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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Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply ...
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Voevodsky's Triangulated Categories of Motives and their Relationships
As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...
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Why presheaves with transfer?
Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH ...
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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.
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Relations between Motivic Galois groups and Motivic t-structure?
What are some relations between the existence of Motivic t-structures and Motivic galois groups?
I heard that indeed the existence of the Motivic t-structure implies the isomorphism between Ayoub's ...
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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 ...
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What are Motivic homotopy types?
There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, ...
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Interesting implications on the theory of motives if the Hodge conjecture holds
For example,
Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as ...
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Quadrics contained in the (complex) Cayley plane
In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...
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What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
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Should all cohomology theories have a smooth proper base change
Let $H$ be a cohomology theory with respect to some Grothendieck topology (e.g. Zariski, analytic, etale, fppf, Nisnevich, etc.)
Does H satisfy smooth proper base?
If yes, does this mean that "...
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Derived version of equivalence between motives and representations of Motivic galois groups?
A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $...
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Feynman diagrams and periods of motives
A recent article in the online science magazine Quanta, Strange Numbers Found in Particle Collisions,
discusses experimental evidence of a connection between Feynman integrals and periods of motives. ...
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On Abhyankar's results cited in a paper of Manin titled "Correspondences, Motifs and Monoidal Transformations"
Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.
Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over ...
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Relationship between motivic Galois groups and Langlands program [duplicate]
I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.
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What does the Jacobian of a curve tell us about the curve?
A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell ...
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A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures
Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite.
A weakening of this conjecture states that the $\ell$-...
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Roadmap to study (Deligne) Algebraic geometry over Tannakian categories
I would like to know the way to proceed in the first lecture of Deligne's Le groupe fondamental de la droite projective moins trois points.
General advices for reading Deligne's paper.
What should I ...
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Motivic proof of Weil-conjectures?
Assuming the standard conjectures (and whatever is needed in addition),
is there a nice proof of the Weil-conjectures written completely in the language of motives?
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What is explicitly, $ \mathcal{P} \mathrm{er} ( X ) / \mathbb{Q} $?
In the following link$^{[1]}$, page $2$, we find the following question :
Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ ...
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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 Motives for modular forms, Scholl ...
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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 ...
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Grothendieck's letter to Faltings: On the yoga of motives and the degeneration of Leray spectral sequence
In Grothendieck's letter to Faltings, he writes
There exists at this time a kind of “yoga des motifs”, which is familiar to a handful of initiates, and in some situations provides a firm support for ...
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Conjecture of relation between residues of Feynman integrals and mixed Tate motives
In many articles (for example in articles given by M.Marcoli) there is statement that there is the following conjecture
Residues of Feynman integrals in scalar field theories are always periods of ...
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"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 ...
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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}}(\...
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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 ...
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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-...
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References for Artin motives
I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
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Is there a yoga of effectivity for motives and their realizations?
Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
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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 ...
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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 $[S] \in \mathbb{Z}[\mathbb{L}]$,...
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Applications of homotopy purity theorem of Morel-Voevodsky
One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...
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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 ...
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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 ...
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The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?
I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory $...
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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 ...
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Correspondences of schemes induced by a finite Galois extension
I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes.
Recall that if $...
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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 ...
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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 ...
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Can the failure of the multiplicativity of archimedean L-factors be corrected?
My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-...
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motive of a modular form
What is the idea behind a "motive of a modular form"? (I know what a motive is and what a Weil cohomology is. I want to know how to get the motive [what is the idea of Scholl?], and why this is ...
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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 ...
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difference between equivalence relations on algebraic cycles
For the definitions of the equivalence relations on algebraic cycles see http://en.wikipedia.org/wiki/Adequate_equivalence_relation.
I want to know how far away from each other the equivalence ...
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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 $\...
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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 ...
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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 $U$.
...
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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 ...
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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) $...
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Progress on the standard conjectures on algebraic cycles
What's the current state of these conjectures?
Who is working on them?
In "Standard conjectures on algebraic cycles" Grothendieck says:
"They would form the basis of the so-called "theory of ...
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