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Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

Yusuke Nemoto, Ken Sato

Abstract

In this paper, we construct higher Chow cycles of type $(2, 1)$ on a family of surfaces related to a product of curves, which are certain degree $N$ abelian covers of $\mathbb{P}^1$ branched over $n+2$ points. We prove that for a very general member, these cycles generate a subgroup of the indecomposable part of $\operatorname{rank} \ge n\cdot \varphi(N)$, where $\varphi(N)$ is Euler's totient function, by computing their images under the transcendental regulator map.

Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

Abstract

In this paper, we construct higher Chow cycles of type on a family of surfaces related to a product of curves, which are certain degree abelian covers of branched over points. We prove that for a very general member, these cycles generate a subgroup of the indecomposable part of , where is Euler's totient function, by computing their images under the transcendental regulator map.
Paper Structure (18 sections, 10 theorems, 75 equations, 1 figure)

This paper contains 18 sections, 10 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notations
  4. The family of surfaces
  5. Construction of higher Chow cycles
  6. Preliminaries
  7. Construction of families of higher Chow cycles
  8. Computation of the regulator
  9. Levine's regulator formula
  10. Construction of topological chains
  11. Picard-Fuchs differential operator
  12. A sheaf of period functions
  13. A relative 2-form $\omega$ on $S$
  14. The differential operator $\mathscr{D}$
  15. Proof of the main theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

For a very general $t\in T$, the cycles $\xi_{c_1}^{(i)},\xi_{c_2}^{(i)},\dots, \xi_{c_n}^{(i)}$ generate a subgroup of rank at least $n\cdot \varphi(N)$ in $\operatorname{CH}^2(S_t,1)_{\mathrm{ind}}$. In particular, we have Here, $\operatorname{CH}^2(S_t, 1)_{\rm ind}$ denotes the indecomposable part of the higher Chow group $\operatorname{CH}^2(S_t, 1)$.

Figures (1)

  • Figure 1: The topological 2-chain $(K_j^l)^{(i)}$

Theorems & Definitions (17)

  • Theorem 1.1: Theorem \ref{['mainthm']}
  • Definition 3.1
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Lemma 5.1
  • proof
  • Proposition 5.2
  • ...and 7 more