I wonder what this wild mess –
Let be a complete algebraically closed extension, and let be the Fargues-Fontaine curve associated with . If is any vector bundle on , the cohomology groups vanish for all and are naturally Banach-Colmez Spaces for . Recall that the latter things are roughly “finite-dimensional -vector spaces up to finite-dimensional -vector spaces”. By a hard and wonderful theorem of Colmez, these Spaces form an abelian category, and they have a well-defined Dimension valued in which is (componentwise-) additive in short exact sequences. The Dimension roughly records the -dimension and the -dimension, respectively. Typical examples are , which has Dimension , and , which has Dimension .
Here I want to record the following beautiful Riemann-Roch formula.
Theorem. If is any vector bundle on , then .
One can prove this by induction on the rank of , reducing to line bundles; the latter were classified by Fargues-Fontaine, and one concludes by an explicit calculation in that case. In particular, the proof doesn’t require the full classification of bundles.
Let be some p-divisible group of dimension d and height h, and let be the rigid generic fiber (over ) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map , where is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module . Note that is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).
The morphism is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by . The structure of the admissible locus is understood in very few cases, and getting a handle on it more generally is a famous and difficult problem first raised by Grothendieck (cf. the Remarques on p. 435 of his 1970 ICM article). About all we know so far is the following:
That’s about it for general results.
To go further, let’s switch our perspective a little. Since is an open and partially proper subspace of , the subset is open and specializing, so its complement is closed and generalizing. Now, according to a very general theorem of Scholze, namely Theorem 2.42 here (for future readers, in case the numbering there changes: it’s the main theorem in the section entitled “The miracle theorems”), if is any diamond and is any locally closed generalizing subset, there is a functorially associated subdiamond with inside . More colloquially, one can “diamondize” any locally closed generalizing subset of , just as any locally closed subspace of for a scheme comes from a unique (reduced) subscheme of .
Definition. The inadmissible/nonadmissible locus is the subdiamond of obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset .
It turns out that one can actually get a handle on in a bunch of cases! This grew out of some conversations with Jared Weinstein – back in April, Jared raised the question of understanding the inadmissible locus in a certain particular period domain for with non-minuscule Hodge numbers, and we managed to describe it completely in that case (see link below). Last night, though, I realized we hadn’t worked out any interesting examples in the minuscule (i.e. p-divisible group) setting! Here I want to record two such examples, hot off my blackboard, one simple and one delightfully bizarre.
Example 1. Take h=4, d=2 and isoclinic. Then is a single classical point, corresponding to the unique filtration on with Hodge numbers which is not weakly admissible. So in this case.
Example 2. Take h=5, d=2 and isoclinic$. Now things are much stranger. Are you ready?
Theorem. In this case, the locus is naturally isomorphic to the diamond , where is an open perfectoid unit disk in one variable over and is the division algebra over with invariant 1/3, acting freely on in a certain natural way. Precisely, the disk arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural -action comes from the natural -action on via the map .
This explicit description is actually equivariant for the -actions on and . As far as diamonds go, is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 here (cf. also the discussion in Section 4.3 here). So even though this beast doesn’t have any points over any finite extension of , it’s still morally a diamondly version of a smooth projective curve!
The example Jared and I had originally worked out is recorded in section 5.5 here. The reader may wish to try adapting our argument from that situation to the cases mentioned above – this is a great exercise in actually using the classification of vector bundles on the Fargues-Fontaine curve in a hands-on calculation.
Anyway, here’s a picture of , with some other inadmissible loci in the background:
*All rigid spaces here and throughout the post are viewed as adic spaces: in the classical language, does not generally correspond to an admissible open subset of , so one would be forced to say that there exists a rigid space together with an etale monomorphism . But in the adic world it really is a subspace.