Questions tagged [algebraic-cycles]

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0answers
136 views

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 ...
3
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1answer
175 views

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 ...
4
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0answers
82 views

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 ...
2
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0answers
206 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 ...
2
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0answers
170 views

(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 ...
0
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1answer
66 views

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 ...
9
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1answer
193 views

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( \...
2
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1answer
225 views

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 ...
4
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2answers
309 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 ...
1
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1answer
160 views

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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89 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 ...
7
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1answer
337 views

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 ...
3
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214 views

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: ...
4
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0answers
267 views

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 ...
3
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0answers
117 views

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 $\...
2
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0answers
124 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
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76 views

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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167 views

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 ...
9
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1answer
598 views

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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0answers
36 views

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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110 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 \...
12
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1answer
547 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 ...
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0answers
44 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:...
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477 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 ...
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1answer
162 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
372 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 $cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
8
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412 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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399 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}(c^...
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0answers
48 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
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1answer
299 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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282 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
267 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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49 views

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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343 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
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1answer
201 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 ...
4
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0answers
57 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
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0answers
396 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
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1answer
389 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
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1answer
858 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
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1answer
198 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 ...
7
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1answer
752 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
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1answer
285 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
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1answer
375 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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436 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
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0answers
214 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 ...
8
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1answer
583 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
253 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
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0answers
140 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
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0answers
181 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
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0answers
181 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 $0\le p\le d$,...