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Higher Chow cycles on Eisenstein K3 surfaces

Ken Sato

TL;DR

This work constructs explicit higher Chow cycles of type $(2,1)$ on Eisenstein K3 surfaces, i.e., K3 surfaces with a non-symplectic automorphism of order 3, and proves their indecomposability for very general members. The authors develop the Eisenstein Jacobian framework and use a degeneration argument to reduce to the starting case $r=18$, where they perform an explicit regulator computation. By analyzing the transcendental regulator and its associated normal function, they show that the indecomposable part of CH^2(X,1) has rank at least 2 for very general Eisenstein K3 surfaces in the considered families. The approach highlights a novel interaction between explicit cycle construction, regulator maps, and period/differential equation methods (notably a Picard-Fuchs operator) in the study of higher Chow groups on K3 surfaces. This provides new insight into the structure of CH^2(X,1) in the presence of order 3 symmetries and demonstrates the effectiveness of degeneration techniques in detecting indecomposability.

Abstract

We construct higher Chow cycles of type (2,1) on some families of K3 surfaces with non-symplectic automorphisms of order 3 and prove that our cycles are indecomposable for very general members. The proof is a combination of some degeneration arguments, and explicit computations of the regulator map.

Higher Chow cycles on Eisenstein K3 surfaces

TL;DR

This work constructs explicit higher Chow cycles of type on Eisenstein K3 surfaces, i.e., K3 surfaces with a non-symplectic automorphism of order 3, and proves their indecomposability for very general members. The authors develop the Eisenstein Jacobian framework and use a degeneration argument to reduce to the starting case , where they perform an explicit regulator computation. By analyzing the transcendental regulator and its associated normal function, they show that the indecomposable part of CH^2(X,1) has rank at least 2 for very general Eisenstein K3 surfaces in the considered families. The approach highlights a novel interaction between explicit cycle construction, regulator maps, and period/differential equation methods (notably a Picard-Fuchs operator) in the study of higher Chow groups on K3 surfaces. This provides new insight into the structure of CH^2(X,1) in the presence of order 3 symmetries and demonstrates the effectiveness of degeneration techniques in detecting indecomposability.

Abstract

We construct higher Chow cycles of type (2,1) on some families of K3 surfaces with non-symplectic automorphisms of order 3 and prove that our cycles are indecomposable for very general members. The proof is a combination of some degeneration arguments, and explicit computations of the regulator map.

Paper Structure

This paper contains 20 sections, 19 theorems, 55 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eisenstein $K3$ surfaces
  4. Higher Chow cycles
  5. The regulator map
  6. Normal functions
  7. Eisenstein Jacobians
  8. Construction of cycles
  9. A triple covering of ${\mathbb{P}}^1\times {\mathbb{P}}^1$
  10. Construction of cycles
  11. The parameter space $\hat{U}_r$
  12. The main theorem
  13. A degeneration lemma
  14. The main theorem
  15. Case $r=18$
  16. ...and 5 more sections

Key Result

Theorem 1.1

For a very general $(X,\sigma)$, the indecomposable parts of the $(2,1)$-cycles $\xi$ and $\sigma_*(\xi)$ are non-torsion and linearly independent. In particular, the rank of the indecomposable parts of ${\mathrm{CH}}^2(X,1)$ is equal or greater than $2$.

Figures (10)

  • Figure 1: The possible invariants $(r,a)$ and the cases we treat in this paper
  • Figure 2: The typical situation of $F_1,F_2$
  • Figure 4: Figure for Case 3
  • Figure 5: The branching locus $B_\lambda$ and curves $F_1,F_2$
  • Figure 6: The topological 2-chain $\Gamma_0, \Gamma_1$ and paths appearing their boundaries
  • ...and 5 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Definition 4.3
  • Proposition 5.1
  • proof
  • ...and 24 more