# 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.

370
questions

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### 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 ...

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182 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 $A$,...

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304 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 ...

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### 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 ...

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**1**answer

177 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 ...

**2**

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**3**answers

347 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 ...

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162 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 ...

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**1**answer

152 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 ...

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86 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 ...

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**1**answer

150 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 ...

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342 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 ...

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184 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**

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**1**answer

820 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-...

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218 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 $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$.
...

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132 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 ...

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193 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 ...

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368 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 ...

**34**

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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 ...

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375 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). ...

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172 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 ...

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373 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 ...

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votes

**1**answer

340 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 ...

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135 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)...

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747 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 ...

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301 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. ...

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488 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 ...

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191 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 ...

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168 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 ...

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**1**answer

216 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) \...

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222 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 ...

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324 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 ...

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90 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 ...

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468 views

### P-adic Volume Conjecture

Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...

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154 views

### Where is smoothness used in Voevodsky's homotopy theory of schemes? [duplicate]

Let $S$ be a smooth noetherian scheme, and let $Sm/S$ be the category of smooth schemes over $S$. Voevodsky constructs the homotopy category of motives (resp. the stable homotopy category of motives) $...

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549 views

### An inverse problem for Grothendieck rings of varieties

Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...

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387 views

### Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...

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190 views

### Etale cohomology of projective spaces in the rigid analytic setting

Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...

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393 views

### Motivic cohomology is universal with respect to what (co)homology theories?

I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
...

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187 views

### holomorphic continuation of motivic $L$-functions

The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...

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**1**answer

346 views

### Motive of a conic without points

Let $C\subset \mathbb{P}^2$ be a smooth conic without $k$-points.
Call the Chow $k$-motive in zero-dimensional if it is a sum of $M\mathbb{L}^n$ where $M$ is an Artin motive, i.e. a part of a motive ...

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344 views

### A question about the vanishing of motivic cohomology in negative Tate twist

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...

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123 views

### Artin-Tate chow motives and graded Galois representations

Consider the category $\mathsf{Chow}_{\mathbb{Q}}$ of rational pure effective chow $k$-motives. The full subcategory of Artin motives (generated by (X,p,0), X smooth projective zero-dimensional, let $...

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878 views

### What is the best reference for motives?

I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...

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222 views

### Reference request: Tate's conjecture for L functions of motives

What's a good reference for the most general form of Tate's conjecture for the order of poles of the L function of a motive? Thanks!

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323 views

### Are exotic affine spaces motivic/whatever equivalent to affine space?

This question is inspired by this MO question; in turn by this MO; in turn by these MO, MO.
An exotic affine space is an affine variety $V$ whose $\mathbb{C}$-points are diffeomorphic to $\mathbb{R}^{...

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**1**answer

247 views

### Inequality in the Grothendieck ring of stacks

In the Grothendieck ring of varieties, there are ways of distinguishing classes of varieties, for example $\ell$-adic cohomology. The Grothendieck ring of stacks is a localization of the Grothendieck ...

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197 views

### Quotient of a motive by a finite group

Given a smooth scheme $X$ over a field $k$, we can consider its motive $M(X)$. It is an object in Voevodsky's triangulated category of motives $DM(k,\Lambda)$ where $\Lambda$ is the ring of ...

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370 views

### Reference request: Motivic Cohomology and Cycle class maps

For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...

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214 views

### Poincare duality for mixed motives

Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality,
\begin{equation}
...

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### A question on Nekovar's paper Belinson's Conjectures

In Section 2 of Nevovar's paper "Beilinson's Conjectures", for a pure motive $M$ of the form $h^i(X)(n)$ where $X$ is a projective smooth variety over $\mathbb{Q}$ and $n$ is an integer such that the ...