Questions tagged [chow-groups]

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Chow ring of simplicial toric varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
3 votes
0 answers
140 views

Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?

Following up on this question which received a negative answer, I wonder if something weaker is true. We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
1 vote
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132 views

Locus where a family of cycles is rationally trivial is closed?

Let $B$ be a smooth quasi-projective variety over a field of characteristic zero. Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
1 vote
0 answers
141 views

Bloch's higher Chow group as relative ordinary Chow group

If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of ...
1 vote
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Universal properties for Bloch's higher Chow groups

I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow ...
2 votes
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232 views

Tate's conjecture for arithmetic schemes

Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
1 vote
2 answers
238 views

Correspondences acting on cohomology groups $H^*(X)$ & splittings

Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $...
2 votes
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55 views

Composition of correspondences pulled back to $\mathrm{CH}_0$

Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition, $$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...
1 vote
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155 views

Interpretation of Tate conjecture using motivic homotopy

For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps $$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$ are surjective. To ...
1 vote
1 answer
419 views

The Ogus conjecture for crystalline cohomology

How is the Ogus conjecture explicitly stated, which is a variant of the Hodge and the Tate conjectures for crystalline cohomology ? How do we build its class cycle map, and how do we formulate its ...
3 votes
0 answers
227 views

Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
3 votes
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242 views

Pushforward and pullback on the level of Chow varieties

Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic ...
1 vote
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246 views

A question on the Chow group on stacks

Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows. Let $\...
0 votes
0 answers
75 views

General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
1 vote
1 answer
190 views

Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
2 votes
0 answers
112 views

About finite dimensionality of Chow groups of zero cycles

Let $S$ be a connected smooth complex projective surface. Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$. Let $Sym^{d,d}(S)=...
4 votes
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104 views

Chow group of different reductions of a smooth projective variety

Let $(R,\mathfrak{m},k)$ be a discrete valuation ring with fraction field $K$. Let $X/K$ be a smooth projective variety. Let $\mathcal{X},\mathcal{X}'$ be smooth projective models of $X$ over $R$. Let ...
1 vote
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About Definition 2 in Roĭtman's Paper

Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero. In Definition 2 of Roĭtman's paper ...
0 votes
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Higher Chow group of complex field

It is well-known fact that there is an isomorphism $$ K_i(\mathbb{C})\simeq \left\{ \begin{array}{ll} \mathbb{Q}/\mathbb{Z} & \text{if } i:odd \\ 0 & \text{if }i:even \end{array} \right. $$ My ...
1 vote
0 answers
201 views

Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
1 vote
1 answer
318 views

Singularities of Chow varieties

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...
1 vote
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Non-torsion infinitely divisible elements in the Chow group

It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex ...
0 votes
0 answers
144 views

Chow countability argument

I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!
7 votes
1 answer
471 views

Motivic $\mathbf{Z}(1)$

I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$: $$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$ How to see ...
0 votes
1 answer
222 views

Notation on a Mumford's paper

I am reading the paper " D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1969) 195 - 204" and I do not understand a notation in bottom of page 197. It ...
3 votes
0 answers
195 views

Are cubical higher Chow groups of field $CH^{n-1}(F,n)$ generated by linear cycles?

In the paper "The linearization of higher Chow cycles of dimension one" W. Gerdes proved that Higher Chow homology group $CH^{n-1}(F,n)=H^{n}(z^{n-1}(F,*))$ are generated by linear cycles. ...
2 votes
0 answers
63 views

Motivic complexes associated to adequate equivalence relations

Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic ...
0 votes
0 answers
158 views

Pullback of algebraic $K$-theory along the surjection of abelian varieties

Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
2 votes
0 answers
249 views

Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
1 vote
1 answer
373 views

How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $?

Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ). Can you tell me how to show that, $$ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq ...
2 votes
0 answers
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Homological and rational adequate equivalences for product of curves

There is a variant of Standard Conjecture D for projective varieties over finite fields. It claims that rational and homological equivalences are equivalent on cycles after tensoring with $\mathbb{Q}$....
5 votes
1 answer
250 views

Moduli of rational equivalent classes of 0-cycles

Let $X$ be a smooth variety over a field $k$ and I'd like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I'm wondering if there is any reasonable ...
0 votes
0 answers
124 views

Injectivity of pushforward of rational Chow groups

I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...
2 votes
1 answer
394 views

Invariance of Chow groups of projective bundles under automorphisms of bundles

Suppose $X$ is a smooth scheme, $E=O_X^{\oplus n}$ and $\varphi\in SL_n(E)$, i.e. $\varphi$ has trivial determinant and is an isomorphism. Is the morphism $$\mathbb{P}(\varphi)^*:CH^{\bullet}(\mathbb{...
5 votes
0 answers
184 views

Applications of Chow rings of classifying spaces in algebraic geometry

For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of Totaro, The Chow ring of a classifying space has been computed in many cases. Is there any interesting ...
3 votes
0 answers
246 views

Descent and Chow groups

One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have $$[X,K(\mathbb{Z}(n),2n)]\...
2 votes
0 answers
213 views

Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups $$Ch^i(Y)\rightarrow Ch^i(X).$$ A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
1 vote
0 answers
87 views

What can be said about the Chow rings of classifying spaces of semi-direct products of groups?

For instance, what can we say about the Chow ring of the classifying space of a semi-direct product $CH^*(B(G\ltimes H))$, in terms of the Chow rings of $CH^*(BG)$, $CH^*(BH)$, and the singular ...
9 votes
0 answers
244 views

Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? ...
4 votes
2 answers
416 views

Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
3 votes
0 answers
700 views

Relations between rational algebraic K-theory and Chow groups

A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic $ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
2 votes
1 answer
538 views

Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph: For any scheme of finite type over a ground field ...
6 votes
0 answers
199 views

Chow ring of $E_7$ varieties

Consider a split algebraic group $G$ of type $E_7$ over a field of characteristic zero. It is known that some subgroups $P_i$ of $G$, which are called parabolic, have the property that the object $G/...
11 votes
1 answer
695 views

Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$

What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$? I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
4 votes
1 answer
236 views

Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m $

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{...
1 vote
0 answers
107 views

Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
2 votes
1 answer
495 views

Question regarding Chow group of a blow-up

Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \...
2 votes
0 answers
308 views

Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers: 1) Is there any example of a smooth projective variety $X$ over a ...
2 votes
0 answers
199 views

Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...
2 votes
0 answers
197 views

Vanishing of Chow groups in high codimension

Let $X$ be a smooth affine variety of dimension $n>2$ over $\mathbb{C}$. From the examples I have seen (admittedly very little) it seems to me that these varieties don't have torsion classes a ...