scholarly journals Zero cycles with modulus and zero cycles on singular varieties

2017 ◽  
Vol 154 (1) ◽  
pp. 120-187 ◽  
Author(s):  
Federico Binda ◽  
Amalendu Krishna
Keyword(s):  
Cartier Divisor ◽  
Chow Group ◽  
Canonical Decomposition ◽  
Smooth Variety ◽  
Affine Surface ◽  
Singular Varieties ◽  
Smooth Affine ◽  
Torsion Free ◽  
Cycle Class ◽  
Albanese Variety

Given a smooth variety$X$and an effective Cartier divisor$D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of$X$along$D$has a canonical decomposition in terms of the Chow group of 0-cycles$\text{CH}_{0}(X)$and the Chow group of 0-cycles with modulus$\text{CH}_{0}(X|D)$on$X$. When$X$is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that$\text{CH}_{0}(X|D)$is torsion-free and there is an injective cycle class map$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$if$X$is affine. For a smooth affine surface$X$, this is strengthened to show that$K_{0}(X,D)$is an extension of$\text{CH}_{1}(X|D)$by$\text{CH}_{0}(X|D)$.

scholarly journals Beilinson's Hodge Conjecture for Smooth Varieties

2013 ◽  
Vol 11 (2) ◽  
pp. 243-282 ◽  
Author(s):  
Rob de Jeu ◽  
James D. Lewis
Keyword(s):  
Projective Variety ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Generic Point ◽  
Higher Chow Group ◽  
Cycle Class

AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

scholarly journals Involutions of varieties and Rost’s degree formula

2018 ◽  
Vol 2018 (745) ◽  
pp. 231-252 ◽  
Author(s):  
Olivier Haution
Keyword(s):  
Algebraic Variety ◽  
Chow Group ◽  
Complete Variety ◽  
Arbitrary Characteristic ◽  
Cycle Class

Abstract To an algebraic variety equipped with an involution, we associate a cycle class in the modulo two Chow group of its fixed locus. This association is functorial with respect to proper morphisms having a degree and preserving the involutions. Specialising to the exchange involution of the square of a complete variety, we obtain Rost’s degree formula in arbitrary characteristic (this formula was proved by Rost and Merkurjev in characteristic not two).

scholarly journals RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

2017 ◽  
Vol 18 (06) ◽  
pp. 1233-1293 ◽  
Author(s):  
Federico Binda ◽  
Shuji Saito
Keyword(s):  
Cartier Divisor ◽  
Explicit Construction ◽  
Chow Groups ◽  
Chow Group ◽  
Deligne Cohomology ◽  
Fundamental Class ◽  
Normal Crossing ◽  
Rham Complex ◽  
Relative Version ◽  
Higher Chow Groups

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$ . When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$ , the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$ , we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$ . For $r=\dim \overline{X}$ , we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$ -cycles with modulus.

On K2 of varieties over number fields

2008 ◽  
Vol 1 (1) ◽  
pp. 175-183
Author(s):  
Cristian D. González-Avilés
Keyword(s):  
Number Field ◽  
Cohomology Group ◽  
Complex Multiplication ◽  
Galois Cohomology ◽  
Number Fields ◽  
Chow Group ◽  
Torsion Subgroup ◽  
Vanishing Theorem ◽  
Finite Exponent ◽  
Torsion Free

AbstractLet k be a number field and let X be a smooth, projective and geometrically integral k-variety. We show that, if the geometric Néron-Severi group of X is torsion-free, then the Galois cohomology group is finite. Previously this group was only known to have a finite exponent. We also obtain a vanishing theorem for this group, showing in particular that it is trivial if X belongs to a certain class of abelian varieties with complex multiplication. The interest in the above cohomology group stems from its connection to the torsion subgroup of the Chow group CH2(X) of codimension 2 cycles on X. In the last section of the paper we record certain results on curves which must be familiar to all specialists in this area but which we have not formerly seen in print.

Affine Lines on Affine Surfaces and the Makar–Limanov Invariant

2008 ◽  
Vol 60 (1) ◽  
pp. 109-139 ◽  
Author(s):  
R. V. Gurjar ◽  
K. Masuda ◽  
M. Miyanishi ◽  
P. Russell
Keyword(s):  
Complex Field ◽  
Picard Number ◽  
Affine Surface ◽  
Proper Maps ◽  
Affine Line ◽  
Smooth Affine

AbstractA smooth affine surface X defined over the complex field C is an ML0 surface if the Makar– Limanov invariant ML(X) is trivial. In this paper we study the topology and geometry of ML0 surfaces. Of particular interest is the question: Is every curve C in X which is isomorphic to the affine line a fiber component of an A1-fibration on X? We shall show that the answer is affirmative if the Picard number ρ(X) = 0, but negative in case ρ(X) ≥ 1. We shall also study the ascent and descent of the ML0 property under proper maps.

Jordan k-Homomorphisms onto Completely Prime ΓN Rings

1970 ◽  
Vol 30 ◽  
pp. 32-40
Author(s):  
Sujoy Charaborty ◽  
Akhil Chandra Paul
Keyword(s):  
Torsion Free

By introducing the notions of k-homomorphism, anti-k-homomorphism and Jordan khomomorphism of Nobusawa Γ -rings, we establish some significant results related to these concepts. If M1 is a Nobusawa Γ1 -ring and M2 is a 2-torsion free completely prime Nobusawa Γ2 -ring, then we prove that every Jordan k-homomorphism θ of M1 onto M2 such that k(Γ1 ) = Γ2 is either a k-homomorphism or an anti-k-homomorphism. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 32-40 DOI: http://dx.doi.org/10.3329/ganit.v30i0.8500  

Continuity of homotopies

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Open Subset ◽  
Small Neighborhood ◽  
Unit Vector ◽  
Zariski Topology ◽  
Simple Point ◽  
Smooth Variety ◽  
Additional Material ◽  
Generic Type ◽  
Zariski Open Subset

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

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