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Classification of subbundles on the Fargues-Fontaine curve

Serin Hong

Abstract

We completely classify all subbundles of a given vector bundle on the Fargues-Fontaine curve. Our classification is given in terms of a simple and explicit condition on Harder-Narasimhan polygons. Our proof is inspired by the proof of the main theorem in [Hon19], but also involves a number of nontrivial adjustments.

Classification of subbundles on the Fargues-Fontaine curve

Abstract

We completely classify all subbundles of a given vector bundle on the Fargues-Fontaine curve. Our classification is given in terms of a simple and explicit condition on Harder-Narasimhan polygons. Our proof is inspired by the proof of the main theorem in [Hon19], but also involves a number of nontrivial adjustments.

Paper Structure

This paper contains 12 sections, 34 theorems, 109 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Overview of the result
  4. Vector bundles on the Fargues-Fontaine curve
  5. The Fargues-Fontaine curve
  6. Slope theory for vector bundles
  7. Moduli of bundle maps
  8. Classification of subbundles
  9. The main theorem and primary reductions
  10. Formulation of the key inequality
  11. Reduction on slopes and ranks
  12. Degeneration of the dual bundles

Key Result

Theorem 1.1.1

Fix a prime number $p$. Let $E$ be a finite extension of $\mathbb{Q}_p$, and let $F$ be an algebraically closed perfectoid field of characteristic $p$. Denote by $X = X_{E, F}$ the Fargues-Fontaine curve associated to the pair $(E, F)$.

Figures (6)

  • Figure 1: Illustration of the condition \ref{['slopewise dominance for subbundles']} in Theorem \ref{['classification of subbundles']}.
  • Figure 2: Illustration of the condition \ref{['slopewise dominance for subbundles']} in Theorem \ref{['classification of subbundles']}.
  • Figure 4: Illustration of the induction step in terms of dual HN polygons
  • Figure 5: Illustration of the maximal slope reduction
  • Figure 6: Vector representation of HN polygons.
  • ...and 1 more figures

Theorems & Definitions (73)

  • Theorem 1.1.1: Fargues-Fontaine FF_curve, Kedlaya Kedlaya_slopefiltrations
  • Theorem 1.2.1
  • Definition 2.1.1
  • Proposition 2.1.2: "GAGA for the Fargues-Fontaine curve", KL15
  • Proposition 2.1.3: FF_curve
  • Remark
  • Definition 2.1.4
  • Definition 2.2.1
  • Proposition 2.2.2: FF_curve
  • Definition 2.2.3
  • ...and 63 more