Extensions of vector bundles on the Fargues-Fontaine curve

Christopher Birkbeck, Tony Feng, David Hansen, Serin Hong, Qirui Li, Anthony Wang, Lynnelle Ye

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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 (HN) 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 HN polygons.

Original languageEnglish
Pages (from-to)487-532
Number of pages46
JournalJournal of the Institute of Mathematics of Jussieu
Issue number2
Early online date14 May 2020
Publication statusPublished - 14 Mar 2022


  • Diamonds
  • Fargues-Fontaine
  • Vector bundles

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