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Questions tagged [algebraic-cycles]

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3
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0answers
76 views

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 \...
10
votes
1answer
397 views

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 ...
2
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0answers
40 views

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:...
8
votes
0answers
358 views

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 ...
-2
votes
1answer
130 views

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 $\...
4
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0answers
218 views

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 $cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
7
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0answers
306 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 ...
3
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0answers
244 views

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}(...
0
votes
0answers
29 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 $\text{CH}^p(X)_{\rm hom}$ ...
2
votes
1answer
206 views

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?
3
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0answers
171 views

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 ...
3
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0answers
125 views

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 ...
3
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0answers
26 views

Tate Conjecture birational invariant?

Is the Tate Conjecture stable under birational equivalence? In particular, is the Tate Conjecture for rational varieties known?
2
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0answers
207 views

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\...
1
vote
1answer
111 views

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 ...
3
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0answers
29 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}_{...
4
votes
0answers
241 views

Pullback of the canonical bundle

Let $f : X\to Y$ be a morphism of smooth projective varieties over a field $k$. Assume $f^*\omega_{Y/k} \simeq \omega_{X/k}$. I'd like to collect a bestiary of the properties $f$ has, or even ...
3
votes
1answer
303 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^{...
11
votes
1answer
795 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}...
4
votes
1answer
173 views

Descent of isomorphisms between irreducible closed subschemes

Let $S$ be an affine scheme, $X$ be a projective $S$-scheme, $W,Z\to X$ two reduced, irreducible closed $S$-subschemes, flat over $S$. Let $S'\to S$ be a faithfully flat map, with $S'$ affine. Assume ...
6
votes
1answer
600 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) $z_n(X,d) :=$...
3
votes
1answer
217 views

Analytic cycles on complex-analytic spaces

If $X$ is a proper smooth complex analytic space, one can define Chow groups of analytic cycles on $X$ the usual way. We have a cycle map $$c^p_X: \text{CH}^p(X) \to \text{H}^{2p}_{D}(X,\mathbf{Z}(...
12
votes
1answer
334 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 ...
11
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0answers
414 views

Sheaf-theoretic Grothendieck groups

Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral ...
2
votes
0answers
166 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 ...
7
votes
1answer
551 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 ...
3
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0answers
193 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-...
3
votes
0answers
129 views

Hodge cycles defined over algebraic extensions of $\mathbf{Q}$

Is it true that the Hodge conjecture for all smooth projective varieties over the complex numbers, follows from the Hodge conjecture for smooth projective varieties defined over $\overline{\mathbf{Q}}$...
2
votes
0answers
149 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\...
2
votes
0answers
159 views

On a class of loci in Chow varieties

Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$. For $0\le p\le d$,...
2
votes
1answer
125 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(...
5
votes
1answer
268 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 ...
4
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0answers
214 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 ...
1
vote
1answer
453 views

A question on heights and Northcott's Theorem

Could anyone please point out some references and different proofs for Northcott's Theorem about finiteness of the number of points of bounded height in projective varieties over number fields, and ...
2
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0answers
95 views

Fibers of reciprocity maps and higher dimensional analogs

Part I. Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$. We have the local Artin map for every finite $v$: $$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...
4
votes
0answers
296 views

Homotopical enhancements of cycle class maps

Fix a smooth projective variety $X$ over the complex numbers. We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups. Notation For a field $k$, recall $\Delta^n_{k} :=...
6
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0answers
267 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 ...
11
votes
3answers
532 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 ...
3
votes
0answers
149 views

Explicit algebraic cycles

Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...
8
votes
1answer
380 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{...
7
votes
1answer
263 views

Higher Chow groups for complete smooth intersections?

Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group ...
1
vote
1answer
327 views

Self-intersection of divisors and Chern class

Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^\ast:H^2(X) \to H^2(Y)$. Then, When is the image of $c_1(\mathcal{O}_X(Y)) \in H^...
6
votes
1answer
610 views

Higher Chow groups revisited

Let $X$ be an algebraic variety over a field $k$. Bloch defines the "algebraic singular complex" using the algebraic simplices $$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
6
votes
1answer
489 views

Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$. In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...
10
votes
1answer
550 views

Blow-ups, pullbacks and proper transforms

Let $X$ be a smooth projective variety, $Z$ a smooth subvariety of $X$, and let $f:\widetilde{X}\to X$ be the blow-up of $X$ along $Z$. Then for a subvariety $V\subset X$, we have two cohomology ...
7
votes
0answers
155 views

Conceptual proof of a theorem of Bloch on $K_2$ of Artinian $\mathbb Q$-algebras

Recall the following theorem of S. Bloch from his paper ($K_2$ of Artinian $\mathbb Q$-algebras, with applications to algebraic cycles, 1975): For any local $\mathbb Q$-algebra $B$ and an augmented $B$...
5
votes
0answers
327 views

Blow-up and the Chow group of zero cycles

Let $\tilde{X}\to X$ be a blow-up of a variety $X$ (over an algebraically closed field). Is it true that the Chow group of zero cycles of $\tilde{X}$ is isomorphic to that of $X$? What if $X$ is a ...
2
votes
0answers
98 views

Existence of a universal $0$-cycle

For a smooth (complex) projective surface $S$ with non trivial Albanese variety, are there simple conditions for the existence a universal codimension $2$ cycle i.e. a $Z\in \mathrm{CH}^2(Alb(S)\times ...
3
votes
0answers
77 views

Taking the inverse of the total Chern class of a subvariety

Let $V,V'\subset X$ be smooth subvarieties of a smooth projective variety. Denoting $i_V, i_{V'}$ the inclusion, is it true that if $i_{V,*}c_i(T_V)=i_{V',*}c_i(T_{V'})$ in $\mathrm{CH}^*(X)$ for all $...
12
votes
0answers
524 views

Refinement of Hodge conjecture

This question deals with the classic Hodge conjecture on projective non-singular complex varieties, or in other words, projective Kähler manifolds. In Deligne's writeup for the Clay Foundation he says ...