Hilbert-Mumford stability on algebraic stacks and applications to G -bundles on curves
´´Epijournal de G´eom´etrie Alg´ebrique epiga.episciences.org
Volume 1 (2017), Article Nr. 11
Hilbert-Mumford stability on algebraic stacks andapplications to G -bundles on curves Jochen Heinloth
Abstract.
In these notes we reformulate the classical Hilbert-Mumford criterion for GITstability in terms of algebraic stacks, this was independently done by Halpern-Leinster [22].We also give a geometric condition that guarantees the existence of separated coarse modulispaces for the substack of stable objects. This is then applied to construct coarse modulispaces for torsors under parahoric group schemes over curves.
Keywords.
Hilbert-Mumford stability; moduli of G -bundles on curves [Fran¸cais]Titre. Stabilit´e de Hilbert-Mumford sur les champs alg´ebriques et applicationsaux G -fibr´es sur les courbesR´esum´e. Dans ces notes, nous reformulons le crit`ere classique de stabilit´e de Hilbert-Mumford pour la th´eorie g´eom´etrique des invariants en termes de champs alg´ebriques ; celaa ´et´e fait ind´ependamment par Halpern-Leinster [22]. Nous donnons ´egalement une conditiong´eom´etrique qui garantit l’existence d’espaces de modules grossiers pour le sous-champ desobjets stables. Nous l’appliquons ensuite pour construire des espaces de modules grossierspour des torseurs sous des sch´emas en groupes parahoriques sur des courbes.
Received by the Editors on October 6, 2016, and in final form on July 23, 2017.Accepted on December 12, 2017.Jochen HeinlothUniversit¨at Duisburg–Essen, Fachbereich Mathematik, Universit¨atsstrasse 2, 45117 Essen, Germany e-mail : [email protected] © by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/ a r X i v : . [ m a t h . AG ] J a n Contents
1. The Hilbert-Mumford criterion in terms of stacks . . . . . . . . . . . . . . . . . . . . .
32. A criterion for separatedness of the stable locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bun G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
The aim of these notes is to reformulate the Hilbert-Mumford criterion from geometric invarianttheory (GIT) in terms of algebraic stacks (Definition 1.2) and use it to give an existence result forseparated coarse moduli spaces.Our original motivation was that for various moduli problems one has been able to guess stabilitycriteria which have then been shown to coincide with stability conditions imposed by GIT construc-tions of the moduli stacks. It seemed strange to me that in these constructions it is often not toodifficult to find a stability criteria by educated guessing, however, in order to obtain coarse modulispaces one then has to prove that the guess agrees with the Hilbert-Mumford criterion from GIT,which often turns out to be a difficult and lengthy task.Many aspects of GIT have of course been reformulated in terms of stacks by Alper [3]. Also Iwanari[31] gave a clear picture for pre-stable points on stacks and constructed possibly non-separated coarsemoduli spaces. The analog of the numerical Hilbert-Mumford criterion has been used implicitly inmany places by several authors. Most recently Halpern-Leinster [22] independently gave a formulation,very close to ours and applied it to construct analogs of the Harder–Narasimhan stratification formoduli problems under a condition he calls Θ-reductivity.Our main aim is to give a criterion that guarantees that the stable points form a separated substack(Proposition 2.6). Once this is available, one can apply general results (e.g., the theorems of Keel andMori [32] and Alper, Hall and Rydh [4]) to obtain separated coarse moduli spaces (Proposition 2.8).As side effect, we hope that our formulation may serve as an introduction to the beautiful picturedeveloped [22].The guiding examples which also provide our main applications are moduli stacks of torsors underparahoric group schemes. Using our method we find a stability criterion for such torsors on curves andconstruct separated coarse moduli spaces of stable torsors (Theorem 3.19). Previously such modulispaces had been constructed for generically trivial group schemes in characteristic 0 by Balaji andSeshadri [7], who obtain that in these cases the spaces are schemes. Also in any characteristic thespecial case of moduli of parabolic bundles has been constructed in [26], but it seems that coarsemoduli spaces for twisted groups had not been constructed before.This problem was the starting point for the current article, because in [26] most of the technicalproblems arose in the construction of coarse moduli spaces by GIT that were needed to prove coho-mological purity results for the moduli stack. The results of this article allow to bypass this issue andapply to the larger class of parahoric groups.The structure of the article is as follows. In Section 1 we state the stability criterion dependingon a line bundle L on an algebraic stack M . As a consistency check we then show that this coincideswith the Hilbert Mumford criterion for global quotient stacks (Proposition 1.8). To illustrate themethod we then consider some classical moduli problems and show how Ramanathan’s criterion forstability of G -bundles on curves can be derived form our criterion rather easily. The same argumentapplies to related moduli spaces, as the moduli of chains or pairs. . Heinloth, GIT-stability for stacks 3J. Heinloth, GIT-stability for stacks 3 In Section 2 we formulate the numerical condition on the pair (M , L) implying that the stablepoints form a separated substack (Proposition 2.6) and derive an existence result for coarse modulispaces (Proposition 2.8). Again, as an illustration we check that this criterion is satisfied in forGIT-quotient stacks and for G -bundles on curves.Finally in Section 3 we apply the method to the moduli stack of torsors under a parahoric groupscheme on a curve. We construct coarse moduli spaces for the substack of stable points of thesestacks. For this we also need to prove some of the basic results concerning stability of parahoricgroup schemes that could be of independent interest. Acknowledgments:
This note grew out of a talk given in Chennai in 2015. The comments afterthe lecture encouraged me to finally revise of an old sketch that had been on my desk for a verylong time. I am grateful for this encouragement and opportunity. I thank M. Olsson for pointingout the reference [22] before it appeared and J. Alper, P. Boalch and the referees for many helpfulcomments and suggestions. While working on this problem, discussions with V. Balaji, N. Hoffmann,J. Martens, A. Schmitt have been essential for me.
1. The Hilbert-Mumford criterion in terms of stacks
Throughout we will work over a fixed base field k . The letter M will denote an algebraic stack over k , which is locally of finite type over k and we will always assume that the diagonal ∆ ∶ M → M × M is quasi-affine, as this implies that our stack is a stack for the fpqc topology ([34, Corollaire 10.7]).We have two guiding examples in mind: First, global quotient stacks [ X / G ] where X is a properscheme and G is an affine algebraic group acting on X and second, the stack Bun G of G -bundles ona smooth projective curve C for a semi-simple group G over k . As the numerical criterion for stability from geometric invariant theory [36, Theorem 2.1] serves asa guideline we start by recalling this briefly. To state it and in order to fix our sign conventions weneed to recall the definition of weights of G m -equivariant line bundles: As usual we denote the multiplicative group scheme by G m ∶= Spec k [ t, t − ] and the affine line by A ∶= Spec k [ x ] . The standard action act ∶ G m × A → A is given by t.x ∶= tx , i.e. on the level of rings act ∶ k [ x ] → k [ x, t, t − ] is given by act ( x ) = tx . Wewrite [ A / G m ] for the quotient stack defined by this action. This stack is called Θ in [22].By definition G m -equivariant line bundles L on A are the same as line bundles on [ A / G m ] . Forany line bundle L on A the global sections are H ( A , L) = k [ x ] ⋅ e for some section e which is uniqueup to a scalar multiple.Thus an equivariant line bundle L on A defines an integer d ∈ Z by act ( e ) = t d e called the weightof L and we will denote it as wt (L) ∶= d. In particular we find that: H ([ A / G m ] , L) = H ( A , L) G m = { k ⋅ x d e if wt (L) = d ≤
00 if wt (L) = d > . To compare the sign conventions in different articles the above equation is the one to keep in mind,because Mumford’s construction of quotients uses invariant sections of line bundles. Similarly, a
1. The Hilbert-Mumford criterion in terms of stacks
1. The Hilbert-Mumford criterion in terms of stacks G m -equivariant line bundle L on Spec k is given by a morphism of underlying modules act ∶ L →L ⊗ k [ t, t − ] with act ( e ) = t d e for some d ∈ Z . The integer d is again denoted by:wt G m (L) ∶= d. Mumford considers a projective scheme X , equipped with an action of a reductive group G and a G -linearized ample bundle L on X . For any x ∈ X ( k ) and any cocharacter λ ∶ G m → G Mumford defines µ L ( x, λ ) ∈ Z as follows. The action of G m on x defines a morphism λ.x ∶ G m → X which extends to anequivariant morphism f x,λ ∶ A → X because X is projective. He defines µ L ( x, λ ) ∶= − wt ( f ∗ x,λ L) . The criterion [36, Theorem 2.1] then reads as follows:A geometric point x ∈ X ( k ) is stable if and only if the stabilizer of x in G is finite and for all λ ∶ G m → G we have µ L ( x, λ ) > , equivalently wt ( f ∗ x,λ L) < . L -stability on algebraic stacks It is easy to reformulate the numerical Hilbert–Mumford criterion in terms of stacks, once we fixsome notation. The quotient stack [ A / G m ] has two geometric points 1 and 0 which are the imagesof the points of the same name in A . For any algebraic stack M and f ∶ [ A / G m ] → M we will write f ( ) , f ( ) ∈ M( k ) for the points given by the images of 0 , ∈ A ( k ) . Definition 1.1. (Very close degenerations)
Let M be an algebraic stack over k and x ∈ M( K ) a geometric point for some algebraically closed field K / k .A very close degeneration of x is a morphism f ∶ [ A K / G m,K ] → M with f ( ) ≅ x and f ( ) /≅ x .Very close degenerations have been used under different names, e.g. in the context of K -stabilitythese are often called test-configurations. Our terminology should only emphasize that f ( ) is anobject that lies in the closure of a K point of M K , which only happens for stacks and orbit spaces,but if X = M is a scheme, then there are no very close degenerations. Definition 1.2. ( L -stability) Let M be an algebraic stack over k , locally of finite type with affinediagonal and L a line bundle on M . A geometric point x ∈ M( K ) is called L -stable if(1) for all very close degenerations f ∶ [ A K / G m,K ] → M of x we havewt ( f ∗ L) < K ( Aut M ( x )) = Remark 1.3. (1) We can also introduce the notion of L -semistable points, by requiring only ≤ in (1) and droppingcondition (2).(2) The notion admits several natural extensions: Since the weight of line bundles extends to aelements of the groups Pic ( B G m ) ⊗ Z R ≅ R , . Heinloth, GIT-stability for stacks 5J. Heinloth, GIT-stability for stacks 5 the above definition can also be applied if L ∈
Pic (M)⊗ Z R . This is often convenient for classicalnotions of stability that depend on real parameters.In [22] Halpern-Leinster uses a cohomology class α ∈ H (M k , Q (cid:96) ) instead of a line bundle. For α = c (L) this gives the same condition. As the condition given above is a numerical one, thisis sometimes more convenient and we will refer to it as α -stability.(3) As the above condition is numerical and H d ([ A / G m ] , Q (cid:96) ) = Q (cid:96) for all d > M the stack of morphisms Mor ([ A / G m ] , M) is again an algebraic stack, locally of finite type. Asweights of G m -actions on line bundles are locally constant in families this implies that L -stabilityis preserved under extension of algebraically closed fields L / K . In particular, if we define an S -valued point M( S ) to be L -stable if the corresponding objects are stable for all geometricpoints of S , we get a notion that is preserved under pull-back and therefore defines an abstractsubstack M s ⊂ M . We will see that in may cases L -stability turns out to be an open conditionand then M s is again an algebraic stack, but this is not true for arbitrary M , L . Notation.
Given a line bundle L on M and x ∈ M( K ) we will denote by wt x (L) the homomorphismwt x (L)∶ X ∗ ( Aut M ( x )) = Hom ( G m , Aut M ( x )) → Z which maps λ ∶ G m → Aut M ( x ) to wt G m (( x, λ ) ∗ L) , where ( x, λ )∶ [ Spec K / G m ] → M is the morphismdefined by x and λ . Example 1.4.
A toy example illustrating the criterion is given by the anti-diagonal action G m × A → A defined as t. ( x, y ) ∶= ( tx, t − y ) . The only fixed point of this action is the origin 0. The quotient ( A − { })/ G m is the affine line with a doubled origin, the first example of a non-separated scheme.Since the latter space is a scheme none of its points admits very close degenerations. This changes ifwe look at the full quotient [ A / G m ] , which contains the additional point [( , )/ G m ] . The inclusionsof the coordinate axes ι x , ι y ∶ A → A define very close degenerations of the points ( , ) and ( , ) and it will turn out (Lemma 1.6) that these constitute essentially the only very close degenerationsin this stack.A line bundle on [ A / G m ] is an equivariant line bundle on A . Since line bundles on A are trivial,all equivariant line bundles are given by a character () d ∶ G m → G m and we find wt ∶ Pic ([ A / G m ]) ≅ Z .Moreover for the corresponding line bundle L d the weights wt ( ι ∗ x L d ) = d, wt ( ι ∗ y L d ) = − d so for each d ≠ ( , ) and ( , ) can be L d -stable.For d = ( , ) , ( , ) , ( , ) are all semistable and these points would be identified inthe GIT quotient. To apply the definition of L -stability, one needs to classify all very close degenerations. The nextlemma shows that these can be described by deformation theory of objects x that admit non-constantmorphisms G m → Aut M ( x ) . Let us fix our notation for formal discs: D ∶= Spec k [[ t ]] , ˚ D ∶= Spec k (( t )) .
1. The Hilbert-Mumford criterion in terms of stacks
1. The Hilbert-Mumford criterion in terms of stacks
Lemma 1.5.
Let M be an algebraic stack locally of finite type over k = k with quasi-affine diagonal. (1) For any very close degeneration f ∶ [ A / G m ] → M the induced morphism λ f ∶ G m = Aut [ A / G m ] ( ) → Aut M ( f ( )) is nontrivial. (2) The restriction functor
M([ A / G m ]) → lim ←— M([ Spec ( k [ x ]/ x n )/ G m ]) is an equivalence of cat-egories.Proof. For part (2) elegant proofs have been given independently by Alper, Hall and Rydh [4, Corol-lary 3.6] and Bhatt, Halpern-Leistner [10, Remark 8.3]. Both proofs rely on Tannaka duality, oneuses an underived version the other a derived version. As the statement produces the point f ( ) outof a formal datum let us explain briefly why this is possible: The composition φ ∶ D = Spec k [[ x ]] → A → [ A / G m ] is faithfully flat, because both morphisms are flat and the map is surjective, because both points 1 , [ A / G m ] are in the image.By our assumptions M is a stack for the fpqc topology ([34, Corollaire 10.7]) we therefore see that M([ A / G m ]) can be described as objects in M( k [[ x ]]) together with a descent datum with respectto φ .Moreover, the canonical map M( D ) ≅ —→ lim ←— M( k [ x ]/( x n )) is an equivalence of categories: Thisfollows for example, because the statement holds for schemes and choosing a smooth presentation X → M one can reduce to this statement, [42, Tag 07X8].In particular this explains already that an element of lim ←— M([ Spec ( k [ x ]/ x n )/ G m ]) will producea k [[ x ]] -point of M . The problem now lies in constructing a descent datum for this morphism, as D × [ A / G m ] D = Spec ( k [[ x ]] ⊗ k [ x ] k [ x, t, t − ] ⊗ k [ y ] k [[ y ]]) where the last tensor product is taken via y = xt . The ring on the right hand side is not completeand the formal descent data coming form an element in lim ←— M([ Spec ( k [ x ]/ x n )/ G m ]) only seems toinduce a descent datum on the completion of the above ring.Here the Tannakian argument greatly simplifies the problem, as it gives a concise way to capturethe information that a G m action induces a grading and therefore allows to pass from power series topolynomials.Let us deduce (1). First note that this holds automatically if M is a scheme, because then f ( ) is a closed point and f ( ) lies in the closure of f ( ) .In general choose a smooth presentation p ∶ X → M . If λ f is trivial, we can lift the morphism f ∣ ∶ [ / G m ] → Spec k → M to ˜ f ∶ [ / G m ] → X . Since p is smooth, we can inductively lift thismorphism to obtain an element in lim ←— X ([ Spec ( k [ x ]/ x n )/ G m ]) . Thus we reduced (1) to the case M = X . ◻ Using this lemma, we can compare L -stability to classical notions of stability, simply by first iden-tifying objects for which the automorphism group contains G m and then studying their deformations.The next subsections illustrate this procedure in examples. Let X be a projective variety equipped with the action of a reductive group G and a G -linearized linebundle L . Again bundles on the quotient stack [ X / G ] are the same as equivariant bundles on X , sowe will alternatively view L as a line bundle on [ X / G ] . . Heinloth, GIT-stability for stacks 7J. Heinloth, GIT-stability for stacks 7 Let us fix some standard notation: Given λ ∶ G m → G we will denote by P λ the correspondingparabolic subgroup, U λ its unipotent radical and L λ the corresponding Levi subgroup, i.e., P λ ( R ) = { g ∈ G ( R )∣ lim t → λ ( t ) gλ ( t − ) exists } ,U λ ( R ) = { g ∈ G ( R )∣ lim t → λ ( t ) gλ ( t − ) = } ,L λ ( R ) = { g ∈ G ( R )∣ λ ( t ) gλ ( t − ) = g } i.e., L λ = Centr G ( λ ) . To compare L -stability on [ X / G ] to GIT-stability on X we first observe that the test objectsappearing in the conditions coincide: Lemma 1.6.
For any cocharacter λ ∶ G m → G and any geometric point x ∈ X ( K ) that is not a fixedpoint of λ the equivariant map f λ,x ∶ A K → X defines a very close degeneration f λ,x ∶ [ A K / G m,K ] →[ X / G ] . Moreover, any very close degeneration in the stack [ X / G ] is of the form f λ,x for some x, λ .Proof. Since f λ,x ( ) is a fixed point of λ and x is not, we have f ( ) /≅ f ( ) , thus f is a very closedegeneration.Conversely let f ∶ [ A / G m ] → [ X / G ] be any very close degeneration. We need to find a G m equivariant morphism A (cid:15) (cid:15) ˜ f (cid:47) (cid:47) X π (cid:15) (cid:15) [ A / G m ] f (cid:47) (cid:47) [ X / G ] . Since π ∶ X → [ X / G ] is a G -bundle, the pull-back p ∶ X × [ X / G ] [ A / G m ] → [ A / G m ] is a G -bundle on [ A / G m ] . To find ˜ f is equivalent to finding a G m -equivariant section of this bundle. This will followfrom the known classification of G m -equivariant G -bundles on the affine line: Lemma 1.7. ( G -bundles on [ A / G m ] ) Let G be a reductive group and P a G -bundle on [ A / G m ] .Denote by P the fiber of P over ∈ A . (1) If there exists x ∈ P ( k ) (e.g. this holds if k = k ). Then there exists a cocharacter λ ∶ G m → G ,unique up to conjugation and an isomorphism of G -bundles P ≅ [( A × G )/( G m , ( act , λ ))] . Moreover, P has a canonical reduction to P λ . (2) Let G ∶= Aut G ( P ) and λ ∶ G m → G the cocharacter defined by P∣ [ / G m ] . Consider the G bundle P G ∶= Isom G (P , P ) on [ A / G m ] . Then P G ≅ [( A × G )/( G m , ( act , λ ))] , i.e. P ≅
Isom G ([( A × G )/( G m , ( act , λ ))] , P ) . Moreover, P G has a canonical reduction to P ,λ ⊂ G . For vector bundles this result is [6, Theorem 1.1] where some history is given. The general case canbe deduced from this using the Tannaka formalism. As we will need a variant of the statement later,we give a slightly different argument.
Proof of Lemma 1.7.
The second part follows from the first, as the G -bundle P G = Isom G (P , P ) has a canonical point id. We added (2), because it gives an intrinsic statement, independent of choices.To prove (1) note that x defines an isomorphism G ≅ —→ Aut G ( P ) and a section [ Spec k / G m ] →P∣ [ / G m ] . This induces a section [ Spec k / G m ] → P/ P λ . As the map π ∶ P/ P λ → [ A / G m ] is smooth, any
1. The Hilbert-Mumford criterion in terms of stacks
1. The Hilbert-Mumford criterion in terms of stacks section can be lifted infinitesimally to Spec k [ x ]/ x n for all n ≥
0. Inductively the obstruction to theexistence of a G m -equivariant section is an element in H ( B G m , T (P/ P λ )/ A , ⊗( x n − )/( x n )) = H ( B G m , T P/ P λ / A , ⊗ ( x n − )/( x n )) . Now by construction G m acts with negative weight on T P/ P λ , = Lie ( G )/ Lie ( P λ ) and it also acts with negative weight onthe cotangent space ( x )/( x ) , so there exists a canonical G m equivariant reduction P λ of P to P λ .Similarly, the vanishing of H implies that we can also find a compatible family of λ -equivariantsections [( Spec k [ t ]/ t n )/ G m ] → P and by Lemma 1.5 this defines a section over [ A / G m ] , i.e. amorphism of G -bundles [( A × G )/ G m , λ ] → P .Here, we could alternatively see this explicitly as follows: Let us consider π ∶ P λ → A as G m equivariant P λ bundle on A and consider the twisted action ⋆∶ G m × P λ → P λ given by t ⋆ p ∶= ( t.p ) ⋅ λ ( t − ) . Our point x is a fixed point for this action by construction, as we usedit to identify P with G and e ∈ G is a fixed point for the conjugation λ ( t ) ⋅ ⋅ λ ( t − ) . Moreover λ actswith non-negative weights on Lie ( P λ ) and also on T A = (( x )/( x )) ∨ . Therefore Bia(cid:32)lynicki-Biruladecomposition [30] implies that there exists a point x ∈ π − ( ) such that lim t → t ⋆ x = x . Thisdefines a λ -equivariant section A → P . ◻ This also completes the proof of Lemma 1.6. ◻ Proposition 1.8.
Let
X, G, L be a projective G -scheme together with a G -linearized line bundle L .A point x ∈ X ( k ) is GIT-stable with respect to L if and only if the induced point x ∈ [ X / G ]( k ) is L -stable.Proof. As X is projective, given x ∈ X ( k ) and a one parameter subgroup λ ∶ G m → G we obtainan equivariant map f λ,x ∶ A → X and thus a morphism f ∶ [ A / G m ] → X . By Lemma 1.6 all veryclose degenerations arise in this way. As wt ( f ∗ L) = wt ( f ∗ L) we therefore find that x satisfies theHilbert-Mumford criterion for stability if and only if it is L -stable. ◻ We want to show how the classical notion of stability for G -bundles arises as L -stability. For the sakeof clarity we include the case of vector bundles first. Let C be a smooth, projective, geometricallyconnected curve over k and denote by Bun dn the stack of vector bundles of rank n and degree d on C . A natural line bundle on Bun dn is given by the determinant of cohomology L det , i.e., for any vectorbundle E on C we have L det , E ∶= det ( H ∗ ( C, E nd (E))) − , and more generally for any f ∶ T → Bun dn corresponding to family E on C × T one defines f ∗ L det ∶= ( det R pr T, ∗ E nd (E)) ∨ . Remark 1.9.
Since any vector bundle admits G m as central automorphisms, to apply our criterionwe need to pass to the rigidified stack Bun dn ∶= Bun dn (cid:40) G m obtained by dividing all automorphismgroups by G m ([1, Theorem 5.1.5]). To obtain a line bundle on this stack, we need a line bundleon Bun dn on which the central G m -automorphisms act trivially. This is the reason why we use thedeterminant of H ∗ ( C, E nd (E)) instead of H ∗ ( C, E) . It is known that up to multiples and bundlespulled back from the Picard variety this is the only such line bundle on Bun dn (see e.g. [11] which alsogives some history on the Picard group of Bun dn ). . Heinloth, GIT-stability for stacks 9J. Heinloth, GIT-stability for stacks 9 To classify maps f ∶ [ A / G m ] → Bun dn we have as in Lemma 1.7. Lemma 1.10.
There is an equivalence
Map ([ A / G m ] , Bun dn ) ≅ ⟨(E , E i ) i ∈ Z ∣E i ⊂ E i − ⊂ E , ∪E i = E , E i = for i ≫ ⟩ , i.e. giving a vector bundle, together with a weighted filtration is equivalent to giving a morphism [ A / G m ] → Bun dn .Proof. We give the reformulation to fix the signs: A vector bundle E on C together with a weightedfiltration E i ⊂ E i − ⊂ ⋅ ⋅ ⋅ ⊂ E the Rees construction Rees (E ● ) ∶= ⊕ i ∈ Z E i x − i defines an O C [ x ] module, i.e.a family on C × A which is G m equivariant for the action defined on the coordinate parameter withRees (E ● )∣ C × ≅ gr (E ● ) .For the converse we argue as in Lemma 1.7: Any morphism f ∶ [ A / G m ] → Bun dn defines a morphism G m → Aut
Bun n ( f ( )) , i.e., a grading on the bundle f ( ) such that the corresponding filtration liftscanonically to the family. ◻ Given a very close degeneration [ A / G m ] → Bun dn we can easily compute wt (L det ∣ f ( ) ) , as follows.We use the notation of the preceding lemma and write E i ∶= E i /(E i + ) so that f ( ) = ⊕E i . Note that G m acts with weight − i on E i . Denoting further µ (E i ) ∶= deg E i rk E i we findwt (L det ∣ f ( ) ) = − wt G m ( det H ∗ ( C, ⊕ H om (E i , E j ))) , because L det was defined to be the dual of the determinant of cohomology. As G m acts with weight ( i − j ) on H om (E i , E j ) , it acts with the same weight on the cohomology groups, taking determinantswe find the weight ( i − j ) χ (H om (E i , E j )) on det H ∗ ( C, H om (E i , E j )) . Thus by Riemann-Roch we findwt (L det ∣ f ( ) ) = ∑ i,j ( j − i ) χ (H om (E i , E j ))= ∑ i,j ( j − i )( rk (E i ) deg (E j ) − rk (E j ) deg (E i ) + rk (E i ) rk (E j )( − g ))= ∑ i < j ( j − i )( rk (E i ) deg (E j ) − rk (E j ) deg (E i )) . As it is more common to express the condition in terms of the subbundles E l instead of the subquotients E i let us replace the factors ( j − i ) by a summation over l with i ≤ l < j :wt (L det ∣ f ( ) ) = ∑ l (∑ i ≤ l rk (E i ))(∑ j > l deg (E j )) − (∑ i ≤ l deg (E i ))(∑ j > l rk (E j ))= ∑ l rk (E l )( n − rk (E l ))( µ (E l ) − µ (E/E l )) . This is < µ (E i ) > µ (E/E i ) for some i . Conversely, if µ (E i ) > µ (E/E i ) for some i , then thetwo step filtration 0 ⊂ E i ⊂ E defines a very close degeneration of positive weight. Thus we find theclassical condition: Lemma 1.11.
A vector bundle E is L det -stable if and only if for all E ′ ⊂ E we have µ (E ′ ) < µ (E) .
1. The Hilbert-Mumford criterion in terms of stacks
1. The Hilbert-Mumford criterion in terms of stacks
Let us formulate the analog for G -bundles, where G is a semisimple group over k and we assume k = k to be algebraically closed. We denote by Bun G the stack of G -bundles on C . For the stability condition we need a line bundle on Bun G . One way to construct a positive linebundle is to choose the adjoint representation Ad ∶ G → GL ( Lie ( G )) , which defines for any G -bundle P its adjoint bundle Ad (P) ∶= P × G Lie ( G ) and set: L det ∣ P ∶= det H ∗ ( C, Ad (P)) ∨ . If G is simple and simply connected, it is known that Pic ( Bun G ) ≅ Z (e.g. [11]). In general L det willnot generate the Picard group, but since our stability condition does not change if we replace L by amultiple of the bundle this line bundle will suffice for us. G -bundles Recall from section 1.D that for a cocharacter λ ∶ G m → G we denote by P λ , U λ , L λ the correspondingparabolic subgroup, its unipotent radical and the Levi subgroup.To understand very close degenerations of bundles will amount to the observation that Lemma1.7 has an extension that holds for families of bundles.The source of degenerations is the following analog of the Rees construction. Given λ ∶ G m → G we obtain a homomorphism of group schemes over G m :conj λ ∶ P λ × G m → P λ × G m ( p, t ) ↦ ( λ ( t ) pλ ( t ) − , t ) . By [30, Proposition 4.2] this homomorphism extends to a morphism of group schemes over A :gr λ ∶ P λ × A → P λ × A in such a way that gr λ ( p, ) = lim t → λ ( t ) pλ ( t ) − ∈ L λ × G m equivariant with respect to the action ( conj λ , act ) on P λ × A .Given a P λ bundle E λ on a scheme X this morphism defines a P λ bundle on X × [ A / G m ] by:Rees (E λ , λ ) ∶= [((E λ × A ) × gr λ A ( P λ × A ))/ G m ] , where × gr λ A denotes the bundle induced via the morphism gr λ , i.e., we take the product over A anddivide by the diagonal action of the group scheme P λ × A / A , which acts on the right factor via gr λ .By construction this bundle satisfies Rees (E λ , λ )∣ X × ≅ E λ andRees (E λ , λ )∣ X × ≅ E λ / U λ × L λ P λ is the analog of the associated graded bundle. Remark 1.12. If λ ′ ∶ G m → P λ is conjugate to λ in P λ , say by an element u ∈ U λ then P ∶= P λ = P ′ λ and gr λ ′ ( p, t ) = gr λ ( upu − , t ) . Therefore we also have Rees (E λ , λ ) ≅ Rees (E λ , λ ′ ) , which tells us that the Rees construction only depends on the reduction to P and the homomorphism λ ∶ G m → Z ( P / U ) ⊂ P / U. In the case G = GL ( V ) this datum is the analog of a weighted filtration on V , whereas λ ∶ G m → P ⊂ GL ( V ) would define a grading on V . . Heinloth, GIT-stability for stacks 11J. Heinloth, GIT-stability for stacks 11 Given a G -bundle E , a cocharacter λ and a reduction E λ of E to a P λ bundle the G -bundleRees (E λ , λ ) × P λ G defines a morphism f ∶ [ A / G m ] → Bun G with f ( ) = E . We claim that all very closedegenerations arise in this way: Lemma 1.13.
Let G be a split reductive group over k . Given a very close degeneration f ∶ [ A / G m ] → Bun G corresponding to a family E of G -bundles on X × [ A / G m ] there exist: (1) a cocharacter λ ∶ G m → G , canonical up to conjugation, (2) a reduction E λ of the bundle E to P λ , (3) an isomorphism E λ ≅ Rees (E λ ∣ X × , λ ) .Proof. Given E we will again denote by E ∶= E∣ X × and E ∶= E∣ X × . We define the group scheme G E ∶= Aut G (P / X ) = E × G, conj G . This is a group scheme over X that is an inner form of G × X .And the morphism f ∣ [ / G m ] induces a morphism λ ∶ G m × X → Aut G (E / X ) = G E .As in Lemma 1.7 (2) it is convenient to replace E by the G E -torsor E ′ ∶= Isom G (E , E ) . We knowthat λ defines a parabolic subgroup P λ ⊂ G E and the canonical reduction of E ′ to P λ lifts uniquelyto a reduction E ′ λ of E ′ by the same argument used in Lemma 1.7. The last step of the proof is thento consder the twisted action ⋆∶ G m × E ′ λ → E ′ λ . Note that the fixed points for the action are simplythe points in the Levi subgroup L λ ⊂ E ′ = G E .The needed analog of the Bia(cid:32)lynicki-Birula decomposition is a result of Hesselinck [30]: By thelemma we already know that for all geometric points x of X and p ∈ E ′ λ ∣ x all limit points lim t → λ ( t )⋆ p exist. On the other hand by [30, Proposition 4.2] the functor whose S -points are morphisms S × A →E ′ λ such that the restriction to S × G m is given by the action of G m on E ′ λ is represented by a closedsubscheme of E ′ λ . Thus the functor is represented by E ′ λ .Thus the twisted action ⋆ on E ′ λ extends to a morphism ⋆∶ A × E ′ λ → E ′ λ . In particular this induces A × E λ, → E λ and thus a morphism of P λ -bundles Rees (E λ, , λ ) → E λ .This proves that the statement of the Lemma holds if we replace G by G E .To compare this with the description given in the lemma note that for any geometric point x ∈ X ( k ) the choice of a trivialization E ,x ≅ G defines an isomorphism G P ≅ G and therefore λ ∣ x defines adefines a conjugacy class of cocharacters λ ∶ G m → G . This conjugacy class is locally constant (andtherefore does not depend on the choice of x ) because we know from [40, Expos´e XI, Corollary 5.2bis]that the scheme parametrizing conjugation of cocharacters Transp G ( λ , λ ) is smooth over X . (Thisis the analog of the statement for vector bundles, that a G m action on E allows to decompose E = ⊕E i as bundles, i.e. the dimension of the weight spaces of the fibers is constant over x .) This defines λ .To conclude we only need to recall that reductions of E ′ to P λ correspond to reductions of E to P λ : Lemma 1.14.
Let
G → X be a reductive group scheme, λ ∶ G m,X → G a cocharacter and E a G -torsorover X . Then a natural bijection between: (1) Reductions E λ of E to P λ , (2) Parabolic subgroups
P ⊂ G E = Aut G (E/ X ) that are locally conjugate to P λ ,is given by E λ ↦ P ∶= Aut P λ (E λ / X ) ⊂ Aut G (E/ X ) .Proof of Lemma 1.14. A reduction of E is a section s ∶ X → E/ P λ . Note that G E acts on E/ P λ andStab G E ( s ) ⊂ G E is a parabolic subgroup that is locally of the same type as P λ , becasuse this holds if E is trivial and s lifts to a section of E . Locally in the smooth topology we may assume these conditions.Similarly given P ⊂ G E locally the action of P on E/ P λ has a unique fixed point and this definesa section. ◻
1. The Hilbert-Mumford criterion in terms of stacks
1. The Hilbert-Mumford criterion in terms of stacks
Using the lemma we find that both sections of E ′ /P λ and sections of E/ P λ correspond to parabolicsubgroups of Aut G (E) = Aut G E (E ′ ) . This proves the lemma. ◻ Remark 1.15.
Note that in the above result we assumed that G is a split group. In general, wesaw that the natural subgroup that contains a cocharacter is Aut G (E ) , which is an inner form of G over X . In particular it may well happen that G does not admit any cocharacter or any parabolicsubgroup.This is apparent for example in the case k = R and G = U ( n ) . For a G -torsor E on C we will finda canonical reduction to a parabolic subgroup P k ⊂ G k but the descent datum will then only be givenfor P ⊂
Aut G (E/ X ) . Finally we have to compute the weight of L det on very close degenerations.The computation is the same as for vector bundles and for the criterion it is sometimes convenientto reduce it to reductions for maximal parabolic subgroups. Let us choose T ⊂ B ⊂ G a maximaltorus and a Borel subgroup and λ ∶ G m → G a dominant cocharacter, i.e. ⟨ λ, α ⟩ ≥ g α ∈ Lie ( B ) . Let us denote by I the set of positive simple roots with respect to ( T, B ) and by I P ∶= { α i ∈ I ∣ λ ( α i ) = } the simple roots α i for which − α i is also a root of P λ . For j ∈ I let us denoteby ˇ ω j ∈ X ∗ ( T ) R the cocharacter defined by ˇ ω j ( α i ) = δ ij . And by P j the corresponding maximalparabolic subgroup.Then λ ∶ G m → Z ( L λ ) ⊂ L λ ⊂ P λ . Thus for any very close degeneration f ∶ [ A / G m ] → Bun G givenby Rees (E λ , λ ) the bundle L det defines a morphismwt L ∶ X ∗ ( Z λ ) ⊂ Aut
Bun G ( f ( )) → Z . Then λ = ∑ j ∈ I − I P a j ˇ ω j for some a j > (L det ∣ f ( ) ) = wt L ( λ ) = ∑ j ∈ I − I P a j wt L ( ˇ ω j ) . For each j we get a decomposition Lie ( G ) = ⊕ i Lie ( G ) i , where Lie ( G ) i is the subspace of the Liealgebra on which ˇ ω j acts with weight i . Each of these spaces is a representation of L λ and also of theLevi subgroups L j of P j . Using this decomposition we find as in the case of vector bundles:wt L ( ˇ ω j ) = − wt G m ( det H ∗ ( C, ⊕E ,λ × L λ Lie ( G ) i ))= ∑ i i ⋅ χ (E ,λ × L λ Lie ( G ) i )= ∑ i i ( deg (E ,λ × L λ Lie ( G ) i ) + dim ( Lie ( G ) i )( − g ))= ∑ i > i ( deg (E ,λ × L λ Lie ( G ) i ) Now deg (E ,λ × L λ Lie ( G ) i ) = ( deg ( det (E ,λ × L λ Lie ( G ) i )) . Since the Levi subgroups of maximalparabolics have only a one dimensional space of characters, all of these degrees are positive multiplesof det ( Lie ( P j )) . Thus we find the classical stability criterion: Corollary 1.16. A G -bundle E is L det -stable if and only if for all reductions E P to maximal parabolicsubgroups P ⊂ G we have deg (E P × P Lie ( P )) < . . Heinloth, GIT-stability for stacks 13J. Heinloth, GIT-stability for stacks 13 Parabolic G -bundles are G -bundles equipped with a reduction of structure group at a finite set ofclosed points.Let us fix notation for these. We keep our reductive group G , the curve C and a finite set ofrational points { x , . . . , x n } ⊂ C ( k ) and parabolic subgroups P , . . . , P n ⊂ G .Bun G,P ,x ( S ) ∶= ⟨(E , s , . . . , s n )∣E ∈ Bun G ( S ) , s i ∶ S → E x i × S / P i sections ⟩ The forgetful map Bun
G,P ,x → Bun G is a smooth proper morphism with fibers isomorphic to ∏ i G / P i .In particular, very close degenerations of a parabolic G bundle are uniquely defined by a veryclose degenerations of the underlying G -bundle.There are more line bundles on Bun G,P ,x , namely any dominant character of χ i ∶ P i → G m definesa positive line bundle on G / P i and this induces a line bundle on Bun G,P ,x .The weight of a this line bundle on a very close degeneration, is given by the pairing of χ i withthe one parameter subgroup in Aut P i (E x i ) .We will come back to this in the section on parahoric bundles (Section 3.E). We briefly include the example of chains of bundles as an easy example of a stability condition thatdepends on a parameter.Again we fix a curve C . A holomorphic chain of length r and rank n ∈ N r + is the datum (E i , φ i ) where E , . . . , E r are vector bundles of rank n i and φ i ∶ E i → E i − are morphisms of O C -modules. Thestack of chains is denoted Chain n . It is an algebraic stack, locally of finite type. One way to seethis is to show that the forgetful map Chain n → ∏ ri = Bun n i is representable. As for the stack Bun dn all chains admit scalar automorphisms G m , so we will need to look for line bundles on which theseautomorphisms act trivially.The forgetful map to ∏ ri = Bun n i already gives a many line bundles on Chain n , as we can takeproducts of the pull backs of the line bundles L det on the stacks Bun n i . Somewhat surprisingly theseare only used in [39], whereas the standard stability conditions (e.g., [5]) arise from the followingbundles:(1) L det ∶= det ( H ∗ ( C, End (⊕E i ))) ∨ (2) Fix any point x ∈ C and i = , . . . r . Set L i ∶= det ( Hom (⊕ j ≥ i E j,x , ⊕ l < i E l,x )) . Remark 1.17.
Note that on all of these bundles the central automorphism group G m of a chain actstrivially and one can check that up to the multiple [ k ( x ) ∶ k ] the Chern classes of the bundles L i donot depend on x . We will not use this fact.The choice of the bundles L i ∈ Pic ( Chain n ) is made to simplify our computations. From a moreconceptual point of view the lines L n i ∶= det (E i,x ) define bundles on Bun n i which are of weight n i withrespect to the central automorphism group G m . The pull backs of these bundles generate a subgroupof Pic ( Chain n ) and the L i are a basis for the bundles of weight 0 in this subgroup.To classify maps [ A / G m ] → Chain n note that composing with forget ∶ Chain n → ∏ Bun n i such amorphism induces morphisms [ A / G m ] → Bun n i , which we already know to correspond to weightedfiltrations of the bundles E i and a lifting of a morphism [ A / G m ] → ∏ Bun n i to Chain n is given byhomomorphisms φ i ∶ E i → E i − that respect the filtration.
2. A criterion for separatedness of the stable locus
2. A criterion for separatedness of the stable locus
Thus we find that a very close degeneration of a chain E ● is a weighted filtration E i ● ⊂ E ● and wealready computed wt (L det ∣ gr (E ● ) ) = ∑ i rk (E i ● )( n − rk (E i ● ))( µ (E ● /E i ● ) − µ (E i ● ))= ∑ i rk (E i ● ) n ( µ (E ● ) − µ (E i ● )) . Further we have: wt (L j ∣ gr (E ● ) ) = ∑ i (∑ l ≥ j rk (E il ) n ) − n i (∑ l ≥ j rk (E i )) . Thus we find that a chain E ● is L det ⊗ L m i i -stable if and only if for all subchains E ′● ⊂ E ● we have ∑ i deg (E ′ i ) + ∑ j m j rk (⊕ l ≥ j E ′ l )∑ i rk (E ′ i ) < ∑ i deg (E i ) + ∑ j m j rk (⊕ l ≥ j E l )∑ i rk (E i ) . This is equivalent to the notion of α − stability used in [5, Section 2.1]. Remark 1.18.
Also for the moduli problem of coherent systems on C , i.e. pairs (E , V ) where E is a vector bundle of rank n on C and V ⊂ H ( C, E) is a subspace of dimension r one recovers thestability condition quite easily: Families over S are pairs (E , V , φ ∶ V ⊗ O C → E) where E is a family ofvector bundles on X × S , V is a vector bundle on S and we drop the condition that φ corresponds toan injective map V → pr S, ∗ E . We denote this stack by CohSys n,r . There are natural forgetful mapsCohSys n,r → Bun n and CohSys n,r → B GL n induced respectively by the bundles E and V . As abovefor any point x ∈ X we obtain a line bundle det (V) n ⊗ det (E x ) − r on CohSys n,r and together with L det one then recovers the classical stabilty condition that one finds for example in [15, Definition 2.2]. Other, more advanced examples can be found in the article [22, Section 4.2]. For example thiscontains an argument, how the Futaki invariant introduced by Donaldson arises from the point ofview of algebraic stacks.
2. A criterion for separatedness of the stable locus
We now want to give a criterion which guarantees that the set of L -stable points is a separatedsubstack if the stack M and the line bundle L satisfy suitable local conditions (Proposition 2.6). Thearticle [35] by Martens and Thaddeus was an important help to find the criterion. Again, the proofsturn out to be quite close to arguments that already appear in Mumford’s book. Let us first sketch the basic idea. Let M be an algebraic stack. For the valuative criterion forseparatedness one considers pairs f, g ∶ D = Spec k [[ t ]] → M together with an isomorphism f ∣ Spec k (( t )) ≅ g ∣ Spec k (( t )) and tries to prove that for such pairs f ≅ g . The basic datum is therefore a morphism f ∪ g ∶ D ∪ ˚ D D → M . Note that the scheme D ∪ ˚ D D is a completed neighborhood of the origins in the affine line with doubledorigin A ∪ G m A ≅ [ A − { }/ G m , ( t, t − )] ⊂ [ A / G m ] . . Heinloth, GIT-stability for stacks 15J. Heinloth, GIT-stability for stacks 15 Thus also the union of two copies of D along their generic point is naturally an open subschemeof a larger stack: D ∪ ˚ D D ⊂ [( A × A D )/ G m ] = [ Spec ( k [ x, y ] ⊗ k [ π ] ,π = xy k [[ π ]])/ G m , ( t, t − )] , where the right hand side inserts a single point [ / G m ] .Further, the coordinate axes Spec k [ x ] , Spec k [ y ] ⊂ Spec k [ x, y ] = A define closed embeddings [ A / G m ] → [( A × A D )/ G m ] that intersect in the origin [ / G m ] . As the weights of the G m action onthe two axes are 1 and − L on this stack the weights of therestriction to the two copies of [ A / G m ] differ by a sign.In terms of L -stability on M this means that whenever a morphism f, g ∶ D ∪ ˚ D D → M extends to [( A × A D )/ G m ] , only one of the two origins can map to an L -stable point.As the complement of the origin [ / G m ] is of codimension 2 in the above stack one could expectthat a morphism f ∪ g ∶ D ∪ ˚ D D → M extends to a morphism of some blow up of [( A × A D )/ G m ] centered at the origin. This will be the assumption that we will impose on M .The basic observation then is that the exceptional fiber of such a blow up can be chosen to be achain of equivariant projective lines and it will turn out that the weight argument indicated abovestill works for such chains if the line bundle L satisfies a numerical positivity condition. Let R be a discrete valuation ring together with a local parameter π ∈ R , K ∶= R [ π − ] the fractionfield and k = R /( π ) the residue field.As for the affine line, the scheme Spec R has a version with a doubled special point ST R ∶= Spec R ∪ Spec K Spec R , the test scheme for separateness. The analog of [ A / G m ] is given as follows.The multiplicative group G m acts on R [ x, y ]/( xy − π ) via t.x ∶= tx, t.y ∶= t − y . Let us denoteST R ∶= [ Spec ( R [ x, y ]/( xy − π ))/ G m ] . As before we have:(1) two open embeddings j x ∶ Spec R (cid:31) (cid:127) ○ (cid:47) (cid:47) ≅ (cid:15) (cid:15) ST R [ Spec R [ x, x − ]/ G m ] ≅ (cid:47) (cid:47) [ Spec R [ x, x − , y ]/( xy − π )/ G m ] (cid:63)(cid:31) ○ (cid:79) (cid:79) j y ∶ Spec R (cid:31) (cid:127) (cid:47) (cid:47) ≅ (cid:15) (cid:15) ST R [ Spec R [ y, y − ]/ G m ] ≅ (cid:47) (cid:47) [ Spec R [ x, y, y − ]/( xy − π )/ G m ] (cid:63)(cid:31) ○ (cid:79) (cid:79) that coincide on Spec K .(2) two closed embeddings: i x ∶ [ A k / G m ] ≅ [ Spec k [ x ]/ G m ] ≅ [ Spec R [ x, y ]/( y, xy − π )/ G m ] ⊂ ST R i y ∶ [ A k / G m ] ≅ [ Spec k [ y ]/ G m ] ≅ [ Spec R [ x, y ]/( x, xy − π )/ G m ] ⊂ ST R and the intersection of these is [ Spec k / G m ] =∶ [ / G m ] .
2. A criterion for separatedness of the stable locus
2. A criterion for separatedness of the stable locus
We will need to understand blow ups of ST R supported in [ Spec k / G m ] . For this, let us introducesome notation. A chain of projective lines is a scheme E = E ∪ ⋅ ⋅ ⋅ ∪ E n where E i ⊂ E are closedsubschemes, together with isomorphisms φ i ∶ E i ≅ —→ P such that E i ∩ E i + = { x i } is a reduced pointwith φ i ( x i ) = ∞ , φ i + ( x i ) =
0. An equivariant chain of projective lines is a chain of projective linestogether with an action of G m such that for each i the action induces the standard action of someweight w i on P = Proj k [ x, y ] , i.e. this is given by t.x = t w i + d x , t.y = t d y for some d .We say that an equivariant chain is of negative weight, if all the w i are negative. In this case forall i the points φ − i ( ) are the repellent fixed points in E i for t → Lemma 2.1.
Let I ⊂ R [ x, y ]/( xy − π ) be a G m invariant ideal supported in the origin Spec k = Spec R [ x, y ]/( x, y, π ) . Then there exists an invariant ideal ˜ I and a blow up p ∶ Bl ˜ I ( ST R ) = [ Bl ˜ I ( Spec R [ x, y ]/( xy − π ))/ G m ] → ST R dominating Bl I such that p − ([ / G m ]) ≅ [ E / G m ] where E is a chain of projective lines of negativeweight.Proof. As I is supported in ( x, y ) there exists n such that ( x, y ) n ⊂ I and as it is G m invariant,it is homogeneous with respect to the grading for which x has weight 1 and y has weight −
1. Let P ( x, y ) = ∑ Ni = l a i x d + i y i be a homogeneous generator of weight d ≥ a l ≠ a i ∈ R ∗ as π = xy . We claim that then x d + l y l ∈ I . As ( x, y ) n ⊂ I we may assumethat d + N < n . But then P ( x, y ) − ( a l + / a l ) xyP ( x, y ) ∈ I is an element for which the coefficient of x d + l + y l + vanishes. Inductively this shows that x d + l y l ∈ I , so that I is monomial.Write I = ( x n , y m , x n i y m i ) i = ,...N with n i < n, m i < m . This ideal becomes principal after sucessivelyblowing up 0 and then blowing up 0 or ∞ in the exceptional P ’s: Blowing up ( x, y ) we get chartswith coordinates ( x, y ) ↦ ( x ′ y, y ) and ( x, y ) ↦ ( x, xy ′ ) . Since x has weight 1 and y has weight − ( x ′ , y ) are ( , − ) and the weights of ( x, y ′ ) are ( , − ) .In the first chart the proper transform of I is ( x ′ n y n , y m , x ′ n i y m i + n i ) i = ,...N . This ideal is principalif m = y k ( y m − k , mixed monomials of lower total degree ) . A similar computation works in the other chart. By induction this shows that the ideal will becomeprincipal after finitely many blow ups and that in each chart the coordinates ( x ( i ) , y ( i ) ) have weights ( w i , v i ) with w i > v i . ◻ Remark 2.2.
Let L be a line bundle on [ P /( G m , act w )] thendeg (L∣ P ) = w ( wt G m (L∣ ∞ ) − wt G m (L∣ )) . Proof.
Write d = deg (L∣ P ) . Let A = Spec k [ x ] , A ∞ = Spec k [ y ] be the two coordinate charts of P and e x ∈ L( A ) , e y ∈ L( A ∞ ) two generating sections such that e y = x d e x on Spec k [ x, x − ] .Then act ( e y ) = t wt G m (L∣ ∞ ) e y and act ( x d e x ) = t wd t wt G m (L∣ ) x d e x . Thus we find t wd = t wt G m (L∣ ∞ )− wt G m (L∣ ) . ◻ Combining the above computations we propose the following definitions:
Definition 2.3.
Let M be an algebraic stack, locally of finite type with affine diagonal. We say that M is almost proper if . Heinloth, GIT-stability for stacks 17J. Heinloth, GIT-stability for stacks 17 (1) For all valuation rings R with field of fractions K and f K ∶ Spec K → M there exists a finiteextension R ′ / R and a morphism f ∶ Spec R ′ → M such that f ∣ Spec K ′ ≅ f K ∣ Spec K ′ and(2) for all complete discrete valuation rings R with algebraically closed residue field and all mor-phisms f ∶ ST R → M there exists a blow up Bl ˜ I ( ST R ) supported at 0 such that f extends to amorphism f ∶ Bl ˜ I ( ST R ) → M .Given a line bundle L on an almost proper algebraic stack M we say that L is nef on exceptionallines iffor all f ∶ ST R → M the extension f from (2) can be chosen such that for all equivariant projectivelines E i in the exceptional fiber of the blow up we have deg (L∣ E i ) ≥ . Remark 2.4.
Note that for schemes (and also algebraic spaces) of finite type the above definitionreduces to the usual valuative criterion for properness: We already saw in Lemma 1.5 that in case X = M is a scheme, any morphism from an equivariant projective line to X must be constant andmore generally if G m acts on Spec R [ x, y ]/( xy − π )) with positive weight on x and negative weight on y the morphism [( Spec R [ x, y ]/( xy − π ))/ G m ] → Spec R is a good coarse moduli space ([3]), so for anyalgebraic space X any morphism Bl ˜ I ( ST R ) → X factors through Spec ( R ) . Therefore the existence ofa morphism as in (2) implies the valuative criterion for properness ([42, Tag 0ARL]).Similarly (2) above could be used to define a notion of almost separatedness for stacks. Remark 2.5.
In the above definition we could have replaced condition (2) by the condition:(2’) For all discrete valuation rings R and all morphisms f ∶ ST R → M there exists a finite extension R ′ / R and a blow up Bl ˜ I ( ST R ′ ) supported at 0 such that f extends to a morphism f ∶ Bl ˜ I ( ST R ′ ) →M .These two conditions are equivalent for stacks locally of finite type, we put (2) because it is sometimesslightly more convenient to check. Proof.
We only need to show that (2) implies (2’). The standard argument for schemes can be adaptedhere with some extra care taking into account automorphisms of objects: Note first that because M is locally of finite type it suffices to prove (2’) in the case where the DVR R has an algebraicallyclosed residue field. In this case we denote by ˆ R the completion of R and by f ˆ R the restriction of f to ST ˆ R . This map has an extension f ˆ R ∶ Bl I ( ST ˆ R ) → M and we have seen in Lemma 2.1 that we mayassume that I is already defined over R , i.e., that Bl I ( ST ˆ R ) = Bl I ( ST R ) ˆ R .As the extension ˆ R / R is faithfully flat, to define an extension f ∶ Bl I ( ST R ) → M of f we need todefine a descent datum for ˆ f , i.e. for the two projections p , ∶ Bl I ( ST R ) ˆ R ⊗ R ˆ R → Bl I ( ST R ) ˆ R we needan element φ ∈ Isom M ( p ∗ ( ˆ f ) , p ∗ ( ˆ f )) that over ST ˆ R ⊗ R ˆ R coincides with the one given by f .Now note that if we denote by ˆ K the fraction field of ˆ R there is a cartesian diagram of rings (seee.g., [27, Lemma 5]) ˆ R ⊗ R ˆ R (cid:47) (cid:47) mult (cid:15) (cid:15) ˆ K ⊗ R ˆ K mult (cid:15) (cid:15) ˆ R (cid:47) (cid:47) ˆ K. Also by our assumptions on the diagonal of M we know that Isom ( p ∗ ˆ f , p ∗ ˆ f ) → Bl I ( ST R ) ˆ R ⊗ R ˆ R is affineand we have canonical sections of this morphism over the diagonal ∆ ∶ Bl I ( ST ˆ R ) → Bl I ( ST R ) ˆ R ⊗ R ˆ R and over Bl I ( ST R ) ˆ K ⊗ R ˆ K = ST ˆ K ⊗ R ˆ K we have the section defined by the descent datum for f . Asthese agree on the intersections ∆ ∶ Spec ˆ K → Spec ˆ K ⊗ R ˆ K the cartesian diagram implies that thesedefine φ . This map is a cocycle, because this holds over the open subscheme ST R . ◻
2. A criterion for separatedness of the stable locus
2. A criterion for separatedness of the stable locus
Recall from Remark 1.3 that L -stable points define an abstract substack M s ⊂ M , for these pointsDefinition 2.3 implies the valuative criterion: Proposition 2.6. (Separatedness of stable points)
Let M be an algebraic stack locally of finitetype over k with affine diagonal and L a line bundle on M . Suppose that M is almost proper and L isnef on exceptional lines. Then the stack of stable points M s ⊆ M satisfies the valuative criterion forseparatedness, i.e., for any complete discrete valuation ring R with fraction field K and algebraicallyclosed residue field any morphism ST R → M s factors through Spec R .Proof. We need to show that for any morphism f ∶ ST R → M such that all points in the image of f are stable we have j x ○ f ≅ j y ○ f .Let us first show that the conditions imply that the morphisms coincide on closed points, i.e., f ( j x (( π ))) ≅ f ( j y (( π ))) ∈ M .If f extends to a morphism f ∶ ST R → M , we have wt ( i ∗ x f ∗ L) = − wt ( i ∗ y f ∗ (L)) . As we assumed thatboth closed points are stable this shows that neither i x ○ f nor i y ○ f can be a very close degeneration,i.e. f ( j x (( π ))) ≅ f ( i x ( )) = f ( i y ( )) ≅ f ( j y (( π ))) .If f does not extend, then by assumption there exists an extension f ∶ Bl I ( ST R ) → M such thatdeg (L∣ E i ) ≥ P ’s contained in the exceptional fiber of the blow up.Now for line bundles L ′ on [ P / G m , act d ] we saw that deg (L ′ ) = d ( wt G m (L∣ ∞ ) − wt G m (L∣ )) .Thus if we order the fixed points x , . . . , x n of the G m -action on the chain E i such that x is thepoint in the proper transform of the x − axis and x i , x i + correspond to 0 , ∞ in E i we find that x n corresponds to the proper transform of the y -axis. Note that x being the repellent fixed point of E we have wt ( i ∗ x f ∗ L) = wt G m ( f ∗ L∣ x ) and similarly wt G m ( f ∗ L∣ x k ) = − wt ( i ∗ y f ∗ L) . Finally the conditiondeg (L∣ E i ) ≥ G m ( f ∗ L∣ x i ) ≥ wt G m ( f ∗ L∣ x i + ) for all i as the exceptional divisors are ofnegative weight. Thus we find:wt ( i ∗ x f ∗ L) = wt G m ( f ∗ L∣ x ) ≥ ⋅ ⋅ ⋅ ≥ wt G m ( f ∗ L∣ x n ) = − wt ( i ∗ y f ∗ L) . Now if f ( j x (( π ))) /≅ f ( x ) is stable, we know that wt G m ( f ∗ L∣ x ) < G m ( f ∗ L∣ x n ) =− wt ( i ∗ y f ∗ (L)) <
0, contradicting stability of f ( j y (( π ))) .Thus we find that deg (L∣ E i ) = i and f ( j x (( π ))) ≅ f ( x ) . Then f ( x ) is stable so that f ∣ E −{ x } must be constant and we inductively find that f ( x ) ≅ ⋅ ⋅ ⋅ ≅ f ( x n ) ≅ f ( j y (( π ))) and that f is constant on the exceptional divisor, so f does extend to ST.To conclude that this implies j x ○ f ≅ j y ○ f choose an affine scheme of finite type p ∶ Spec ( A ) → M such that p is smooth and f ( j x (( π ))) = f ( ) = f ( j y (( π ))) ∈ Im ( p ) . Choose moreover a lift x ∈ Spec ( A ) of f ([ / G m ]) . By smoothness we can inductively lift f ∣ [( Spec R [ x,y ]/( π n ,xy − π ))/ G m ] to a morphism˜ f n ∶ [( Spec R [ x, y ]/( π n , xy − π ))/ G m ] → Spec A . All of these maps factor through their coarse modulispace [( Spec R [ x, y ]/( π n , xy − π ))/ G m ] → Spec R /( π n ) → Spec A , defining a map Spec R → Spec A and thus ˜ f ∶ ST R → Spec R → Spec A that lifts both j x ○ f and j y ○ f , so these maps coincide. ◻ Remark 2.7.
For semistable points that are not stable the above computation also suggests to definea notion of S-equivalence, as it shows that in an almost proper stack any two semistable degenerationscould be joined by a chain of projective lines. If L is nef on exceptional lines the line bundle wouldrestrict to the trivial bundle on such a chain. In the examples this reproduces the usual notion ofS-equivalence.Before giving examples let us note that the Keel-Mori theorem now implies the existence of coarsemoduli spaces for M s in many situations: . Heinloth, GIT-stability for stacks 19J. Heinloth, GIT-stability for stacks 19 Proposition 2.8.
Let ( M , L ) be an almost proper algebraic stack with a line bundle L that is nefon exceptional lines and suppose that M s ⊂ M is open. Then the stack M s admits a coarse modulispace M s → M , where M is a separated algebraic space.Proof. By Proposition 2.6 we know that ∆ ∶ M s → M s × M s is proper and we assumed it to be affine,so it is finite. Therefore by the Keel-Mori theorem [32] [19, Theorem 1.1] the stack M s admits acoarse moduli space in the category of algebraic spaces. ◻ Proposition 2.9.
Let X be a proper scheme with an action of a reductive group G and let L be a G -linearized bundle that is numerically effective, then ([ X / G ] , L) is an almost proper stack and L isnef on exceptional lines.Proof. As any morphism Spec K → [ X / G ] can, after passing to a finite extension K ′ / K be lifted to X the stack [ X / G ] satisfies the first part of the valuative criterion.Let f ∶ ST R = Spec R ∪ Spec K Spec R → [ X / G ] be a morphism. Since R is complete with algebraicallyclosed residue field and X → [ X / G ] is smooth we can lift j x ○ f and j y ○ f to morphisms ˜ f x , ˜ f y ∶ Spec R → X . Now since the morphism f defines an isomorphim φ K ∶ f x ∣ K ≅ f y ∣ K there exists g K ∈ G ( K ) suchthat ˜ f y ∣ K = g K ˜ f x .Using the Cartan decomposition G ( K ) = G ( R ) T ( K ) G ( R ) we write g K = k y λ ( π ) k y with k y , k x ∈ G ( R ) and some cocharacter λ ∶ G m → G . Replacing ˜ f x , ˜ f y by k x ˜ f x , k − y ˜ f x respectively, we obtainmay assume that ˜ f y ∣ k = λ ( π ) ˜ f x , i.e. for this choice the isomorphism φ K is defined by the element λ ( π ) ∈ G ( K ) .This defines a ( G m , λ ) -equivariant morphism F ∶ Spec ( R [ x, y ]/( xy − π )) − { } → X that we candescribe explicitly by λ × f x ∶ G m,R = Spec ( R [ x, x − ]) → X and similarly by λ − × f y on Spec ( R [ y, y − ]) ,which glues because λ ( x ) f x = λ ( y − π ) f x = λ − ( y ) f y on the intersection.This morphism is a lift of f , i.e. fits into a commutative diagramSpec ( R [ x, y ]/( xy − π )) − { } F (cid:47) (cid:47) (cid:15) (cid:15) X (cid:15) (cid:15) [ Spec ( R [ x, y ]/( xy − π )) − { }/ G m ] = ST R f (cid:47) (cid:47) [ X / G ] , because taking the standard sections s x , s y ∶ Spec R → Spec ( R [ x, y ]/( xy − π )) − { } given by x = y = R with the quotient appearing in the above diagram is induced by themorphism of groupoids [ Spec K (cid:47) (cid:47) (cid:47) (cid:47) Spec R ∐ Spec R ] → [ Spec ( R [ x, y ]/( xy − π )) − { }/ G m ] given by π ∈ G m ( K ) , s x , s y and by construction F maps π to λ ( π ) = φ K .Since X is proper the morphism F extends after an equivariant blow up and since L is nef thenumerical condition will automatically be satisfied. ◻ G -bundles on curves Proposition 2.10.
Let G be a reductive group and C a smooth projective, geometrically connectedcurve. The stack Bun G is almost proper and the line bundle L det is nef on exceptional lines.
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves
Proof.
We follow the same strategy as for GIT-quotients, replacing the projective atlas X by theBeilinson-Drinfeld Grassmannian p ∶ GR G → Bun G , i.e.,GR G ( S ) = ⎧⎪⎪⎨⎪⎪⎩(E , D, φ ) RRRRRRRRRRR E ∈ Bun G ( S ) , D ∈ C ( d ) ( S ) for some dφ ∶ E∣ C × S − D ≅ —→ G × ( C × S − D ) ⎫⎪⎪⎬⎪⎪⎭ . It is known that GR G is the inductive limit of projective schemes, that L det defines a line bundle onGR G that is relatively ample with respect to the morphism to ∐ d C ( d ) and that the forgetful mapGR G → Bun G is formally smooth. Moreover this morphism admits sections locally in the flat topology([9, Section 5.3], [20]).To show that Bun G is almost proper take f ∶ ST R → Bun G . After extending k we may assume that R = k [[ π ]] . This defines bundles E x , E y on C × Spec R together with an isomorphism E x ∣ C × Spec K ≅E x ∣ C × Spec K . By the properties of GR G we can find lifts ˜ f x , ˜ f y ∶ Spec R → GR G . In particular wefind a divisor D = D x ∪ D y ⊂ C R such that E x ∣ C R − D ≅ G × ( C R − D ) ≅ E y ∣ C R − D . Through ˜ f x , ˜ f y theisomorphism E x ∣ C K ≅ E y ∣ C K therefore defines an element g ∈ G ( C K − D ) ⊂ G ( K ( C )) . As in Proposition2.9, we would like to apply Cartan decomposition now for the field K ( C ) that comes equipped with adiscrete valuation induced by the valuation of R , i.e. the valuation given by the codimension 1 pointSpec k ( C ) ∈ C R . Its ring of integers O K are meromorphic functions on C K that extend to an opensubset on the special fiber C k .Using this we can write g = k λ ( π ) k with k i ∈ G (O K ) . Now each of the k i defines an element k i ∈ G ( U ) for some open subset U ⊂ C R that is dense in the special fiber, so after enlarging D we mayassume k i ∈ G ( C R − D ) .The elements k i allow us to modify the lifts ˜ f x , ˜ f y such that with respect to these new maps wefind g = λ ( π ) .As before this datum defines a G m , λ -equivariant morphismSpec ( R [ x, y ]/( xy − π )) − { } → GR G and by ind-projectivity this can be extended after a suitable blow up to a G m , λ -equivariant morphism,which defines Bl I ( ST ) → Bun G . Finally, as the G m -action preserves the forgetful map GR G → C ( d ) and L det is relatively ample with respect to this morphism we see that L det is nef on exceptionallines. ◻
3. Torsors under parahoric group schemes on curves
In this section we give our main application to moduli of torsors under Bruhat-Tits group schemeson curves as introduced by Pappas and Rapoport [37]. It will turn out that the notion of stability wefind is a variant of the one introduced by Balaji and Seshadri in the case of generically split groupschemes. We will then apply this to construct coarse moduli spaces for stable torsors over fields ofarbitrary characteristic.
We fix a smooth projective, geometrically connected curve C / k and G → C a parahoric Bruhat-Titsgroup scheme in the sense of [37], i.e., G is a smooth affine group scheme with geometrically connectedfibers, such that there is an open dense subset U ⊂ C such that G∣ U is reductive and such that for all p ∈ C − U the restriction G∣ Spec O C,p is a parahoric group scheme as in [17] (see Appendix 4 for details).We will denote by Ram (G) ⊂ C the finite set of closed points for which the fiber G x is not a reductivegroup.We will denote by Bun G the moduli stack of G -torsors on C . As usual we will often denote baseextensions by an index, i.e. for a k -scheme X we abbreviate X C ∶= X × C . . Heinloth, GIT-stability for stacks 21J. Heinloth, GIT-stability for stacks 21 Example 3.1.
It may be helpful to keep the following examples in mind:(1) (Parabolic structures) Let G / k be a reductive group, B ⊂ G be a Borel subgroup and p , . . . , p n ∈ C ( k ) rational points. We define G p,B to be the smooth groupscheme over C that comes equippedwith a morphism G p,B → G C such that for all i the image G p,B (O C,p i ) = { g ∈ G (O C,p i )∣ g mod p i ∈ B } is the Iwahori subgroup. Since G p,B (O C,p i ) is the subgroup of automorphism groupof the trivial G torsor that fixes the Borel subgroups B ⊂ G × p i torsors under this group schemeare G -bundles equipped with a reduction to B at the points p i .(2) (The unitary group) Suppose char ( k ) ≠ π ∶ ˜ C → C is a possibly ramified Z / Z -coveringthen the group scheme π ∗ GL n, ˜ C admits an automorphism, given by the () t, − on the group andthe natural action on the coefficients O ˜ C . The invariants with respect to this action is calledthe unitary group for the covering. Torsors under this group scheme can be viewed as vectorbundles on ˜ C that under the involution become isomorphic to their dual. Bun G As observed by Pappas and Rapoport ([37],[27]) there are many natural line bundles on Bun G :(1) We define L det to be the determinant line bundle given by L det ∣ E ∶= det ( H ∗ ( C, Ad (E)) − . (2) For every x ∈ Ram (G) we have a homomorphism X ∗ (G x ) ↪ Pic ( Bun G ) induced from the pullback via the canonical map Bun G → B G x given by E ↦ E∣ x and the canonical morphism X ∗ (G x ) = Hom (G x , G m ) → Mor ( B G x , B G m ) ≅ Pic ( B G x ) , which is surjective on isomorphism classes. We write L χ x for the line bundle corresponding to χ x ∈ X ∗ (G x ) .(3) We will abbreviate L det ,χ ∶= L det ⊗ ⊗ x ∈ Ram (G) L χ x . As before, positivity of L will be checked on affine Grassmannians. Let us fix the notation. For apoint x ∈ X we denote by Gr G ,x the ind-projective scheme classifying G -torsors on X together with atrivialization on C − { x } . Its k -points are G( K x )/G( ̂O x ) . It comes with a forgetful mapglue x ∶ Gr G ,x → Bun G x . By definition, the bundles obtained from glue x are canonically trivial outside x , so the bundles L χ x pull back to the trivial line bundle on Gr G y for y ≠ x .To check that L is nef on exceptional lines, we will need a line bundle L = L det ,χ such that for all x ∈ X the bundle pulls back to a positive line bundle on the corresponding affine Grassmannian. Remark 3.2. If G η is simply connected, absolutely almost simple and splits over a tame extensionthe positivity condition can be given explicitly, as for example computed in [44, Section 4]. As thisrequires some more notation we only note that L det always satisfies this numerical condition as thisis the pull back of a determinant line bundle on a Grassmanian Gr GL N ,x .The proof of Proposition 2.10 now applies to G , as the proof only uses a group theoretic decom-position at the generic point of C where G is reductive. We therefore find:
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves
Proposition 3.3.
Let G be a parahoric Bruhat–Tits group on C and let L det ,χ be chosen such thatfor all x ∈ Ram (G) the bundle glue ∗ x L det ,χ is nef on Gr G ,x . Then the pair ( Bun G , L det ,χ ) satisfies thevaluative criterion ( ⋆ ). To obtain coarse moduli spaces we now have to show that the stable locus is an open subset offinite type. For this we will need analogs of the basic results on stability for G -torsors. To do this wefirst need to rephrase L det ,χ -stability in terms of reductions of structure groups. As before, very close degenerations of G -bundles will give us cocharacters G m,C → Aut G/ C (E) =∶ G E .In order to describe these in terms of reductions of structure group we first need some general resultson cocharacters and analogs parabolic subgroups of Bruhat–Tits group schemes.Let us first consider the local situation: Let R be a discrete valuation ring with fraction field K , π ∈ R a uniformizer and k = R /( π ) the residue field. Let G →
Spec R parahoric Bruhat-Tits groupscheme.Given λ ∶ G m,R → G we denote by(1) P λ ( S ) ∶= { g ∈ G( S )∣ lim t → λ ( t ) gλ ( t ) − exists in G( S )} the concentrator scheme of the action of G m .(2) L λ ( S ) ∶= { g ∈ G( S )∣ λ ( t ) gλ ( t ) − = g } the centralizer of λ .(3) U λ ( S ) ∶= { g ∈ G( S )∣ lim t → λ ( t ) gλ ( t ) − = } These are analogs of parabolic subgroups and Levi subgroups of G . Alternatively one could definesuch analogs by taking the closure of parabolics in the generic fiber G K . The following Lemma showsthat this leads to an equivalent notion: Lemma 3.4.
Let R be a discrete valuation ring and G R a parahoric Bruhat–Tits group scheme over Spec R . (1) Given a 1-parameter subgroup λ ∶ G m,R → G the group P λ is the closure of P K,λ ⊂ G K , L λ is theclosure of the Levi subgroup L K,λ ⊂ G K and U λ is the closure of the unipotent radical of P K,λ .The group L λ is again a Bruhat–Tits group scheme. (2) Let P K ⊂ G K be a parabolic subgroup and denote by P ⊂ G the closure of P K in G . Then thereexists a 1-parameter subgroup λ ∶ G m,R → G R such that P = P λ . (3) Let P K ⊂ G K be a parabolic subgroup and λ K ∶ G m,K → G K be a cocharacter such that P K = P λ K .Then there exist an element u ∈ U ( K ) such that λ uK ∶= uλ K u − extends to a morphism λ u ∶ G m →P . The class of u in U ( K )/U ( R ) is uniquely determined by λ K . Before proving the lemma let us note that part (3) will be useful to define a Rees construction for G -bundles. In the global setup of a group scheme G/ C on a curve we cannot expect that every parabolicsubgroup P k ( C ) ⊂ G k ( C ) can be defined by a globally defined cocharacter G m,C → G , as G may notadmit any non-trivial cocharacters. However, given λ k ( C ) part (3) will give us a canonical inner formof G for which λ extends. Remark 3.5.
In the setup of the above lemma given two parabolic subrgoups P K , P ′ K ⊂ G K thatare conjugate over G K , their closures P , P ′ ⊂ G need not be conjugate. Roughly this is because theclosure contains information about the relative position of the generic parabolic and the parahoricstructure in the special fiber. More precisely, the Iwasawa decomposition we know that for any Borelsubgroup B K ⊂ G K we have G( K ) = B( K ) W G( R ) ([16, Propositions 4.4.3, 7.3.1]). This implies thatthere are only finitely many conjugacy classes P of closures of generic parabolic subgroups. . Heinloth, GIT-stability for stacks 23J. Heinloth, GIT-stability for stacks 23 Proof of Lemma 3.4.
To show the first part of (1) we have to show that the generic fibers of L λ , U λ and P λ are dense. As G is smooth over R the fixed point scheme L λ and the concentrator scheme P λ are both regular ([30, Theorem 5.8] ). Let x ∈ L λ ( R /( π )) ⊂ G( R /( π )) be a closed point of the specialfiber then T G ,x → T Spec R, is surjective and equivariant, so there exists an invariant tangent vectorlifting the tangent direction in 0. Thus L λ is smooth over R and therefore the generic fiber is dense.The morphism P λ → L λ is an affine bundle ([30, Theorem 5.8]), so the generic fiber of P λ must alsobe dense and U λ is even an affine bundle over Spec R .Let us prove (2) and (3). Any parabolic subgroup P K of the reductive group G K is of the form P K = P K ( λ ) for some λ ∶ G m,K → G K [41, Lemma 15.1.2]. The image of λ is contained in a maximalsplit torus of G K and these are all conjugate over K [41, Theorem 15.2.6]. Fixing a maximal splittorus T R ⊂ G R we therefore find g ∈ G ( K ) such that gλg − ∶ G m → G K factors through T . By Iwasawadecomposition we can write g = kwu with k ∈ G( R ) , w ∈ N (T ) , p ∈ U ( K ) . Thus we can conjugate λ by an element of U ( K ) such that it extends to G R . Applying (1) to this subgroup we find (2). It alsoshows the existence statement in (3). To show uniqueness assume that λ ∶ G m,R → P λ is given andthat u ∈ U ( K ) is such that uλu − still defines a morphism over R . Recall that U K has a canonicalfiltration U K, ≥ r ⊂ ⋅ ⋅ ⋅ ⊂ U K, ≥ = U K such that U K,i /U K,i + ≅ ∏ α ∣ α ○ λ = i U α . Write U i ∶= ∏ α ∣ α ○ λ = i U α anddecompose u = u ⋅ ⋅ ⋅ ⋅ ⋅ u r .We know that uλ ( a ) u − ∈ P ( R ) for all a ∈ R ∗ , i.e. U ( R ) ∋ uλ ( a ) u − λ − ( a ) . The image of thiselement in U ≥ /U ≥ is u au − a − which can only be in U ( R ) for all a if u ∈ U ( R ) , but then we canreplace u by u . . . u r and conclude by induction.Finally we need to show that the group scheme L λ in (1) is a Bruhat–Tits group scheme. Byconstruction it suffices to show this after an ´etale base change Spec R ′ → Spec R , so we may assumethat G R is quasi-split, i.e. that G K contains a maximal torus T K and that G contains the connectedN´eron model T of T K .As conjugation by elements of G( R ) produces isomorphic group schemes we may assume as abovethat λ ∶ G m,R → T ⊂ G . The scheme G R is given by a valued root system and the restriction of thisto the roots of L λ,K defines a Bruhat–Tits group scheme with generic fiber L λ,K , contained in L λ .Finally, by definition the special fiber of L λ is the centralizer of a torus, so it is connected. Thus thesmooth scheme L λ has to be equal to this Bruhat–Tits scheme. ◻ Let us translate this back to our global situation: As before let G/ C be a Bruhat–Tits groupscheme over our curve C and denote by η ∈ C the generic point of C . Lemma 3.6.
Let λ η ∶ G m,η → G η be a cocharacter and P η,λ the corresponding parabolic, U η,λ its unipo-tent radical and U λ , P λ the closures of U η,λ , P η,λ in G . Then: (1) P λ , U λ are smooth group schemes over C . The quotient P λ /U λ =∶ L λ is a Bruhat–Tits groupscheme. (2) The morphism λ η extends to λ ∶ G m × C → Z (L λ ) ⊂ L λ . (3) There exist a canonical U λ -torsor U u together with an isomorphism U u ∣ k ( C ) ≅ U k ( C ) such that λ extends to λ ∶ G m → U u × U , conj G .Proof. First note that λ η extends canonically to a Zariski open subset U ⊂ C (e.g. [40, Expos´e XI,Proposition 3.12 (2)]). Let us denote this morphism λ U ∶ G m,U → G U . Over U the subgroup P λ,U ⊂ G U is a parabolic subgroup of the reductive group scheme G U , the groups U λ,U , L λ,U are the correspondingunipotent radical and Levi subgroup. Part (1) and (2) can thus be checked locally around all points x ∈ C − U and there Lemma 3.4 gives the result. Since a U -torsor together with an isomorphism U u ∣ k ( C ) ≅ U k ( C ) is given by a finite collection of elements u x ∈ U ( k ( C ))/ U (O C,x ) for some x ∈ C thelast part also follows from Lemma 3.4 (3). ◻
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves
This lemma allows us to generalize the Rees construction: Given λ η ∶ G m,η → G η the above lemmaconstructs an inner form P u ⊂ G u such that λ ∶ G m,C → P u extends. As we proved that fiberwiseconjugation by λ contracts P u to L λ we again obtain the morphism of group schemes over C × G m conj λ ∶ P u × G m → P u × G m ( p, t ) ↦ ( λ ( t ) pλ ( t ) − , t ) . By [30, Proposition 4.2] this homomorphism extends togr λ ∶ P u × A → P u × A in such a way that gr ( p, ) = lim t → λ ( t ) pλ ( t ) − ∈ L λ × P u torsor E u we can define the Rees construction:Rees (E u , λ ) ∶= (E u × A ) × gr λ (P u × A ) . Given a G torsor E together with a reduction E P to a parabolic subgroup P and a cocharacter λ η ∶ G m → P η defining P we take the associated P u -torsor E u ∶= Isom P/ C (E P , U u × U P) and defineRees (E P , λ η ) ∶= Isom P u / C ( Rees (E u , λ ) , U u × U P) . As before, this construction only depends on P and the composition λ ∶ G m,η → L η . Remark 3.7.
Note that the adjoint bundle of Rees (E P , λ η ) is the vector bundle on C × [ A / G m ] thaton the generic fiber corresponds to the very close degeneration given by Ad (E P ) η and the cocharactergiven by Ad ( λ ) . As we know that this already defines the very close degeneration, the passage to P u -torsors was only needed in order to give a formula for the very close degeneration in terms of G torsors. G -bundles We can now classify very close degenerations f ∶ [ A / G m ] → Bun G . Such a morphism defines a G -torsor E over C × [ A / G m ] and f ( ) defines a non-trivial cocharacter λ ∶ G m × C → Aut G/ C (E ) =∶ G E .This defines L λ , P λ ⊂ G E . As L λ is the concentrator scheme of the G m action we again get that themorphism: conj λ ∶ P λ × G m → P λ × G m ( p, t ) ↦ ( λ ( t ) pλ ( t ) − , t ) extends to gr λ ∶ P λ × A → P λ × A in such a way that gr ( p, ) = lim t → λ ( t ) pλ ( t ) − ∈ L λ ×
0. And so we can define the Rees constructionfor P λ -bundles F λ : Rees (F λ , λ ) ∶= [((F λ × A ) × gr λ (P λ × A ))/ G m ] . The G torsor E is determined by the G E -torsor E ′ ∶= Isom (E , E ) . As before Isom (E , E ) beingthe trivial G E torsor E ′ comes equipped with a canonical reduction to L λ and as in Lemma 1.7 thecorresponding reduction to P λ lifts canonically to C × [ A / G m ] . We denote this reduction by E ′ λ . Nowwe can argue as in Lemma 1.13 to identify E ′ λ ≅ Rees (E ′ λ, , λ ) . Heinloth, GIT-stability for stacks 25J. Heinloth, GIT-stability for stacks 25 and thus E ≅
Isom G E (E ′ , E ) ≅ Isom G E ( Rees (E ′ λ, , λ ) × P λ G E , E ) . Formulating stability in these terms would have the annoying aspect that all possible groupschemes G E would appear in the formulation. This can be avoided this by restricting to the genericfiber as follows:By the description of G -bundles over [ A / G m ] (Lemma 1.7) we haveIsom (E , E ) η ×[ A / G m ] = E ′ ∣ η ×[ A / G m ] ≅ [ A × G E η / G m ] . In particular E ,η ≅ E ,η . Therefore the canonical reduction of E ′ to P λ corresponds to a reduction of E ,k ( C ) to a parabolic P λ K ⊂ G E .Thus given the cocharacter λ ∶ G m → G E we can find a cocharacter λ K ∶ G m → G E such that thecanonical reduction of E ′ to P λ ⊂ G E defines a reduction E P ′ λ of E to P ′ λ ∶= P λ K ⊂ G E .Given the reduction of E to P ′ λ we already defined the corresponding Rees construction. Thus wefind: Lemma 3.8.
Let
G → C be a Bruhat–Tits group scheme, E ∈
Bun G and G E ∶= Aut G/ C (E) the corre-sponding inner form of G .Then any morphism f ∶ [ A / G m ] → Bun G with f ( ) = E can be obtained from the Rees constructionapplied to a generic cocharacter λ η ∶ G m → G E k ( C ) . In the above we described reductions by cocharacters λ ∶ G m → G E k ( C ) because this description worksover any field. As in [8] this is more suitable to study rationality problems for canonical reductionsof G -bundles. Over algebraically closed fields we can also reformulate this in terms of reductions toparabolic subgroups of G k ( C ) as follows: Lemma 3.9.
Let k = k be an algebraically closed field, G → C be a Bruhat–Tits group scheme, E ∈
Bun G ( k ) . Then any morphism f ∶ [ A / G m ] → Bun G with f ( ) = E can be obtained from the Reesconstruction applied to a reduction of E to a subgroup P λ ⊂ G which is defined by a generic cocharacter λ ∶ G m → G k ( C ) .Proof. By Lemma 3.8 we know that f can be defined by applying the Rees construction to a cocharac-ter λ η ∶ G m → G E k ( C ) . Let P E λ ⊂ G E be the closure of the parabolic subgroup defined by λ η . As explainedin the appendix (Proposition 5.1) any G bundle on C k can be trivialized over an open subset U ⊂ C k which contains Ram (G) ⊂ U . Choose such a trivialization ψ ∶ E∣ U ≅ G∣ U . This induces an isomorphism G E k ( C ) ≅ G k ( C ) and this defines λ ∶ G m → G k ( C ) . Denote by P λ ⊂ G the closure of the parabolic subgroupdefined by λ . By construction we know that P E λ ∣ U ≅ P λ ∣ U and since G is a reductive group schemeover C − U the groups P λ and P E λ are also isomorphic in a neighborhood of the points in C − U . Thusthe reduction of E to P E λ defines a reduction of E to P λ . This proves our claim. ◻ Remark 3.10.
In the case of G -bundles there are only finitely many conjugacy classes of parabolicsubgroups P ⊂ G and therefore any very close degeneration of a bundles E is induced from viewing E as lying in the image of a morphism Bun P → Bun G . To show that the semistable points of Bun G form an open substack the fact that one needs only to consider finitely many such P is helpful.From Lemma 3.9 we can now conclude that the analogous result also holds for G -bundles: Suppose P , P ′ ⊂ G are closures of parabolic subgroups in the generic fiber. If P and P ′ happen to be conjugatein G at the generic point of C they are also conjugate locally on C − Ram (G) as parabolic subgroupsof the same type are conjugate in reductive groups. Also locally around any point x ∈ Ram (G) we sawin Remark 3.5 that there are only finitely many conjugacy classes of closures of parabolic subgroups
P ⊂ G . Thus up to local conjugation in G there are only finitely many closures of parabolic subgroups P ⊂ G .
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves
As in the case of G -bundles, if P , P ′ ⊂ G are locally conjugate over C , then the transporterTransp G (P , P ′ ) of elements of G that conjugate P into P ′ is a P − P ′ bi-torsor (because parabolicsubgroups are equal to their normalizer over C − Ram (G) and a local section of G normalizes P if andonly it normalizes the generic fiber). This defines a commutative diagramBun P ≅ (cid:47) (cid:47) (cid:36) (cid:36)
Bun P ′ (cid:122) (cid:122) Bun G identifying reductions to the structure groups P and P ′ . Let us fix our group scheme G and a line bundle L ∶= L det ,χ as in Section 3.B. In this section wewant to describe L -stability condition in terms of degrees of bundles. Lemma 3.9 and the definitionof L -stability (1.2) imply: Remark 3.11. A G -bundle E on C k is L -stable if and only if for all parabolic subgroups P ⊂ G k ( C ) with closure P ⊂ G , all dominant cocharacters λ ∶ G m → P and all reductions of E to P we havewt L ( Rees (E P , λ )) > . As in the case of G -bundles we want to express the above weight in terms degrees of line bundlesattached to reductions of E . We start out with the intrinsic formulation of reductions as in Lemma3.8, but in the end this reduces to a computation on the adjoint bundle Ad (E) .Fix E a G -bundle, E B a reduction to a Borel subgroup B ⊂
Aut G (E) . As before let us denote by U ⊂ B the closure of the unipotent radical over the generic point η ∈ C and T = B/U the maximaltorus quotient. Fix S ⊂ T η ⊂ B η a maximal split torus in a lifting of the maximal torus at η . We willdenote by Φ = Φ (G η , S ) the roots of G η and Φ +B will be the roots that are positive with respect to B .For any point x ∈ Ram (G) we obtain a character χ B x ∶ B x → G m as composition χ B x ∶ B E x → G E x χ —→ G m . This morphism factors through T E x . For any λ ∶ G m → T we will write ⟨ χ B x , λ ⟩ ∶= ⟨ χ B x , λ ∣ x ⟩ .The Rees construction applied to a generic dominant 1-parameter subgroup gives us a bundle E ,that is induced from the T bundle E B /U =∶ E T .To compute the weights, we can decompose the adjoint bundle of E into weight spaces:ad (E ) = ad (E T ) ⊕ a ∈ Φ (G η ) u E a and the Rees construction induces a filtration of ad (E) such that the associated graded pieces are u E a ≅ u E a .For any 1-parameter subgroup λ ∶ G m → G E η which is dominant with respect to B we then have:wt E B ( λ ) ∶= wt E ( λ ) = ∑ a ∈ Φ χ ( u E a )⟨ a, λ ⟩ + ∑ x ∈ Ram (G) ⟨ χ B x , λ ⟩ . As rk ( u a ) = rk ( u − a ) we can further compute:wt E B ( λ ) = ∑ a ∈ Φ χ ( u E a )⟨ a, λ ⟩ + ∑ x ∈ Ram (G) ⟨ χ B x , λ x ⟩= ∑ a ∈ Φ deg ( u E a )⟨ a, λ ⟩ + ∑ x ∈ Ram (G) ⟨ χ B x , λ x ⟩ . Heinloth, GIT-stability for stacks 27J. Heinloth, GIT-stability for stacks 27 In the case of unramified, constant group schemes, the degree deg ( u a ) is a linear function in theroot a . This does no longer hold for parahoric group schemes, but we still have relations:At the generic point η of C we get a canonical isomorphism k a ∶ u E− a,η ≅ —→ ( u E a,η ) ∨ from the Killing form and this extends to an isomorphism at all points c ∈ C where G c is reductive.Therefore, the determinant of k a defines a divisor D a = ∑ x ∈ Ram (G) f B a,x x. With this notation we have deg ( u E− a ) = − deg ( u E a ) − ∑ x ∈ Ram (G) f B a,x . Here we denote the coefficients by the letter f because for a global torus T ⊂ G with valuated rootsystems f a,x (see Section 4) these numbers are −( f a,x + f − a,x ) . As any two tori are conjugate, wealways find RRRRRRRRRRR f B a,x rk ( u a ) RRRRRRRRRRR ≤ . Thus we find wt E B ( λ ) = ∑ a ∈ Φ deg ( u E a )⟨ a, λ ⟩ + ∑ x ∈ Ram (G) ⟨ χ B x , λ ⟩= ∑ a ∈ Φ + deg ( u E a )⟨ a, λ ⟩ + ∑ x ∈ Ram (G) ⟨ χ B x + ∑ a ∈ Φ +B f B a,x a, λ ⟩ (3.1)To compare this to the usual (parabolic)-degree let us fix a norm on the set of all 1-parametersubgroups. A convenient choice for us will be to fix for any maximal torus containing a maximal splittorus S η ⊂ T η ⊂ G η the canonical invariant bilinear form on X ∗ ( T η ) : ( , ) ∶= ∑ α ∈ Φ (G η ) ⟨ , α ⟩⟨ , α ⟩ We will denote the restriction of ( , ) to X ∗ ( T ) by the same symbol and we will denote by ∣∣ ⋅ ∣∣ ∶=√(⋅ , ⋅) the induced norm. Remark 3.12.
The bilinear form on X ∗ ( T η ) also induces a form on the cocharacter groups X ∗ (T x ) for all x ∈ C closed. Remark 3.13. (1) If G = G × X is a split group scheme we remarked above that deg ( u E a ) is a linear function in a , denoted deg as this is the usual degree of the T -bundle E T = E B /U . Then the above formulareads wt E B ( λ ) = ( deg , λ ) . This expresses the weight in terms of the degree that is classically used to define stability for G -torsors.(2) If G → G = G × X is G is obtained as the parahoric subgroup defined by the choice of parabolicsubgroups P x ⊂ G ,x = G the numbers f a,x are 0 , , − B x and the image of G E x → (G E ) x . By the previous point this then again gives the same relation ofthe weight and the parabolic degree.
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves (3) As the difference deg ( u E a ) − deg ( u E− a ) is always an integer, the formula also shows that as inthe case of parabolic bundles the weight cannot be 0 if for at least one x ∈ Ram (G) the groupscheme G∣ O x is an Iwahori group scheme and χ x is chosen generically, e.g. such that the numbers ⟨ χ x , ˇ α i ⟩ for some basis ˇ α i of one parameter subgroups of T are rational numbers with sufficientlylarge denominators.(4) For generically split groups G in [7, Definition 6.3.4] Balaji and Seshadri describe stability interms of ( Γ , G ) bundles. In their setup the reductions E P are computed from taking invariantsof an invariant parabolic reduction of a G -bundle on a covering ˜ C → C ([7, Proof of Proposition6.3.1]). The numerical invariants defining stability are then expressed as the parabolic degreesdefined by characters of P attached to a reduction of E P . To compare this with our conditionone can proceed as in (1) by using (3.1). In this case, as G is generically split, all u E a are linebundles, which are the invariant direct images of the corresponding bundles of the reductionto P on ˜ C . The parabolic weight then turns out to be related to the contribution form theramification points which depends only on the relative position of the generic parabolic and thevaluation of the root system defining G . As the precise relation requires a careful recollectionon the relation of ( Γ , G ) -bundles and valued root systems we leave this to a later time. G -torsors In this section we want to check that the canonical reduction of G -bundles introduced by Behrendalso exists for G -bundles. We will then use this to deduce that the stack of stable and semistable G -bundles are open substacks of finite type of Bun G .In [22] Halpern-Leistner gives general criteria for the existence and uniqueness of canonical reduc-tions for θ -reductive stacks. Unfortunately, Bun G does not satisfy this condition, so that we have togive a separate argument. It will turn out that once we formulate the classical approach for G -bundles(see [24],[8]) in a suitable way, most of the arguments generalize to this framework. This was alsoexplained by Gaitsgory and Lurie in [21, Section 10] for a notion of stability induced from G -bundles.Note that Harder and Stuhler also introduced the concept of canonical reductions for Bruhat–Titsgroups in the adelic description of the points of the moduli stack [25].Let us fix our group scheme G and a line bundle L ∶= L det ,χ as in Section 3.B. In general, todefine canonical destabilizing 1-parameter subgroups one needs to fix a norm on the set of all suchsubgroups. We will simply use the invariant form ( , ) on X ∗ ( T η ) from the previous section.As for parabolic bundles, we will need the following assumption on χ . We will call L admissible if 2 ⟨ χ x , ˇ a ⟩ ≤ rk u a ⟨ a, ˇ a ⟩ for all roots a of G x .As in [22] a canonical reduction of a G -torsor should be a 1-parameter subgroup λ ∈ X ∗ (G E η ) suchthat wt ( λ )∣∣ λ ∣∣ is maximal. First of all this number is bounded: Lemma 3.14.
For every G torsor E there exists c E > such that wt ( λ )∣∣ λ ∣∣ ≤ c for all λ ∶ G m,η → G E η .Proof. As in the classical case for every reduction E P of E to a parabolic subgroup we have that H ( ad (E P )) ⊂ H ( ad (E)) . By Riemann-Roch this implies that the degree of the unipotent radicaldeg (E P × P u P ) is uniformly bounded above for all reductions. In turn this gives for every reductionto a Borel subgroup B ⊂ G E an upper bound for the weight wt ( ω i ) for all dominant coweights ω i . Asany dominant λ is a positive linear combination of these, this gives the required bound. ◻ Notation.
For any G -torsor E over an algebraically closed field we define µ max (E) ∶= sup λ wt E ( λ )∣∣ λ ∣∣ . . Heinloth, GIT-stability for stacks 29J. Heinloth, GIT-stability for stacks 29 To characterize the canonical reduction we need to compare the weights of different reductions.Suppose B , B ′ ⊂ G E are two Borel subgroups. Any two such subgroups share a maximal split torus S ⊂ T η ⊂ B η ∩ B ′ η ([13, Proposition 4.4]).Let { a , . . . a n } be the positive simple roots of B .We say that B , B ′ are neighboring reductions if there exists i such that − a i is a positive simpleroot of B ′ and a i are positive roots of B ′ for all i ≠ i .In this case B , B ′ generate a parabolic P that is minimal among the parabolic subgroups that arenot Borel subgroups. Denote by L the corresponding Levi quotient. Lemma 3.15.
Let B , B ′ be neighboring parabolic subgroups. Then we have: (1) wt B ( λ ) = wt B ′ ( λ ) for all λ ∈ Z (L I ) . (2) wt B ( ˇ α i ) + wt B ′ (− ˇ α i ) ≤ Proof.
As the weight only depends on the Rees construction for E we may replace E by the Reesconstruction applied to any λ that is dominant for B and B ′ . Thus we may assume that E = E L × L G E is induced from a L torsor. In this case (1) is immediate from the definition, as the Rees constructionfor λ ∈ Z ( L ) does not change E L .Moreover, L is of semisimple rank 1 and B ∶= B ∩ L and B ′ ∩ L are Borel subgroups that areopposite over the generic point η ∈ C . Let us denote by u , u ′ the Lie algebras of the correspondingunipotent radicals in L and by u − ∶= Lie (L )/ Lie (B ) and similarly u ′ , − . Then we get an injectivehomomorphism E B × B u → ad (E L ) → ad (E L )/ ad (E B ′ ) ≅ E B ′ × B ′ u − , ′ . If a is a multipliable root then the homomorphism respects the filtration on u , u ′ , − given by the roots a, a . So we find: deg ( u B a ) ≤ deg ( u B ′ a ) = − deg ( u B ′ − a ) − ∑ x ∈ Ram (G) f B ′ − a,x − deg ( u B a ) − ∑ x ∈ Ram (G) f B a,x = deg ( u B− a ) ≥ deg ( u B ′ − a ) an the same hods for u a if 2 a is also a root.Thus: 2 ( deg ( u B a ) + deg ( u B ′ − a )) ≤ ∑ x ∈ Ram (G) f B ′ − a,x + f B a,x . Moreover, the map u B a → u B ′ a is an isomorphism at x if and only if the groups B E x and B ′ , E x are oppositein G E x . And in this case ⟨ χ B x , ˇ α i ⟩ = −⟨ χ B ′ x , − ˇ α i ⟩ . If the morphism is not an isomorphism, then theBorel subgroups are parallel in x , so that ⟨ χ B x , ˇ α i ⟩ = ⟨ χ B ′ x , − ˇ α i ⟩ . Thus if 2 ⟨ χ x , ˇ a ⟩ ≤ rk u a ⟨ a, ˇ a ⟩ for allroots a we find: 2 ( deg ( u B a ) + deg ( u B ′ − a ))⟨ a, ˇ a ⟩ ≤ ⟨ ∑ x ∈ Ram (G) f B ′ − a,x + f B a,x + χ B x − χ B ′ x , ˇ a ⟩ . And this means: wt B ( ˇ α i ) + wt B ′ (− ˇ α i ) ≤ χ is admissible.
3. Torsors under parahoric group schemes on curves
3. Torsors under parahoric group schemes on curves
To compute wt B ( ˇ α i ) + wt B ′ (− ˇ α i ) we note that E P ≅ E B × B P ≅ E B ′ × B ′ P and the unipotentradical of Lie (P ) has a filtration by L -invariant subspaces such that over the generic point theassociated graded pieces are isomorphic to the unipotent groups u b ∗ ,η = ⊕ c = b + na u B c,η = ⊕ c = b − na u B ′ c,η .Moreover since a is a positive root for B and a negative root for B ′ over C the isomorphic bundles E B × B u b ∗ and E B ′ × B ′ u b ∗ come with canonical filtrations that are opposite at the generic point.Therefore we find again that ∑ c = b + na ( deg ( u B c,η ) − deg ( u B ′ c,η ))⟨ c, ˇ α i ⟩ ≤ b we obtain the result. ◻ We define
L − deg (E B ) by the formula (L − deg , λ ) ∶= − wt E B ( λ ) . Then the previous lemma shows that
L − deg defines a complementary polyhedron as defined byBehrend [8]. As in [26, Section 4.3] we therefore obtain:
Proposition 3.16.
Suppose L det ,χ is an admissible line bundle and E a G -torsor. Then there existsa reduction λ ∶ G m → G E η such that E P λ is a reduction for which wt ( λ )∣∣ λ ∣∣ is maximal and such that forevery other such reduction to a parabolic subgroup Q λ ′ we have Q λ ′ ⊂ P λ . Lemma 3.17. (Semicontinuity of instability)
Let L det ,χ be an admissible line bundle on Bun G .Let R / k be a discrete valuation ring with fraction field K and residue field κ and let E R be a G -torsoron C R . (1) If E K is unstable and the canonical reduction is defined over K , then E κ is unstable and µ max (E K ) ≤ µ max (E κ ) . The equality is strict, unless the canonical reduction of E K extends to R . (2) If E K is semistable but not stable then E κ is also not stable.Proof. The first part follows as in [26, Lemma 4.4.2]. This also shows that if E K admits a reductionof weight 0, then E κ also cannot be stable.Finally suppose dim Aut (E K ) >
0. We know that G E R p —→ C R is an affine group scheme of finitetype over C R . The group of global automorphisms of this group scheme is Spec p ∗ (O G E R ) , so thegeneric fiber of this is not a finite K -algebra. But then by semi continuity also the special fiber willnot be finite. ◻ G -torsors With the canonical reduction at hand we can now deduce:
Proposition 3.18.
Let L det ,χ be an admissible line bundle on Bun G . Then the stacks Bun st G ⊂ Bun sst G ⊂ Bun G of L det ,χ -(semi)-stable G -torsors are open substacks of finite type.Proof. By Lemma 3.17 instability and strict semistability are stable under specialization. Thereforewe only need to show that Bun sst G is constructible and contained in a substack of finite type. Againwe argue as in [8]. First we show that for any c ≥ G -torsors of fixed degree satisfying µ max (E) ≤ c is of finite type. To prove this we may suppose that χ =
0, as the linear function λ ↦ ⟨ χ, λ ⟩ only changes wt ( λ )∣∣ λ ∣∣ by a finite constant. . Heinloth, GIT-stability for stacks 31J. Heinloth, GIT-stability for stacks 31 By [8] the claim holds if
G = G is a reductive group scheme over C . Moreover if G ′ → G is aparahoric group scheme mapping to G such that G ′ is an Iwahori group scheme at all x ∈ Ram (G ′ ) then the morphism Bun G ′ → Bun G is a smooth morphism with fibers isomorphic to a product of flagvarieties G ,x /B x . Moreover the cokernel of Lie (G ′ ) → Lie (G ) is of finite length. Thus there existsa constant d such that for any E is a G ′ torsor we have that if E × G ′ G admits a reduction of slope µ (E × G ′ G ) > c + d then this reduction induces a reduction of E of slope µ (E) > c .Therefore the result also holds for G ′ . Now any parahoric group scheme contains an Iwahori groupscheme so that by the same reasoning the result also holds if G is any parahoric Bruhat–Tits groupscheme such that G∣ C − Ram (G) admits an unramified extension.Now choose π ∶ ˜ C → C a finite Galois covering with group Γ such that π ∗ G∣ C − Ram (G) is a genericallysplit reductive group scheme on ˜ C − π − ( Ram (G)) , equipped with a Γ-action. We can extend this groupscheme to a Bruhat–Tits group scheme ˜ G that is Γ equivariant and admits an morphism π ∗ (G) → ˜ G that is an isomorphism over ˜ C − π − ( Ram (G)) . We already know the result for Bun ˜ G . Now if E isa G -torsor then ˜ E ∶= π ∗ E × π ∗ G ˜ G is a Γ-equivariant ˜ G torsor. Now if ˜ E admits a canonical reductionto a parabolic subgroup ˜ P this will define an equivariant reduction of π ∗ E∣ C − Ram (G) and therefore areduction of E .Again we can compare the weights of the reductions because π ∗ ad ( π ∗ (E)) = ad (E) ⊗ π ∗ (O ˜ C ) .Therefore the determinant of the cohomology det ( H ∗ ( ˜ C, ad ( π ∗ E)) = det H ∗ ( C, ad (E)⊗ π ∗ O ˜ C ) definesa power of L det on Bun G . This implies that the weight of the reduction of π ∗ E is a just deg ( π ) -timesthe weight of the induced reduction of E . So again a very destabilizing canonical reduction of ˜ E induces a destabilizing reduction of E .We are left to show that the stable and semistable loci are constructible. We saw that unstablebundles admit a canonical reduction to some P ⊂ G , i.e., they lie in the image of the natural morphismBun P → Bun G . From Remark 3.10 we know that it suffices to consider the image of this morphismfor finitely many P .To conclude, we need to see that for every substack of finite type U ⊂ Bun G only finitely manyconnected components of Bun P contain canonical reductions that map to U . We just proved that on U the slope µ max is bounded above.We claim that for the canonical reduction of a G -bundle with µ max ≤ c the degree of the corre-sponding P -bundle deg ∈ ( X ∗ (P)) ∨ lies in a finite set. By construction the canonical reduction wasobtained from a complementary polyhedron and therefore the L -degree of the reduction that was de-fined from the weight of the reduction that is bounded below when evaluated at fundamental weights.As we bounded the µ max this will also be bounded above. Now we saw that the from equation 3.1that the weight of the reduction can be computed from the degree of the reduction and a local termonly depending on the group P . Therefore we find obtain our bound for the degree of the P -bundle.To conclude we use that the number of connected components of Bun P of a fixed degree is finite.This is known for G -bundles (see Proposition 5.2 ). From this we can deduce our statement by lookingat the Levi quotient P → L and the morphism Bun P → Bun L , which is smooth with connected fibers,because the kernel U → P has a filtration by additive groups.For the stack of stable points we can argue in the same way, oserving that strictly semistablebundles admit a reduction to a parabolic with maximal slope equal to 0. ◻ G -torsors Theorem 3.19.
Let G be a parahoric Bruhat–Tits group scheme that splits over a tamely ramifiedextension ˜ k ( C )/ k ( C ) and let L = L det ,χ be an admissible line bundle on Bun G . Then the stack of L -stable G bundles Bun st G admits a separated coarse moduli space of finite type over k .Proof. This now follows from Proposition 2.8, because the stack of stable G -torsors Bun st G is an opensubstack by Proposition 3.18 and it satisfies the conditions of Proposition 2.6 by Proposition 3.3. ◻
4. Appendix: Fixing notations for Bruhat–Tits group schemes
4. Appendix: Fixing notations for Bruhat–Tits group schemes
Remark 3.20.
For admissible line bundles
L = L det ,χ the results on the existence of canonical re-ductions for G -bundles allow to copy the proof of the semistable reduction theorem using Langton’salgorithm from [28] and [29]. In particular in those cases where L -semistability is equivalent to L -stability this then implies that the coarse moduli spaces are proper.
4. Appendix: Fixing notations for Bruhat–Tits group schemes
In this appendix we collect the results from [17] on the structure of Bruhat–Tits group schemes thatwe use. As the definition is local let us fix a discrete valuation ring R with fraction field K and residuefield k and G K a reductive group over K . In general the groups are defined by descent starting fromthe quasi-split case over an unramified extension of R . In our applications we can always extend thebase field and if k = k the group G K is quasi-split by the theorem of Steinberg [12, Section 8.6]. Wewill therefore assume that G K is quasi-split.We choose a maximal split torus S K ⊂ G K and denote T K ∶= Z ( S K ) the centralizer of S which isa maximal torus of G K because G K was quasi-split. To construct models of G K over R Bruhat–Tits first extend the torus T K to a scheme over R andthen the root subgroups of G K using a pinning of G K that they upgrade to a Chevalley-Steinbergvaluation ϕ . ([17] Section 4.2.1 and 4.1.3). Let us recall these notions: If G K is split, a Chevalley–Steinberg system is simply a pinning of our group, i.e., an identifica-tion x α ∶ G a ≅ —→ U α which is compatible for α, − α in the sense that it comes from an embedding of ζ α ∶ SL ,K → G K identifying G a with the strict upper and lower triangular matrices [17, Paragraph3.2.1] and is compatible with reflections [17, Paragraph 3.2.2, (Ch1),(Ch2)].If G K is not split, we can split it over a Galois extension ˜ K and choose an equivariant pinning[17, Paragraph 4.1.3]: Let Γ ∶= Gal ( ˜ K / K ) and denote by ˜Φ the roots with respect to T ˜ K of G ˜ K andΦ the roots for S . Then X ∗ ( S ) = X ∗ ( S ˜ K ) and the elements of Φ are the restrictions of elements of˜Φ to S .The root subgroups U a can then be described as follows. For any root ray a ∈ Φ denote ˜∆ a ⊂ ˜Φthe set of simple roots that restrict to a . The analog of the SL defined by a root is a morphism ζ a ∶ G aK → G K . If a is not a multiple root we have G aK ≅ Res L α / K SL , where L α ⊂ ˜ K is the field obtained by the stabilizer of any α ∈ ˜∆ a . In this case a pinning is anisomorphism Res L α / K G a ≅ —→ U a , which is again assumed to be compatible for a and − a . ( [17, Paragraph 4.1.7,4.1.8])If a is a multiple root ray G a is the Weil restriction of a unitary group . In this case for any pair α, α ′ ∈ ˜∆ a such that α + α ′ is a root let L α = ˜ K Stab α which is a quadratic extension of L α + α ′ =∶ L . Thisextension defines the unitary group SU ( L α / L )) over L (with respect to the standard hermitionform). Then G a = Res L / K SU ( L α / L ) . ↑ In [7] Balaji and Seshadri study the case where S K = T K is a split maximal torus, i.e. the case where G K is asplit reductive group, which already shows many interesting features. ↑ This happens if the Galois group interchanges neighboring roots in the Dynkin diagram. . Heinloth, GIT-stability for stacks 33J. Heinloth, GIT-stability for stacks 33
In this case the root subgroups U a , U − a are of the form U a ( L ) = { x a ( u, v ) = ⎛⎜⎝ − u σ − v u ⎞⎟⎠ } , U − a ( L ) = { x − a ( u, v ) = ⎛⎜⎝ u − v − u σ ⎞⎟⎠ } with v + v σ = uu σ . This has a filtration U a ≅ { v ∈ L α / L ∣ tr ( v ) = } and U a / U a ≅ L α .The Chevalley pinning induces valuations on the groups U a . For non multipliable roots one sets φ α ( x α ( u )) ∶= ∣ u ∣ and for multiple roots one defines φ a ( x a ( u, v )) ∶= ∣ v ∣ and φ a ( x a ( , v )) ∶= ∣ v ∣ .(These choices define a valued root system and identify the standard appartment A ≅ X ∗ ( S ) R inthe Bruhat–Tits building of G ). With this notation, we can recall the construction of an open part of parhoric group schemes.For the torus T K Bruhat and Tits choose a version of the N´eron model as extension. In order tobe consistent with the conventions from [37] we choose the connected
N´eron model T R as an extensionover Spec R .For the unipotent groups U a , the valuations introduced above can be used to define extensions of U a to group schemes U a,k over Spec R for any k ∈ R . For non-multiple roots the pinning identifies theabstract group U a,k ∶= { u ∈ U a ( K )∣ φ a ( u ) ≤ k } ≅ { u ∈ L a ∣∣ u ∣ ≤ k } and this can be equipped with thestructure of a groups scheme isomorphic to Res R a / R G a .For multiple roots this is slightly more complicated to spell out, but again these group schemesalways correspond to free R modules [17, 4.3.9].The open subset of a parahoric group scheme will be of the form ([17, Theorem 3.8.1]): ∏ a ∈ Φ − U a,f ( a ) × T × ∏ a ∈ Φ + U a,f ( a ) . Now a facet Ω ⊂ A defines a valuation of the root system (4.6.26) f ( a ) ∶= inf { k ∈ R ∣ a ( x ) + k ≥ ∀ a ∈ Ω } , here we used our Chevalley-Steinberg valuation, which defines an isomorphism − φ ∶ A ≅ X ∗ ( S ) R .The product described above carries a birational group law ([33, Propostion 5.12]) and thus onecan use [14, Theorem 5] to construct a the group scheme G Ω (denoted by G ○ Ω in [17]), containing theproduct as an open neighborhood of the identity. For our computations this is sufficient as these onlyuse the Lie-algebra of G .
5. Appendix: Basic results on
Bun G As in Section 3 we fix a smooth projective geometrically connected curve C over an algebraicallyclosed field k and G → C a parahoric Bruhat-Tits group scheme. In this appendix we collect someresults on the stack of G -bundles Bun G for which we could not find a reference.The basic tool will be the Beilinson–Drinfeld Grassmannian GR G i.e., the ind-projective schemerepresenting the functor of G -bundles together with a trivialization outside of a finite divisor on C :GR G ( T ) = ⎧⎪⎪⎨⎪⎪⎩(E , D, φ ) RRRRRRRRRRR E ∈ Bun G ( T ) , D ∈ C ( d ) ( T ) for some d φ ∶ E∣ C × T − D ≅ —→ G × C ( C × T − D ) ⎫⎪⎪⎬⎪⎪⎭ . It comes equipped with a natural forgetful maps p ∶ GR G → Bun G ,
5. Appendix: Basic results on Bun G
5. Appendix: Basic results on Bun G supp ∶ GR G → ∐ d C ( d ) . For Bruhat-Tits groups it will be useful to consider the open subfunctor, where the divisor D does not intesect some fixed finite subset of C , e.g., the ramification points of the group G . Given S ⊂ C finite, we will denote by GR G ,C − S ∶= supp − ( ∐( C − S ) ( d ) ) , i.e. the ind-scheme parametrizing G bundles together with a trivialization outside of a divisor D that is disjoint form S . Similarly wedenote by Gr G ,x ∶= supp − ( x ) , the space parametrizing G -bundles equipped with a trivialization on C − { x } .The following is a probably well-known geometric version of a weak approximation theorem: Proposition 5.1.
For any Bruhat-Tits scheme
G → C and any finite subscheme S ⊂ C the naturalforgetful map p C − S ∶ GR G ,C − S → Bun G is surjective, i.e., after possibly passing to a flat extension, any G -bundle admits a trivialization on an open neighborhood of S .Proof. For S = ∅ this follows from a theorem of Steinberg and Borel-Springer [12], stating that for anyalgebraically closed field K any G -bundle on C K is trivial over the generic point K ( C ) . As any suchtrivialization is defined over an open subset this allows to deduce that p ∶ GR G → Bun G is surjective inthe fppf-topology.To deduce the result for p C − S we can argue as in [27, Theorem 5]: As GR G ,C − S ⊂ GR G is a denseopen subfunctor, it follows that the generic point of any connected component of Bun G is in theimage of p C − S . Let E ∈
Bun G be any bundle. For the inner form G E ∶= Aut G (E) of G we can applythe same reasoning and find that the image of the map GR G E ,C − S → Bun G E ≅ Bun G also containsan open subset of every connected component. In particular there exist G -bundles E ′ in the imagethat are also in the image of p C − S . For such any such bundle E ′ there exist divisors D , D on C − S such that E ′ ∣ C − D ≅ G∣ C − D and E ′ ∣ C − D ≅ E∣ C − D . Composing these isomorphisms we find that E∣ C −( D ∪ D ) ≅ G∣ C −( D ∪ D ) , i.e., E is also in the image of p C − S . Thus p C − S is surjective on geometricpoints. As it is the restriction of a flat morphism to an open subfunctor this implies thst it is againfppf surjective. ◻ To formulate properties of the set of connected components of Bun G we will need some more notation.The generic point of C will be denoted by η = Spec k ( C ) and η will be a geometric generic point. From[37, Theorem 0.1] we know that for any x ∈ C there is a natural isomorphism π ( Gr G ,x ) ≅ π (G η ) I where I = Gal ( K x / K x ) is the inertia group at x and π (G η ) is the algebraic fundamental group, i.e.,the quotient of the coweight lattice by the coroot lattice of G η .We denote by X ∗ (G) = Hom (G , G m,C ) the group of characters of G . As G is a smooth groupscheme with connected fibers X ∗ (G) ≅ X ∗ (G η ) = X ∗ (G η ) Gal ( η / η ) . As any character χ ∶ G → G m definesa morphism Bun G → Pic X given by E ↦ E( χ ) ∶= E × G G m it defines a degree d ∶ Bun G → X ∗ (G) ∨ by E ↦ deg (E( χ )) and we will denote by Bun d G ⊂ Bun G the substack of bundles of degree d , which isopen and closed because the degree of line bundles is locally constant in families. Proposition 5.2. (1)
The natural map π (G η ) Gal ( η / η ) → π ( Bun G ) is surjective. (2) For any d ∈ Hom ( X ∗ (G) , Z ) the stack Bun d G has finitely many connected components.Proof. The first part follows from the surjectivity of GR G → Bun G and the description of the connectedcomponents π ( Gr G ,x ) ≅ π (G η ) I from [37] as follows. Let GR d G denote the components of GR G mapping to C ( d ) . As the projection GR d G is ind-projective every connected component intersects thefibers over the diagonal C ⊂ C ( d ) . The preimage of the diagonal is isomorphic to GR G . For GR G the fiber wise isomorphism π ( Gr G ,x ) ≅ π (G η ) I is given by the Kottwitz homomorphism which byconstruction is induced from a Galois-equivariant map π (G η ) to the sheaf of connected components . Heinloth, GIT-stability for stacks 35J. Heinloth, GIT-stability for stacks 35 of the fibers of p . Thus this induces a surjection π (G η ) Gal ( η / η ) → π ( GR G ) → π ( Bun G ) , which proves(1).This implies (2), because the map Hom ( X ∗ (G) , Z ) I → Hom ( X ∗ (G) , Z ) I induces an isomorphismup to torsion. ◻ References [1] D. Abramovich, A. Corti, and A. Vistoli,
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