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We completely classify all quotient bundles of a given vector bundle on the Fargues-Fontaine curve. As consequences, we have two additional classification results: a complete classification of all vector bundles that are generated by a fixed number of global sections and a nearly complete classification of subbundles of a given vector bundle. For the proof, we combine the dimension counting argument for moduli of bundle maps developed in [BFH+17] with a series of reduction arguments based on some reinterpretation of the classifying conditions.
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 care
We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which exte
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 theore
We give a new definition, simpler but equivalent, of the abelian category of Banach-Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues-Fontaine curve. One goes from one cat
We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.