Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces

Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan.Comment: 22 pages, to appear in the Journal of the London Mathematical Societ

Classification of quotient bundles on the Fargues-Fontaine curve

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.Comment: 40 pages, 15 figure

On the Hodge-Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.Comment: 31 page

On the Hodge-Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge–Newton reducible if its Newton polygon and Hodge polygon have a nontrivial contact point. Katz proved that Hodge–Newton reducible F-crystals admit a canonical filtration called the Hodge–Newton filtration. The notion of Hodge–Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge–Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz’s result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre–Tate deformation theory for local Shimura data of Hodge type

On nonemptiness of Newton strata in the BdR+B_\mathrm{dR}^+-Grassmannian for GLn\mathrm{GL}_n

We study the Newton stratification in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$ associated to an arbitrary (possibly nonbasic) element of $B(\mathrm{GL}_n)$. Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in $B(\mathrm{GL}_n)$, our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues-Fontaine curve.Comment: 20 pages, 8 figure

Extensions of vector bundles on the Fargues-Fontaine curve II

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we completely classify all vector bundles which arise as their extensions.Comment: 15 page

Classification of subbundles on the Fargues-Fontaine curve

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 theorem in [Hon19], but also involves a number of nontrivial adjustments.Comment: 26 page