All Questions
40 questions
42
votes
1
answer
6k
views
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 ...
user55909
39
votes
4
answers
10k
views
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 ...
user19475
31
votes
2
answers
3k
views
On Grothendieck's idea on his Standard Conjecture B
Let me recall the Standard Conjecture B (see [1,2] below):
The $\Lambda$-operation of Hodge theory is algebraic.
It more or less says that the partial inverse to “cupping with the class of a ...
- 5,504
20
votes
1
answer
902
views
Is there any publication of Bombieri about the standard conjectures on algebraic cycles?
In "Standard conjectures of algebraic cycles" Grothendieck says:
"... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and ...
user51486
17
votes
2
answers
1k
views
Hodge standard conjecture for étale cohomology
It is known that Hodge standard conjecture is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\...
- 797
12
votes
2
answers
1k
views
What exactly do the standard conjectures in characteristic zero refer to?
As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an ...
- 5,901
12
votes
3
answers
1k
views
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 ...
- 2,923
12
votes
1
answer
407
views
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
11
votes
1
answer
967
views
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
10
votes
2
answers
1k
views
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'...
- 101
9
votes
1
answer
643
views
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
8
votes
1
answer
989
views
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
8
votes
1
answer
432
views
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{...
user97068
8
votes
0
answers
574
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 ...
- 592
7
votes
1
answer
919
views
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 ...
- 596
7
votes
0
answers
729
views
Integral decomposition of the diagonal (Chow motives)
Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...
- 4,015
6
votes
0
answers
334
views
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 ...
- 61
5
votes
2
answers
2k
views
Connections between Standard, Hodge and Tate conjectures on algebraic cycles?
What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
user56558
5
votes
1
answer
482
views
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 ...
user92332
5
votes
0
answers
397
views
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 ...
user119470
4
votes
1
answer
589
views
Is the scalar extension functor for Chow motives conservative?
Denote $CHM(F)$ to be the category of Chow motives over a field $F$.
Let's consider an algebraic exension $E/F$, then
there is a natural extension of scalars functor $CHM(F) \to CHM(E)$.
I was ...
- 1,783
4
votes
0
answers
92
views
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
4
votes
0
answers
306
views
What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?
Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...
user55909
3
votes
1
answer
568
views
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
3
votes
1
answer
360
views
Chow groups of locally trivial affine fibrations
Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$.
A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...
- 1,121
3
votes
1
answer
971
views
Algebraic equivalence vs linear equivalence
Maybe the question is too general, but nevertheless:
under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence?
What are typical classes of ...
- 133
3
votes
0
answers
175
views
Boundedness indices in Voevodsky's smash nilpotence conjecture in family
Let $X$ be a smooth projective variety over an algebraically closed field $k$. Voevodsky introduced the following notion : an algebraic cycle $Z$ in $X$ is smash nilpotent if there exist $N>0$ such ...
- 7,300
3
votes
0
answers
81
views
Tate Conjecture birational invariant?
Is the Tate Conjecture stable under birational equivalence?
In particular, is the Tate Conjecture for rational varieties known?
user120812
3
votes
0
answers
307
views
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-...
user92332
2
votes
1
answer
172
views
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(...
user92332
2
votes
0
answers
169
views
Reference for facts used in Bloch, "Algebraic cycles and L-functions II"
The proof of lemma 1.1 in [1] does not give references for a few statements it uses.
In the setting of the proof, $X$ is a smooth projective variety over a number field $k$ with a fixed embedding to $\...
- 531
2
votes
0
answers
239
views
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
2
votes
0
answers
261
views
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\...
user113452
2
votes
0
answers
476
views
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 ...
- 16.9k
1
vote
0
answers
107
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 ...
- 533
1
vote
0
answers
179
views
Which "concrete" morphisms of varieties and motives induce bijections of their lower Chow groups?
This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?
What examples are known of morphisms of varieties and Chow motives (say, over ...
- 16.9k
0
votes
1
answer
388
views
Articles about Weil cohomology theory by Grothendieck and Artin
In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...
user55909
0
votes
1
answer
468
views
Pull-back of algebraic cycles under holomorphic maps
Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...
- 1,040
0
votes
0
answers
88
views
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
0
votes
0
answers
288
views
What can one say about zero-cycle groups for products of Chow motives
What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
- 16.9k