|May 24, 2019
|University of Michigan, Ann Arbor
Vector bundles on the Fargues-Fontaine curve play a pivotal role in recent development of p-adic Hodge theory and related fields, as they provide geometric interpretations of many constructions in these fields. The most striking example is the geometrization of the local Langlands correspondence due to Fargues where the correspondence is stated in terms of certain sheaves on the stack of vector bundles over the Fargues-Fontaine curve. In this talk, we give several classification theorems regarding vector bundles over the Fargues-Fontaine curve. Our main result is a complete classification of all quotient bundles of a given vector bundle. As its consequences, we also get a complete classification of globally generated vector bundles and a classification of almost all sub-bundles of a given bundle. Our proof is based on dimension counting of certain moduli spaces of bundle maps using Scholze's theory of diamonds.