Questions tagged [algebraic-cycles]
The algebraic-cycles tag has no usage guidance.
177
questions
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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 ...
- 737
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97
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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 ...
- 533
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635
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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 ...
- 1,964
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364
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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:
...
user39380
4
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480
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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 ...
- 5,851
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121
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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 $\...
- 93
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204
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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 ...
- 737
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131
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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}...
- 1,546
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176
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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 ...
- 22.7k
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884
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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 ...
- 1,014
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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[...
- 141
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120
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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 \...
- 1,573
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679
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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 ...
user134132
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62
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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:...
- 17.5k
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561
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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 ...
user125208
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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 $\...
- 141
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610
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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
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
- 806
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553
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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 ...
- 592
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541
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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^...
user95222
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79
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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 $\text{CH}^p(X)_{\rm hom}$ ...
user97068
2
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427
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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?
user92332
4
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1
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629
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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 ...
user119470
3
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478
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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 ...
user119470
3
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78
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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?
user120812
2
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599
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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\...
user119470
1
vote
1
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392
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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 ...
- 3,938
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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}_{...
user120812
4
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708
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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 ...
user119470
3
votes
1
answer
541
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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^{...
user120812
11
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945
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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}...
user97068
4
votes
1
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250
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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 ...
user92332
7
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953
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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) :=$...
user97068
3
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1
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372
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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}(...
user113452
12
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1
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401
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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 ...
user92332
11
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0
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473
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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 ...
user113393
2
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0
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233
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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 ...
user113393
9
votes
1
answer
637
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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 ...
user113393
3
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301
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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-...
user92332
4
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277
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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 $\overline{\mathbf{Q}}$...
user92332
2
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0
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257
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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\...
user113452
2
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0
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221
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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 $0\le p\le d$,...
user95222
2
votes
1
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167
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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(...
user92332
5
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466
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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 ...
user92332
5
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378
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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 ...
user119470
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1k
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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 ...
user92332
2
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0
answers
99
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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^{\...
user95222
4
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0
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348
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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} :=...
user97068
6
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0
answers
329
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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 ...
- 61
12
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3
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1k
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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 ...
- 2,863
3
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0
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171
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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 ...
- 31