There are expanded notes prepared for a talk in a learning seminar on Fargues' UChicago notes Geometrization of the local Langlands correspondence, January 2016 at Columbia. Our main goal is to state the classification theorem of vector bundles on the Fargues-Fontaine curve and give a sketch of the proof. To put things in context, we first review the moduli space of vector bundles on curves and discuss the analogy between the Fargues-Fontaine curve and .
Let be a smooth projective curve. There is a nice moduli space parameterizing isomorphism classes of line bundles on , its the Picard variety. Unlike the case of line bundles, isomorphism classes vector bundles of higher rank in general do not form nice moduli space, e.g., the jump phenomenon shows that it is not even separated. To resolve this issue, one can either remove the word "isomorphism classes'' and work directly with the moduli stack of vector bundles . Or more concretely, restrict one's attention to those vector bundles which are semi-stable and construct a nice moduli space of semi-stable vector bundles using Mumford's GIT. The latter coincides with the coarse moduli space of the open substack of consisting of semi-stable vector bundles.
We briefly recall the notion of (semi-)stability and the Harder-Narasimhan filtration.
The following facts are not so difficult to prove:
Let us look at the case to illustrate these notions.
We will see soon that the classification of vector bundles on the Fargues-Fontaine curve remarkably resembles that of (and one may even think the Fargues-Fontaine curve as a "twisted "!).
Let be a discretely valued non-archimedean field with uniformizer and residue field . Let be a perfectoid field with uniformizer . We have constructed the Fargues-Fontaine curve (a.k.a. the fundamental curve of -adic Hodge theory) . Recall:
For any integer , we constructed the line bundle on . Geometrically it is given by , where acts on by . Its global section is then given by . We defined the schematic curve It is a scheme over , noetherian, regular, dimensional one but not of finite type.
From now on assume is algebraically closed. Let us see the first resemblance of to by computing the Picard group of . We claim that the degree map gives an isomorphism In fact, let be a section whose divisor is a closed point . Then It turns out (requires some work) that is a PID. It then follows that
Let us see another resemblance to by showing the "genus" of is zero, i.e., . We have an affine covering and an infinitesimal neighborhood of . The cohomology of coherent sheaf on can be computed by the Cech complex Namely For , since and , it reads The latter has to do with the fact that is almost Euclidean.
Let be the degree unramified extension. Notice if we replace by then stays the same but the Frobenius changes since the residue field of changes. Thus we have a natural degree unramified cover
We have following easy properties analogous to the case:
Now we can state the main classification theorem.
Notice (a) implies (b) by the third property in Proposition 1; (a,b) together implies (c).
The main goal today is to reduce to the classification theorem 2 to the following two statements about modification of vector bundles on the Fargues-Fontaine curve.
The other direction is harder. We reduce to the following lemma.
For the other direction, we need to show that every semi-stable vector bundle is a direct sum of . It turns out that is semi-stable stable if and only if is semi-stable and is such a direct sum of if and only if is a direct sum . So we may assume that by pulling back along . Twisting by the line bundle we may assume . We need to show that .
Let us only consider the case (i.e., ). Let be the sub line bundle of maximal degree. It has degree since is semi-stable of degree 0. Write . We know that If , then by the third property. If , then by assumption that , hence there is an injection , which contradicts the maximality of . More generally, if , then . There is an injection . Pullback the exact sequence we obtain a new exact sequence Hence by assumption , which gives an injection , i.e. , which contradicts the maximality of . ¡õ