All Questions
Tagged with motives ag.algebraic-geometry
359 questions
3
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Semisimplicity conjecture
In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...
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1
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735
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Pull-back of algebraic cycles
Since today is the Chow-variety day, I'm going to ask my question here.
Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\...
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8
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1
answer
414
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Sha finiteness vs $\ell$-primary torsion
Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$...
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Codimension restrictions on intersections
This is a question I stumbled across earlier this week. I see a similar one has been asked here.
Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
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1
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172
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Effective cycles of codimension 1 and field extensions
Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension.
For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(...
- 9,061
5
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1
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Around algebraic equivalence of cycles
Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.
The Tate conjecture asserts surjectivity of the cycle ...
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Vector bundles vs algebraic cycles
For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence.
What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...
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4
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1
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Borel regulator and Bloch-Beilinson regulators
Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real ...
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463
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Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
- 13.6k
12
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3
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2k
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Motivic vs Deligne cohomology
Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch ...
- 35.2k
29
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3
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2k
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$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
- 18.4k
12
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Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
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513
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Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
CommunityBot
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Current state of Serre's Motives conjectures in Seattle
It would be worth if we have a current state of the conjectures of
Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. J P Serre. In Motives, Seattle
And ...
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3
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Why linearization leads to arithmetization?
Sorry for this question, but I think it is really important the intuition here.
Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
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Chow Groups of varieties over number fields
I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely ...
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1
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A question on Voevodsky´s categories
I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...
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1
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Tate twists and cohomology of $\mathbf{P}^1$
I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to ...
- 13.6k
2
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1
answer
164
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Full lattice images and Hodge decomposition
Let $X$ be a smooth and proper complex analytic space, or a Kahler complex manifold. Is the image $\Lambda$ of the finitely generated abelian group $H^{2i}(X,\mathbf{Z})$ into $J := H^{2i}(X,\mathbf{C}...
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1
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432
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Finiteness aspects of Deligne cohomology
Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.
For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{...
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403
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relations between nori motives and pure motives
The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) ...
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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 $\...
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0
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Are Generalized Involution Varieties hyperlane sections of Generalized Severi Brauer Varieties?
Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.
Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.
...
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71
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1
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What is the relationship between motivic cohomology and the theory of motives?
I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...
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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 ...
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28
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3
answers
2k
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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 ...
- 1,720
4
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0
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537
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Why do motivic stacks make sense?
In the paper "Motivic model categories and motivic derived algebraic geometry", Yuki Kato, whose email-address I sadly couldn't find out, describes a procedure to "motivy" the objects of any $(\infty,...
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18
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2
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1k
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Grothendieck ring of "varieties carrying a function"
Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme
of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not
constructible!) function on $X$.
I want to consider a ...
- 3,202
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3
answers
4k
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How are motives related to anabelian geometry and Galois-Teichmuller theory?
In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...
- 1,307
6
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1
answer
1k
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Intuition for polarized Hodge structures
A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The ...
- 35.2k
10
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1
answer
1k
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Motives associated to a Number Field
Suppose $k$ is a number field, i.e. an extension of $\mathbb{Q}$ of finite degree, so we have a natural inclusion $\mathbb{Q} \rightarrow k$, which induces a morphism,
\begin{equation}
\text{Spec}\,k \...
- 9,061
19
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3
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2k
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$p$-adic periods
For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism
$$
H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}.
$$
If we pick $\mathbb{Q}$-bases for ...
- 9,061
3
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1
answer
438
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Virtual Lefschetz motive
Hi there,
I have a question which popped up while reading papers on motives.
Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then $\...
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1
answer
477
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Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov
There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their ...
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2
answers
1k
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What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?
I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I don'...
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0
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244
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Effectivity and Lower Shriek for Voevodsky Motives
I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{...
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1
answer
529
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Quadrics in the Grothendieck ring
Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...
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0
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699
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Motivic Galois theory and Betti realizations?
Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
- 1,720
29
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1
answer
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Is there a higher Grothendieck ring of varieties?
Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor ...
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2
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2k
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Why would the category of Motives be Tannakian?
After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify?
I decided to read up on Tannakian formalism.
Given the ...
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6
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0
answers
400
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Hodge Realisation of Mixed Tate Motives
For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,
https://www.uni-due.de/~bm0032/publ/TateMotives.pdf
There exists an abelian category of mixed Tate ...
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3
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2
answers
525
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Clarification on Hanamura's work on $t$ structure of triangulated category of mixed motives
In Hanamura's paper Mixed Motives and Algebraic Cycles III
http://intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0001/MRL-1999-0006-0001-a005.pdf
He proved that if assume Grothendieck'...
- 1,088
6
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1
answer
314
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Virtual Motives Infinitely Divisible by Lefschetz Motive
Let $K_0(Var_k)$ be the grothendieck group of the category of $k$-varieties, and call its elements virtual motives. $\mathbb{L}:=[\mathbb{A}^1_k]$ is called the Lefschetz motive. I think that if a ...
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4
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1
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Explicit description of Verdier quotient of effective motives
Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory ...
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3
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330
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Nearby Cycle Functor and the Limit of a Variation of Hodge Structures
I am reading Ayoub's paper, The Motivic Nearby Cycles and the Conservation Conjecture,
http://user.math.uzh.ch/ayoub/PDF-Files/Leiden.pdf
In section 2.3, he talks a little about the limit of a ...
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0
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892
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Stack of Tannakian categories? Galois descent?
I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
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1
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Generalised Hodge Conjecture
Further to my question,
A Naive Question on Mixed Motives and Mixed Hodge Structures
that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on ...
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1
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A Naive Question on Mixed Motives and Mixed Hodge Structures
As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated!
The category of mixed motives ...
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26
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1
answer
4k
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Voevodsky's counterexample to the existence of a motivic t-structure
I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts".
Voevodsky in "...
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2
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6k
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Derived Algebraic Geometry and Chow Rings/Chow Motives
I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all ...
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