# Questions tagged [algebraic-cycles]

The algebraic-cycles tag has no usage guidance.

150
questions

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### Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...

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### Algebraic deformation retract of subvarieties

Let $X$ be a complex projective variety. Let $V_1$ and $V_2$ be two closed subvarieties of $X$. We say $V_2$ admits an algebraic deformation retract to $V_1$ in $X$, if the closed immersion $V_1\...

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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 ...

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### Torsion in homology of Chow variety

For the sake of this question for simplicity you can assume $X$ is a hypersurface in the complex projective space (not necessarily smooth). Let $C_{r,d}(X)$ be the quasi-projective Chow variety ...

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### Motives based on Hodge cycles vs algebraic cycles

I am not a specialist of motives. I am afraid my questions are rather naive.
We have the category of (pure) motives based on Hodge cycles by Deligne. In his articles with Milne, morphisms between ...

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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 ...

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### 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 :)!

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### Subrings of Chow rings

Let $X$ be a smooth projective variety over $\mathbf{F}_p$, call $\overline{X}$ the base change to $\overline{\mathbf{F}}_p$, and denote by $F$ the base change to $\overline{X}$ of the absolute ...

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### Cycle of non-equidimensional scheme

In Fulton's intersection theory, example 1.7.1, he mentioned an example that contradicts to the splitting of cycles with respect to irreducible components. Consider the subscheme $X$ in $\mathbb{A}^3$ ...

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### Definition of rational equivalence

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 247, definition 9.4, says:
Definition 9.4.
The ...

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### Closed algebraic subset dominating a curve

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge
studies in advanced mathematics 76 (2002)" page 228 says:
Let $X$ be a smooth projective variety. If ...

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1
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### 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 ...

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### 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.
...

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### $0$-cycles and $0$-cycles of degree $0$

$\DeclareMathOperator\Rat{Rat}$Let $S$ be a surface, $\Rat_0(S)$ the set of $0$-cycles rationally equivalent to zero on $S$, and $\Rat_0^0(S)$ the set of $0$-cycles of degree $0$ rationally equivalent ...

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### algebras of algebraic cycles satisfying Hard Lefschetz

June Huh and Botong Wang considired (arxiv:1609:08808) algebras of algebraic classes generated by (all) divisor classes and constructed examples of smooth projective varieties for which this algebra ...

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### Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of ...

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### Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $

Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?
Here is the formula, which ...

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### 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 ...

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### (1,1)-form that does not come from a divisor

Let $M$ be projective complex manifold. The Lefschetz (1,1)-theorem says that the cycle map
$$
\text{cl}:\operatorname{Pic}(M) \to \text{Hod}^1(M)
$$
is surjective.
Question. Is there an interesting ...

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### Find the representative cycle location in the persistence diagram

I know that some toolbox such as Dionysus can also return the representative points which are on the boundaries of the cycles (topological features of a point cloud). I can clearly extract these ...

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### Matrix obtained by recursive multiplication and a cyclic permutation

Have you ever seen this matrix? Each row is obtained from the previous one by multiplying each element by the corresponding element of the next cyclic permutation of $(a_1,\dots, a_n)$:
$$\left(
\...

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1
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### Meaning of "combinatorial data"

I saw several times that often some data describing certain
algebraic objects,
eg the set of cells of a simplical complex
or a Cech cycle of a chosen coving of a variety
are called "combinatorial ...

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### 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 ...

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### Confusion regarding a definition of cycles

For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo ...

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### 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 ...

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### Why can Euler systems constructed from algebraic cycles only be anticyclotomic?

In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ...

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### Rational equivalence and ''$\mathbb{A}^1$-equivalence" for zero cycles

Let $k$ be a field, let $X\subset\mathbb{P}^r$ be a smooth projective algebraic variety defined over $k$.
We define an equivalence relation on the group of zero cycles $Z_0(X)$ in the following way:
...

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### Chern classes of torsion-free sheaves

Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by ...

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### Extra Algebraic $(1,1)$ cycles on a complex surface

Suppose $x,y,w,z$ are homogeneous coordinates of $\mathbb{CP}^3$, and
\begin{eqnarray}
X_t := \left(F_t = x f_2 +y g_2 +t F_3 = 0 \right)
\end{eqnarray}
be a family of degree 3 hypersurfaces in $\...

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### 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 ...

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### Chow group of a pair

In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows.
We consider the sub-complex $z^{*}(X;.)_{Y}...

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### Topological vs algebraic intersection forms

Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...

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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 ...

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### Calculations of residue homomorphisms in cycle modules

In the proof of Proposition 2.2 and Theorem 2.3 in Chow groups with coefficients https://eudml.org/doc/233731 written by M. Rost, he wrote
$\mathbb{A}^{1}={\rm Spec}F[u], \mathbb{A}^{2}={\rm Spec}F[...

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### Degeneration of cycle class map

Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...

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### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

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### Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching.
The starting point:...

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### On Topological Hochschild Homology

Nowdays, I hear talking about Topological Hochschild Homology more and more often, and I was wondering if someone could point out references to explain why it's important and interesting, and what ...

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### Meaning of notations in Rost's cycle modules

In chapter 2 of markus Rost's "Chow Groups with Coefficients", I encountered the notations $c_{\kappa(v)\vert F}$ and $r_{\kappa(v)\vert F}$ in formula (2.1.0) and in the homotopy property for $\...

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### Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...

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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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### Chern class map and the exponential sequence

Let $X$ be a smooth projective variety over the complex numbers, and
$$c^1_X : \text{NS}(X)\to H^2_{\rm Betti}(X,\mathbf{Z}(1))$$
the first cycle map to Betti cohomology. The cokernel $\text{coker}(c^...

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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 $\text{CH}^p(X)_{\rm hom}$ ...

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### Closure of quasi projective scheme

If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a smooth $S$-scheme with $X$ an open subscheme?

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### Picard group of quasi-projective varieties

Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field.
Is $\text{Pic}(X)$ a finitely generated abelian group?
I'm tempted to just ...

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### Picard group finitely generated

Let $X$ be a smooth projective variety over a finite field $k$. Is the Picard group finitely generated? Equivalently, is $\text{Pic}^0(X)$ finitely generated?
(I am not assuming $k$ is separably ...

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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?

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### Specialization maps for Chow groups

Let $S$ be a finite type regular integral affine scheme of finite type over $\text{Spec}(\mathbf{Z})$, and $\mathcal{X}\to S$ a smooth projective morphism. Let $\eta$ be the generic point of $S$, $s\...

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### Properties of codimension under pull back

If I pull back a cycle of codimension $c$ along a morphism of schemes I can easily see that the codimension can stay the same or the codimension can drop to all the way to $0$. But intuitively it ...

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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}_{...