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4
votes
1answer
103 views

Pullback of $0$-cycle by generically finite rational map

Let $f:Y\dashrightarrow X$ be a generically finite (and separable) rational map of smooth projective $k$-varieties and $x,y\in U(k)$ be two rational points, where $U\subset X$ is an open subset of $X$ ...
4
votes
1answer
162 views

$l$-dependence of the group of homologically zero cycles

Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
0
votes
0answers
234 views

Is the localization sequence exact in the middle mod. algebraic equivalence?

Let $X$ be a smooth projective $k$-variety ($k=\bar k$) and $U\subset X$ an nonempty open subset. Is it true that a cycle algebraically equivalent to zero in $U$ comes from a cycle of $X\backslash U$ ...
1
vote
0answers
68 views

Endomorphism of Chow goup induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example ...
0
votes
0answers
99 views

Cycle with integral coefficients from cycle with $\mathbb Z_l$-coefficients

Let $X$ be a $n$-dimensional ($n>2$) smooth projective variety over $k=\bar k$ of positive characteristic. Take a divisor $D\in Pic(X).$ Suppose we know that $\frac{[D]^2}{2}\in H^4(X,\mathbb ...
3
votes
0answers
122 views

Commuting diagram, algebraic cycles and K-theory

What is the easiest way to see the veracity of the following commutative diagram?$$\require{AMScd} \begin{CD} K(X) \otimes K(Y) @>\text{ch}(-) \otimes \text{ch}(-)>> A(X, \mathbb{Q}) \otimes ...
2
votes
0answers
101 views

Cycle map and flat cycle

Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...
3
votes
2answers
152 views

Obstruction to get a galois invariant cycle

Let $X$ be a smooth projective variety over a finite field $k$, $G=Gal(\bar{k}/k)$ and $\Gamma\in CH^i(\bar{X})$ such that: $cl(\Gamma) \in H_{et}^{2i}(\bar{X},\mathbb{Z}_l(i))^G$ and $\exists$ ...
1
vote
1answer
156 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 ...
2
votes
1answer
184 views

Schemes associated to algebraic cycles and local complete intersection

We know that for an effective divisor on a smooth projective variety there is a natural way of associating to it a scheme, in particular using the Cartier divisor. Can we do the same for higher ...
9
votes
2answers
452 views

Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?

Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ ...
23
votes
2answers
1k 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 ...
3
votes
1answer
243 views

cycle class as Chern class

Let $X$ be a smooth projective complex variety and $Z \subset X$ a codimension $p$ closed algebraic subvariety. Then one can define the class of $Z$ in singular cohomology $[Z] \in H^{2p}(X, ...
2
votes
1answer
264 views

Degeneracy locus and flatness over local Artinian ring

Let $X$ be a projective scheme flat over a local Artinian ring $A$, the residue field of $A$ is algebraically closed, and the special fiber of $X$ (under the natural morphism from $X$ to $A$) is ...
-1
votes
1answer
55 views

How to prove embedded copies of a curve using different base points in its Jacobian are algebraically equivalent

Let $X$ be a smooth projective curve over $k\subset\mathbb{C}$, and $p,q\in X(k)$. Let $X_p$ (resp. $X_q$) be the embedded copy of $X$ in the Jacobian $Jac(X)$ using the base point $p$ (resp. $q$). Is ...
0
votes
0answers
106 views

Chow group of a product

Let $X$ and $Y$ be smooth varieties over $k$. I was wondering if there is a decomposition of the Chow group $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$ similar to the Kunneth decomposition of ...
2
votes
1answer
386 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 ...
0
votes
0answers
151 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 ...
1
vote
0answers
156 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 ...
2
votes
0answers
140 views

Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...
4
votes
0answers
246 views

Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...
1
vote
1answer
210 views

Reference request: Deligne's conjecture (cycles)

In "The work of Tate" Milne says: "The relation between the two conjectures has been greatly clarified by the work of Deligne. He defines the notion of an absolute Hodge class on a (complete ...
4
votes
2answers
769 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?
3
votes
0answers
191 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 ...
21
votes
1answer
1k 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 ...
0
votes
1answer
256 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 ...
2
votes
1answer
319 views

Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves?

In "Standard conjectures of algebraic cycles" nLab says: "... They were also followed by “Beilinson’s dream” on motivic (complexes of) sheaves which comprise so called standard conjectures of ...
15
votes
1answer
441 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 ...
3
votes
1answer
666 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 ...
0
votes
0answers
109 views

Question about the “middle” intermediate Jacobian

Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$. Does it come from (i.e. is ...
1
vote
1answer
313 views

intersection pairing and cup product

Let $X$ be a smooth quasi-projective algebraic variety over $\mathbb{C}$ and $A^k(X)$ be the Chow group of codimension-$k$ algebraic cycles on $X$. let $\mathrm{cl}$ be the cycle map from $A^k(X)$ ...
3
votes
0answers
127 views

Cycle class map in non-smooth family of projective varieties

Let $\pi:\mathcal{X} \to T$ be a family of smooth projective complex varieties. Assume $T$ is quasi-projective, reduced, irreducible but not smooth and of positive dimension. Let $\mathcal{Z}$ be a ...
4
votes
1answer
372 views

Abel-Jacobi map

Let $X$ be a smooth projective variety defined over a number field $F$ and consider the Abel-Jacobi map $\mathrm{AJ}_k:\mathrm{CH}_0^k(X_{\overline{\mathbb{Q}}})\rightarrow \mathrm{Jac}^{2k-1}(X)$, ...
4
votes
1answer
148 views

Comparison of cycle maps

Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from ...
5
votes
0answers
162 views

Cycle classes that are killed by pushing forward from normalization

Let $X$ be a non-normal algebraic variety and $f \colon X' \to X$ its normalization. Is there a general description $\mathrm{ker}\left(\mathrm{CH}_k(X') \to \mathrm{CH}_k(X)\right)$? Are there ...
0
votes
1answer
203 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: ...
14
votes
0answers
506 views

Is there a functor of points approach to algebraic cycles and intersection theory?

Motivation Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is ...
1
vote
1answer
220 views

Why does a homologically trivial cycle have trivial projections?

Let $X$ be a smooth curve over a field. Let $Y$ be the triple product $X \times X \times X$. Let $\gamma$ be a homologically trivial codimension $2$ cycle. In the text [Zhang, p. 76] that I am ...
3
votes
1answer
352 views

An example of an affine variety with non-zero Chow groups

Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ...
1
vote
0answers
94 views

When the class of a complex is necessarily equi-dimensional

Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes ...
7
votes
1answer
373 views

Degree zero zero-cycles on the square of a curve

A well-known mathematician once explained the following conjecture to me, as "an example of how little we know about cycles of codimension $\geq 2$." Let $C$ be a curve defined over a number field ...
7
votes
1answer
786 views

Higher K theory and algebraic cycles in representation theory?

Can anybody talk about how Higher K theory and algebraic cycles play roles in representation theory? I am more interested in how they play roles in Kazhdan-Lusztig conjectures. Of course K_0 plays ...
4
votes
1answer
515 views

Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on ...
3
votes
1answer
300 views

Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...
4
votes
1answer
306 views

What is the Exceptional Locus of a flopping contraction between threefolds?

Hi, I'm trying to understand the group of cycles (modulo numerical equivalence) contracted by a flopping contraction $f$. More precisely, I'm in the setup of Definition 2.12 of this paper by ...
8
votes
1answer
440 views

Do we know the Chow groups of spheres?

Let $k$ be a field (of char. not $2$) and $X_k=\text{Spec} (k[x_1,\cdots,x_n]/(x_1^2+\cdots +x_n^2-1))$. Do we know the Chow groups $A_i (X_k)$? I could not find any references, even for $X_{\mathbb ...
7
votes
0answers
474 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, ...
9
votes
4answers
544 views

Algebraic cycles of dimension 2 on the square of a generic abelian surface

I would like to know, what is known on algebraic cycles of dimension 2 modulo algebraic or rational equivalence on the square of a generic abelian surface. First, let $A$ be a generic abelian surface ...
19
votes
4answers
3k views

Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
2
votes
1answer
2k views

Algebraic equivalence VS Numerical Equivalence - An Example.

This question is arose from the question Difference between equivalence relations on algebraic cycles and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface. ...