A CONSTRUCTION OF SURFACES WITH LARGE HIGHER CHOW GROUPS

2018 ◽  
Vol 236 ◽  
pp. 311-331
Author(s):  
TOMOHIDE TERASOMA
Keyword(s):  
Power Series ◽  
Linear Algebra ◽  
Chow Groups ◽  
Chow Group ◽  
Spectral Sequences ◽  
Configurations Of Lines ◽  
Higher Chow Group ◽  
Extension Class ◽  
Higher Chow Groups

In this paper, we construct surfaces in $\mathbf{P}^{3}$ with large higher Chow groups defined over a Laurent power series field. Explicit elements in higher Chow group are constructed using configurations of lines contained in the surfaces. To prove the independentness, we compute the extension class in the Galois cohomologies by comparing them with the classical monodromies. It is reduced to the computation of linear algebra using monodromy weight spectral sequences.

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.

Algebraic K-theory and higher Chow groups of linear varieties

2001 ◽  
Vol 130 (1) ◽  
pp. 37-60 ◽  
Author(s):  
ROY JOSHUA
Keyword(s):  
General Condition ◽  
Chow Groups ◽  
Closed Field ◽  
Spectral Sequences ◽  
Motivic Cohomology ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Base Scheme ◽  
Kunneth Formula ◽  
Higher Chow Groups

The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.

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 The Abel–Jacobi map for higher Chow groups

2006 ◽  
Vol 142 (02) ◽  
pp. 374-396 ◽  
Author(s):  
Matt Kerr ◽  
James D. Lewis ◽  
Stefan Müller-Stach
Keyword(s):  
Chow Groups ◽  
Higher Chow Groups

Polygonal cycles in higher Chow groups of Jacobians

2004 ◽  
Vol 183 (3) ◽  
pp. 387-399 ◽  
Author(s):  
J.C. Naranjo ◽  
G.P. Pirola ◽  
F. Zucconi
Keyword(s):  
Chow Groups ◽  
Higher Chow Groups

DGA-Structure on Additive Higher Chow Groups

2013 ◽  
Vol 2015 (1) ◽  
pp. 1-54 ◽  
Author(s):  
Amalendu Krishna ◽  
Jinhyun Park
Keyword(s):  
Chow Groups ◽  
Higher Chow Groups
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