a continuous homomorphism $\pi_F:\mathrm{Gal}(\overline{E}/E)\to \mathrm{Gal}(F/E)$.
Let $X$ be a smooth projective ...

**4**

votes

**0**answers

253 views

### Explicit linear object underlying $l$-adic cohomology for almost all $l$

If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...

**8**

votes

**0**answers

169 views

### Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...

**3**

votes

**1**answer

134 views

### Are “strongly finite dimensional” homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...

**2**

votes

**0**answers

96 views

### Analytic properties of motivic L-functions twisted by Dirichlet characters

Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic ...

**2**

votes

**0**answers

205 views

### Why the scissor relations in Grothendieck rings?

Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. One type of relation which defines $K_0(V_k)$ is the following: if $A$ is a $k$-variety and $C$ a closed subset of $A$,...

**5**

votes

**0**answers

332 views

### What is the motive of $\operatorname{Bun}_G(X)$?

$\DeclareMathOperator\Bun{Bun}$Let $X$ be a scheme over algebraically closed field $k$, $G$ a reductive group and $\Bun_G(X)$ the stack of $G$ bundles on $X$. Write $[\Bun_G(X)]\in K_\text{st}$ for ...

**2**

votes

**0**answers

102 views

### Adjoining data about singularities to “correct” the category of pure motives?

There are a few well known constructions of potential categories of pure motives for smooth projective varieties over a field. My understanding is that modulo the standard conjectures these should be ...

**2**

votes

**1**answer

234 views

### Arc space & formal loops in motivic integration

One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...

**3**

votes

**3**answers

551 views

### Motivation for Karoubi envelope/ idempotent completion

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...

**3**

votes

**0**answers

171 views

### Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy

In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...

**1**

vote

**1**answer

173 views

### Pseudo-Abelian Completion in the constrution of Motifs (by Y. Manin)

I reading Yuri Manin's famous paper on "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS" and struggle with his definition for so called pseudo-abelian completion given on page 453 by a reason I ...

**1**

vote

**0**answers

89 views

### Why is $\Delta - p_0 - p_{2}$ a projector?

I apologize in advance, since I am probably doing a very naive mistake in my computation. I am learning about pure (Chow / Grothendieck) motives. One of the first steps is to consider the category ...

**4**

votes

**1**answer

185 views

### Definition field of weight homomorphism and moduli interpretation of Shimura varieties

In "Canonical models of Shimura curves" by J.S. Milne (avaliable at https://www.jmilne.org/math/articles/2003a.pdf), he explains the definition of quaternion Shimura curve, and explains the modern ...

**7**

votes

**1**answer

389 views

### How much of the category of motives can be recovered from automorphisms of the Betti functor

Say we are working with schemes over a field $k\subset \mathbb{C}.$ A motive in the sense of Voevodsky is a functor $Sch\to D^bVect$ from (an appropriate category of) schemes to the DG category of ...

**1**

vote

**0**answers

196 views

### Motivic integration of an Abelian variety and its dual are same?

Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are ...

**21**

votes

**1**answer

930 views

### 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-...

**2**

votes

**0**answers

222 views

### Brauer groups and del Pezzo surfaces

Let $k$ be a field of characteristic $0$ und let $X$ be a del Pezzo surface over $k$. Note that $X$ may not have points.
Let us consider $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$.
...

**4**

votes

**0**answers

146 views

### Derived weight filtration on motivic Galois representations

Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...

**3**

votes

**1**answer

209 views

### Motivic class of mixed Tate motive

Let $k$ be a field (of characteristic zero), $R$ be a ring and let $X\in DM(k;R)$ be a Tate motive. By definition, this means that $X$ is a summand of an object of the smallest strictly full ...

**7**

votes

**1**answer

399 views

### Motivation for Suslin’s Rigidity Conjecture

Suslin Rigidity conjecture states that motivic cohomology
$$
H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n))
$$
of the field $F$ coincides with motivic cohomology for the subfield of ...

**35**

votes

**0**answers

1k views

### 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 ...

**4**

votes

**2**answers

398 views

### Why is the triangulated category of motives easier than the abelian one?

There are several expository articles with the title "You could have invented [insert something mysterious here]" (a notable one being about spectral sequences, possibly it even started this genre). ...

**1**

vote

**0**answers

185 views

### Does the pure motive determine the Voevodsky motive?

I do not quite understand the construction of Voevodsky motives yet. Let $k$ be a field (possibly not algebraically closed), $X$ be a connected smooth projective $k$-scheme. Does the motive of $X$ in ...

**4**

votes

**2**answers

412 views

### Current status of independence of Betti numbers for different Weil cohomology theories

Previous problem: Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Let $X$ be an smooth projective variety over a field $k$. For any Weil cohomology theory for ...

**6**

votes

**1**answer

495 views

### $l$-adic periods?

For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...

**2**

votes

**0**answers

156 views

### Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

Let $X$ be a smooth projective variety over $\mathbb{Q}$. The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $GL(H^k_{\bullet}(X)...

**6**

votes

**3**answers

855 views

### Motives and homotopy theories of algebraic varieties

The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a ...

**6**

votes

**1**answer

319 views

### Motives of complex-analytic spaces

In any setting where we have a notion of space and a notion of cohomology theory, we in principle could ask "what are motives in this setting?". In some settings the question can be interesting (i.e. ...

**9**

votes

**1**answer

597 views

### Does a conservativity conjecture imply the standard conjectures?

Does a conservativity conjecture (e.g. Conjecture 2.1 of http://user.math.uzh.ch/ayoub/PDF-Files/Article-for-Steven.pdf) imply the standard conjectures? Specifically I am confused with Beilinson's ...

**3**

votes

**1**answer

201 views

### How to cook up an Artin motive from a positive-dimensional variety

I am trying to make sense of the paper "Eigenvalues of Frobenius and Hodge Numbers" (Kisin--Lehrer). I have not succeeded after some hours of intent staring at the screen.
In the proof of Corollary ...

**3**

votes

**0**answers

181 views

### Motivic strong bellows conjecture

There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...

**4**

votes

**1**answer

243 views

### $p$-adic realisation of Kummer motive and Frobenius matrix

Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$
\begin{equation}
0 \rightarrow \mathbb{Q}(1) \...

**8**

votes

**0**answers

240 views

### Searching for hypergeometric motives that split

Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...

**7**

votes

**1**answer

332 views

### Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?

Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.
Is it ...

**1**

vote

**0**answers

96 views

### Motivic Knot Embedding

I've been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine's 'An Overview of Motivic Homotopy Theory.' I think the subject is fascinating, and I've ...