# Extensions of Vector Bundles on the Fargues-Fontaine Curve

## TL;DR

This paper classifies all possible extensions between semistable vector bundles on the Fargues-Fontaine curve using Harder-Narasimhan polygons, employing moduli spaces and combinatorial geometry techniques.

## Contribution

It provides a complete classification of extensions between semistable bundles on the Fargues-Fontaine curve based on a simple polygon condition, utilizing diamonds and geometric analysis.

## Key findings

- Classification criterion based on Harder-Narasimhan polygons
- Reduction of the problem to combinatorial and geometric analysis
- Use of diamonds and moduli spaces in the classification

## Abstract

We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the euclidean geometry of Harder-Narasimhan polygons.