On certain extensions of vector bundles in $p$-adic geometry

Abstract

Given two arbitrary vector bundles on the Fargues–Fontaine curve, we give an explicit criterion in terms of Harder–Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in $\href{https://doi.org/10.1017/S1474748020000183}{[1]}$.