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.
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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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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 ...
user138661
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458
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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. ...
user138661
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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 ...
- 1,014
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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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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 ...
user138661
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$p$-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) \...
- 2,961
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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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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 ...
- 6,158
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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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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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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) $...
- 2,969
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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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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 ...
user134132
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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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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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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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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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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)[...
- 761
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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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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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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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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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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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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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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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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 ...
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Mixed motives and motivic cohomology
In Scholl's paper "Remarks on special values of $L$-functions", he defines that an object $M$ of $\textbf{MM}_{\mathbb{Q}}$ (the conjectured abelian category of mixed motives with coefficients $\...
- 2,961
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Lefschetz standard conjecture under specialization/generization
Let $S$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field.
Let $f: \mathcal{X}\to S$ be a smooth projective ...
user92332
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Coniveau in étale motivic cohomology
Let $X$ be a smooth variety over a field.
Is there a spectral sequence:
$$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...
user92332
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Multiplicative structure on Deligne cohomology
Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...
user92332
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Spectral sequence in Betti cohomology
Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X_{\rm an}\to X_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $...
user119470
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0
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Torsion homologically trivial cycles
Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?
user119470
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0
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Filtrations and the Betti cycle map
Let $X$ be a smooth projective complex variety.
Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...
user119470
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0
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Cycles modulo homological equivalence
Let $\text{CH}^p(X)_{\rm hom}$ be the abelian group of codimension $p$ algebraic cycles on a smooth projective variety over a field $k$, modulo homological equivalence. Is $\text{CH}^p(X)_{\rm hom}$ ...
user97068
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Absolute Hodge cycles over $\mathbf{Q}$
In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
user120812
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1
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Blowup formula for motivic cohomology
If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula
$$H^j(X'_{et},\mathbf{...
user95222
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1
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Two motivic complexes, compared
Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
user120812
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1
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Is this Mayer-Vietoris sequence motivic?
Suppose $Y$ is a variety defined over $\mathbb{Q}$ and $pt$ is a rational point of $Y$. Let $\pi:X \rightarrow Y$ be the blow up of $Y$ at $pt$ and $D$ be the exceptional divisor. For simplicity let's ...
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Tate Conjecture birational invariant?
Is the Tate Conjecture stable under birational equivalence?
In particular, is the Tate Conjecture for rational varieties known?
user120812
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Locus of Hodge classes
Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...
user120812
3
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1
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Absolute Hodge cycles
Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...
user120812
6
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1
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Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus
Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...
- 7,902
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$\mathbf{A}^1$- contractibility
Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...
user119470
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1
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How to think about infinite generatedness of motivic cohomology
In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
user97068
7
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1
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953
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How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
A refresher (you may skip to the question at the bottom)
One defines
(1) $z_n(X,d) :=$...
user97068
12
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1
answer
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Precise formulation of conjectures on orders of vanishing?
Let $X$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z})$.
C. Soulé has conjectures about special values of the completed zeta function of $X$, $\zeta(X,s)$, which were first reformulated ...
user92332
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0
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Group completion of Chow varieties
Let $X$ be a quasi-projective variety over a perfect field $k$.
Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...
user113393
9
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1
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Torsion in Deligne cohomology
Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.
What ...
user113393