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Notes on acceptable bundles

Osamu Fujino, Taro Fujisawa, Takashi Ono

TL;DR

The work provides a detailed, self-contained treatment of acceptable bundles on the punctured disk and their prolongations by increasing orders, extending Simpson–Mochizuki framework to the one-dimensional setting. It introduces and analyzes the $\gamma$ invariant, determinant and duality relations, and establishes coherence of prolongations, as well as compatibility with filtered-bundle structures. The paper develops $L^2$-$\overline{\partial}$ techniques to obtain growth control and proves Simpson's key lemma, then studies the behavior under cyclic covers to build a robust, explicit picture of parabolic data. These results lay groundwork for explicit handling of one-dimensional prolongations and parabolic structures within the broader theory of Mochizuki–Simpson.

Abstract

This paper provides a detailed study of acceptable bundles on a punctured disk.

Notes on acceptable bundles

TL;DR

The work provides a detailed, self-contained treatment of acceptable bundles on the punctured disk and their prolongations by increasing orders, extending Simpson–Mochizuki framework to the one-dimensional setting. It introduces and analyzes the invariant, determinant and duality relations, and establishes coherence of prolongations, as well as compatibility with filtered-bundle structures. The paper develops - techniques to obtain growth control and proves Simpson's key lemma, then studies the behavior under cyclic covers to build a robust, explicit picture of parabolic data. These results lay groundwork for explicit handling of one-dimensional prolongations and parabolic structures within the broader theory of Mochizuki–Simpson.

Abstract

This paper provides a detailed study of acceptable bundles on a punctured disk.

Paper Structure

This paper contains 17 sections, 64 theorems, 505 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas for functions on a punctured disk
  4. Prolongation of acceptable line bundles
  5. On growth estimates
  6. Prolongation of acceptable vector bundles
  7. Basic properties
  8. On filtered prolongation of acceptable bundles
  9. Some elementary inequalities
  10. Simpson's key lemma
  11. On cyclic covers
  12. On determinant bundles
  13. On dual bundles
  14. Examples of filtered bundles
  15. Dual bundles revisited
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $(E, h)$ be an acceptable vector bundle on $\Delta^*$. Then ${}_a E$ is a holomorphic vector bundle for every $a \in \mathbb{R}$.

Theorems & Definitions (136)

  • Definition 1.1: Acceptable bundles, see Definition \ref{['p-def2.1']}
  • Definition 1.2: Prolongation by increasing orders, see Definition \ref{['p-def2.3']}
  • Theorem 1.3: Simpson, see simpson1 and simpson2
  • Corollary 1.4: see Section \ref{['p-sec7']}
  • Proposition 1.5: Proposition \ref{['p-prop4.1']}
  • Theorem 1.6: Theorem \ref{['p-thm4.4']}
  • Corollary 1.7: Duality for line bundles, see Corollary \ref{['p-cor4.3']}
  • Corollary 1.8
  • Theorem 1.9: see Definition \ref{['p-def7.4']}, Corollary \ref{['p-cor7.6']}, Theorem \ref{['p-thm7.13']}, and Theorem \ref{['p-thm12.3']}
  • Theorem 1.10: Determinant bundles, see Theorem \ref{['p-thm7.5']}
  • ...and 126 more