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Arithmetic Cycles with Modulus

Souvik Goswami, Rahul Gupta

TL;DR

The paper introduces arithmetic Chow groups with modulus, $\widehat{\mathrm{CH}}^p(X|D)$, by attaching analytic data that vanishes appropriately along a simple normal crossing divisor $D$ to cycles with modulus. It develops the analytic apparatus (differential forms, currents, Green currents with modulus), the modulus Chow groups $\mathrm{CH}^p(X|D)$, and the arithmetic enhancements, establishing functoriality, exact sequences, and a module/action framework tying $\widehat{\mathrm{CH}}^*(X)$ to $\widehat{\mathrm{CH}}^*(X|D)$. A central result is the isomorphism between the relative hermitian Picard group $\widehat{\mathrm{Pic}}(X,D)$ and $\widehat{\mathrm{CH}}^1(X|D)$, mirroring the classical Picard/chow correspondence in the modulus setting. The framework paves the way toward an arithmetic motivic theory with modulus by integrating Green currents that respect modulus vanishing conditions and establishing a robust product-action structure with modulus constraints. Overall, this work extends arithmetic intersection theory to incorporate zariski and analytic moduli, enabling refined arithmetic intersections on quasi-projective varieties with boundary data."

Abstract

We add analytic components to algebraic cycles with modulus and define an arithmetic Chow group with modulus that resembles the classical arithmetic Chow groups by Gillet and Soulé. The analytic component is dictated by imposing a vanishing condition on the cohomology class of a cycle with modulus. We prove several natural properties of this group as a consequence.

Arithmetic Cycles with Modulus

TL;DR

The paper introduces arithmetic Chow groups with modulus, , by attaching analytic data that vanishes appropriately along a simple normal crossing divisor to cycles with modulus. It develops the analytic apparatus (differential forms, currents, Green currents with modulus), the modulus Chow groups , and the arithmetic enhancements, establishing functoriality, exact sequences, and a module/action framework tying to . A central result is the isomorphism between the relative hermitian Picard group and , mirroring the classical Picard/chow correspondence in the modulus setting. The framework paves the way toward an arithmetic motivic theory with modulus by integrating Green currents that respect modulus vanishing conditions and establishing a robust product-action structure with modulus constraints. Overall, this work extends arithmetic intersection theory to incorporate zariski and analytic moduli, enabling refined arithmetic intersections on quasi-projective varieties with boundary data."

Abstract

We add analytic components to algebraic cycles with modulus and define an arithmetic Chow group with modulus that resembles the classical arithmetic Chow groups by Gillet and Soulé. The analytic component is dictated by imposing a vanishing condition on the cohomology class of a cycle with modulus. We prove several natural properties of this group as a consequence.
Paper Structure (23 sections, 23 theorems, 111 equations)

This paper contains 23 sections, 23 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Layout
  5. Notations
  6. Analytic component
  7. Differential forms and currents
  8. Extension lemma and relative forms
  9. Arithmetic Chow groups
  10. Conjugate linear involution $F_{\infty}$ and arithmetic rings
  11. Arithmetic Chow groups
  12. Picard group of Hermitian line bundles
  13. Chow groups with modulus
  14. Relative Picard Group and cycles with modulus of codimension $1$
  15. Action of Chow ring
  16. ...and 8 more sections

Key Result

Theorem 1

(Theorem thm:Surj) Let $Z\in \mathcal{Z}^{p}(X|D)$. Then there exists a Green current $g_{Z}$ which satisfies the differential equation $\partial \overline{\partial}g_{Z}+\delta_{Z}=[\omega_{Z}]$, such that $\omega_{Z}\in \Sigma_{E}A^{p,p}(X)$, is real, satisfies $F^{\ast}_{\infty}\omega_{Z}=(-1)^{p

Theorems & Definitions (58)

  • Theorem 1
  • Definition A
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4: Extension Lemma
  • proof
  • Lemma 2.5
  • ...and 48 more