Degenerating vector bundles in p-adic Hodge theory

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1 Degenerating vector bundles in p-adic Hodge theory David Hansen July 3, 2017 Abstract We compute the closure relations among the individual Harder-Narasimhan strata in the moduli stack of rank n vector bundles on the Fargues-Fontaine curve. The proof combines adynamicalargumentonbanach-colmezspaceswithapreciseexistencetheorem(provedin [BFH + 17]) for certain parabolic reductions of vector bundles. Contents 1 Introduction Background and main results Acknowledgments Preliminaries Small v-stacks Relative Banach-Colmez spaces as diamonds The stack Bun n Dynamics on Banach-Colmez spaces The space S Q Orbit closures References 18 1 Introduction 1.1 Background and main results The fields of p-adic geometry and p-adic Hodge theory have undergone tremendous development in recent years, largely on account of two parallel developments: Fargues and Fontaine s discovery of the fundamental curve of p-adic Hodge theory (also known as the Fargues-Fontaine curve), and Scholze s discovery of the theory of perfectoid spaces. One of the most fascinating outcomes of this development is Fargues s conjectural geometrization of the local Langlands correspondence for a connected reductive group G over a non-archimedean local field E, in terms of -adic sheaves on the stack Bun G of G-bundles on the Fargues-Fontaine curve [Far16]. Even more recently, Scholze Department of Mathematics, Columbia University, 2990 Broadway, New York NY 10027; hansen@math.columbia.edu 1

2 has announced a natural construction associating a semisimple L-parameter ϕ π with any smooth irreducible representation π of G(E), which relies crucially on the geometry and étale sheaf theory of Bun G. In this article, we study some basic geometry of this stack in the case where G =GL n. To explain our main result, fix an algebraic closure F p,andletperf Fp denote the site of perfectoid spaces over F p with its v-topology. For any characteristic p perfectoid space S, letx S denote the relative Fargues-Fontaine curve over S. LetBun n Perf Fp denote the fibered category whose fiber over S Perf Fp is the groupoid of rank n vector bundles on X S. This stack is a basic example of a small v-stack in the sense of [Sch17, Def. 12.4]. In particular, Bun n has enough geometric structure that it comes with a naturally associated topological space Bun n. According to a fundamental theorem of Fargues and Fontaine, the underlying point set of Bun n is canonically identified with the set P n of Harder-Narasimhan polygons of width n. For any P P n,letbun P n (resp. Bun P n )denotethesubstackparametrizingbundlese/x S such that for every geometric point x S, theharder-narasimhanpolygonofe x lies above or on (resp. below or on) P with the same endpoints as P.ByresultsofKedlaya-Liu,Bun P n and Bun P n are closed and open substacks of Bun n, respectively, and so the individual Harder-Narasimhan strata Bun P n =Bun P n Bun P n are locally closed substacks. (We will see that small v-stacks admit reasonable notions of open and (locally) closed substacks, cf. Definition 2.2.) Each individual stratum Bun P n is a gerbe, and the associated topological spaces Bun P n Bun n are singletons. Our main result computes the closure of Bun P n inside Bun n. The precise statement is as follows. Theorem 1.1. For any n 2 and any P P n, we have Bun P n =Bun P n as substacks of Bun n. More precisely, Bun P n is the minimal closed substack of Bun n containing Bun P n, and Bun P n = Bun P n as subsets of Bun n. We note that in the classical setting of vector bundles on a connected smooth projective curve over an algebraically closed field, the analogue of Theorem 1.1 holds for curves of genus zero and one, but fails in higher genus [FM02, Sch15]. Theorem 1.1 is thus related to the heuristic idea that the Fargues-Fontaine curve has genus between zero and one. Let us sketch the proof of Theorem 1.1 in some detail. As we ve already mentioned, the inclusion Bun P n Bun P n is known, so it suffices to demonstrate the opposite inclusion. This is not formal, and roughly amounts to constructing well-behaved families of vector bundles whose HN polygons degenerate from a given polygon P to any specified Q P. To produce the necessary families, we introduce certain auxiliary moduli spaces S Q /Spd F p parametrized by Q P n.precisely,foranygivenq, letλ 1 < <λ k denote the slopes of Q, and let m i N >0 (1 i k) bethemultiplicitiessuchthatq = HN( 1 i k O(λ i ) mi ).Wethendefine S Q Perf Fp as the category fibered in groupoids whose fiber category over T Perf Fp has objects given by tuples E,F E = {0 =F 0 E F 1 E F k E = E},r = {r i : O(λ i ) mi Fi E/F i 1 E} 1 i k where E/X T is a rank n vector bundle and the remaining data has the evident meaning ( a filtration together with a rigidification of its graded pieces ), and whose morphisms (E,...) (E,...) are given by isomorphisms f : E E which are compatible with the filtrations and such that gr i f r i = ri. One easily checks that this fibered category is a v-stack, and that objects of S Q have no automorphisms, i.e. that S Q Perf Fp is a category fibered in setoids in the terminology of [Sta17]. There is thus no harm in replacing S Q with its associated sheaf of sets. Having done this, it turns out 2

3 that S Q is a small v-sheaf, and the map S Q Spd F p is representable in locally spatial diamonds and moreover is partially proper, cf. Proposition 3.1. Furthermore, the natural map π Q : S Q Bun n given by forgetting the filtration and rigidification induces a continuous map S Q Bun n ;by general properties of slope filtrations, this map factors through the inclusion Bun Q n Bun n. We now appeal to the following crucial theorem, which is more or less the main result of [BFH + 17]. Theorem 1.2 ([BFH + 17, Theorem 1.1.4]). For any Q P n, the map π Q : S Q Bun Q n a surjective map S Q Bun Q n. induces In particular, pulling back the HN stratification of Bun n along π Q induces a stratification S Q = P Q SQ P by locally closed sub-v-sheaves such that every stratum is nonempty. We observe that S Q Q consists of a single point s Q S Q,andinfactS Q Q Spd F p,sincetheq-filtration splits and rigidifies the HN-filtration on this stratum. The key observation is that s Q is contained in the closure of any stratum: Theorem 1.3. Any open neighborhood of s Q in S Q meets every stratum SQ P,P Q. Equivalently, the closure of SQ P in S Q contains s Q for every P Q. From here, the proof of Theorem 1.1 is immediate: if P and Q P are fixed, then either Bun Q n Bun P n or 1 Bun Q n Bun P n = ; but if the latter holds, we can find some open subset U Bun n containing Bun Q n and disjoint from Bun P n, in which case π Q 1 (U) S Q is a nonempty open neighborhood of s Q disjoint from SQ P, contradicting Theorem 1.3. Let us sketch the argument for Theorem 1.3. Consider the locally profinite group J Q def = 1 i k GL mi (D λi ). Any element j =(j i ) 1 i j J Q defines an automorphism of S Q by sending an object (E,F E,r ) as before to the altered object (E,F E,r j) where we abbreviate r j = {r i j i : O(λ i ) mi Fi E/F i 1 E} 1 i k. This formula defines a right J Q -action on S Q ;notethatthestratasq P are stable under this action. The intuitive idea now is that S Q is something like an iterated tower of H 1 s, and the action of J Q should move a point of S Q all around inside these Q p -vector spaces. In particular, since s Q is roughly the point corresponding to the product of the zero classes in these H 1 s, one might hope that s Q lies in the closure of the J Q -orbit of any x S Q, which is a strictly stronger statement than Theorem 1.3. For example, take n =2and Q = HN(O O(1)); thens Q is just the sheafification of the presheaf sending S Perf Fp to the Q p -vector space H 1 (X S, O( 1)), andanelement(a, b) of J Q = Q p Q p acts by sending f H 1 (X S, O( 1)) to b 1 a f. This intuition turns out to be correct in general: Proposition 1.4. For any point x S Q, the closure of the orbit xj Q S Q contains s Q. 1 This dichotomy follows easily from the definition of stack-theoretic closure in our setting, together with the fact that each stratum Bun Q n is a gerbe. 3

4 Note that this is equivalent to the statement that the only J Q -stable open neighborhood of s Q is the entirety of S Q,cf.Lemma3.2. The proof of Proposition 1.4 runs by an induction on the number of slopes of Q. Note that when Q has a single slope, S Q = Spd Fp is a single point, and Proposition 1.4 is trivial. To explain the induction step, fix a general Q = HN( 1 i k O(λ i ) mi ) as above, and let Q = HN( 1 i k 1 O(λ i ) mi ) be the truncated polygon obtained by removing the side of largest slope from Q. Thereisanatural map q : S Q S Q defined by sending an object (E,F E,r ) as before to F k 1 E equipped with the obvious truncated filtration and rigidification. We will see (in the proof of Proposition 3.1) that q is representable in locally spatial diamonds and is partially proper. Moreover, this map admits a canonical section σ : S Q S Q sending (E,F E,r ) to E O(λ k ) m k equipped with the obvious k-step filtration and rigidification. Writing J Q = JQ GL mk (D λk ), the map q is then J Q -equivariant for the evident actions on its source and target; in particular, the fibers of q are stable under the GL mk (D λk )-action. We now argue as follows. By induction, we may assume that Proposition 1.4 is known for S Q ; this implies that the J Q -orbit closure of any point lying in the subset S Q S Q has the desired property. The key observation is that it now suffices to check that for any point x S Q, the orbit closure xgl mk (D λk ) meets S Q. This is a much more tractable problem, since the orbits in question lie in individual fibers of the map q, andthefibersofthismaparecloselyrelatedto Banach-Colmez spaces. Intuitively, the fibration structure of S Q S Q together with the product structure of the group J Q allow us to prove the desired property of J Q -orbit closures by a two-step procedure: first we use the GL mk (D λk )-action to show that orbit closures meet S Q, andthenwe use the induction hypothesis to show that the closure of the J Q -orbit of any point in S Q contains s Q. We note that it seems hard to directly prove Theorem 1.3 by an inductive argument like this, since the fibration S Q S Q interacts quite poorly with the Harder-Narasimhan stratifications of its source and target. 1.2 Acknowledgments Many of the ideas in this paper were also known to Laurent Fargues and Peter Scholze, and it s a pleasure to thank them both for some very interesting conversations over the years. I would also like to sincerely thank my coauthors on [BFH + 17] fortheirenthusiasticparticipationinthatproject. Finally, I m grateful to Johan de Jong for some helpful conversations, and Michael Thaddeus for several interesting conversations and for drawing my attention to the paper [FM02]. 2 Preliminaries 2.1 Small v-stacks Let Perf denote the site of characteristic p perfectoid spaces with its v-topology. In [Sch17, 12], Scholze defines the extremely general notion of a small v-stack on Perf. By definition, a small 4

5 v-stack X is a stack in groupoids on Perf admitting some surjective map U X from a small v-sheaf such that R = U X U is also a small v-sheaf. Equivalently, a small v-stack is a v-stack on Perf which can be presented as the quotient stack [U/R] associated with some groupoid in small v-sheaves (U, R, s, t, c). Small v-stacks are presumably the most general class of v-stacks on Perf with some reasonable geometric meaning. If X is a small v-stack, a point of X is an equivalence class of maps Spd(K, K + ) X for some perfectoid field K with an open bounded valuation subring K + ;heretwomapsspd(k i,k + i ) X (i =1, 2) areequivalentifthereexistsurjectivemapsspd(k 3,K 3 + ) Spd(K i,k + i ) for i =1, 2 such that the diagram Spd(K 3,K + 3 ) Spd(K 2,K + 2 ) Spd(K 1,K + 1 ) X is 2-commutative (as in [Sta17, Tag 04XF],one checks that this defines an equivalence relation). We write X for the set of points of X. The set of points admits a canonical topology: Proposition 2.1 ([Sch17, Prop. 12.7]). Let X be a small v-stack with presentation X [U/R]. Then X = U / R, and the quotient topology on X induced by the surjection U X is independent of the choice of presentation. For any map X Yof small v-stacks, the associated map X Y is continuous. In [Sch17, Def.10.7],Scholzedefinesopenandclosedimmersionsofsmallv-sheaves.Weextend this notion to small v-stacks as follows. Definition 2.2. Given a small v-stack X, an open (resp. closed) substack of X is a strictly full subcategory Z Xsuch that Z X W W is an open (resp. closed) immersion of small v-sheaves for any small v-sheaf W with a map W X. One easily checks that any open or closed substack of a small v-stack X is itself a small v-stack. Moreover, there is an equivalence between open substacks of X and open subsets of X, cf.[sch17, Prop. 12.9]. For closed substacks, a weaker result holds. Proposition 2.3. Let X be a small v-stack, and let Z X be a closed substack. i. The natural map Z X is a closed embedding. ii. There is a natural identification Z = X X Z, in the sense that an arbitrary map of small v-stacks Y Xfactors over the inclusion Z Xifand only if Y X factors through Z X. Proof. 1. By the strict fullness of Z and the definition of points, one easily checks that Z X is an injection. Moreover, for any small v-sheaf T with a map T X,wehave Z X T = Z X T as subsets of T : onetheonehand, Z X T Z X T is surjective by [Sch17, Prop ], while on the other hand the composite map is a closed embedding. Z X T Z X T T 5

6 Now, let U X be surjective map from a small v-sheaf. Then Z X U U is a closed embedding, since Z X U U is a closed immersion of small v-sheaves. But the map Z X U U identifies with the pullback of Z X along the quotient map U X,sowegettheclaim. 2. Only if is easy. For if, consider a perfectoid space S with a map S Y,correspondingto some y Ob(Y S ).WeneedtocheckthattheinducedobjectofX S is an object of the full subcategory Z S. Choose a surjective map U X from a small v-sheaf as before and set V = U X Z and T = S X U,sowegetacommutativediagram T U V S X Z of small v-stacks. Now, since S X factors over Z, theinducedmap T U factors over V = U X Z U, sot U factors over a map ψ : T V.ButT Sis a surjective map of small v-sheaves, so after passing to some v-cover {S i S} we can choose sections fitting into a diagram S i s i T ψ U V S X Z Going around the diagram via s i and ψ, weseethaty Si induces an object of Z Si for each i. Since Z is a stack, we conclude that y induces an object of Z S,asdesired. For a general small v-stack, not every closed subset of X arises as the topological space of a closed substack. For example, if X is a locally spatial diamond, the subsets of X associated with closed sub-diamonds of X are exactly those subsets of X which are closed and generalizing. This makes the notion of stack-theoretic closure slightly delicate. In particular, the existence of Z in the following definition is not automatic. Definition 2.4. Let X be a small v-stack, and let Z Xbe a small sub-v-stack. Suppose there exists a closed sub-v-stack Z Xsuch that the inclusion Z Xfactors via Z Z X,such that Z is initial among closed sub-v-stacks with this property. Then Z (which is unique if it exists) is the closure of Z in X. Proposition 2.5. Let X be a small v-stack, and let Z X be a small sub-v-stack such that X X Z is a closed substack of X. Then X X Z is the closure of Z in X. Proof. Let Y Xbe any closed substack such that the inclusion Z Xfactors over Y. Then Z X factors over an inclusion Z Y. Since X X Z = Z, Proposition2.3.ii implies that the inclusion X X Z X factors over a map X X Z Y. This shows that X X Z has the required universal property. 2.2 Relative Banach-Colmez spaces as diamonds Given any perfectoid space S/F p,wehavethe(adic)relativefargues-fontainecurvex S. In this section we make a detailed study of the cohomology groups H i (X S, E) for E avectorbundleons,. 6

7 in the language of diamonds. When S =SpaC is a tilted geometric point for some C/Q p,these are usually known as Banach-Colmez spaces. Awordonterminology:SupposegivenS together with a vector bundle E/X S as above. By the slopes of E, wemeantheset {λ Q λ is a slope of HN(E x ) for some x S}. When S is quasicompact, this is a finite set by [KL15, Prop.7.4.6]. Definition 2.6. Given a perfectoid space S Perf and a vector bundle E/X S,wedefinefunctors H i (E) S for i =0, 1 as the pro-étale sheafifications of the presheaves Perf /S Sets (T S) H i (X T, E T ), where E T is the pullback of E along the canonical map X T X S. We will sometimes write HS i (E) if we need to emphasize the base space S. These are sheaves of Q p -vector spaces over S, sothezerovectorcorrespondstoasections : S H i (E) of the structure morphism. Note that the sheafification of T H i (X T, E T ) vanishes for any i 2 by [KL15, Theorem ]. In particular, applying the H i s to a short exact sequence of vector bundles on X S induces a six-term long exact sequence of sheaves of Q p -vector spaces over S in the obvious manner. Note also that H i (E 1 E 2 ) = H i (E 1 ) S H i (E 2 ). Proposition 2.7. i. If E has only negative slopes, then H 0 (E) =S via the zero section. ii. If E has only nonnegative slopes, then H 1 (E) =S via the zero section. Proof. Part i. is immediate from Corollary and Theorem in [KL15]. Part ii. is local on S, sowemayassumes is affinoid and that E has constant rank and degree. After passing to a further rational covering of S, if necessary, Lemma and Corollary of [KL15] guaranteetheexistenceofashortexactsequenceofvectorbundles 0 G F E 0 over X S such that GO( 1) n and FO m after pullback along any geometric point x S. In particular, F and G(1) are pointwise-étale at all points of S. Bythesheaf-theoreticsurjectivityof H 1 (F) H 1 (E), itsufficestoprovethath 1 (F) =S via the zero section. This can be checked proétale-locally on S. Afterpassingtoanaffinoidpro-étalecoverS S, wecanchooseisomorphisms F S O m and G S O( 1) n.by[sch17, Lemma7.18],wemayassume,afterpassingtoafurther affinoid pro-étale cover of S,thatanysurjectiveétalemapV S admits a section. By Theorems and in [KL15], H 1 (X S, F S ) H 1 proet(s, Q m p ), so we re reduced to the claim that Hproet(S 1, Q p )=0for S chosen as above. Since Hproet(S 1, Q p ) = lim H 1 n proet(s, Z/p n Z) [ 1 p ], this reduces further to the vanishing of H 1 proet(s, Z/p n Z). The sheaf Z/p n Z on S proet is pulled back from S et, so[sch17, Prop.14.8]givesanisomorphism H 1 proet(s, Z/p n Z) H 1 et(s, Z/p n Z). But H 1 et(s, Z/p n Z)=0,sinceanyétalecoverofS splits, and the result follows. 7

8 It turns out that H 0 (E) is well-behaved in all generality. Proposition 2.8. The functor H 0 (E) is a locally spatial diamond, and the structure map H 0 (E) S is partially proper. Proof. This is local on S, sowecanassumes is affinoid. Applying [KL15, Theorem ], we can choose (locally on some rational covering of S) an exact sequence 0 E O(m 1 ) N1 i O(m 2 ) N2 for some N 1,N 2 0 and 0 m 1 m 2 (we learned this device from [Far16]). Applying H 0 then presents H 0 (E) as the fiber product H 0 (E) = H 0 O(m 1 ) N1 i,h0 (O(m 2) N 2 ),s S, so it suffices to prove the result in the case where E = O(m) N. This reduces further to E = O(m), which can be proved as in e.g. [BFH + 17, Prop.3.3.2]. For partial properness, the valuative criterion is obvious, so we need to check that the relative diagonal is closed. Writing it as the pullback of the zero section S H 0 (E) along H 0 (E) S H 0 (E) (f,g) f g H 0 (E), it suffices to check that S H 0 (E) is closed. Again, choose an injection E O(m) N for some large m and N, sowegetaninjectivemaph 0 (E) H 0 (O(m) N ) compatible with the zero sections of the source and target. This reduces us to the case where E = O(m) N, which again reduces to the case E = O(m), in which case the result follows from [Far17,Lemme2.10]. Proposition 2.9. If E has only negative slopes, the functor H 1 (E) is a diamond, and the structure map H 1 (E) S is partially proper. Proof. To check that H 1 (E) is a diamond, we can (by [Sch17, Prop. 11.6])workpro-étale-locallyon S. ArguingasintheproofofProposition2.7.ii, we can find a short exact sequence 0 O( 1) n O m E 0 of vector bundles locally on some pro-étale cover of S. Dualizing this sequence, passing to the associated long exact sequence of H i s, and applying Proposition 2.7, wegetashortexactsequence 0 H 0 (O m ) Q m p H 0 (O(1) n ) H 1 (E) 0 of Q p -vector diamonds over S. Thus we get an isomorphism H 1 (E) H 0 (O(1) n )/Q m p, which presents H 1 (E) as the quotient of a diamond by a quasi-pro-étale equivalence relation. Therefore H 1 (E) is a diamond by [Sch17, Prop.11.8]. For partial properness, the valuative criterion is obvious, so we again need to check that the diagonal is closed. Writing it as the pullback of the zero section S H 1 (E) along H 1 (E) S H 1 (E) (f,g) f g H 1 (E) 8

9 as before, it suffices to check that the zero section s : S H 1 (E) is closed. For this we first argue on presheaves. More precisely, suppose given a perfectoid space T S and an element c H 1 (X T, E T ), with associated extension bundle F/X T. Using [KL15, Corollary ], one checks that the presheaf H 1 (E) :T H 1 (X T, E T ) is separated in the sense of [Sta17, Tag00WA]. Moreover, if x = Spa(K, K + ) T is any point, then the pullback of c to H 1 (X x, E x ) vanishes if and only if the point (1, 0) lies on or below the HN polygon of F x. By semicontinuity, the locus of such points is closed and generalizing in T ; itthereforecorrespondstoaclosedimmersionof diamonds X T. It s then easy to see (using separatedness) that X T satisfies the correct universal property: if g : T T is any map of perfectoid spaces, the pullback of c to H 1 (X T, E T ) vanishes if and only if g factors through a map T X. Therefore T c,h 1 (E),s S is representable by a closed subdiamond of T, and in particular is already a sheaf. Thus X = T c,h 1 (E),s S = T c,h 1 (E),s S T is a closed immersion of diamonds. (Here we use that sheafification commutes with finite limits.) In other words, we ve shown that zero section s : S H 1 (E) pulls back to a closed immersion of diamonds along any map T H 1 (E) which factors through the canonical map H 1 (E) H 1 (E). To conclude the general case, choose any perfectoid space T with a map f : T H 1 (E). Since H 1 (E) is separated, we can find (by [Sta17, Tags00W9&00WB])apro-étalecover T T such that the composite map f : T H 1 (E) factors (uniquely) as T H 1 (E) H 1 (E), in which case the base change T f,h 1 (E),s S T T T = T f,h1 (E),s S T is a closed immersion by our arguments so far. Since T T is surjective as a map of v-sheaves, we deduce that T H1 (E) S T is a closed immersion by [Sch17, Prop i],sotheresult follows. Our next goal is the local spatiality of H 1 (E) for any E with only negative slopes. Although one can likely deduce this directly from the presentation of H 1 (E) used in the proof of Proposition 2.9, it seems a little bit tricky to write this deduction out transparently. Our strategy instead is to show that H 1 (E) admits a separated quasi-pro-étale map to a diamond whose local spatiality can be checked by hand, which implies the desired result by [Sch17, Corollary 11.28]. This argument relies on some auxiliary results which turn out to be very useful in their own right. Theorem Suppose that E has only nonnegative slopes, and let X H 0 (E) be any open subdiamond such that X S is surjective. Then X S is surjective as a map of pro-étale sheaves. This result is extremely useful in practice: the surjectivity of X S more or less amounts to the non-emptiness of X S s for all geometric points s = Spa(C, C + ) S, andmanynaturalx s of interest are much more accessible when the base is a geometric point. Proof. Passing to an open-closed decomposition of S, we may assume E has constant rank and degree. The claim is clearly pro-étale-local on S, so replacing S by a suitable pro-étale cover and arguing as in the proof of Proposition 2.7.ii, we can then find a short exact sequence 0 O( 1) m O n E 0. 9

10 Taking the associated long exact sequence of H i s and applying Proposition 2.7, wegetashortexact sequence 0 H 0 (O n ) Q n i p H 0 (E) H q 1 (O( 1) m ) 0 of Q p -vector diamonds over S. Clearly H 0 (E) Q n p (f,a) (f,f+i(a)) H 0 (E) H 1 (O( 1) m ) H 0 (E) is an isomorphism, so we get a cartesian diagram H 0 (E) Q n p (f,a) f H 0 (E) (f,a) f+i(a) H 0 (E) q q H 1 (O( 1) m ) of diamonds; note that the upper horizontal arrow is clearly a Q n p -torsor in the sense of [Sch17, Def ], and the vertical copy of q is surjective as a map of pro-étale sheaves. Thus the map q is a Q n p -torsor by [Sch17, Lemma10.13];inparticular,q is universally open and quasi-pro-étale. Let Y H 1 (O( 1) m ) be the open subdiamond associated with the open subset im H 0 (E) H 1 (O( 1) m ) ( X ) H 1 (O( 1) m ). The map H 0 (E) H 1 (O( 1) m ) then induces a map X Y which is universally open and quasipro-étale, and such that X Y is surjective. It s easy to check that any such map of locally spatial diamonds is surjective as a map of pro-étale sheaves. It thus remains to check that the induced map Y S is surjective as a map of pro-étale sheaves. In fact, we claim this map admits sections étale-locally on S. This is local on S, sowemayassume that S is affinoid, in which case it admits a map a : S Spd Q cyc p. The sheafification of the presheaf Perf /Spd Q cyc p Sets T H 1 (X T, O( 1)) is then representable by the sheaf quotient A 1, Q /Q p,cf. theproofofproposition2.13 below. Base cyc p changing back to S along a gives H 1 S(O( 1)) = A 1, Q cyc p /Q p Spd Q cyc S = A 1, /Q p p where S is the untilt of S specified by a. Pulling back the inclusion Y HS 1 (O( 1)) along A 1, A 1, /Q S S p gives an open subdiamond of A 1,, corresponding to an open adic subspace S W A 1 S such that W S is surjective. We re now reduced to the existence of a section S W over some étale cover S S. This is local on S,sowecanassumethatS = Spa(A, A + ) is affinoid perfectoid. Assume for the moment that f : A 1 S S is open. Writing W as a union of its quasicompact open subspaces W i,thequasicompactopensubsetsf(w i ) cover S,andbyquasicompacityofS this covering can be refined to a cover by f(w i ) for some finite set of i s. Replacing W with the union S 10

11 of these finitely many W i s reduces us to the case where W is quasicompact. Rescaling W A 1 S by a large power of p, wecanassumethatw is contained in B 1 S def = Spa(A T,A + T ) A 1 S. As in [KL15, 2.6], we can write A as the completed direct limit of a directed system of strongly Noetherian Tate Q p -algebras A i along submetric transition maps. Setting X i = Spa(A i,a + i ) and B 1 X i = Spa(A i T,A + i T ), wehavenaturalcompatiblehomeomorphisms S = lim X i and B 1 S = lim i B 1 X i.byspectralityofallspacesandmaps,thesubset W B 1 S is the preimage of a quasicompact open W i B 1 X i for some large i; bytakingi sufficiently large, we can also assume that W i X i is surjective. Then W i X i is a smooth surjective map of locally Noetherian adic spaces, so there exists an étale cover X i X i and a section s : Xi W i. Base changing the diagram W i s X i along the map S X i then gives the desired section. It still remains to check that A 1 S S is open. Since A 1 S = n 1 Spa(A p n T,A + p n T ), it suffices to check that B 1 S S is open. By the limit argument above, this reduces to openness of X i, which is a special case of [Hub96, Prop.1.7.8]. B 1 X i As a first application, we have the following result. Theorem Suppose S is an affinoid perfectoid space over F p, and let E/X S be a vector bundle of constant rank r and degree d whose slopes are all positive. Let N be any positive integer such that 1 N is less than the smallest slope of E. Then, after pullback along some affinoid pro-étale cover S S, we can find a short exact sequence X i Note that necessarily m = Nd r. 0 O m O( 1 N )d E S 0. Proof. Fix N as in the theorem. Regarding H 0 (O( 1 N )d ) E as parametrizing bundle maps O( 1 N )d E,let Surj O( 1 N )d, E H 0 (O( 1 N )d ) E be the subfunctor whose T -points parametrize surjective bundle maps O( 1 N )d E T on X T,and let Surj O( 1 N )d, E ss be the subfunctor of Surj O( 1 N )d, E whose T -points parametrize surjective bundle maps O( 1 N )d E T whose kernel is pointwise-semistable at all points of T. These are clearly pro-étale sheaves. We claim that it suffices to prove that the structure map Surj O( 1 N )d, E ss S is surjective as a map of pro-étale sheaves. Indeed, if this map is surjective, then we can choose apro-étalecovers S together with an S -point of Surj O( 1 N )d, E ss lying over S S and corresponding to a short exact sequence 0 F O( 1 N )d E S 0 11

12 of vector bundles on X S such that F is pointwise-semistable of slope zero and degree m. Passing to a further pro-étale cover S S such that F S O m and S S is affinoid pro-étale, the theorem follows. By [BFH + 17,Prop ],Surj O( 1 N )d, E is an open subfunctor of H 0 (O( 1 N )d ) E (strictly speaking, this result is only proved in loc. cit. when the base is a geometric point, but the proof works verbatim in our more general setup). By semicontinuity, Surj O( 1 N )d, E ss is an open subfunctor of Surj O( 1 N )d, E. Thus Surj O( 1 N )d, E ss is an open subdiamond of H 0 (O( 1 N )d ) E.Bythe previous theorem, it now suffices to check that Surj O( 1 N )d, E ss S is surjective. We first prove this in the special case where S = Spa(C, C + ) is a geometric point. For this, choose a decomposition E i E i where E i is semistable of rank r i and degree d i. By [BFH + 17, Theorem 1.1.2], we can choose short exact sequences 0 O Ndi ri O( 1 N )di E i 0 for each i, and taking the termwise direct sum of these sequences over i gives a short exact sequence as in the theorem. In particular, this shows that Surj O( 1 N )d, E ss S admits a section, so Surj O( 1 N )d, E ss S is surjective. Now let S be any affinoid perfectoid space. For any given point s S, wecanfindamapfrom ageometricpoint x = Spa(C, C + ) S such that the topological image of the closed point in x is s. Foranysuchs and x, wegetacartesiandiagram Surj O( 1 N )d, E ss S x t x Surj O( 1 N )d, E ss S of diamonds. By the argument in the previous paragraph, t admits a section r. Passingtotopological spaces and going around the resulting diagram via r implies that Surj s im O( 1 N )d, E ss S, and so the claim follows upon varying s and x. Corollary Let S be any perfectoid space, and let E be any vector bundle on X S with only positive slopes. Then the structure morphism H 0 (E) S is smooth. Proof. This is pro-étale-local on S, sowecanassumethats is affinoid and that E has constant rank r and degree d and moreover fits into a short exact sequence 0 O m O( 1 N )d E 0 as in the previous theorem. Passing to H i s and applying Proposition 2.7, wegetashortexact sequence 0 H 0 (O m ) Q m p H 0 (O( 1 N )d ) q H 0 (E) 0 of Q p -vector diamonds over S. Arguing as in the proof of Theorem 2.10 gives that q is a pro-étale Q m p -torsor. In particular, choosing any open compact subgroup K Q m p,weget(by[sch17, Lemma 10.13]) a surjective separated étale map H 0 (O( 1 N )d )/K H q 0 (O( 1 N )d )/Q m p H 0 (E). 12

13 The diamond H 0 (O( 1 N )d ) is smooth over S, sinceit srepresentedbyanopend-variable polydisk over S. Since K is a pro-p group acting freely and continuously on H 0 (O( 1 N )d ),wededucethat H 0 (O( 1 N )d )/K is smooth over S, andthusthetargetofq is smooth as well, since smoothness is étale-local on the source. Finally, we need the following seed result. Proposition For any N 1 and any S, the functor H 1 (O( 1 N )) is a locally spatial diamond. Proof. Arguing locally on S, wecanassumethats admits a map a : S Spa Q cyc p,soit senough to show that H 1 Spa Q (O( 1 cyc p N )) is a locally spatial diamond. By [Sch17, Prop.13.4.ii-iv],itsuffices to show that HSpa 1 C p (O( 1 N )) is a locally spatial diamond. Next, observe that over X Spa Cp we can choose a short exact sequence of coherent sheaves 0 O( 1 N ) ON i C p 0, where i :SpaC p X Spa Cp is the inclusion of the point at infinity. Note that for a general S with a map S Spd C p,thepullbackofi C p along X S X Spa Cp is i O S, where i : S X S is the closed immersion of the specified untilt of S into X S,cf. [Han16, 2.3]. In particular, the functor H 0 (i C p ) on perfectoid spaces over Spd C p is represented by the diamond (A 1 C p ). Therefore, passing to H i s and applying 2.7, wegetashortexactsequence 0 H 0 (O N ) Q N p α H 0 (i C p ) (A 1 C p ) H β 1 (O( 1 N )) 0. Note that α corresponds to an embedding of Q N p as a closed subgroup of A 1 C p consisting of classical points; here of course we give A 1 C p the usual additive group structure. Now, writing A 1 C p = n 1 Spa(C p p n T, O Cp p n T ) = U n as a union of quasicompact open subgroups, we get an associated covering of HSpd 1 C p (O( 1 N )) by open subdiamonds HSpd 1 C p (O( 1 N )) n 1Un /(α(q N p ) Un ). Since α(q N p ) U n Z N p is profinite and U n is spatial, we re now reduced to the following lemma, whose proof we leave as an exercise for the reader. Lemma Let G be a profinite group, and let X be a spatial diamond with a G-action. Then X/G is a spatial diamond. Theorem Suppose that E has only negative slopes. Then H 1 (E) is a locally spatial diamond. Proof. This is local on S, sowecanassumethats is affinoid and that E has constant rank r and degree d. WealreadyknowthatH 1 (E) is a diamond with separated structure map H 1 (E) S, so by [Sch17, Prop. 13.4],thelocalspatialityofH 1 (E) can be checked pro-étale-locally on S. Applying Theorem 2.11, we may choose(after replacing S by some affinoid pro-étale cover) a short exact sequence 0 O m O( 1 N )d E 0 13

14 of vector bundles on S for some fixed large N. Dualizing this sequence, passing to the associated long exact sequence of H i s, and applying Proposition 2.7, wegetashortexactsequence 0 H 0 (O m ) Q m p i H 1 (E) q H 1 O( 1 N )d 0 of Q p -vector diamonds over S. Clearly H 1 (E) Q m p (f,a) (f,f+i(a)) is an isomorphism, so we get a cartesian diagram H 1 (E) H1 (O( 1 N )d ) H1 (E) H 1 (E) Q m p (f,a) f H 1 (E) (f,a) f+i(a) H 1 (E) q q H 1 O( 1 N )d of diamonds; note that the upper horizontal arrow is clearly separated and quasi-pro-étale. In particular, (the lower horizontal copy of) q becomes separated and quasi-pro-étale after base changing along the v-cover given by (the right vertical copy of) q, so q is separated and quasi-pro-étale by [Sch17, Prop v]. Since the target of q is locally spatial, we can now conclude the local spatiality of its source by applying [Sch17, Corollary 11.28]. 2.3 The stack Bun n In this section we check that the stack Bun n is a small v-stack. Proposition The fibered category Bun n is a v-stack. Proof. This follows from Proposition and Lemma of [SW15] Proposition The diagonal map :Bun n Bun n Bun n is representable in locally spatial diamonds. Proof. Let S be a perfectoid space with a map S Bun n Bun n,correspondingtoapairofvector bundles E 1 and E 2 over X S.WeneedtocheckthatBun n Bunn Bun n S is a locally spatial diamond. By definition, this fiber product is the sheaf Isom S (E 1, E 2 ):Perf /S Sets T S O XT module isomorphisms E 1,T E2,T. For any two bundles E, F on X S,letHom S (E, F) = HS 0 (E F) be the functor on Perf /S sending T S to the set of O XT -module maps E T F T. By Proposition 2.8, thisisalocallyspatial diamond. Note that the identity map E Edefines a distinguished section id : S Hom S (E, E) of the structure morphism to S. Wethenconcludebyobservingtheisomorphism Isom S (E 1, E 2 ) = (Hom S (E 1, E 2 ) S Hom S (E 2, E 1 )) γ,homs (E 1,E 1) S Hom S (E 2,E 2),id 2 S, where γ is the map sending (f,g) Hom S (E 1, E 2 ) S Hom S (E 2, E 1 ) to (g f,f g) Hom S (E 1, E 1 ) S Hom S (E 2, E 2 ). 14

15 It remains to construct reasonable charts for Bun n.tofacilitatethis,notethatbun n decomposes as the disjoint union of open and closed substacks Bun d n Bun n parametrizing rank n vector bundles of constant degree d. It thus suffices to find small v-sheaves X d together with surjective maps X d Bun d n for each d. There are several options for how to do this; in particular, one can build suitable X d s from affine Grassmannians and prove a Beauville-Laszlo type uniformization, or one can build X d s inspired by the theory of Quot schemes. We take the latter approach, following an idea of Fargues. For any fixed m 0, considerthefunctor X d,m = Surj O(m) mn+n d, O(m + 1) mn d on perfectoid spaces over Q p. By the proof of Proposition 2.11, thisisalocallyspatialdiamond. An easy calculation shows that for any complete algebraically closed field C/F p and any surjection q : O(m) mn+n d O(m + 1) mn d of vector bundles over X Spa C,thebundleker q has rank n, degree d, andmaximalhnslopeatmostm. Moreover, every vector bundle E satisfying these three numerical conditions arises as the kernel of such a surjection: after replacing E by E (m), this becomes the statement that any vector bundle of rank n with (positive) degree e and with all HN slopes non-negative can be realized as the cokernel of an injection O( 1) e O e+n, which again follows from Lemma and Corollary of [KL15]. In particular, the natural map π m : X d,m Bun d n (q : O(m) mn+n d O(m + 1) mn d ) ker q factors through the inclusion of the open substack Bun d,max.slope m n parametrizing bundles with maximal slope m. LetS be a perfectoid space with a map f : S Bun d,max.slope m n,corresponding to a bundle E/X S. Replacing E with E (m) and arguing as in the proof of Proposition 2.7.ii, we can find a pro-étale cover S S such that the composite map S Bun d,max.slope m n lifts to an S -point of X d,m. In particular, the map π m : X d,m Bun d,max.slope m n is surjective as a map of v-stacks. Setting X = m> d X d,m, theevidentmapx Bun d n is then surjective as a map of v-stacks, and the source is a locally spatial diamond, so we conclude. 3 Dynamics on Banach-Colmez spaces 3.1 The space S Q Proposition 3.1. The map S Q Spd F p is representable in locally spatial diamonds and is partially proper. Proof. We argue by induction on the number of slopes of Q. WhenQ has one slope, S Q = Spd Fp, so we may assume Q has two or more slopes. Write Q = HN( 1 i k O(λ i ) mi ) as in the introduction. Notation as in the introduction, it then suffices to show that q : S Q S Q is representable in locally spatial diamonds and partially proper. Let T be a perfectoid space with amapf : T S Q,correspondingtoabundleE /X T with filtration and rigidification. One then 15

16 checks directly from the definitions that the sheaf of sets on Perf /T is represented by the functor S Q SQ T H 1 ((O(λ k ) m k ) E ). By [BFH + 17, Corollary ], the maximal slope of Ex at any point x T is at most λ k 1,so (O(λ k ) m k ) E has only negative slopes. We then conclude by Proposition 2.9 and Theorem Orbit closures In this section we fill in the details of the proof of Proposition 1.4. We begin with some easy lemmas. Lemma 3.2. Let X be a topological space with an action of a group G, and let x X be a G-fixed point. Then x yg for all y X if and only if X is the unique G-stable open neighborhood of x. Proof. The existence of a G-stable open neighborhood U of x with U X is clearly equivalent to the existence of a non-empty G-stable closed subset V X with x/ V.Buttheexistenceofsuch a V is clearly equivalent to the existence of a G-orbit yg with x/ yg (one direction is obvious; for the other direction, write V = y V yg). Lemma 3.3. Let X and Y be topological spaces with actions of a group G, and let f : Y X be a continuous G-equivariant map. Then for any G-fixed point y Y and any y Y such that y y G, we have f(y) f(y )G. Proof. Observe that f(y) f(y G) f(y G)=f(y )G, where the middle containment follows from continuity. Lemma 3.4. Let G be a group with a product decomposition G = H K, and let X be a topological space with a G-action. Let x X be a G-fixed point, and let S X be a K-stable subspace containing x. Suppose that every H-orbit closure in X meets S and that every K-orbit closure of a point of S contains x. Then every G-orbit closure in X contains x. Proof. Let x X be any point. By assumption, we may choose some s S with s x H. Then sk x HK,so sk x HK = x HK = x G, where the middle equality follows from the general identity i I V i = i I V i for any collection of subsets V i of any topological space X. Sincex sk by assumption, the result follows. We now return to the problem at hand. 16

17 Proof of Proposition 1.4. Let x S Q be any point. We need to prove that s Q xj Q. As in the introduction, we have the fibration q : S Q S Q with its canonical section σ : S Q S Q,sowe can regard S Q as a closed subspace of S Q via σ; notealsothatσ(s Q )=s Q. By induction, we can assume that the the J Q -orbit closure of any point in S Q S Q contains s Q.ByLemma3.4, it then suffices to check that q(x) xgl mk (D λk ). To verify this, choose a complete algebraically closed extension C/Q p and some open bounded valuation subring C + C together with a map Spd(C, C + ) S Q such that the topological image of the unique closed point of Spd(C, C + ) is q(x). LetE /X Spa(C,C + ) be the vector bundle (with k 1-step filtration and rigidication) defined by this map. Set S = S Q q, SQ Spd(C, C+ ),so S = H 1 (E O( λ k ) m k ) is a locally spatial Q p -vector diamond over Spd(C, C + ) by the arguments in the previous sections. Let 0 S be the topological image of the unique closed point in Spd(C, C + ) along the zero section, so we get a natural GL mk (D λk )-equivariant commutative diagram S π S Q {0} {q(x)} such that the image of π contains x. ByLemma3.3, itsufficestocheckthatanygl mk (D λk )-orbit closure in S contains 0. To proceed, note it suffices to prove that for any given F/X Spa(C,C + ) with only negative slopes, the p Z -orbit (for the scaling action of p Z Q p )ofanypointx H 1 (F) has the point 0 in its closure. Indeed, for the particular F of interest to us, the scaling action of a Q p corresponds to the action of the element diag(a, a,..., a) GL mk (D λk ),andsothep Z -orbit of any x H 1 (F) is contained in the GL mk (D λk )-orbit of x. To check this claim about p Z -orbit closures, we observe that F can be written as the kernel of a surjection O m O(1) n for some m, n. Takingcohomologyoftheassociatedshortexactsequence 0 F O m O(1) n 0, we get a surjection of vector diamonds H 0 (O(1) n ) H 1 (F) over Spd(C, C + ). Applying Lemma 3.3 again reduces us to checking that the closure of any p Z -orbit in H 0 (O(1) n ) contains 0. This statement, finally, can be checked by hand. Indeed, there is a natural identification of H 0 (O(1) n ) with the n-variable open perfectoid unit disk D n =Spa C + [[T 1/p 1,...,Tn 1/p ]],C + [[T 1/p 1,...,Tn 1/p ]] η over Spa(C, C + ), matching the scaling action of p with the Frobenius operator ϕ : T i T p i. Moreover, the point 0 identifies with the (unique) point x 0 lying over the closed point of Spa(C, C + ) and whose associated valuation sends each T i to 0. 17

18 By Lemma 3.2, we renowreducedtocheckingthattheonlyϕ-stable open neighborhood of x 0 in D n is the entirety of D n, which is easy. Indeed, it suffices to check that if U D n is an open neighborhood of x 0,then j0 ϕ j (U) = D n ;butthesubsets V m = x D n T i x p m x 1 i n are cofinal among open neighborhoods of x 0,andclearly j0 ϕ j (V m )= D n for any m. References [BFH + 17] Christopher Birkbeck, Tony Feng, David Hansen, Serin Hong, Qirui Li, Anthony Wang, and Lynnelle Ye, Extensions of vector bundles on the Fargues-Fontaine curve, preprint. [Far16] [Far17] [FM02] [Han16] [Hub96] Laurent Fargues, Geometrization of the local Langlands correspondence: an overview, preprint., Simple connexité des fibres d une application d Abel-Jacobi et corps de classe local, preprint. Robert Friedman and John W. Morgan, On the converse to a theorem of Atiyah and Bott, J. Algebraic Geom. 11 (2002), no. 2, MR David Hansen, Moduli of local shtukas and Harris s conjecture, I, preprint. Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, MR [KL15] Kiran S. Kedlaya and Ruochuan Liu, Relative p-adic Hodge theory: foundations, Astérisque (2015), no. 371, 239. MR [Sch15] [Sch17] Simon Schieder, The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld s compactifications, SelectaMath.(N.S.) 21 (2015), no. 3, MR Peter Scholze, Étale cohomology of diamonds, preprint. [Sta17] The Stacks Project Authors, stacks project, [SW15] Peter Scholze and Jared Weinstein, p-adic geometry, preprint. 18