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Moduli Spaces for Principal Bundles in Arbitrary Characteristic

Tomas L. Gomez, Adrian Langer, Alexander H.W. Schmitt, Ignacio Sols

Abstract

In this article, we solve the problem of constructing modulispaces of semistable principalbundles (and singular versions of them) over smooth projective varieties over algebraically closedground fields of positive characteristic. Together with thesemistable reduction theorem that hasjust been proved by Jochen Heinloth, these findings bring Ramanathan’s result on the existenceof projective moduli spaces of semistable principal bundles with connected semisimple structuregroup on a smooth projective curve to arbitrary characteristic.

Contents

1 Introduction 21.1 Faithful representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 41.2 The semistable reduction theorem . . . . . . . . . . . . . . . . . . .. . . . . . . . 6

2 Preliminaries 72.1 GIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 SomeG-linearized sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 An extension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 15

3 Fundamental results on semistable singular principal bundles 163.1 The basic formalism of singular principal bundles . . . . .. . . . . . . . . . . . . . 163.2 Semistable rational principalG-bundles . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Boundedness of the family of semistable singular principal bundles with fixed numerical data 19

4 Dispo sheaves 204.1 The basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 204.2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 224.3 S-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 234.4 The main theorem on dispo sheaves . . . . . . . . . . . . . . . . . . . .. . . . . . 244.5 The proof of the main theorem on dispo sheaves . . . . . . . . . .. . . . . . . . . . 24

5 The proof of the main theorem 375.1 Associated dispo sheaves . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 375.2 S-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 38

5.3 Moduli spaces forδ -semistable pseudoG-bundles . . . . . . . . . . . . . . . . . . . 395.4 Semistable singular principal bundles . . . . . . . . . . . . . .. . . . . . . . . . . 425.5 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 45

1

http://arxiv.org/abs/math/0506511v2

2 T.L. Gomez, A. Langer, A.H.W. Schmitt, I. Sols

6 Examples 456.1 Stable orthogonal and symplectic sheaves . . . . . . . . . . . .. . . . . . . . . . . 456.2 Stable algebra sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 46

1 Introduction

In this article, we introduce a formalism for dealing with principal bundles on projective manifoldsdefined over an algebraically closed ground field of arbitrary characteristic that enables us to constructand partially compactify the moduli space of Ramanathan-stable principal bundles. As a major ap-plication, we obtain the solution of the long-standing problem of constructing the projective modulispace of semistable principal bundles (with semisimple structure group) on a (smooth, projective)algebraic curve over a ground field of positive characteristic.

The theory of (semi)stable principalG-bundles starts for the structure groupG= GLr(C) as thetheory of (semi)stable vector bundles. Based on his development of Geometric Invariant Theory,David Mumford proposed the notion of a (semi)stable vector bundle on a Riemann surface [25]. Atabout the same time, Narasimhan and Seshadri made the fundamental discovery that stable vectorbundles on the Riemann surfaceX are precisely those arising from irreducible unitary representationsof the fundamental groupπ1(X) [27]. (Recall that the relationship between vector bundlesand repre-sentations of the fundamental groups was first investigatedby A. Weil [39].) Finally, Seshadri gavethe GIT construction of the moduli space of stable vector bundles on a Riemann surface together withits compactification by S-equivalence classes of semistable vector bundles [36]. This constructioneasily generalizes to ground fields of arbitrary characteristic.

Since its beginnings, the study of stableG-bundles has widely developed and interacted with otherfields. The scope of the theory has been progressively enlarged by eliminating limitations on the “threeparameters” of the theory, i.e., the structure groupG, the base manifoldX, and the ground fieldk. First,in the work of Gieseker [8] and Maruyama [24], the theory of stable vector bundles was enlarged toa theory of semistable torsion free sheaves on projective manifolds over fields of characteristic zero.Later, Simpson brought this theory into its final form [37]. In the work [19] and [20], the barriers ofextending Simpson’s results to fields of positive characteristic were finally removed. The argumentsgiven there improve the formalism even in characteristic zero.

At the time when the results of Gieseker and Maruyama were published, Ramanathan had alsotreated the theory of principalG-bundles on a compact Riemann surfaceX for an arbitrary connectedreductive structure groupG. In the paper [30], he introduced the notion of (semi)stability for a princi-palG-bundleP on the Riemann surfaceX and generalized the results of the paper [27], i.e., linked thetheory of semistable principal bundles onX to the study of representations of the fundamental groupin a compact real formK of G. More important to us is the main result of his PhD thesis, finished atthe Tata Institute in 1976. There, Ramanathan provides an ingenious GIT construction for the modulispace of semistable principalG-bundles on a compact Riemann surfaceX. Due to the untimely deathof the author, this important result appeared in the posthumous publication [31]. At that time, thesubject had become of general interest to mathematicians and physicists.

In the recent papers [10], [11], [32], and [34] two independent—although related—methods forgeneralizing Ramanathan’s theory to the case of higher dimensional base manifolds defined over thecomplex numbers were presented. More precisely, the modulispace of Ramanathan stable bundleswas constructed and compactified with certain “generalized” principal bundles, satisfying a Giesekertype semistability condition.

It thus seemed natural to join the forces of the four authors to cope with the problem of bringing

Principal Bundles in Arbitrary Characteristic 3

these recent developments to base fields of arbitrary characteristic. In the present paper, we rewritethe theory of the paper [32] from scratch. We will see that theresults of that paper are, in fact, truein positive characteristic. Furthermore, some of the fundamental discoveries of the papers [10], [11],and [34] also remain valid over any algebraically closed field. Using the boundedness of Ramanathan-semistable rational principalG-bundles which was obtained by the third author in [21], we manageto construct our moduli spaces as open subschemes of the moduli spaces of “δ -semistable pseudoG-bundles”. In a separate publication [9], we will explain howthe approach via the adjoint representationof a connected reductive groupG of [31] and [11], or more generally via faithless representations withkernel in the the center ofG, may be generalized to positive characteristic.

The main change of philosophy which made the progress possible is the following. Classically,as suggested by the work of Ramanan and Ramanathan [29], one studied semistability of principalbundles by relating it to the semistability of associated vector bundles. This works well in characteris-tic zero but makes the assumption of sufficiently high characteristic of the base field necessary whileworking over fields of positive characteristic. In the more recent work quoted above, we linked thesemistability of a principal bundle to the semistability ofan associateddecoratedvector bundle. Un-fortunately, the theory of polynomial representations of the general linear group is more complicatedin positive characteristic (see the book [12]), so that the set-up of [32] and [34] cannot be directlycopied. Nevertheless, the basic idea of that work makes perfect sense over fields of positive char-acteristic. Thanks to the results of [19] and [20], one may adapt the fundamental arguments fromcharacteristic zero.

Let us introduce a piece of notation, so that we may state our result in a precise form. In thispaper, we will deal with moduli functors of the form

M(s)s: Schk −→ Set

S 7−→

Equivalence classes of familiesof (semi)stable objects

.

In each case, we define S-equivalence on the set of isomorphism classes of semistable objects (e.g.,semistable sheaves or principalG-bundles with fixed numerical data) which, restricted to stable ob-jects, reduces to isomorphism. Assuming we have the moduli functor and S-equivalence, we introducethe following convenient terminology. Acoarse moduli schemefor the functors M(s)s consists of ascheme Mss, an open subscheme Ms ⊆ Mss, and natural transformations of functors

ϑ (s)s: M(s)s −→ hM(s)s

with the following properties:

1. The space M(s)s corepresents M(s)s with respect toα = ϑ (s)s. It does so uniformly, if Char(k)>0, and universally, if Char(k) = 0.(See [18], Definition 2.2.1. Observe that “uniformly” refers to the base change property forflatmorphismsϕ in that definition.)

2. The mapϑs(k):Ms(k)−→Ms(k) is a bijection between the set of isomorphism classes of stableobjects and the closed points of Ms.

3. The mapϑss(k):Mss(k)−→Mss(k) induces a bijection between the set of S-equivalence classesof semistable objects and the closed points of Mss.

The difference between positive and zero characteristic inthe above definition comes from our use ofGeometric Invariant Theory, as GIT quotients in positive characteristic are not necessarily universal


categorical. ForG= GL(V), one can in fact show that, in positive characteristic, the moduli space ofstable sheaves universally corepresents the moduli functor (see [19], Theorem 0.2). This follows fromthe fact that stable sheaves are simple. However, even in characteristic zero, the sheaves correspondingto stable principalG-bundles on a curve are no longer simple (see [30], Remark 4.1), so this prooffails in general. We now come to the more detailed presentation of the contents of our work.

1.1 Faithful representations

Let G be a connected semisimple group. Fix a faithful representation ρ :G −→ GL(V), such thatρ(G) ⊆ SL(V). In characteristic zero, a theory for semistable singularG-bundles based on such arepresentation was developed in [32] and [34]. However, some characteristic zero gadgets such asthe Reynolds operator, some properties of the instability flag, and normal forms for homogeneouspolynomial representations were used. In this paper, we will rewrite the theory from the beginning,such that it becomes independent of the characteristic of the base field and works without decoratedsheaves.

We will look at pairs(A ,τ) with a torsion free ØX-moduleA that has rank dimk(V) and trivialdeterminant and a homomorphism of ØX-algebrasτ :S ym⋆(A ⊗V)G −→ ØX which is non-trivial inthe sense that the induced sectionσ :X −→ S pec(S ym⋆(A ⊗V)G) is not the zero section. Such apair is called apseudo G-bundle, and if, furthermore,σU(U) ⊂ I som(V ⊗ØU ,A

∨|U)/G, U := UA

being the maximal open subset whereA is locally free, we speak of asingular principal G-bundle1.In the case of a singular principalG-bundle(A ,τ), we get a principalG-bundleP(A ,τ) overU ,defined by means of base change:

P(A ,τ) //

I som(V ⊗ØU ,A∨|U)

Uσ|U

// I som(V ⊗ØU ,A∨|U )/G.

We now define the notion of semistability for a singular principal G-bundle (A ,τ). For this, letλ :Gm(k) −→ G be a one parameter subgroup ofG. This yields a parabolic subgroupQG(λ ) (see(4) below) and a weighted flag(V•(λ ),α•(λ )) in the vector spaceV (see Section 2.1). Then, areduction of(A ,τ) to λ is a sectionβ :U ′ −→ P(A ,τ)|U ′/QG(λ ) over an open subsetU ′ ⊆ Uwith codimX(X \U ′) ≥ 2. This defines a weighted filtration(A•(β ),α•(β )) of A . Here,α•(β ) =(αt , ...,α1) if α•(λ ) = (α1, ...,αt), and the filtrationA•(β ): 0( A1 ( · · · ( At ( A is obtained asfollows. The section

β ′: U ′ β−→ P(A ,τ)|U ′/QG(λ ) → I som(V ⊗ØU ′,A ∨

|U ′)/QGL(V)(λ )

yields a filtration0( A

′1 ( · · ·( A

′t ( A

∨|U ′

of A ∨|U ′ by subbundles with rk(A ′

i ) = dimk(Vi), i = 1, ..., t. This is becauseQGL(V)(λ ) is the GL(V)-stabilizer of the flagV•(λ ) and, thus,I som(V ⊗ØU ′,A ∨

|U ′)/QGL(V)(λ ) −→U ′ is the bundle of flagsin the fibers ofA ∨

|U ′ having the same dimensions as the members of the flagV•(λ ). We defineA ′′i :=

ker(A|U ′ −→ A ′∨t+1−i), i = 1, ..., t, so that we obtain a filtration

0( A′′

1 ( · · ·( A′′

t ( A|U ′

1Here, we deviate from the original terminology in [32] and [34].


of A|U ′ by subbundles. Letι :U ′ −→X be the inclusion and defineAi as the saturation ofA ∩ ι⋆(A ′′i ),

i = 1, ..., t. This is the filtration we denote byA•(β ). It is worth noting that, ifλ ′ = g·λ ·g−1 for someg∈G, then any reduction toλ may also be interpreted as a reduction toλ ′. Now, we say that a singularprincipal G-bundle(A ,τ) is (semi)stable, if for every one parameter subgroupλ :Gm(k) −→ G andevery reductionβ of (A ,τ) to λ , we have

M(A ,τ ;β ) := M(A•(β ),α•(β )

)()0,

where, for every weighted filtration(A•,α•) of A , we set

M(A•,α•

):=

t

∑i=1

αi ·(rkAi ·P(A )− rkA ·P(Ai)

).

Finally, there is a notion of S-equivalence which will be explained in Section 5.2 and Remark 5.4.3.We have the implications

P(A ,τ) is Ramanathan-stable=⇒ (A ,τ) is stable=⇒ (A ,τ) is semistable=⇒ P(A ,τ) is Ramanathan-semistable.

More precisely, in our language, Ramanathan’s notion of (semi)stability becomes

L(A ,τ ;β

):= L

(A•(β ),α•(β )

):=

t

∑i=1

αi ·(rkAi ·deg(A )− rkA ·deg(Ai)

)(≥)0 (1)

for every one parameter subgroupλ :Gm(k) −→ G and every reductionβ of (A ,τ) to λ . Here, degstands for the degree with respect to the chosen polarization.

Remark.It is easy to check from the definition that the condition of semistability has to be verifiedonly for the indivisible one parameter subgroups that definemaximal parabolic subgroups.

For a fixed Hilbert polynomialP, we define the moduli functors

M(s)sP (ρ): Schk −→ Set

S 7−→

Equivalence classes of families of(semi)stable singular principalG-bundles(A ,τ), such thatP(A ) = P

.

MAIN THEOREM. The coarse moduli space for the functorsM(s)sP (ρ) exists as a quasi-projective

schemeMssP (ρ).

This moduli space contains the moduli space for Ramanathan-stable principal bundles (whoseassociated vector bundle has Hilbert polynomialP) as an open subscheme. In particular, we haveconstructed the moduli spaces for Ramanathan-stable principal bundles in any characteristic.

The moduli space MssP (ρ) will be constructed inside a larger moduli space Mδ-ss

P (ρ) of δ -semi-stable pseudoG-bundles, so that it always comes with a natural compactification.

Example.Let us mention some examples which will be explained in more detail in Section 6.i) If G is one of the classical groups On(k), SOn(k), and Spn(k), we recover, for the natural

choices ofρ , the moduli of semistable orthogonal sheaves, special orthogonal sheaves, and symplecticsheaves, respectively, constructed in [10]. LetG be for instance Spn(k). Giving a singular principal


G-bundle onX, for the fundamental representationρ :Spn(k) → SLn(k), is equivalent to giving asymplectic sheaf(A ,ϕ

A) of rankn, i.e., a torsion free sheafA of this rank, decorated with a skew-

symmetric homomorphismϕA

:A ⊗A −→ ØX, such thatA −→ A ∨ is a monomorphism. For anysubsheafA ′ ⊆ A , theorthogonal complementA ′⊥ ⊆ A is defined as the kernel ofA −→ A ∨ −→A ′∨. Our notion of (semi)stability then translates into the condition that, for any isotropic subsheaf0⊆ A ′ ⊆ A ′⊥ ⊆ A , one has

P(A ′)+P(A ′⊥)()P(A ).

For ρ :On(k) → GLn(k), one obtains the analogous definition and moduli of semistable orthogonalsheaves; and forρ :SOn(k) → SLn(k) one obtains the same definition of semistability, and corre-sponding moduli, but for special orthogonal sheaves, i.e.,for orthogonal sheaves(A ,ϕ

A) together

with an isomorphismψA

:detA ∼= ØX, such thatψ2A

= det(ϕA). In all these cases, the moduli space

MssP (ρ) is projective (see Example 6.1).

ii) For the realizationsρ of simple groups as groups of automorphisms of a non-associative algebra(V,ϕ :V ⊗V → V)—such as the algebra of octonions forG2—, singular principalG-bundles areϕ-algebra sheaves(A ,ϕ

A) of rank dimk(V), i.e., torsion free sheavesA of this rank, decorated with

a homomorphismϕA :A ⊗A → A ∨∨, such that, at any pointx ∈ X whereA is locally free, thealgebra structureϕA (x) is isomorphic toϕ . Its (semi)stability translates to

∑i∈Z

(rk(Ai) ·P(A )− rk(A ) ·P(Ai)

)()0,

for any balanced algebra filtrationA• ⊂ A (i.e.,Z-indexed filtration with∑ i(rkAi − rkAi−1) = 0andϕA (Ai ,A j)⊆ Ai+ j ). We thus obtain quasi-projective moduli spaces for these objects.

1.2 The semistable reduction theorem

The projectivity of the moduli spaces is not built into our new approach. The remaining questionis thus under which assumptions (on the representation or the characteristic of the base field), themoduli space Mss

P (ρ) is projective. Since any connected semisimple group is overk isomorphic toone defined over the integers, one may assume thatG is itself defined over the integers. Then, there isalso a faithful representationρ :G−→ GL(V) that is defined over the integers. Under this assumption,one may develop an elegant formalism which provides projective moduli spaces in any dimension,provided that the characteristic of the base field is either zero or greater than a constant which dependson ρ . These results will appear in [9]. As remarked before, the moduli spaces will also be projective,if G is one of the classical groups andρ its standard representation.

Until very recently, the most general result in that direction was contained in the work of Balaji andParameswaran [1] where the existence of moduli spaces of semistable principalG-bundles on a smoothprojective curve was established under the assumptions that G is semisimple and the characteristic ofthe base field issufficiently large. Heinloth, however, managed in [16] to adapt Langton’s algorithm[22] to the setting ofG-bundles. His new approach is to work with the affine Graßmannian (see [7]for a discussion of this object in positive characteristic), and the precise result is the following.

THEOREM (HEINLOTH). Suppose X is a curve, G is a semisimple group and that one of thefollowingconditions holds:

a. The simple factors of G are of type A, D, E.

b. Char(k) 6= 2 and none of the simple factors of G is of type G2.


c. Char(k) 6∈ 2,3,5,7.

Then, given a semistable principal G-bundleGK over X×Spec(K) where K is the spectrum of thecomplete discrete valuation ring R, there exists a finite extension R⊆ R′, such that the pullbackGK′ ofGK to X×Spec(K′), K′ being the fraction field of R′, extends to a semistable principal G-bundleGR′

over X×Spec(R′).

This theorem applies, in particular, to any simple group of typeE in any characteristic.

COROLLARY. Under the assumption of the above theorem, the moduli spaceMssP (ρ) is projective. In

particular, the moduli spaceMss(G, t) of semistable principal G-bundles (as defined by Ramanathan)of “topological type” t ∈ π1(G) exists as a projective scheme over k.

This is a generalization of the work of Ramanathan [31] and Balaji and Parameswaran [1].

Notation

We work over the algebraically closed fieldk of characteristicp ≥ 0. A schemewill be a locallynoetherian scheme overk. For a vector bundleE over a schemeX, we setP(E ) := Proj(S ym⋆(E )),i.e.,P(E ) is the projective bundle of hyperplanes in the fibers ofE . An open subsetU ⊆ X is saidto bebig, if codimX(X \U) ≥ 2. The degree deg(E ) and the Hilbert polynomialP(E ) of a torsionfree coherent ØX-moduleE are taken with respect to the fixed polarization ØX(1). We set[x]+ :=max0,x, x∈R.

2 Preliminaries

In this section, we collect different results which will be needed throughout the construction of themoduli space for singularG-bundles for a semisimple groupG via a faithful representationG −→GL(V).

2.1 GIT

We recall some notation and results from Geometric Invariant Theory. LetG be a reductive groupover the fieldk and κ :G −→ GL(W) a representation on the finite dimensionalk-vector spaceW.This yields the action

α :G×W −→ W

(g,w) 7−→ κ(g)(w).

Recall that a one parameter subgroup is a homomorphism

λ :Gm(k)−→ G.

Such a one parameter subgroup defines a decomposition

W =⊕

γ∈ZWγ

withWγ =

w∈W

∣∣κ(λ (z))(w) = zγ ·w, ∀z∈Gm(k), γ ∈ Z.


Let γ1 < · · ·< γt+1 be the integers withWγ 6= 0 andγ•(λ ) := (γ1, ...,γt+1). We define the flag

W•(λ ) : 0 (W1 :=Wγ1 (W2 :=Wγ1 ⊕Wγ2 ( · · ·(Wt :=Wγ1 ⊕·· ·⊕Wγt (Wt+1 :=W

and the tupleα•(λ ) := (α1, ...,αt) of positive rational numbers with

αi :=γi+1− γi

dimk(W), i = 1, ..., t.

If λ is a one parameter subgroup of the special linear group SL(W), we will refer to(W•(λ ),α•(λ ))as theweighted flag ofλ . For a pointw∈W \0, we define

µκ(λ ,w) := max

γi∣∣w has a non-trivial component inWγi , i = 1, ..., t +1

.

Note that, forG= SL(V) andλ j :Gm(k)−→ G, j = 1,2,

µκ(λ1,w) = µκ(λ2,w), if(V•(λ1),α•(λ1)

)=

(V•(λ2),α•(λ2)

). (2)

(See [26], Proposition 2.7, Chapter 2. Note that we take the weighted flags inV and not inW.)Suppose we are given a projective schemeX, a G-action σ :G×X −→ X, and a linearization

σ :G×Ł −→ Ł of this action in the line bundle Ł. For a pointx∈ X and a one parameter subgroupλ ,we get the point

x∞(λ ) := limz→∞

σ(λ (z),x

).

This point is fixed under the actionGm(k)×X −→ X, (z,x) 7−→ σ(λ (z),x). Therefore,Gm(k) acts onthe fiber Ł〈x∞(λ )〉. This action is of the forml 7−→ zγ · l , z∈Gm(k), l ∈ Ł〈x∞(λ )〉, and we set

µσ (λ ,x) :=−γ .

For a representationκ of G as above, we obtain the action

σ :G×P(W∨) −→ P(W∨)(g, [w]

)7−→

[κ(g)(w)

]

together with an induced linearizationσ in ØP(W∨)(1). One checks that

µκ(λ ,w) = µσ(λ , [w]

), ∀w∈W \0,λ :Gm(k)−→ G. (3)

Finally, we recall that a one parameter subgroupλ :Gm(k)−→ G gives the parabolic subgroup

QG(λ ) :=

g∈ G∣∣ lim

z→∞λ (z) ·g·λ (z)−1 exists inG

. (4)

The unipotent radical ofQG(λ ) is the subgroup

Ru(QG(λ )

):=

g∈ G

∣∣ limz→∞

λ (z) ·g·λ (z)−1 = e.

Remark2.1.1. In the book [38], one defines the parabolic subgroup

PG(λ ) :=

g∈ G∣∣ lim

z→0λ (z) ·g·λ (z)−1 exists inG

,

i.e.,PG(λ ) = QG(−λ ).

Therefore, every parabolic subgroup ofQ is of the shapeQG(λ ) for an appropriate one parametersubgroupλ of G. We have chosen a different convention, because it is compatible with our GITnotation.


Actions on homogeneous spaces. —Let H be a reductive algebraic group,G a closed reductivesubgroup, andX := H/G the associated affine homogeneous space. Then, the following holds true.

PROPOSITION2.1.2. Suppose that x∈X is a point andλ :Gm(k)−→ H is a one parameter subgroup,such that x0 := limz→∞ λ (z) ·x exists in X. Then, x∈ Ru(QH(λ )) ·x0.

Proof. We may assumex0 = [e], so thatλ is a one parameter subgroup ofG. Define

Y :=

y∈ X∣∣ lim

z→∞λ (z) ·y= x0

.

This set is closed and invariant under the action ofRu(QH(λ )). Note that viewingX as a varietywith Gm(k)-action, x0 is the unique point inY with a closedGm(k)-orbit, and by the first lemmain Section III of [23] (or Lemma 8.3 in [3], or 3.1 in [17]), there is aGm(k)-equivariant morphismf :X −→ Tx0(X) which mapsx0 to 0 and is etale inx0. Obviously, f mapsY to

v∈ Tx0X

∣∣ limz→∞

λ (z) ·v= 0= uH(λ )/uG(λ )⊂ h/g. (5)

Here,uH(λ ) anduG(λ ) are the Lie algebras ofRu(QH(λ )) andRu(QG(λ )), respectively, andh andg are the Lie algebras ofH andG, respectively. Note thath andg receive theirG-module structuresthrough the adjoint representation ofG, and, moreover, by definition,

uH(λ ) =

v∈ h∣∣ lim

z→∞λ (z) ·v= 0

.

This yields the asserted equality in (5).On the other hand, the dimension ofuH(λ )/uG(λ ) equals the one of theRu(QH(λ ))-orbit of x0 at

X. By [17], Theorem 3.4,f mapsY isomorphically ontouH(λ )/uG(λ ). Therefore, sinceRu(QH(λ )) ·x0 ⊆Y, the subsetY must agree with the closed orbitRu(QH(λ )) ·x0, and we are done.

The proof of the above result was communicated to us by Kraft and Kuttler (cf. [34]). Its purposeis to characterize one parameter subgroups ofG among the one parameter subgroups of SL(V), givena faithful representationρ :G−→ GL(V).

Some specific quotient problems. — A key of understanding classification problems for vectorbundles together with a section in an associated vector bundle is to study the representation definingthe associated vector bundle. In our case, we have to study a certain GIT problem which we will nowdescribe.

As before, we fix a representationρ :G−→ GL(V) on the finite dimensionalk-vector spaceV. Welook at the representation

R:GLr(k)×G −→ GL(kr ⊗V)

(g,g′) 7−→(

w⊗v∈ kr ⊗V 7−→ (g·w)⊗ρ(g′)(v)).

The representationRprovides an action ofG×GLr(k) on(V ⊗kr)∨ = Hom(kr ,V∨) and P

(Hom(kr ,V∨)∨

)

and induces a GLr(k)-action on the categorical quotients

H := Hom(kr ,V∨)//G and H := P(Hom(kr ,V∨)∨

)//G=

(H\0)//Gm(k).


The coordinate algebra ofH is Sym⋆(kr ⊗kV)G. Fors> 0, we set

Ws :=s⊕

i=1

Ui , Ui :=(Symi(kr ⊗kV)G)∨, i ≥ 0.

If s is so large that⊕s

i=0 Symi(kr ⊗kV)G contains a set of generators for the algebra Sym⋆(kr ⊗kV)G,then we have a GLr(k)-equivariant surjection of algebras

Sym⋆(W

∨s

)−→ Sym⋆(kr ⊗kV)G,

and, thus, a GLr(k)-equivariant embedding

ιs:H →Ws.

SetI := Isom(kr ,V∨)/G (∼= GLr(k)/G). This is a dense open subset ofH. The semistability ofpointsιs(h), h∈H, with respect to the action of thespeciallinear group SLr(k) is described by thefollowing result.

LEMMA 2.1.3. i)Every pointιs(i), i ∈ I, is SLr(k)-polystable.ii) A point ιs(h), h∈H\ I, is notSLr(k)-semistable.

Proof (compare Lemma4.1.1in [34]). Ad i). We choose a basis forV∨. This provides us with the(SLr(k)×G)-invariant functiond:Hom(kr ,V∨) −→ k, f 7−→ det( f ), which descends to a (non con-stant) function onH, called againd. For any i ∈ I, we clearly haved(ιs(i)) 6= 0, so thatιs(i) isSLr(k)-semistable. Furthermore, for anyf ∈ Isom(kr ,V∨), the(SLr(k)×G)-orbit of f is just a levelsetd−1(z) for an appropriatez∈ Gm(k). In particular, it is closed. The image of this orbit is theSLr(k)-orbit of i := [ f ] in H which is, therefore, closed. Sinceιs is a closed, SLr(k)-equivariantembedding, the orbit ofιs(i) is closed, too.

Ad ii). It is obvious from the construction that the ring of SLr(k)-invariant functions onH isgenerated byd. This makes the asserted property evident.

A key result is now the following.

PROPOSITION2.1.4. Fix a basis for V in order to obtain a faithful representationρ :G−→ GLr(k)and aGLr(k)-equivariant isomorphism

ϕ :GLr(k)/G−→ Isom(kr ,V∨)/G.

Suppose that x= ιs(i) for some i= ϕ(g) ∈ I. Then, for a one parameter subgroupλ :Gm(k) −→SLr(k), the following conditions are equivalent:

i) µκs(λ ,x) = 0, κs being the representation ofSLr(k) onWs.

ii) There is a one parameter subgroupλ ′:Gm(k)−→ g·G ·g−1 with

(V•(λ ),α•(λ )

)=

(V•(λ ′),α•(λ ′)

).

Proof. We may clearly assumeg= Er . We first show “ii)=⇒i)”. SinceG is the GLr(k)-stabilizer ofx, we haveµ(λ ′,x) = 0 for any one parameter subgroupλ ′:Gm(k)−→ G. Now, Formula (2) impliesthe claim.


We turn to the implication “i)=⇒ii)”. By Lemma 2.1.3, i), there exists an elementg′ ∈ SLr(k),such that

x′ := limz→∞

λ (z) ·x= ϕ(g′).

By Proposition 2.1.2, we may chooseg′ ∈ Ru(QSLr(k)(λ )). In particular, the elementg′ fixes the

flag V•(λ ). Sinceλ fixes x′, it lies in g′ ·G · g′−1. Settingλ ′ := g′−1 · λ · g′, we obviously have(V•(λ ),α•(λ )) = (V•(λ ′),α•(λ ′)), andλ ′ is a one parameter subgroup ofG.

Next, we look at the categorical quotient

H= Proj(Sym⋆(kr ⊗kV)G).

For any positive integerd, we define

Sym(d)(kr ⊗kV)G :=∞⊕

i=0

Symid(kr ⊗kV)G.

Then, by the Veronese embedding,

Proj(Sym⋆(kr ⊗kV)G)∼= Proj

(Sym(d)(kr ⊗kV)G).

We can chooses, such that

a) Sym⋆(kr ⊗kV)G is generated by elements in degree≤ s.

b) Sym(s!)(kr ⊗kV)G is generated by elements in degree 1, i.e., by the elements inthe vector spaceSyms!(kr ⊗kV)G.

SetVs :=

⊕

(d1,...,ds):di≥0,∑ idi=s!

(Symd1

((kr ⊗kV)G

)⊗·· ·⊗Symds

(Syms(kr ⊗kV)G

)). (6)

Obviously, there is a natural surjectionVs −→ Syms!(kr ⊗kV)G and, thus, a surjection

Sym⋆(Vs)−→ Sym(s!)(kr ⊗kV)G.

This defines a closed and GLr(k)-equivariant embedding

ιs:H → P(Vs).

We also defineØH

(s!) := ι⋆s(ØP(Vs)(1)

).

Note thatØH

((s+1)!

)= Ø

H

(s!)⊗(s+1). (7)

LEMMA 2.1.5. Let s be a positive integer, such thata)andb) as above are satisfied, and f∈ Hom(kr ,V∨) a G-semistable point. Set h:= ιs([ f ]) and h := ιs([ f ]). Then, for any one parameter subgroupλ :Gm(k)−→ G, we have

µκs(λ ,h) > (= / <) 0 ⇐⇒ µσs(λ ,h)> (= / <) 0.

Here,κs is the representation ofSLr(k) onWs, andσs is the linearization of theSLr(k)-action onHin Ø

H

(s!). In particular, h isSLr(k)-semistable, if and only if f∈ Isom(kr ,V∨).


Proof. Note that we have the following commutative diagram

Hom(kr ,V∨)//G

Gm(k)- quotient

ιs//Ws\0

α

P

(Hom(kr ,V∨)∨

)//G ιs

//P(Vs).

The morphismα factorizes naturally over the quotient with respect to theGm(k)-action onWs whichis given onUi by scalar multiplication withz−i, i = 1, ...,s, z∈ Gm(k). The explicit description ofαis as follows. An element(l1, ..., ls) ∈Ws with

l i :Symi(kr ⊗kV)G −→ k, i = 1, ...,s,

is mapped to the class [⊕

d=(d1,...,ds):di≥0,∑ idi=s!

ld

]:Vs −→ k

with

ld:u1⊗·· ·⊗us ∈ Symd1((kr ⊗kV)G)⊗·· ·⊗Symds

(Syms(kr ⊗kV)G) 7−→ l1(u1)

d1 · ... · ls(us)ds.

With this description, one easily sees

µκs(λ ,h)> (= / <) 0 ⇐⇒ µσs

(λ ,α(h)

)> (= / <) 0

for all λ :Gm(k) −→ SLr(k) and allh∈Ws\0. Together with the above diagram, this implies theclaim.

Let us conclude this section with a formula for theµ-function. Note thatVs is a GLr(k)-submodule of

Ss :=⊕

(d1,...,ds):di≥0,∑ idi=s!

(Symd1

(kr ⊗kV

)⊗·· ·⊗Symds

(Syms(kr ⊗kV)

)).

SinceGm(k) is a linearly reductive group, the weight spaces insideVs with respect to any one param-eter subgroupλ :Gm(k) −→ SLr(k) are the intersection of the weight spaces forλ insideSs with thesubspaceVs.

The moduleSs is a quotient module of(W⊗s!)⊕N, W := kr . Therefore, the weight spaces insideSs

with respect to a one parameter subgroupλ :Gm(k)−→ SLr(k) are the projections of the correspond-ing weight spaces in(W⊗s!)⊕N. The latter may be easily described. Given a one parameter subgroupλ :Gm(k)−→ SLr(k), we obtain the decomposition

W =t+1⊕

i=1

Wi

into eigenspaces and the corresponding weightsγ1 < · · · < γt+1. Set I := 1, ..., t + 1×s! , and for(i1, ..., is!) ∈ I define

Wi1,...,is! :=Wi1 ⊗·· ·⊗Wis.


Then, all the weight spaces inside(W⊗s!)⊕N may be written as direct sums of some subspaces of theform (Wi1,...,is! )

⊕N.If we let W•

i1,...,is! be the image of(Wi1,...,is! )⊕N in Ss andW⋆

i1,...,is! the intersection ofW•i1,...,is! with

Vs, (i1, ..., is!) ∈ I , we understand that all the weight spaces insideVs are direct sums of subspaces ofthe formW⋆

i1,...,is! , (i1, ..., is!) ∈ I .This enables us to compute the weights inVs in terms of the weighted flag(W•(λ ),α•(λ )). In-

deed, we defineWi1 · · · · ·Wis!

as the image of(Wi1 ⊗·· ·⊗Wis! )⊕N under the projection map(W⊗s!)⊕N −→ Ss and

Wi1 ⋆ · · ·⋆Wis! :=(Wi1 · · · · ·Wis!

)∩Vs, (i1, ..., is!) ∈ I .

Altogether, we compute for a point[l ] ∈ P(Vs) and a one parameter subgroupλ :Gm(k) −→ SLr(k)with weighted flag(W•(λ ),α•(λ )) as before

µσs

(λ , [l ]

)=−min

γi1 + · · ·+ γis!

∣∣(i1, ..., is!) ∈ I : l|Wi1⋆···⋆Wis!6≡ 0

. (8)

Good quotients. — Suppose the algebraic groupG acts on the schemeX. In the framework of hisGIT, Mumford defined the notion of agood quotient[26]. Moreover, auniversal (uniform) categoricalquotientis a categorical quotient(Y,ϕ) for X with respect to the action ofG, such that, for every (everyflat) base changeY′ −→Y, Y′ is the categorical quotient forY′×Y X with respect to the inducedG-action. In particular,Y×Z is the categorical quotient forX×Z with respect to the givenG-action onthe first factor.

Example2.1.6. i) Mumford’s GIT produces good, uniform (universal, if Char(k) = 0) categoricalquotients (see [26], Theorem 1.10, page 38).

ii) If G andH are algebraic groups and we are given an action ofG×H on the schemeX, suchthat the good, universal, or uniform categorical quotientsX//G and(X//G)//H exist, then

X//(G×H) = (X//G)//H.

This follows from playing around with the universal property of a categorical quotient. For goodquotients, one might also use the argument from [28].

The following lemma is well known (see [8], Lemma 4.6, and [31], Lemma 5.1). We recallthe proof for the convenience of the reader (and because there is a small additional difficulty withRamanathan’s proof: It uses the Reynolds operator2).

LEMMA 2.1.7. Let G be a reductive linear algebraic group acting on the schemes X1 and X2, andlet ψ :X1 −→ X2 be an affine G-equivariant morphism. Suppose that there exists a good quotientX2 −→ X2//G. Then, there also exists a good quotient X1 −→ X1//G, and the induced morphismψ :X1//G−→ X2//G is affine. Moreover, the following holds:

1. If ψ is finite, thenψ is also finite.

2. If ψ is finite and X2//G is a geometric quotient, then X1//G is also a geometric quotient.

2This is only one instance where the Reynolds operator had to be replaced.


Proof. If X2//G is affine, thenX1 andX2 are also affine, and the existence ofX1//G is well known(see [26], Theorem A.1.1). In general, the existence ofX1//G affine overX2//G is an easy exercise ongluing affine quotients (see [31], proof of Lemma 5.1). The only non-trivial statement in the lemmais 1. It follows from the last part of [26], Theorem A.1.1. Thepoint is that, ifψ is finite, thenX1 isthe spectrum of the sheafψ⋆(ØX1) of ØX2-algebras which is coherent as an ØX2-module. Hence, bythe theorem cited above,(ψ⋆ØX1)

G is a coherent ØX2//G-module, which is also an ØX2//G-algebra thespectrum of which isX1//G. Hence,ψ is a finite morphism.

2.2 SomeG-linearized sheaves

Assume thatρ :G−→ GL(V) is any representation. LetB be a scheme andA a coherent ØB-module.Equip B with the trivial G-action. We obtain theG-linearized sheafA ⊗V. It follows easily fromthe universal property of the symmetric algebra ([13], Section (9.4.1)) thatS ym⋆(A ⊗V) inherits aG-linearization. Note that the algebraS ym⋆(A ⊗V) is naturally graded and that theG-linearizationpreserves this grading. LetS ym⋆(A ⊗V)G be the sub-algebra ofG-invariant elements inS ym⋆(A ⊗V). TheG-linearization provides aπ-equivariant action ofG on

H om(A ,V∨⊗ØB

):= S pec

(S ym⋆(A ⊗V)

),

π:S ym⋆(A ⊗V) −→ B being the natural projection. Then, the categorical quotient of the schemeS pec(S ym⋆(A ⊗V)) by theG-action is given through

S pec(S ym⋆(A ⊗V)

)//G= S pec

(S ym⋆(A ⊗V)G

) π−→ B.

LEMMA 2.2.1. For any base change morphism f:B′ −→ B, there is a natural isomorphism

S ym⋆(

f ⋆A ⊗V)G ∼= f ⋆(S ym⋆(A ⊗V)G).

Proof. It suffices to check that the natural isomorphism

u:S ym⋆(

f ⋆A ⊗V)∼= f ⋆(S ym⋆(A ⊗V)

)

is compatible with theG-linearizations. Note that the linearization off ⋆(A )⊗V on B′ is just thepullback of the linearization ofA ⊗V on B by f , so that the naturality properties of the symmetricalgebra immediately imply thatu is compatible with theG-linearizations.

Another important observation is the following.

LEMMA 2.2.2. Let ψ :A ′ −→ A be a surjective map ofØB-modules. Then, the induced homomor-phism

S ym⋆(A ′⊗V)G −→ S ym⋆(A ⊗V)G

of ØB-algebras is surjective as well.

Proof. We have to check the surjectivity of the induced map between the stalks at a pointb ∈ B.The formation of the symmetric algebra commutes with localization ([13], (9.4.2)). By Nakayama’slemma and Lemma 2.2.1, we may tensorize withk〈b〉 := ØB,b/mb. Let s1, ...,sm be a basis forAb⊗


k〈b〉 ands1, ...,sn a basis forA ′b ⊗ k〈b〉, such thatsi projects tosi, i = 1, ...,m. Then, for anyd ≥ 0,

we have an isomorphism ofG-modules ([13], (9.4.1))

Symd(Ab⊗ (V ⊗k k〈b〉)

)∼=

⊕

(d1,...,dm), di≥0:d1+···+dm=d

Symd1(V ⊗k k〈b〉

)⊗·· ·⊗Symdm

(V ⊗k k〈b〉

)

Symd(A

′b ⊗ (V ⊗k k〈b〉)

)∼=

⊕

(d1,...,dn), di≥0:d1+···+dn=d


)⊗·· ·⊗Symdn

(V ⊗k k〈b〉

).

The submodule⊕

(d1,...,dn):di≥0, d1+···+dn=ddm+1=···=dn=0


)⊗·· ·⊗Symdn

(V ⊗k k〈b〉

)

of Symd(A ′

b ⊗ (V ⊗k k〈b〉))

clearly projects isomorphically onto Symd(Ab⊗ (V ⊗k k〈b〉)

), so that

Symd(A

′b ⊗ (V ⊗k k〈b〉)

)G−→ Symd(

Ab⊗ (V ⊗k k〈b〉))G

is surjective for alld ≥ 0.

2.3 An extension property

PROPOSITION2.3.1. Let S be a scheme andES a vector bundle on S×X. Let Z⊂ S×X be a closedsubset, such thatcodimX(Z∩ (s×X)) ≥ 2 for every point s∈ S. Denote byι :U := (S×X) \Z ⊆S×X the inclusion. Then, the natural map

ES−→ ι⋆(ES|U

)

is an isomorphism.

Proof (after Maruyama[24], page 112).Since this is a local question, we may clearly assumeES=ØS×X. Note thatZ is “stable under specialization” in the sense of [14], (5.9.1), page 109. By [14],Theorem (5.10.5), page 115, one has to show that infx∈Z depth(ØS×X,x) ≥ 2. SinceX is smooth, themorphismπS:S×X −→ X is smooth. Thus, by [15], Proposition (17.5.8), page 70,

dim(ØS×X,x)−depth(ØS×X,x) = dim(ØS,s)−depth(ØS,s)

for every pointx∈ S×X ands := πS(x). This implies

depth(ØS×X,x)≥ dim(ØS×X,x)−dim(ØS,s) = dim(Øπ−1S (s),x). (9)

Since for any pointx ∈ π−1S (s), one has dim(Øπ−1

S (s),x) = codimπ−1S (s)(x), we derive the desired

estimate depth(ØS×X,x)≥ 2 for every pointx∈ Z from the fact that codimπ−1S (s)(Z∩π−1

S (s)) ≥ 2 and(9).

COROLLARY 2.3.2. Suppose S is a scheme,ES is an S-flat family of torsion free coherent sheaves onX parameterized by S, andFS is a locally free sheaf on S×X. Let U⊆ S×X be the maximal opensubset whereES is locally free. Then, for any homomorphismϕS:ES|U −→ FS|U , there is a uniqueextension

ϕS:ES−→ FS


to S×X. In particular, for a base change morphism f:T −→ S, we have

ϕT = ( f × idX)⋆(ϕS).

Here,ϕT is the extension of( f × idX)|( f×idX)−1(U)⋆(ϕS).

Proof. An extension is given by

ϕS:ES−→ ι⋆(ES|U

) ι⋆(ϕS)−→ ι⋆

(FS

) Theorem2.3.1= FS.

SinceES can be written as the quotient of a locally free sheaf, the uniqueness also follows fromTheorem 2.3.1. The final statement is clearly a consequence of the uniqueness property.

3 Fundamental results on semistable singular principal bundles

In this section, after reviewing some elementary properties, we present two crucial results on semistablesingular principal bundles. The first one concerns the fact that a singular principal bundle(A ,τ) isslope semistable in the sense that has been defined in the introduction if and only if the associatedrational principalG-bundle(U,P(A ,τ)) is semistable in the sense of Ramanathan. This observationenables us to establish the second fundamental result, namely the boundedness of semistable singularprincipal bundles(A ,τ) with fixed Hilbert polynomial, by applying the results from [21].

3.1 The basic formalism of singular principal bundles

SinceG is a semisimple group, the basic formalism of pseudoG-bundles in positive characteristic isexactly the same as in characteristic zero. Therefore, we may refer the reader to [32], Section 3.1,for more details (be aware that in this reference the term “singular principalG-bundle” is used forour “pseudoG-bundle”). We fix a faithful representationρ :G−→ GL(V). Then, apseudo G-bundle(A ,τ) consists of a torsion free coherent ØX-moduleA of rank dimk(V) with trivial determinantand a homomorphismτ :S ym⋆(A ⊗V)G −→ ØX which is non-trivial in the sense that it is not justthe projection onto the degree zero component. LetU ⊆ X be the maximal open subset whereA islocally free. Sinceρ(G)⊆ SL(V), we have the open immersion

I som(A|U ,V

∨⊗ØU)/G⊂ H om

(A ,V∨⊗ØX

)//G.

Recall the following alternatives.

LEMMA 3.1.1. Let (A ,τ) be a pseudo G-bundle and

σ :X −→ H om(A ,V∨⊗ØX

)//G

the section defined byτ . Then, either

σ(U)⊂ I som(A|U ,V

∨⊗ØU)/G

orσ(U)⊂

(H om

(A|U ,V

∨⊗ØU)//G

)∖(I som

(A|U ,V

∨⊗ØU)/G

).

Proof. See [32], Corollary 3.4.


In the former case, we call(A ,τ) a singular principal G-bundle. We may form the base changediagram

P(A ,τ) //

H om(A|U ,V

∨⊗ØU)

Uσ|U

// H om(A|U ,V

∨⊗ØU)//G.

If (A ,τ) is a singularG-bundle, thenP(A ,τ) is a principalG-bundle overU in the usual sense, i.e.,a rational principalG-bundle onX in the sense of Ramanathan (see Section 3.2).

A family of pseudo G-bundles parameterized by the scheme Sis a pair(AS,τS)which consists of anS-flat family ASof torsion free sheaves onS×X and a homomorphismτS:S ym⋆(AS⊗V)G −→ØS×X.We say that the family(A 1

S ,τ1S) is equivalent to the family(A 2

S ,τ2S), if there is an isomorphism

ψS:A 1S −→ A 2

S , such that the induced isomorphismS ym⋆(A 1S ⊗V)G −→ S ym⋆(A 2

S ⊗V)G carriesτ2

S into τ1S. The base change properties for singular principalG-bundles have been outlined in Section

2.2.

3.2 Semistable rational principalG-bundles

We now review the formalism introduced by Ramanathan and compare it with our setup. Arationalprincipal G-bundle on Xis a pair(U,P) which consists of a big open subsetU ⊆ X and a principalG-bundleP onU . Such a rationalG-bundle is said to be(semi)stable, if for every open subsetU ′ ⊆Uwhich is big inX, every parabolic subgroupP of G, every reductionβ :U ′ −→P|U ′/P of the structuregroup ofP to P overU ′, and every antidominant character (see below)χ onP, we have

deg(L (β ,χ)

)(≥)0.

Note that the antidominant characterχ and theP-bundleP|U ′ −→ P|U ′/P define a line bundle onP|U ′/P. Its pullback toU ′ via β is the line bundleL (β ,χ). SinceU ′ is big in X, it makes sense tospeak about the degree ofL (β ,χ).

We fix a pair(B,T) which consists of a Borel subgroupB⊂ G and a maximal torusT ⊂ B. If PandP′ are conjugate inG, a reductionβ of a principalG-bundle toP may equally be interpreted as areduction toP′. Thus, it suffices to consider parabolic subgroups of the type PG(λ ), λ :Gm(k) −→ Ta one parameter subgroup, which containB. Here, we use the convention

PG(λ ) := QG(−λ ) =

g∈ G∣∣ lim

z→0λ (z) ·g·λ (z)−1 exists inG

.

Let X⋆(T) andX⋆(T) be the freeZ-modules of one parameter subgroups and characters ofT, respec-tively. We have the canonical pairing〈., .〉:X⋆(T)×X⋆(T) −→ Z. SetX⋆,K(T) := X⋆(T)⊗ZK andX⋆K

(T) := X⋆(T)⊗Z

K, and let〈., .〉K

:X⋆,K(T)×X⋆K

(T) −→K be theK-bilinear extension of〈., .〉,K=Q,R. Finally, suppose(., .)⋆:X⋆

R

(T)×X⋆R

(T)−→R is a scalar product which is invariant underthe Weyl groupW(T) = N (T)/T. This also yields the product(., .)⋆:X⋆,R(T)×X⋆,R(T)−→R. Weassume that(., .)⋆ is defined overQ.

The datum(B,T) defines the set of positive rootsR and the setR∨ of coroots (see [38]). LetC ⊂ X⋆,R(T) be the cone spanned by the elements ofR∨ andD ⊂ X⋆

R

(T) the dual cone ofC withrespect to〈., .〉

R

. Equivalently, the coneD may be characterized as being the dual cone of the conespanned by the elements inR with respect to(., .)⋆. Indeed, one has〈 · ,α∨〉

R

= 2( · ,α)⋆/(α ,α)⋆,α ∈ R. Now, a characterχ ∈ X⋆(T) is calleddominant, if it lies in D , andantidominant, if −χ lies in


D . A characterχ of a parabolic subgroup containingB is called(anti)dominant, if its restriction toTis (anti)dominant.

In the definition of semistability, we may clearly assume that (χ ,α)⋆ > 0 for everyα ∈ R with〈λ ,α〉 > 0, if P = PG(λ ). Otherwise, we may chooseλ ′, such that(χ ,α)⋆ > 0 if and only if〈λ ′,α〉 > 0. Then,PG(λ ′) is a parabolic subgroup which containsPG(λ ), χ is induced by a char-acter χ ′ on PG(λ ′), the reductionβ defines the reductionβ ′:P|U ′/PG(λ ) −→ P|U ′/PG(λ ′), andŁ(β ,χ) = Ł(β ′,χ ′). Every one parameter subgroupλ of T defines a characterχλ of T, such that〈λ ,χ〉= (χλ ,χ)⋆ for all χ ∈ X⋆(T). Finally, observe that the coneC ′ of one parameter subgroupsλof T, such thatB⊆ PG(λ ), is dual to the cone spanned by the roots. Thus,

∀λ ∈ X⋆(T) : B⊆ PG(λ ) ⇐⇒ χλ ∈ D .

If one of those conditions is verified, then(PG(λ ),χλ ) consists of a parabolic subgroup containingBand a dominant character on it. Similarly, ifQG(λ ) containsB, thenχλ is an antidominant characteron QG(λ ). Our discussion shows.

LEMMA 3.2.1. A rational principal G-bundle(U,P) is (semi)stable, if and only if for every opensubset U′ ⊆U which is big in X, every one parameter subgroupλ ∈C ′, and every reductionβ :U ′ −→P|U ′/QG(λ ) of the structure group ofP to QG(λ ) over U′, we have

deg(L (β ,χλ )

)(≥)0.

For any one parameter subgroupλ of G, we may find a pair(B′,T ′) consisting of a Borel subgroupB′ of G and a maximal torusT ′ ⊂ B′, such thatλ ∈ X⋆(T ′) andB′ ⊆ QG(λ ). Then, there exists anelementg∈ G, such that(g·B ·g−1,g·T ·g−1) = (B′,T ′), and we obtain

(., .)⋆ ′:X⋆R

(T ′)×X⋆R

(T ′) −→ R

(χ ,χ ′) 7−→(χ(g−1 · . ·g),χ ′(g−1 · . ·g)

)⋆.

Since(., .)⋆ is invariant underW(T), the product(., .)⋆′ does not depend ong. In particular, we obtaina characterχλ onQG(λ ). This character does not depend on(B′,T ′) (see [34], (2.31)). We conclude.

LEMMA 3.2.2. A rational principal G-bundle(U,P) is (semi)stable, if and only if for every opensubset U′ ⊆U which is big in X, every one parameter subgroupλ :Gm(k)−→ G, and every reductionβ :U ′ −→ P|U ′/QG(λ ) of the structure group ofP to QG(λ ) over U′, we have

deg(L (β ,χλ )

)(≥)0.

Suppose thatρ :G −→ GL(V) is a faithful representation. We may assume thatT maps to themaximal torusT ⊂ GL(V), consisting of the diagonal matrices. The character groupX⋆(T) is freelygenerated by the charactersei :diag(λ1, ...,λn) 7−→ λi, i = 1, ...,n. We define

(., .)⋆T:X⋆R

(T)×X⋆R

(T) −→ R

( n

∑i=1

xi ·ei ,n

∑i=1

yi ·ei

)7−→

n

∑i=1

xiyi .

The scalar product(., .)⋆T

is clearly defined overQ and invariant under the Weyl groupW(T). Theproduct(., .)⋆

Ttherefore restricts to a scalar product(., .)⋆ onX⋆

R

(T) with the properties we have asked

for. We find a nice formula for deg(L (β ,χλ )

). Indeed, if(U,P) is a rationalG-bundle, and ifE


is the vector bundle onU associated toP by means ofρ , then we have, for every one parametersubgroupλ :Gm(k)−→ G, the embedding

ι :P/QG(λ ) → I som(V ⊗ØX,E )/QGL(V)(λ ).

As usual, we obtain a weighted filtration(V•(λ ),α•(λ )) of V, and, for every reductionβ :U ′ −→P|U ′/QG(λ ) over a big open subsetU ′ ⊆U , the reductionι β corresponds to a filtration

E•(β ) : 0( E1 ( · · ·( Et ( E|U ′

by subbundles with rk(Ei) = dimk(Vi), i = 1, ..., t. With the weighted filtration(E•(β ),α•(λ )) of E|U ′ ,we find

deg(L (β ,χλ )

)= L

(E•(β ),α•(λ )

)=

t

∑i=1

αi(rk(Ei) ·deg(E )− rk(E ) ·deg(Ei)

).

To see this, observe that the characterχλ is, by construction, the restriction of a characterχ of T, sothat L (β ,χλ ) = L (ι β ,χ). The degree of the latter line bundle is computed in Example 2.15 of[35] and gives the result stated above. Thus, we conclude.

LEMMA 3.2.3. A rational principal G-bundle(U,P) is (semi)stable, if and only if, for every opensubset U′ ⊆U which is big in X, every one parameter subgroupλ :Gm(k)−→ G, and every reductionβ :U ′ −→ P|U ′/QG(λ ) of the structure group ofP to QG(λ ) over U′, we have

L(E•(β ),α•(λ )

)(≥)0.

Remark3.2.4. If (U,P) is given as a singular principalG-bundle(A ,τ), then, in the notation of theintroduction, we have

A|U = E∨, Ai|U = ker

(E

∨ −→ E∨

t+1−i

)and αi;β = αt+1−i , i = 1, ..., t.

Then, one readily verifiesL(A ,τ ;β

)= L

(E•(β ),α•(λ )

).

This proves that the definition of slope semistability (1) given in the introduction is the original defini-tion of Ramanathan. We have arrived at our notion of semistability, by replacing degrees with Hilbertpolynomials. Thus, our semistability concept is a “Gieseker version” of Ramanathan semistability.

3.3 Boundedness of the family of semistable singular principal bundles with fixed nu-merical data

THEOREM 3.3.1. Fix a Hilbert polynomial P. Then, the set of isomorphism classes of torsion freecoherentØX-modulesA with Hilbert polynomial P which occur in semistable singular principalbundles(A ,τ) is bounded.

Proof. By the boundedness result [19], Theorem 4.2, we have to find a constant, such that

µmax(A )≤C

holds for any semistable singular principalG-bundle(A ,τ) with P(A ) = P.


If Char(k) = p> 0, denote byF l :X −→ X the l -th power of the Frobenius morphism,l > 0. Forany rational vector bundle(U,E ), i.e., pair that consists of a big open subsetU ⊆ X and a vectorbundleE onU , one defines in this case

Lmin/max(E ) :=

[liml→∞

µmin/max(F l ⋆E

)

pl

]

+

.

Theorem 8.4 of [21] specialized to our situation gives the following result.

THEOREM 3.3.2. Let G be a connected semisimple group andρ :G −→ GL(V) a representation.Then, there are the following cases.

i) Assume eitherChar(k) = 0 or µmax(ΩX) ≤ 0. If (U,P) is a semistable rational principal G-bundle on X and(U,E ) is the rational vector bundle with fiber V associated to(U,P), then(U,E )is (strongly) slope semistable.

ii) If Char(k) = p> 0 andµmax(ΩX)> 0, there is a constant C(ρ), depending only onρ , such thatfor any semistable rational principal G-bundle(U,P) on X with associated rational vector bundle(U,E ), one finds

0≤ Lmax(E )−Lmin(E )≤C(ρ) ·Lmax(ΩX)

p.

In the case of a singular principalG-bundle(A ,τ), we have seen in the preceding section that(U,P(A ,τ)) is a Ramanathan-semistable rational principalG-bundle whose associated rational vec-tor bundle is(U,A ∨

|U ). If Case i) applies, we find thatA ∨|U and consequentlyA is slope semistable,

and we are done. Using deg(A ) = 0 and the consequenceLmax(A∨|U ) ≥ 0, the theorem gives in the

second case

µmax(A ) =−µmin(A∨|U)≤−Lmin(A

∨|U )≤C :=C(ρ) ·

Lmax(ΩX)

p.

This is the estimate that we needed to establish the contention of the theorem.

4 Dispo sheaves

In the papers [32] and [34], the theory of decorated sheaves was used to construct projective modulispaces for singular principalG-bundles in characteristic zero. Due to the more difficult representationtheory of general linear groups in positive characteristic, this approach is not available in (low) positivecharacteristic. Nevertheless, one may still associate to any singular principal bundle a more specificobject than a decorated sheaf, namely a so-called “dispo sheaf”. The moduli theory of these disposheaves may be developed along the lines of the theory of decorated sheaves in [33] and [10], makingseveral non-trivial modifications.

4.1 The basic definitions

For this section, we fix the representationρ :G −→ SL(V) ⊆ GL(V) and a positive integers as inSection 2.1, “Some specific quotient problems”.

Suppose thatA is a coherent ØX-module of rankr := dimk(V). Then, thesheaf of invariants inthe symmetric powers ofA is defined as

Vs(A ) :=⊕

(d1,...,ds):di≥0,∑ idi=s!

(S ymd1

((A ⊗kV)G)⊗·· ·⊗S ymds

(S yms(A ⊗kV)G)).


A dispo3 sheaf(of type(ρ ,s)) is a pair(A ,ϕ) that consists of a torsion free sheafA of rank r withdet(A )∼= ØX onX and a non-trivial homomorphism

ϕ :Vs(A )−→ ØX.

Two dispo sheaves(A1,ϕ1) and (A2,ϕ2) are said to beequivalent, if there exists an isomorphismψ :A1 −→ A2, such that, with the induced isomorphismVs(ψ):Vs(A1)−→Vs(A2), one obtains

ϕ1 = ϕ2Vs(ψ).

A weighted filtration(A•,α•) of the torsion free sheafA consists of a filtration

0( A1 ( · · ·( At ( At+1 = A

of A by saturated subsheaves and a tupleα• = (α1, ...,αt) of positive rational numbers. Given such aweighted filtration(A•,α•), we introduce the quantities

M(A•,α•) :=t

∑j=1

α j ·(rk(A j) ·P(A )− rk(A ) ·P(A j)

),

L(A•,α•) :=t

∑j=1

α j ·(rk(A j) ·deg(A )− rk(A ) ·deg(A j)

).

Next, let(A ,ϕ) be a dispo sheaf and(A•,α•) a weighted filtration ofA . Fix a flag

W•:0(W1 ( · · ·(Wt (W := kr with dimk(Wi) = rk(Ai), i = 1, ..., t.

We may find a small open subsetU , such that

• ϕ|U is a surjection onto ØU ;

• there is a trivializationψ :A|U −→W⊗ØU with ψ(Ai|U) =Wi ⊗ØU , i = 1, ..., t.

In presence of the trivializationψ , the homomorphismϕ|U provides us with the morphism

β :U −→P

(Vs(A )

)Vs(ψ)−→ P(Vs)×U −→P(Vs).

(Consult Section 2.1 for the notation “Vs”.) Finally, let λ :Gm(k) −→ SL(W) be a one parametersubgroup with(W•,α•) as its weighted flag. With these choices made, we set

µ(A•,α•;ϕ

):= max

µσs

(λ ,β (x)

) ∣∣x∈U.

(The linearizationσs has been introduced in Lemma 2.1.5.) As in [33], p. 176, one checks that thequantityµ

(A•,α•;ϕ

)depends only on the data(A•,α•) andϕ .

Remark4.1.1. Let us outline another, intrinsic way for defining the numberµ(A•,α•;ϕ

). First,

observe thatVs(A ) is a submodule of

Ss(A ) :=⊕

(d1,...,ds):di≥0,∑ idi=s!

(S ymd1

(A ⊗kV

)⊗·· ·⊗S ymds

(S yms(A ⊗kV)

))

3decorated with invariants in symmetric powers


and thatSs(A ) is a quotient of(V⊗s!)⊕N. Let (A•,α•) be a weighted filtration ofA . Set I :=1, ..., t +1×s! andAt+1 := A . For(i1, ..., is!) ∈ I , define

Ai1 · · · · ·Ais!

as the image of the subsheaf(Ai1 ⊗·· ·⊗Ais! )⊕N of (V⊗s!)⊕N in Ss and

Ai1 ⋆ · · ·⋆Ais! :=(Ai1 · · · · ·Ais!

)∩Vs(A ).

Thestandard weight vectorsare

γ(i)r :=(i − r, ..., i − r︸︷︷︸

i×

, i, ..., i︸︷︷︸(r−i)×

), i = 1, ..., r −1.

Given a weighted filtration(A•,α•) of the torsion free sheafA , we obtain theassociated weightvector

(γ1, . . . ,γ1︸︷︷︸(rkA1)×

, γ2, . . . ,γ2︸︷︷︸(rkA2−rkA1)×

, . . . ,γt+1, . . . ,γt+1︸︷︷︸(rkA −rkAt)×

):=

t

∑j=1

α j · γ(rkA j )r .

(We recoverα j = (γ j+1 − γ j)/r, j = 1, ..., t.) For a dispo sheaf(A ,ϕ) and a weighted filtration(A•,α•) of A , we finally find with (8)

µ(A•,α•;ϕ

)=−min

γi1 + · · ·+ γis!

∣∣(i1, ..., is!) ∈ I : ϕ|Ai1⋆···⋆Ais!6≡ 0

. (10)

Fix a positive polynomialδ ∈Q[x] of degree at most dim(X)−1. Now, we say that a dispo sheaf(A ,ϕ) is δ -(semi)stable, if the inequality

M(A•,α•)+δ ·µ(A•,α•;ϕ

)()0

holds for any weighted filtration(A•,α•) of A .If δ is a non-negative rational number, we call a dispo sheaf(A ,ϕ) δ -slope (semi)stable, if the

inequalityL(A•,α•)+δ ·µ

(A•,α•;ϕ

)(≥)0

holds for any weighted filtration(A•,α•) of A . Note that, forδ = δ/(dim(X)−1)! ·xdim(X)−1+ · · ·(wheren= dimX), we have

(A ,ϕ) is δ -semistable =⇒ (A ,ϕ) is δ -slope semistable. (11)

4.2 Boundedness

PROPOSITION4.2.1. Fix a Hilbert polynomial P, the representationρ and the integer s as above, andthe non-negative rational numberδ . Then, the set of isomorphism classes of torsion free sheaves A

on X with Hilbert polynomial P for which there do existδ -slope semistable dispo sheaves(A ,ϕ) oftype(ρ ,s) is bounded.

Proof. Let (A ,ϕ) be aδ -slope semistable dispo sheaf of type(ρ ,s). Let B be any non-trivial propersaturated subsheaf ofA . We look at the weighted filtration(A•,α•) := (0( B ( A ,(1)) of A . In

the notation of Remark 4.1.1, the associated weight vector is justγ(rk(B))r . It easily follows that

µ(A•,α•;ϕ)≤ s! ·(r − rk(B)

).


We remind the reader that deg(A ) = 0, so that the semistability condition translates into

−r ·deg(B)+δ ·s! ·(r − rk(B)

)≥−r ·deg(B)+δ ·µ(A•,α•;ϕ)≥ 0

that is

µ(B)≤δ ·s! ·

(r − rk(B)

)

r · rk(B)≤

δ ·s! · (r −1)r

.

This meansµmax(A )≤ δ ·s! · (r −1)/r, so that the result follows from Theorem 4.2 of [19].

4.3 S-equivalence

An important issue is the correct explanation of S-equivalence of properly semistable dispo sheaves.For this, suppose we are given aδ -semistable dispo sheaf(A ,ϕ) and a weighted filtration(A•,α•)of A with

M(A•,α•)+δ ·µ(A•,α•;ϕ

)≡ 0.

We want to define theassociated admissible deformationdf(A•,α•)(A ,ϕ) = (Adf,ϕdf). Of course, wesetAdf =

⊕ti=0Ai+1/Ai . Let U be the maximal (big!) open subset whereAdf is locally free. We

may choose a one parameter subgroupλ :Gm(k) −→ SLr(k) whose weighted flag(W•(λ ),α•(λ )) inkr satisfies:

• dimk(Wi) = rkAi , i = 1, ..., t, in W•(λ ) : 0(W1 ( · · ·(Wt ( kr ;

• α•(λ ) = α•.

Then, the given filtrationA• corresponds to a reduction of the structure group ofI som(Ø⊕rU ,A|U ) to

Q(λ ). On the other hand,λ defines a decomposition

Vs =U γ1 ⊕·· ·⊕U γu+1, 0= γ1 < · · ·< γu+1.

Now, observe thatQ(λ ) fixes the flag

0(U1 :=U γ1(U2 := (U γ1

⊕U γ2)( · · ·(Uu := (U γ1 ⊕·· ·⊕U γu)(Vs. (12)

Thus, we obtain aQ(λ )-module structure on

u⊕

i=0

Ui+1/Ui∼=Vs. (13)

Next, we writeQ(λ ) = Ru(Q(λ ))⋊L(λ ) whereL(λ ) ∼= GL(W1/W0)×·· ·×GL(kr/Wt) is the cen-tralizer of λ . Note that (13) is an isomorphism ofL(λ )-modules. The process of passing fromA to Adf corresponds to first reducing the structure group toQ(λ ), then extending it toL(λ ) viaQ(λ )−→ Q(λ )/Ru(Q(λ ))∼= L(λ ), and then extending it to GLr(k) via the inclusionL(λ )⊂GLr(k).By (12),Vs(A|U) has a filtration

0( U1 ( U2 ( · · ·( Uu (Vs(A|U ),

and, by (13), we have a canonical isomorphism

Vs(A|U )∼=u⊕

i=0

Ui+1/Ui .


Now, the restrictionϕ1 of ϕ|U to U1 is non-trivial, and thus we may defineϕdf asϕ1 on U1 and aszero on the other components. This makes sense, becauseU1 is also a submodule ofVs(A|U). Then,we finally obtain

ϕdf:Vs(A )−→ ι⋆(Vs(A|U )

) ι⋆(ϕdf)−→ ι⋆

(ØU

)= ØX,

ι :U −→ X being the inclusion. A dispo sheaf(A ,ϕ) is said to beδ -polystable, if it is δ -semistableand equivalent to every admissible deformation df(A•,α•)(A ,ϕ) = (Adf,ϕdf) associated to a filtration(A•,α•) of A with

M(A•,α•)+δ ·µ(A•,α•;ϕ

)≡ 0.

By the GIT construction of the moduli space that will be givenin Section 4.5, one has the following.

LEMMA 4.3.1. Let(A ,ϕ) be aδ -semistable dispo sheaf. Then, there exists aδ -polystable admissibledeformationgr(A ,ϕ) of (A ,ϕ). The dispo sheafgr(A ,ϕ) is unique up to equivalence.

In general, not every admissible deformation will immediately lead to a polystable dispo sheaf,but any iteration of admissible deformations (leading to non-equivalent dispo sheaves) will do so afterfinitely many steps. We call twoδ -semistable dispo sheaves(A ,ϕ) and (A ′,ϕ ′) S-equivalent, ifgr(A ,ϕ) and gr(A ′,ϕ ′) are equivalent.

Remark4.3.2. Another way of looking at S-equivalence is the following. With the notation as above,we may choose an open subsetU ⊆ X (no longer big), such thatϕ is surjective overU and we havean isomorphismψ :A|U

∼= kr ⊗ØU with ψ(Ai) =Wi ⊗ØU for i = 1, ..., t. For such a trivialization, weobtain, fromϕ|U , the morphism

β :U −→P

(Vs(A|U)

) Vs(ψ)∼= P(Vs)×U −→P(Vs).

For the morphismβdf:U −→P(Vs) associated toϕdf|U , we discover the relationship

βdf(x) = limz→∞

λ (z) ·β (x), x∈U. (14)

4.4 The main theorem on dispo sheaves

With the definitions that we have encountered so far, we may introduce the moduli functors

Mδ-(s)sP (ρ ,s):Schk −→ Sets

S 7−→

Equivalence classes of families ofδ -(semi)stable dispo sheavesAS oftype(ρ ,s) with Hilbert polynomialPthat are parameterized by the schemeS

for δ -(semi)stable dispo sheaves(A ,ϕ) of type(ρ ,s) on X with Hilbert polynomialP(A ) = P.

THEOREM 4.4.1. Given the input data P,ρ , s, andδ as above, then the moduli spaceMδ-ssP (ρ ,s) for

δ -semistable dispo sheaves(A ,τ) of type(ρ ,s) with P(A ) = P exists as aprojective scheme.

4.5 The proof of the main theorem on dispo sheaves

In this section, we will outline how a GIT construction may beused for proving the main auxiliaryresult Theorem 4.4.1. Once one has the correct set-up, the details become mere applications of thetechniques of the papers [33] or [10] and [20].


Construction of the parameter space. — As we have seen in Proposition 4.2.1, there is a constantC, such thatµmax(A )≤C for everyδ -semistable dispo sheaf(A ,ϕ) with P(A ) = P, i.e.,A lives ina bounded family. Thus, we may choose ann0 ≫ 0 with the following properties. For every sheafA

with Hilbert polynomialP andµmax(A )≤C and everyn≥ n0, one has

• H i(A (n)) = 0 for i > 0;

• A (n) is globally generated.

We also fix ak-vector spaceU of dimensionP(n). Let Q be the quasi-projective scheme which pa-rameterizes quotientsq:U ⊗ØX(−n)−→ A whereA is a torsion free sheaf with Hilbert polynomialP andH0(q(n)) an isomorphism. Let

qQ:U ⊗π⋆X

(ØX(−n)

)−→ AQ

be the universal quotient. Setting

Vs(U) :=⊕

(d1,...,ds):di≥0,∑ idi=s!

(Symd1

((U ⊗kV)G)⊗·· ·⊗Symds

(Syms(U ⊗kV)G)),

there is a surjectionVs(U)⊗π⋆

X

(ØQ×X(−s!n)

)−→Vs(AQ),

as can be seen with Lemma 2.2.2. For a point[q:U ⊗ØX(−n) −→ A ] ∈ Q, any homomorphismϕ :Vs(A )−→ ØX is determined by the induced homomorphism

Vs(U)−→ H0(ØX(s!n))

of vector spaces. Hence, our parameter space should be a subscheme of

D0 :=Q×P(

Hom(Vs(U),H0(ØX(s!n))

)∨)

︸︷︷︸=:P

.

Note that, overD0×X, there is the universal homomorphism

ϕ ′′′ :Vs(U)⊗ØD0×X −→ H0(ØX(s!n))⊗π⋆

P

(ØP

(1)).

Let ϕ ′′ = evϕ ′′′ be the composition ofϕ ′′′ with the evaluation map

ev:H0(ØX(s!n))⊗ØD0×X −→ π⋆

X

(OX(s!n)

).

We twistϕ ′′ by idπ⋆X(ØX(−s!n)) in order to obtain

ϕ ′:Vs(U)⊗π⋆X

(ØX(−s!n)

)−→ π⋆

P

(ØP

(1)).

SetAD0 := π⋆Q×X(AQ×X). We have the surjectionS:Vs(U)⊗π⋆

X(ØX(−s!n)) −→ Vs(AD0). There-fore, we may define a closed subschemeD of D0 by the condition thatϕ ′ factorizes overVs(AD0) onD. DeclaringAD := (AD0)|D×X, there is thus the homomorphism

ϕD:Vs(AD)−→ π⋆P

(ØP

(1))

with ϕ|D×X = ϕD S. The family(AD,ϕD) is theuniversal family of dispo sheaves parameterized byD. By its construction, it has the features listed below.

PROPOSITION4.5.1 (LOCAL UNIVERSAL PROPERTY). Let S be a scheme and(AS,ϕS) a family ofδ -semistable dispo sheaves with Hilbert polynomial P parameterized by S. Then, there exist a coveringof S by open subschemes Si , i ∈ I, and morphismsβi :Si −→D, i ∈ I, such that the family(AS|Si

,ϕS|Si)

is equivalent to the pullback of the universal family onD×X byβi × idX for all i ∈ I.


The group action. — There is a natural action of GL(U) on the quot schemeQ and onD0. Thisaction leaves the closed subschemeD invariant, and therefore yields an action

Γ:GL(U)×D−→D.

PROPOSITION 4.5.2 (GLUING PROPERTY). Let S be a scheme andβi :S−→ D, i = 1,2, two mor-phisms, such that the pullback of the universal family viaβ1 × idX is equivalent to its pullback viaβ2× idX. Then, there is a morphismΞ:S−→ GL(U), such thatβ2 equals the morphism

SΞ×β1−→ GL(U)×D

Γ−→D.

Good quotients of the parameter space. — For a pointz∈D, we let(Az,ϕz) be the dispo sheafobtained from the universal family by restriction toz×X. It will be our task to show that the setDδ-(s)s parameterizing those pointsz∈ D for which (Az,ϕz) is δ -(semi)stable are open subsets ofD that possess a good or geometric quotient. This can be most conveniently done by applying GIT.To this end, we first have to exhibit suitable linearizationsof the group action. We will use here theapproach by Gieseker in order to facilitate the computations. The experienced reader should have noproblem in rewriting the proof in Simpson’s language.

There is a projective subschemeA → Pic(X), such that the morphism det:Q−→ Pic(X), [q:U ⊗ØX(−n) −→ A ] 7−→ [det(A )] factorizes overA. We choose a Poincare sheafPA onA×X. Then,there is an integern1, such that for every integern≥ n1 and every line bundle Ł onX with [Ł]∈A, thebundle Ł(rn) is globally generated and satisfieshi(Ł(rn)) = 0 for all i > 0. For such ann, the sheaf

G := πA⋆

(PA⊗π⋆

X

(ØX(rn)

))

is locally free. We then form the projective bundle

G1 := P(H om

( r∧U ⊗ØA,G

)∨)

over the schemeA. For our purposes, we may always replace the Poincare sheafPA by its tensorproduct with the pullback of the dual of a sufficiently ample line bundle onA, so that we can achievethat Ø

G1(1) is ample. The homomorphism

r∧(qQ⊗ idπ⋆

X(ØX(n))

):

r∧U ⊗ØQ×X −→ det(AQ)⊗π⋆

X

(ØX(rn)

)

defines a GL(U)-equivariant and injective morphism

Q−→G1.

We declareG2 := P

(Hom

(Vs(U),H0(ØX(s!n))

)∨)and G :=G1×G2.

Then, we obtain the injective and SL(U)-equivariant morphism

Gies:D−→G.

The ample line bundles ØG

(ν1,ν2), ν1,ν2 ∈ Z>0, are naturally SL(U)-linearized, and we chooseν1

andν2 in such a way thatν1

ν2=

p−s! ·δ (n)r ·δ (n)

. (15)


THEOREM 4.5.3. There exists n7 ∈Z>0, such that for all n≥ n7 the following property is verified. Fora point z∈D, the Gieseker pointGies(z) ∈ G is (semi)stable with respect to the above linearizationif and only if(Az,ϕz) is a δ -(semi)stable dispo sheaf of type(ρ ,s).

In the following, we will prove the theorem in several stages. As the first step, we establish thefollowing result.

PROPOSITION4.5.4. There is an n2 > 0, such that the following holds true. The setS of isomorphismclasses of torsion free sheavesA with Hilbert polynomial P for which there exist an n≥ n2 and apoint z= ([q:U ⊗ØX(−n)−→ A ],ϕ) ∈D, such thatGies(z) is semistable with respect to the abovelinearization, is bounded.

Proof. We would like to find a lower bound forµmin(A ) for a sheafA as in the Proposition. Then,we may conclude with Theorem 4.2 of [19].

Let Q = A /B be a torsion free quotient sheaf ofA . We have the exact sequence

0 −−−→ H0(B(n)

)−−−→ H0

(A (n)

)−−−→ H0

(Q(n)

).

Let λ :Gm(k)−→ SL(U) be a one parameter subgroup with weighted flag(U•(λ ) : 0(U1 := H0(q(n)

)−1(H0(B(n))

)(U, α•(λ ) = (1)

).

DefineB′ := q(U1⊗ØX(−n)). If Gies(z) = ([M], [L]), then

µ(λ , [M]) = P(n) · rk(B′)−h0(B(n)) · r ≤ P(n) · rk(B)−h0(B(n)) · r.

Similarly to the proof of Proposition 4.2.1, one finds

µ(λ , [L]) ≤ s! ·(P(n)−h0(B(n))

).

The assumption that Gies(z) is semistable thus gives

0 ≤ν1

ν2·µ(λ , [M])+µ(λ , [L])

≤P(n)−s! ·δ (n)

r ·δ (n)·(P(n) · rk(B)−h0(B(n)) · r

)+s! ·

(P(n)−h0(B(n))

)

=P(n)2 · rk(B)

r ·δ (n)−

P(n) ·h0(B(n))δ (n)

−s! ·P(n) · rk(B)

r+s! ·P(n).

We multiply this byr ·δ (n)/P(n) and find

P(n) · rk(B)− r ·h0(B(n))+δ (n) ·s!(r−1)≥P(n) · rk(B)− r ·h0(B(n))+δ (n) ·s!(r− rk(B)

)≥ 0.

The first exact sequence impliesh0(B(n))≥P(n)−h0(Q(n)). This enables us to transform the aboveinequality into

h0(Q(n))r

≥P(n)

r−

δ (n) ·s! · (r −1)rk(Q) · r

≥P(n)

r−

δ (n) ·s! · (r −1)r

. (16)

For a semistable sheafE with µ(E )≥ 0, [20] provides the estimate

h0(E )

rk(E )≤ deg(X) ·

( µ(E )deg(X) + f (r)+dim(X)

dim(X)

)≤

deg(X)

dim(X)!·( µ(E )

deg(X)+ f (r)+dim(X)

)dim(X). (17)


If µ(E ) < 0, we have of courseh0(E ) = 0. The right hand sideR(n) of (16) is a positive polyno-mial of degree dim(X) with leading coefficient deg(X)/dim(X)!. We can bound it from below by apolynomial of the form

deg(X)

dim(X)!·( C

deg(X)+ f (r)+dim(X)+n

)dim(X).

Assume thatn2 is so large that the value of this polynomial is positive and smaller thanR(n) for alln≥ n2. Then, (16), applied to the minimal destabilizing quotientQ of A , together with (17) yields

µmin(A )≥C,

and we are done.

THEOREM 4.5.5. There is an n3, such that for every n≥ n3 and every point z∈D with (semi)stableGieseker pointGies(z) ∈G, the dispo sheaf(Az,ϕz) is δ -(semi)stable.

Proof. As in [35], Proposition 2.14, one may show that there is a finite set

T =(r j

•,α j•)

∣∣ r j• = (r j

1, ..., rjt j) : 0< r j

1 < · · ·< r jt j< r;

α j• = (α j

1, ...,αj

t j) : α j

i ∈Q>0, i = 1, ..., t j , j = 1, ..., t,

depending only on the GLr(k)-moduleVs, such that the condition ofδ -(semi)stability of a dispo sheaf(A ,ϕ) of type(ρ ,s) with P(A ) = P has to be verified only for weighted filtrations(A•,α•) with

((rk(A1), ..., rk(At)),α•

)∈ T .

We may prescribe a constantC′. Then, there exists a constantC′′, such that for every dispo sheaf(A ,ϕ) of type (ρ ,s) with P(A ) = P and[A ] ∈S and every weighted filtration(A•,α•), such that((rk(A1), ..., rk(At)),α•) ∈ T and

µ(Ai)≤C′′, for oneindexi ∈ 1, ..., t , (18)

one hasL(A•,α•)>C′.

It is easy to determine a constantC′′′ that depends only onVs with

µ(A•,α•;ϕ)≥−C′′′

for any weighted filtration(A•,α•) of a sheafA as above with((rk(A1), ..., rk(At)),α•) ∈ T . WechooseC′ ≥ δ ·C′′′. Then, for a dispo sheaf(A ,ϕ) of type (ρ ,s) with [A ] ∈ S and a weightedfiltration (A•,α•), such that((rk(A1), ..., rk(At)),α•) ∈ T and (18) holds, one has

L(A•,α•)+δ ·µ(A•,α•;ϕ)>C′−δ ·C′′′ ≥ 0,

so that alsoM(A•,α•)+δ ·µ(A•,α•;ϕ)≻ 0.

Thus, the condition ofδ -(semi)stability has to be verified only for weighted filtrations (A•,α•) with((rk(A1), ..., rk(At)),α•) ∈ T for which (18) fails. But these live in bounded families. We conclude.


COROLLARY 4.5.6. There is a positive integer n4 ≥ n3, such that any n≥ n4 has the following prop-erty. For every dispo sheaf(A ,ϕ) of type(ρ ,s) for which [A ] belongs to the bounded familyS, theconditions stated below are equivalent.

1. (A ,ϕ) is δ -(semi)stable.

2. For any weighted filtration(A•,α•) with ((rk(A1), ..., rk(At)),α•) ∈ T , such thatA j(n) isglobally generated and hi(A j(n)) = 0 for all i > 0, j = 1, ..., t, one has

t

∑j=1

α j ·(h0(A (n)) · rk(A j)−h0(A j(n)) · rk(A )

)+δ (n) ·µ(A•,α•;ϕ)(≥)0.

We assume thatn≥ n4. Now, letz= ([q:U ⊗ØX(−n)−→A ],ϕ)∈D be a point with (semi)stableGieseker point Gies(z). Then,[A ] belongs to the bounded familyS. Therefore, it suffices to checkCriterion 2. in Corollary 4.5.6 for establishing theδ -(semi)stability of(A ,ϕ).

Let, more generally,(A•,α•) be a weighted filtration ofA , such that

• A j(n) is globally generated,j = 1, ..., t;

• hi(A j(n)) = 0, i > 0, j = 1, ..., t.

SinceH0(q(n)) is an isomorphism, we define the subspaces

U j := H0(q(n))−1(

H0(A j(n)))(U, j = 1, ..., t.

Define thestandard weight vectors

γ(i)p :=(i − p, ..., i − p︸︷︷︸

i×

, i, ..., i︸︷︷︸(p−i)×

), i = 1, ..., p−1,

and choose a basisu= (u1, ...,up) of U , such that

〈u1, ...,ul j 〉=U j , l j = dimk(U j) = h0(A j(n)), j = 1, ..., t.

These data yield the weight vector

γ = (γ1, ...,γp) :=t

∑j=1

α j · γ(l j )p

and the one parameter subgroupλ := λ (u,γ):Gm(k)−→ SL(U) with

λ (z) ·p

∑i=1

ci ·ui =p

∑i=1

zγi ci ·ui , z∈Gm(k).

Similarly, we define the one parameter subgroupsλ j := λ (u,γ(l j )p ), j = 1, ..., t. Let

L:Vs(U)−→ H0(ØX(s!n))

be a linear map that represents the second component of Gies(z). We wish to computeµ(λ , [L]). First,we note that the choice of the basisu provides an identification

gr(U) :=t+1⊕

j=1

U j ∼=U, U j :=U j/U j−1, j = 1, ..., t +1,


which we will use without further mentioning in the following. DefineI := 1, ..., t + 1×s! . Inanalogy to the considerations at the very end of Section 2.1,“Some specific quotient problems”, weintroduce the subspaces

U⋆i1,...,is! ⊂Vs(U), (i1, ..., is!) ∈ I .

As before, we check that all weight spaces with respect to theone parameter subgroupλ insideVs(U)are direct sums of some of these subspaces. In addition, the subspacesU⋆

i1,...,is! are eigenspaces for theone parameter subgroupsλ 1, ...,λ t . More precisely,λ j acts onU⋆

i1,...,is! with the weight

s! · l j −ν j(i1, ..., is!) · p, (i1, ..., is!) ∈ I , j = 1, ..., t.

In that formula, we have used

ν j(i1, ..., is!) := #

ik ≤ j |k= 1, ...,s!.

Thus, we find

µ(λ , [L]) =−min t

∑j=1

α j(s! · l j −ν j(i1, ..., is!) · p

)∣∣W⋆i1,...,is! 6⊆ ker(L), (i1, ..., is!) ∈ I

. (19)

Fix an index tuple(i01, ..., i0s!) ∈ I for which the minimum is achieved.

Let

M:r∧

U −→ H0(det(A )(rn))

represent the first component of Gies(z). It is well-known that

µ(λ , [M]) =t

∑j=1

α j(h0(A (n)) · rk(A j)−h0(A j(n)) · rk(A )

)

=t

∑j=1

α j(p· rk(A j)−h0(A j(n)) · r

).

Since we assume Gies(z) to be (semi)stable, we have

0 (≤) ν1ν2·µ(λ , [M])+µ(λ , [L])

= ν1ν2·∑t

j=1 α j(p· rk(A j)−h0(A j(n)) · r

)−∑t

j=1α j(s! · l j −ν j(i01, ..., i

0s!) · p

)

= p−s!·δ (n)r ·δ (n) ·∑t

j=1α j(p· rk(A j)−h0(A j(n)) · r

)−∑t

j=1α j(s! · l j −ν j(i01, ..., i

0s!) · p

)

l j=h0(A j (n))= ∑t

j=1 α j

(p2·rk(A j )

r ·δ (n) −p·s!·rk(A j )

r −p·h0(A j (n))

δ (n)

)+∑t

j=1 α j ·ν j(i01, ..., i0s!) · p.

We multiply this inequality byr ·δ (n)/p. This leads to the inequality

t

∑j=1

α j(p· rk(A j)−h0(A j(n)) · r

)+δ (n) ·

(−

t

∑j=1

α j(s! · rk(A j)−ν j(i

01, ..., i

0s!) · r

))(≥)0.

To conclude, we have to verify

µ(A•,α•;ϕ)≥−t

∑j=1


01, ..., i

0s!) · r

). (20)


If γ is the weight vector with the distinct weightsγ1 < · · ·< γt+1 which is associated to(A•,α•) as inRemark 4.1.1, then one easily checks that

γi01+ · · ·+ γi0s!

=t

∑j=1


01, ..., i

0s!) · r

).

In view of (10), it remains to show that

ϕ|Ai01⋆···⋆A

i0s!

6≡ 0. (21)

To this end, note that, up to a scalar,L is given as

(ι :Vs(U)−→ H0(

Vs(A )(s!n))) H0(ϕ⊗idØX(s!n))

−→ H0(ØX(s!n)).

The image ofU⋆i01,...,i

0s!

lies in the subspaceH0((Ai01⋆ · · ·⋆Ai0s!

)(s!n)) and that shows (21).

We now turn to the converse direction in the proof of Theorem 4.5.3. Again, we need a preparatoryresult.

PROPOSITION4.5.7. There is a positive integer n5, such that anyδ -(semi)stable dispo sheaf(A ,ϕ)of type(ρ ,s) with P(A ) = P satisfies

t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

)+δ (n) ·µ(A•,α•;ϕ)(≥)0

for every weighted filtration(A•,α•) of A and every n≥ n5.

Proof. By Proposition 4.2.1, we know that there is a bounded familyS′ of torsion free sheaves withHilbert polynomialP, such that[A ] ∈S′ for anyδ -semistable dispo sheaf(A ,ϕ) of type(ρ ,s) withP(A ) = P. We choose a constantC, such thatµmax(A )≤C for every torsion free sheafA onX with[A ] ∈S′. Given an additional positive constantC′, we subdivide the class of torsion free sheavesB

that might occur as saturated subsheaves of ØX-modulesA with [A ] ∈S into two classes:

A. µ(B)≥−C′

B. µ(B)<−C′.

By a Lemma of Grothendieck’s ([18], Lemma 1.7.9), the sheaves B falling into Class A live again inbounded families, so that we may always assume that ourn is large enough, such that any such sheafB satisfieshi(B(n)) = 0, i > 0.

If E is a torsion free sheaf onX with Harder-Narasimhan filtration 0=: E0 ( E1 ( · · · ( Et (

Et+1 := E , then

h0(E )≤t+1

∑i=1

h0(Ei/Ei−1

),

so that (17) gives, withF(r) := max f (i) | i = 1, ..., r and rk(E )≤ r,

h0(E ) ≤(rk(E )−1

)·

deg(X)

dim(X)!·(µmax(E )

deg(X)+F(r)+dim(X)

)dim(X)

+deg(X)

dim(X)!·( µ(E )

deg(X)+F(r)+dim(X)

)dim(X).


For any sheafA with [A ] ∈S′ and any saturated subsheafB of A which belongs to the Class B, wethus find

h0(B(n)) ≤(rk(B)−1

)·

deg(X)

dim(X)!·( C

deg(X)+F(r −1)+dim(X)+n

)dim(X)

+deg(X)

dim(X)!·( −C′

deg(X)+F(r −1)+dim(X)+n

)dim(X)(22)

=: R(rk(B),C′

)(n).

We chooseC′ so large that

P· rk(B)−h0(B(n)) · r ≥ P · rk(B)−R(rk(B),C′

)· r = K ·xdim(X)−1+ · · · ≻ δ ·s! · (r −1). (23)

For alln≫ 0, (23) remains true when evaluated atn.Now, let (A•,α•) be any weighted filtration ofA . Write 1, ..., t = IA ⊔ IB with i ∈ IA/B if and

only if Ai belongs to the Class A/B. Let 1≤ iA/B1 < · · · < iA/B

tA/B≤ t be the indices inIA/B and define

the weighted filtrations(A A/B• ,αA/B

• ) with

AA/B• : 0( A

A/B1 := A

iA/B1

( · · ·( AA/B

tA/B:= A

iA/BtA/B

( A ,

αA/B• = (αA/B

1 , ...,αA/BtA/B

) := (αiA/B1

, ...,αiA/BtA/B

).

It is easy to see that

µ(A•,α•;ϕ

)≥ µ

(A

A• ,αA

• ;ϕ)−s! · (r −1) ·

tB

∑j=1

αBj . (24)

Now, we compute

t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

)+δ (n) ·µ(A•,α•;ϕ)

(24)≥

tA

∑j=1

αAj

(P(n) · rk(A A

j )−h0(A Aj (n)) · r

)+δ (n) ·µ(A A

• ,αA• ;ϕ)+

+tB

∑j=1

αBj

(P(n) · rk(A B

j )−h0(A Bj (n)) · r

)−δ (n) ·s! · (r −1) ·

tB

∑j=1

αBj

n≫0= M

(A

A• ,αA

•

)(n)+δ (n) ·µ(A A

• ,αA• ;ϕ)+

+tB

∑j=1

αBj

((P(n) · rk(A B

j )−h0(A Bj (n)) · r

)−δ (n) ·s! · (r −1)

)

(23)&n≫0≥ M

(A

A• ,αA

•

)(n)+δ (n) ·µ(A A

• ,αA• ;ϕ)

n≫0(≥) 0.

The last estimate results from the condition ofδ -(semi)stability, applied to the weighted filtration(A A

• ,αA• ). We still have to justify that, in this last estimate,n can be uniformly chosen for all poly-

nomials of the formM(A•,α•)+ δ · µ(A•,α•;ϕ) where all the members of the filtrationA• belongto the Class A. We use again the setT that has been introduced in the proof of Theorem 4.5.5.


Then, any polynomial of the above form can be written as a positive rational linear combination ofpolynomialsM(A i

• ,α i•)+δ ·µ(A i

• ,α i•;ϕ) where((rk(A i

1), ..., rk(Ai

ti ),αi•) ∈ T and all the members

of the filtration A i• belong to the Class A,i = 1, ...,u. By the boundedness of the set of isomor-

phism classes of sheaves in the Class A, it now follows that there are only finitely many polynomialsof the formM(A•,α•)+ δ · µ(A•,α•;ϕ) where all the members ofA• belong to the Class A and((rk(A1), ..., rk(At),α•) ∈ T . This proves our last claim and the proposition.

THEOREM 4.5.8. There exists a positive integer n6, enjoying the following property. If n≥ n6 and(A ,ϕ) is a δ -(semi)stable dispo sheaf of type(ρ ,s) with P(A ) = A , then, for a point z∈D of theform z= ([q:U ⊗ØX(−n) −→ A ],ϕ), the associated Gieseker pointGies(z) is (semi)stable for thegiven linearization.

Proof. Let λ :Gm(k)−→SL(U) be a one parameter subgroup and suppose Gies(z) = ([M], [L]). Then,we have to verify that

ν1

ν2·µ(λ , [M])+µ(λ , [L])(≥)0.

The one parameter subgroupλ provides the weighted flag(U•(λ ),β•(λ )) with

U•(λ ) : 0=: U0 (U1 ( · · ·(Uτ (Uτ+1 :=U ; β•(λ ) = (β1, ...,βτ ).

For eachh∈ 1, ...,τ , we letAh be the saturated subsheaf that is generically generated byq(Uh⊗

ØX(−n)). There may be improper inclusions among theAh’s. After clearing these, we obtain thefiltration

A• : 0=: A0 ( A1 ( · · ·( At ( At+1 := A .

For j = 1, ..., t, we define

T( j) :=

h∈ 1, ...,τ ∣∣Ah = A j

andα j := ∑

h∈T( j)

βh.

This gives the weighted filtration(A•,α•) of A . By Proposition 4.5.7,

t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

)+δ (n) ·µ(A•,α•;ϕ)(≥)0. (25)

Recall from (10) that

µ(A•,α•;ϕ

)=−min

γi1 + · · ·+ γis!

∣∣(i1, ..., is!) ∈ I : ϕ|Ai1⋆···⋆Ais!6≡ 0

. (26)

Let (i01, ..., i0s!) ∈ I = 1, ..., t +1×s! be an index tuple that computes the minimum. With

ν j(i1, ..., is!) := #

ik ≤ j |k= 1, ...,s!,

one calculates

γi01+ · · ·+ γi0s!

=t

∑j=1


01, ..., i

0s!) · r

).


Thus (25) transforms into

t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

)+δ (n) ·

(−

t

∑j=1


01, ..., i

0s!) · r

))(≥)0.

A computation as in the proof of Theorem 4.5.5, but performedbackwards, shows that this implies

ν1

ν2·( t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

))−

t

∑j=1

α j(s! ·h0(A j(n))−ν j(i

01, ..., i

0s!) · p

)(≥)0. (27)

First, we see that

µ(λ , [M]) =τ

∑h=1

βh(P(n) · rk(Ah)−dimk(Uh) · r

)≥

t

∑j=1

α j(P(n) · rk(A j)−h0(A j(n)) · r

). (28)

We need a little more notation. Forj = 0, ..., t +1, we introduce

h( j) := min

h= 1, ...,τ |Ah = A j

; U j :=Uh( j)

h( j) := max

h= 1, ...,τ |Ah = A j

; U j :=Uh( j),

as well asU j :=U j/U j−1, j = 1, ..., t +1.

For an index tuple(i1, ..., is!) ∈ I , we find the vector space

Ui1,...,is! :=(Ui1 ⊗·· ·⊗Uis!

)⊕N.

Using a basisu of U that consists of eigenvectors for the one parameter subgroup λ , we identify thesespaces with subspaces of(U⊗s!)⊕N. Then,U•

i1,...,is! stands for the image ofUi1,...,is! in

⊕

(d1,...,ds):di≥0,∑ idi=s!

(Symd1

(U ⊗kV

)⊗·· ·⊗Symds

(Syms(U ⊗kV)

))

andU⋆i1,...,is! for the intersection ofU•

i1,...,is! with Vs(U). A similar construction, generalizing the oneat the end of Section 2.1, associates to a collection of subspacesY1, ...,Ys! of U , (i1, ..., is!) ∈ I , thesubspaceY1⋆ · · · ⋆Ys! of (U⊗s!)⊕N. Note that

λ = λ (u,γ) with γ =τ

∑h=1

βh · γ(dimk(Uh))p .

We defineλ h := λ (u,γ(dimk(Uh))p ), h= 1, ...,τ . The effect of our definition is that the spacesU⋆

i1,...,is! ,(i1, ..., is!)∈ I , are weight spaces forλ as well as forλ 1, ...,λ τ . We associate to an indexh∈ 1, ...,τ the index j(h) ∈ 1, ..., t with Ah = A j(h). Then,h( j) ≤ h holds if and only if j ≤ j(h), and one

verifies thatλ acts onU⋆i01,...,i

0s!

with the weight

−τ

∑h=1

βh(s! ·dimk(Uh)−ν j(h)(i

01, ..., i

0s!) · p

)≥−

t

∑j=1

α j(s! ·h0(A j(n))−ν j(i

01, ..., i

0s!) · p

). (29)


In view of the estimate (27), (28), and (29), it is now sufficient to ascertain that the restriction ofLto U⋆

i01,...,i0s!

is non-trivial. If it were trivial, then there would have to be an index tuple(i′1, ..., i′s!) with

i′l ≤ i0l , l = 1, ...,s!, at least one inequality being strict, such that

L|U i′1⋆···⋆U i′s!

6≡ 0. (30)

This is becauseL restricts to a non-zero map onU i01⋆ · · · ⋆U i0s!

, asϕ|Ai01⋆···⋆A

i0s!

is non-trivial. Now, if

(30) holds true, then we must also have

ϕ|Ai′1⋆···⋆Ai′s!

6≡ 0. (31)

(Compare the arguments at the end of the proof of Theorem 4.5.5.) But, then the tuple(i01, ..., i0s!)

would not give a minimum in (26), a contradiction.

By Theorem 4.5.3, the subsetsDδ-(s)s that correspond toδ -(semi)stable dispo sheaves are thepreimages of the sets of GIT-(semi)stable points inG under the Gieseker morphism. Therefore, theyare open subsets.

PROPOSITION4.5.9. The restricted Gieseker morphism

Gies|Dδ-ss:Dδ-ss−→G

ss

is proper. Since it is also injective, it is finite.

Proof. This is pretty standard, so we can be a bit sketchy. We apply the valuative criterion ofproperness. LetQ be the closure ofQ in the quot scheme ofU ⊗ØX(−n). Then, the parame-ter spaceD may also be compactified toD −→ Q. Given a discrete valuation ringR, a morphismη :C := Spec(R)−→G

ss which lifts overC⋆ := Spec(Quot(R)) to a morphismη⋆:C⋆ −→Dδ-ss, wemay first extendη⋆ to a morphismη:C −→D. This morphism is associated with a family

(qC:U ⊗π⋆

X(ØX(−n))−→ AC,ϕC:Vs(AC)−→ ØC×X)

on C×X where the restriction ofAC to the special fiber0 ×X may have torsion. LetZ be thesupport of that torsion. Then, the familyqC may be altered to a familyqC:U ⊗π⋆

X(ØX(−n)) −→ AC

whereAC is now aC-flat family of torsion free sheaves,qC agrees withqC on(C×X)\Z, but qC|0×X

may fail to be surjective in points ofZ. Let ι :(C×X)\Z−→C×X be the inclusion. Define

ϕC:Vs(AC)−→ ι⋆(Vs(AC|(C×X)\Z)

)= ι⋆

(Vs(AC|(C×X)\Z)

) ι⋆(ϕC|(C×X)\Z)−→ ι⋆

(Ø(C×X)\Z

)= ØC×X.

The family(qC, ϕC) also defines a morphism toG which coincides withη . Let (q:U ⊗ØX(−n) −→A , ϕ) be the restriction of the new family to0×X. One checks the following results.

• H0(q(n)) must be injective;

• since(q, ϕ) defines a semistable point,A belongs to a bounded family (this is an easy adapta-tion of the proof of Proposition 4.5.4);

• the techniques of the proof of Theorem 4.5.5 may also be used to show that(A , ϕ) must beδ -semistable. In particular, the higher cohomology groups of A (n) vanish, so thatH0(q(n)) isindeed an isomorphism.


The family (q, ϕ) is thus induced by a morphismη which lifts η and extendsη⋆. This finishes theargument.

SinceGss possesses a projective quotient, Proposition 4.5.9 and Lemma 2.1.7 show that the goodquotient

Mδ-ssP (ρ) :=Dδ-ss//SL(U)

exists as a projective scheme. Likewise, the geometric quotient

Mδ-sP (ρ) :=Dδ-s/SL(U)

exists as an open subscheme of Mδ-ssP (ρ) By Proposition 4.5.1, Proposition 4.5.2, and the universal

property of a categorical quotient, the space Mδ-ssP (ρ) is indeed a coarse moduli space.

Remark4.5.10(S-equivalence).Recall that two points inDδ-ss are mapped to the same point in thequotient if and only if the closures of their orbits intersect in Dδ-ss. Given a pointy ∈ Dδ-ss, lety′ ∈Dδ-ss be the point whose orbit is the unique closed orbit inSL(U) ·y(⊆Dδ-ss). Then, there is aone parameter subgroupλ :Gm(k)−→SL(U) with limz→∞ λ (z) ·y∈SL(U) ·y′. For this one parametersubgroup, one has of courseµ(λ ,y) = 0. Thus, the equivalence relation that we have to consider onthe geometric points ofDδ-ss is generated byy ∼ limz→∞ λ (z) · y for all one parameter subgroupsλ :Gm(k)−→ SL(U) with µ(λ ,y) = 0.

If one looks carefully at the arguments given in the proofs ofTheorem 4.5.5 and 4.5.8, one seesthat, for a pointy= ([q:U ⊗ØX(−n)−→ A ],ϕ) ∈Dδ-ss, the following observations hold true.

• If λ :Gm(k)−→SL(U) verifiesµ(λ ,y) = 0, then its weighted flag(U•(λ ),α•(λ )) has the prop-erty that the weighted filtration(A•,α(λ )) with A j := q(U j ⊗ØX(−n)), j = 1, ..., t, satisfies

M(A•,α•(λ )

)+δ ·µ

(A•,α•(λ );ϕ

)≡ 0.

• Given a weighted filtration(A•,α•) of A with

M(A•,α•

)+δ ·µ

(A•,α•;ϕ

)≡ 0,

one can assumehi(A j(n)) = 0, i > 0, and thatA j(n) is globally generated,j = 1, ..., t. Hence,there is a unique flagU• in U , such thatH0(q(U j)) mapsU j onto H0(A j(n)), j = 1, ..., t.Then, any one parameter subgroupλ :Gm(k) −→ SL(U) with weighted flag(U•,α•) satisfiesµ(λ ,y) = 0.

• For a one parameter subgroupλ with µ(λ ,y) = 0, y′ := limz→∞ λ (z) ·y, and induced weightedfiltration (A•,α•) onA , the dispo sheaf(Ay′ ,ϕy′) is equivalent to df(A•,α•)(A ,ϕ).

This shows that the equivalence relation induced by the GIT process on the closed points ofDδ-ss isjust S-equivalence of dispo sheaves as introduced in Section 4.3.

Remark4.5.11(Decorated vector bundles on curves).In [33], given a homogeneous representationκ :GLr(k)−→GL(V) and a smooth projective curveX over the complex numbers, the moduli problemof classifying triples(E,L,ϕ) consisting of a vector bundle of rankr on X, a line bundleL on X, anda non-trivial homomorphismϕ :Eκ −→ L, Eκ being associated toE via κ , was solved by a GITprocedure similar to the one presented above. WriteV = V ⊗det(V)⊗−c whereV is a homogeneouspolynomial GLr(k)-module, say, of degreea. The only characteristic zero issue that is necessary forthe construction in [33] is the fact thatV can be written as the quotient of(V⊗a)⊕b for an appropriate


positive integerb. In characteristicp > 0, this is only true whenp> a. However, we may use thedivided powers

Di(V) :=(Symi(V∨)

)∨

andD

a(V) :=⊕

(d1,...,dt ):t≥1,di≥0∑di=a

Dd1(V)⊗·· ·⊗Ddt (V).

Then,V is a quotient ofDa(V)⊕b for an appropriate integerb> 0, [12]. Now,Da(V) is a subrepre-sentation of(V⊗a)⊕N. This shows that the arguments given in the present paper maybe used to dealwith decorated vector bundles on smooth projective curves in any characteristic.

5 The proof of the main theorem

This section is devoted to the proof of the main theorem. In fact, we will prove a slightly strongertheorem which is the exact analog to the main result of [32] inarbitrary characteristic. To do so, werecall the necessary notions ofδ -semistable pseudoG-bundles and so on.

5.1 Associated dispo sheaves

The notion of a “pseudoG-bundle” has been recalled in Section 3.1. Now, we relate pseudoG-bundlesto dispo sheaves.

LetSbe a scheme, and(AS,τS) a family of pseudoG-bundles parameterized byS. Let ι :U ⊆S×Xbe the maximal open subset whereAS is locally free. The locally free sheafAS|U and the GLr(k)-moduleVs give rise to the vector bundleVs(AS|U ), and there is a surjection

S ym⋆(Vs(AS|U)

)−→ S ym(s!)(AS|U ⊗V)G.

Defineτs as the restriction ofτS|U to the subalgebraS ym(s!)(AS|U ⊗V)G. Then,τs is determined bya homomorphism

ϕ ′:Vs(AS|U)−→ ØU .

Thus,τS|U gives rise to the homomorphism

ϕS:Vs(AS)−→ ι⋆(Vs(AS|U)

)−→ ι⋆

(ØU

)= ØS×X,

by Corollary 2.3.2. Therefore, we can associate to the family (AS,τS) of pseudoG-bundles the family(AS,ϕS) of dispo sheaves of type(ρ ,s).

The map which associates to a pseudoG-bundle a dispo sheaf is injective on equivalence classes.More precisely, we find.

LEMMA 5.1.1. Suppose that(A ,τ) and(A ,τ ′) are two pseudo G-bundles, such that the associateddispo sheaves are equal. Then, there is a root of unityζ ∈ k, such thatζ · idA yields an equivalencebetween(A ,τ) and(A ,τ ′).

Proof. Ford > 0, letτd,τ ′

d:S ymd(A ⊗V)G −→ ØX


be the degreed component ofτ andτ ′, respectively. Note thatτ is determined by⊕s

d=1 τd. Let

τs :⊕

(d1,...,ds);di≥0,∑ idi=s!

(S ymd1

((A ⊗V)G)⊗·· ·⊗S ymds

(S yms(A ⊗V)G))−→ ØX

be the map induced byτ1,...,τs, and defineτ ′s in a similar way. By definition,τs|U = τs. Our assumption

thus grants that(A , τs) and(A , τ ′s) are equal. This implies that, for 1≤ d ≤ s,

S yms!d (τd) = S ym

s!d (τ ′

d).

Restricting this equality to the generic point, it follows that there is an(s!/d)-th root of unityζd with

τ ′d = ζd · τd, d = 1, ...,s.

It remains to show that there is ans!-th root of unity ζ , such thatζd = ζ d. To see this, letA be therestriction ofA to the generic point. Then,τs andτ ′

s, restricted to the generic point, define the samepoint

x∈P := P( ⊕

(d1,...,ds);di≥0,∑ idi=s!

Symd1((A⊗V)G)⊗·· ·⊗Symds

(Syms(A⊗V)G)).

On the other hand,⊕s

d=1 τd and⊕s

d=1 τ ′d define points

y,y′ ∈B :=( s⊕

d=1

Symd(A⊗V)G)∨

.

By our assumption,yandy′ map both toxunder the quotient map followed by the Veronese embedding

B\0 −→(B\0

)//Gm(K) →P.

Putting all the information we have gathered so far together, we find the claim about theζi and fromthat the one of the lemma.

Let δ ∈Q[x] be a positive polynomial of degree at most dim(X)−1. We choose ansas before anddefineδ := δ/s!. A pseudoG-bundle is said to beδ -(semi)stable, if the associated dispo sheaf(A ,ϕ)of type (ρ ,s) is δ -(semi)stable. Similarly, given a non-negative rational numberδ ⋆, we define thepseudoG-bundle(A ,τ) to beδ ⋆-slope (semi)stable, if the associated dispo sheaf(A ,ϕ) is (δ ⋆/s!)-slope (semi)stable.

Remark5.1.2. i) The definition ofδ -(semi)stability is the same as the one given in [32], p. 1192.ii) Using (7), it follows that the notion ofδ -(semi)stability does not depend on the choice ofs.

This is why we threw in the factor 1/s!.

5.2 S-equivalence

We fix a stability parameterδ , i.e., a positive rational polynomial of degree at most dim(X)− 1.Suppose(A ,τ) is a δ -semistable pseudoG-bundle with associated dispo sheaf(A ,ϕ) and(A•,α•)is a weighted filtration with

M(A•,α•)+δs!·µ(A•,α•;ϕ)≡ 0.


The construction used for defining an associated admissibledeformation of a dispo sheaf can be easilyextended to give the construction of the associated admissible deformation df(A•,α•)(A ,τ). As before,

we let S-equivalencebe the equivalence relation “∼S” on δ -semistable pseudoG-bundles(A ,τ)generated by

(A ,τ)∼S df(A•,α•)(A ,τ).

The injectivity of the map which assigns to the equivalence class of a pseudoG-bundle the equivalenceclass of the associated dispo sheaf (Lemma 5.1.1) and the definitions of semistability for the respectiveobjects easily imply that for two pseudoG-bundles(A ,τ) and(A ′,τ ′) with associated dispo sheaves(A ,ϕ) and(A ′,ϕ ′) one has

(A ,τ)∼S (A′,τ ′) ⇐⇒ (A ,ϕ)∼S (A

′,ϕ ′). (32)

In Remark 5.4.3 below, we will give a nice description of S-equivalence on semistable singular prin-cipal G-bundles.

5.3 Moduli spaces forδ -semistable pseudoG-bundles

An immediate consequence of the definition of semistabilityof pseudoG-bundles and Proposition4.2.1 is that, for a given Hilbert polynomialP, the set of torsion free sheavesA with Hilbert poly-nomial P for which there exists aδ -(semi)stable pseudoG-bundle(A ,τ) is bounded. Finally, theconstruction carried out in Section 5.1 and Corollary 2.3.2give a natural transformation

AD:M δ-(s)sP (ρ)−→ Mδ-(s)s

P (ρ ,s)

of the functor Mδ-(s)sP (ρ) which assigns to a schemeS the set of equivalence classes of families of

δ -(semi)stable pseudoG-bundles with Hilbert polynomialP parameterized byS into the functorMδ-(s)s

P (ρ ,s) which assigns to a schemeS the set of equivalence classes of families ofδ -(semi)stabledispo sheaves of type(ρ ,s) with Hilbert polynomialP parameterized byS.

THEOREM 5.3.1. Fix the stability parameterδ and the Hilbert polynomial P. Then, there is aprojec-

tive schemeM δ-ssP (ρ) that is a coarse moduli space for the functorsM δ-ss

P (ρ).

Remark5.3.2. This theorem generalizes the main theorem of [32] to arbitrary characteristic.

Construction of the parameter space. — There is a constantC, such thatµmax(A )≤C for everyδ -semistable pseudoG-bundle(A ,τ) with P(A ) = P, i.e.,A lives in a bounded family. Thus, wemay choose the integers in such a way thatS ym⋆(A ⊗V)G is generated by elements in degree atmosts for all suchA . We choose ann0 ≫ 0 with the following properties. For every sheafA withHilbert polynomialP andµmax(A )≤C and everyn≥ n0, one has

• H i(A (n)) = 0 for i > 0;

• A (n) is globally generated;

• the construction of the moduli space of(δ/s!)-semistable dispo sheaves of type(ρ ,s) can beperformed with respect ton.


We choose ak-vector spaceU of dimensionP(n). Let Q be the quasi-projective scheme which pa-rameterizes quotientsq:U ⊗ØX(−n)−→ A whereA is a torsion free sheaf with Hilbert polynomialP and µmax(A ) ≤ C (so thatS ym⋆(A ⊗V)G is generated by elements of degree at mosts) andH0(q(n)) is an isomorphism. Let

qQ:U ⊗π⋆X

(ØX(−n)

)−→ AQ

be the universal quotient. By Lemma 2.2.2, there is a surjection

S ym⋆(U ⊗π⋆

X(ØX(−n))⊗V)G

−→ S ym⋆(AQ⊗V

)G.

For a point[q:U ⊗ØX(−n) −→ A ] ∈ Q, any homomorphismτ :S ym⋆(A ⊗V)G −→ ØX of ØX-algebras is determined by the composite homomorphism

s⊕

i=1

S ymi(U ⊗π⋆X(ØX(−n))⊗V

)G−→ ØX

of ØX-modules. Noting that

S ymi(U ⊗π⋆

X(ØX(−n))⊗V)G ∼= Symi(U ⊗V)G⊗π⋆

X

(ØX(−in)

),

τ is determined by a collection of homomorphisms

ϕi :Symi(U ⊗V)G⊗ØX −→ ØX(in), i = 1, ...,s.

Sinceϕi is determined by the induced linear map on global sections, we will construct the parameterspace inside

Y0 :=Q×s⊕

i=1

Hom(

Symi(U ⊗V)G,H0(OX(in)

)).

Note that, overY0×X, there are universal homomorphisms

ϕ i : Symi(U ⊗V)G⊗OY0×X −→ H0(OX(in))⊗OY0×X, i = 1, ...,s.

Let ϕ i = ev ϕ i be the composition ofϕ i with the evaluation map ev:H0(ØX(in))⊗ØY0×X −→π⋆

X(OX(in)), i = 1, ...,s. We twistϕ i by idπ⋆X(ØX(−in)) and put the resulting maps together to the homo-

morphism

ϕ :VY0 :=s⊕

i=1

S ymi(U ⊗π⋆X(OX(−n))⊗V

)G−→ OY0×X.

Next,ϕ yields a homomorphism ofOY0×X-algebras

τY0:S ym⋆(VY0

)−→ OY0×X.

On the other hand, there is a surjective homomorphism

β :S ym⋆(VY0)−→ S ym⋆(π⋆(AQ)⊗V

)G

of graded algebras where the left hand algebra is graded by assigning the weighti to the elements inS ymi(U ⊗π⋆

X(ØX(−n))⊗V)G. Here,π:Y0×X −→Q×X is the natural projection. The parameter


spaceY is defined by the condition thatτY0 factorizes overβ , i.e., settingAY := (π⋆(AQ))|Y×X ,there is a homomorphism

τY:S ym⋆(AY⊗V)G −→ OY×X

with τY0|Y×X = τY β . Formally,Y is defined as the scheme theoretic intersection of the closedsubschemes

Yd :=

y∈Y0∣∣ τd

Y0|y×X:ker(β d|y×X

)−→ ØX is trivial

, d ≥ 0.

The family(AY,τY) is theuniversal family of pseudo G-bundles parameterized byY.

PROPOSITION5.3.3 (LOCAL UNIVERSAL PROPERTY). Let S be a scheme and(AS,τS) a family of

δ -semistable pseudo G-bundles with Hilbert polynomial P parameterized by S. Then, there exist acovering of S by open subschemes Si , i ∈ I, and morphismsβi :Si −→ Y, i ∈ I, such that the family(AS|Si

,τS|Si) is equivalent to the pullback of the universal family onY×X byβi × idX for all i ∈ I.

The group action. — There is a natural action of GL(U) on the quot schemeQ and onY0. Thisaction leaves the closed subschemeY invariant, and therefore yields an action

Γ:GL(U)×Y−→Y.

PROPOSITION 5.3.4 (GLUING PROPERTY). Let S be a scheme andβi :S−→ Y, i = 1,2, two mor-phisms, such that the pullback of the universal family viaβ1 × idX is equivalent to its pullback viaβ2× idX. Then, there is a morphismΞ:S−→ GL(U), such thatβ2 equals the morphism

SΞ×β1−→ GL(U)×Y

Γ−→Y.

Remark5.3.5. The universal family is equipped with a GL(U)-linearization. If one fixes, in theabove proposition, an isomorphism between its pullbacks via β1× idX andβ2× idX, then there is aunique morphismΞ:S−→ GL(U) which satisfies the stated properties and, in addition, thatthe givenisomorphism is induced by pullback via(Ξ×β1× idX) from the linearization of(AY,τY). This fact

simply expresses that the moduli stack forδ -(semi)stable pseudoG-bundles will be the quotient stackof an appropriate open subscheme of the parameter spaceY.

Conclusion of the proof. — Suppose we knew that the points([q:U ⊗ØX(−n) −→ A ],τ) in

the parameter spaceY for which (A ,τ) is δ -semistable form an open subschemeYδ-ss. Then, it

suffices to show thatYδ-ss possesses a (good, uniform) categorical quotient by the action of GL(U).Indeed, Proposition 5.3.3 and 5.3.4 and the universal property of the categorical quotient then imply

that Mδ-ssP (ρ) :=Yδ-ss//GL(U) has the desired properties. We have the natural surjectionGm(k)×

SL(U)−→ GL(U), (z,m) 7−→ z·m, and obviously

Yδ-ss//GL(U) =Yδ-ss//(Gm(k)×SL(U)

).

By Example 2.1.6, ii), we may first form

Yδ-ss

:=Yδ-ss//Gm(k)

and then

Yδ-ss

//SL(U).


We can easily form the quotientY := Y//Gm(k). SinceGm(k) is linearly reductive,Y is a closedsubscheme of

Q×

( s⊕

i=1

Hom(

Symi(U ⊗V)G,H0(OX(in)

))//Gm(k)

).

In particular,Y is projective overQ. LetD −→Q be the parameter space for dispo sheaves of type(ρ ,s) constructed above. If we apply the construction described in Section 5.1 to the universal family(AY,τY), we get an SL(U)-equivariant andGm(k)-invariant morphism

ψ :Y−→D

and, thus, a proper SL(U)-equivariant morphism

ψ :Y−→D.

By Lemma 5.1.1,ψ is injective, so that it if evenfinite. Now, there are open subsetsDδ-(s)s, δ :=δ/s!, which parameterize theδ -(semi)stable dispo sheaves of type(ρ ,s), such that the good, uniformcategorical quotient

Mδ-ssP (ρ ,s) =Dδ-ss//SL(U)

exists as a projective scheme and the geometric, uniform categorical quotient

Mδ-sP (ρ ,s) =Dδ-s/SL(U)

as an open subscheme of Mδ-ssP (ρ ,s). By definition of semistability,

ψ−1(Dδ-ss)=Yδ-ss,

whenceψ−1(Dδ-ss)=Yδ-ss//Gm(k).

Now, Lemma 2.1.7 implies that the quotient

M δ-ssP (ρ) :=Yδ-ss//GL(U) = ψ−1(Dδ-ss)//SL(U)

exists as a projective scheme. Likewise, the open subscheme

M δ-sP (ρ) :=Yδ-s/GL(U) = ψ−1(Dδ-s)/SL(U)

is a uniform (universal) geometric quotient and an open subscheme of Mδ-ssP (ρ).

5.4 Semistable singular principal bundles

THEOREM 5.4.1. Fix a Hilbert polynomial P. There is a positive polynomialδ0 of degreedim(X)−1,such that for every polynomialδ ≻ δ0 and every singular principal G-bundle(A ,τ) with P(A ) = P,the following properties are equivalent.

i) (A ,τ) is (semi)stable.

ii) (A ,τ) is δ -(semi)stable.


We will use the following result.

LEMMA 5.4.2. Let (A ,τ) be a singular principal G-bundle with associated dispo sheaf (A ,ϕ).Then, for a weighted filtration(A•,α•) of A , the condition

µ(A•,α•;ϕ) = 0

is satisfied, if and only if(A•,α•) =

(A•(β ),α•(β )

)

for some reductionβ of (A ,τ) to a one parameter subgroupλ of G.

Proof. We show that the first condition implies the second one, the converse being an easy exercise.Let λ ′:Gm(k)−→ SLr(V) be a one parameter subgroup, such that the associated weighted flag

(V•(λ ′):0(V1 ( · · ·(Vt ′ (V,α•(λ ′)

)

satisfiest ′ = t, dimk(Vi) = rkA ′i , A ′

i = ker(A ∨ −→ A ∨t+1−i), i = 1, ..., t, andα•(λ ′) = (αt , ...,α1), if

α• = (α1, ...,αt). Then, the weighted filtration(A•,α•) is associated to a reductionβ ′ of the GL(V)-principal bundleI som(V,A ∨

|U ′) to λ ′ with U ′ the maximal open subset whereA is locally free and

all theA ′i are subbundles. We may choose an open subsetU ⊆ X, such that there is a trivialization

ψ :A ∨|U

−→V⊗ØU with ψ(A ′i ) =Vi ⊗ØU , i = 1, ..., t. By definition of the numberµ

(A•,α•;ϕ), (8),

and Proposition 2.1.4, we see that there is a one parameter subgroupλ :Gm(k)−→ G, such that(V•(λ ),α•(λ )

)=

(V•(λ ′),α•(λ ′)

).

To the principal bundlesP(A ,τ) andI som(V,A ∨|U ), we may associate group schemesG ⊂ G L (V)

overU . Now,G L (V) acts onI som(V,A ∨|U )/QGL(V)(λ ), and the stabilizer of the sectionβ ′:U ′ −→

I som(V,A ∨|U )/QGL(V)(λ ) is a parabolic subgroupQ ⊂ G L (V)|U ′ , such that

G L (V)|U ′/Q = I som(V,A ∨|U ′)/QGL(V)(λ ).

The intersectionQG := Q ∩ G|U ′ is a parabolic subgroup. This follows if one applies the abovereasoning on weighted flags to the geometric fibers ofG ⊂ G L (V) overU ′. Furthermore,G|U ′/QG =P(A ,τ)|U ′/QG(λ ). This can be seen as follows. LetC be the set of conjugacy classes of parabolicsubgroups ofG. There is a schemeParp(G|U ′/U ′) overU ′, such that giving a parabolic subgroupQG of G|U ′ the fibers of which belong top∈C is the same as giving a sectionU ′ −→ Par(G|U ′/U ′)([5], p. 443ff). It is easy to see thatParp(G|U ′/U ′) ∼= P(A ,τ)|U ′/Qp, Qp being a representativefor p (compare [29], p. 281). Finally,G|U ′/QG

∼= Parp(G|U ′/U ′) ([5], Corollaire 3.6, page 445).Therefore, we have the commutative diagram

QG

// _

Q _

G|U ′

// G L (V)|U ′ .

Taking QG -quotients in the left hand column andQ-quotients in the right hand column yields thecommutative diagram

U ′

β

U ′

β ′

P(A ,τ)|U ′/QG(λ ) // I som(V,A ∨

|U )/QGL(V)(λ )

and settles the claim.


Proof of Theorem5.4.1. The implication “ii)=⇒i)” results from Proposition 5.4.2. For the converse,we first observe that Lemma 2.1.3 and 2.1.5 implyµ(A•,α•;ϕ) ≥ 0 for every weighted filtration(A•,α•) of A . Moreover, for a weighted filtration(A•,α•) with µ(A•,α•;ϕ) = 0, the conditionM(A•,α•)+ (δ/s!) · µ(A•,α•;ϕ)()0 follows from the definition of semistability given in the in-troduction and Proposition 5.4.2. Thus, we only have to dealwith weighted filtrations for whichµ(A•,α•;ϕ)> 0. Standard considerations (e.g., [35], Proposition 2.14)show that there is a finite set(depending only on the GLr(k)-moduleVs, whence not onδ )

T =(r j

•,α j•)

∣∣ r j• = (r j

1, ..., rjt j) : 0< r j

1 < · · ·< r jt j< r;

α j• = (α j

1, ...,αj

t j) : α j

i ∈Q>0, i = 1, ..., t j , j = 1, ..., t,

such that the condition ofδ -(semi)stability has to be checked only against those weighted filtrations(A•,α•) with (

(rkA1, ..., rkAt),α•

)∈ T .

By Theorem 3.3.1, there is a constantC, such thatµmax(A )≤C for every semistable singular principalG-bundle(A ,τ). Thus, it is clear that there is a negative constantC′, such that

L(A•,α•)>C′

for every semistable singular principalG-bundle(A ,τ) and every weighted filtration(A•,α•) with((rkA1, ..., rkAt),α•) ∈ T . We setδ0 := (−(dim(X)−1)!s!C′) ·xdim(X)−1. Then, for a polynomial

δ =(δ ⋆/(dim(X)−1)!

)·xdim(X)−1+ · · · ≻ δ0,

a semistable singular principalG-bundle(A ,τ), and a weighted filtration(A•,α•) of A , such that((rkA1, ..., rkAt),α•) ∈ T and µ(A•,α•;ϕ) > 0, the leading coefficient of the polynomialM(A•,

α•)+ (δ/s!) ·µ(A•,α•;ϕ) is some positive multiple of

L(A•,α•)+δ ⋆

s!·µ(A•,α•;ϕ)>C′−C′ = 0,

so that

M(A•,α•)+δs!·µ(A•,α•;ϕ)≻ 0,

and we are done.

Remark5.4.3 (S-equivalence for semistable singular principal G-bundles). Let (A ,τ) be a semi-stable singular principalG-bundle. By Lemma 5.4.2, an admissible deformation is associated witha reductionβ :U ′ −→ P(A ,τ)|U ′/QG(λ ), such thatM(A•(β ),α•(β )) ≡ 0. The structure of therational principal bundleP(df(A•,α•)(A ,τ)) may be described in the following way. The reductionβdefines aQG(λ )-bundleQ overU ′, such thatP(A ,τ)|U ′ is obtained fromQ by means of extendingthe structure group viaQG(λ ) ⊂ G. Extending the structure group ofQ via QG(λ ) −→ LG(λ ) ⊂ Gyields the principal bundleP(df(A•,α•)(A ,τ))|U ′ . Thus, our notion of S-equivalence naturally extendsthe one considered by Ramanathan (see, e.g., [31]).

Fix a Hilbert polynomialPand a stability parameterδ . The most important basic fact that has to bekept in mind is that aδ -semistable pseudoG-bundle(A ,τ) with P(A ) = P which is S-equivalent toa semistable singular principalG-bundle(A ′,τ ′) is itself a semistable singular principalG-bundle. Inother words, the class of semistable singular principalG-bundles with Hilbert polynomialP is closedunder S-equivalence inside the class ofδ -semistable pseudoG-bundles with Hilbert polynomialP.


5.5 Proof of the main theorem

We fix a stability parameterδ ≻ δ0 (see Theorem 5.4.1) and use the notation of Section 5.3. Byelimination theory, the points in the parameter spaceY corresponding to singular principalG-bundles

form an open subsetH. By Theorem 5.4.1,H(s)s := Yδ-ss∩H is the open subset corresponding to(semi)stable singular principalG-bundles. It suffices to show that the quotients

M(s)sP (ρ) := H(s)s//GL(U)

exist as open subschemes of Mδ-(s)sP (ρ). This will follow immediately, if we show thatHss is a GL(U)-

saturatedopen subset ofYδ-ss. This means that for every pointx∈Hss the closure of the GL(U)-orbit

in Yδ-ss is entirely contained inHss. Since the points with closed GL(U)-orbit inYδ-ssare mapped to

the points with closed SL(U)-orbit in Yδ-ss

(see [28], proof of Proposition 1.3.2), it suffices to showthat

Hss

:= Hss//Gm(k)

is an SL(U)-saturated open subset ofYδ-ss

. Let y,y′ ∈ Yδ-ss

, such thaty′ lies in the closure of theSL(U)-orbit of y. Then,ψ(y′) lies in the closure of the SL(U)-orbit of ψ(y). We may assume that theorbit of y′ and hence ofψ(y′) is closed. By the Hilbert-Mumford criterion, one knows thatthere existsa one parameter subgroupλ :Gm(k)−→SL(U) with limx→∞ λ (z) ·ψ(y)∈SL(U) ·ψ(y′). Note that theinjectivity of ψ thus implies limz→∞ λ (z) ·y∈SL(U) ·y′. Suppose thaty andψ(y) represent(A ,τ) and(A ,ϕ), respectively. Now, from the GIT constructions in Section 4.5, in particular Remark 4.5.10,and 5.3, one infers thatλ corresponds to a filtration(A•,α•) with

M(A•,α•)+δs!·µ(A•,α•;ϕ)≡ 0

and that a point in the orbit ofψ(y′) represents df(A•,α•)(A ,ϕ), so that a point in the orbit ofy′

represents df(A•,α•)(A ,τ), by Section 5.2. Together with Remark 5.4.3, this shows

y∈ Hss

=⇒ y′ ∈ Hss,

and this is what we wanted to prove.

6 Examples

As mentioned in the introduction, interesting definitions of (semi)stability and corresponding modulispaces are obtained for particular reductive groupsG and representationsρ . These we sketch now, thedetails being mere exercises.

6.1 Stable orthogonal and symplectic sheaves

Let G = Spn(k) andρ :Spn(k) → SL(V) be the standard faithful representation in ann-dimensionalvector spaceV with non-degenerate skew-symmetric formϕ :V⊗V → k. It is easy to check that a oneparameter subgroupγ of SL(V), factoring through Spn(k), amounts to a decomposition

V =⊕

i∈Z

V i


with Gm(k) acting on the factorsV• with weightsγ• (order them increasingly), and eachV i beingorthogonal to allV j but V−i ∼= (V i)∨, with γ−i = −γi. The associated parabolic subgroupQG(λ ) ⊆SL(V) is the stabilizer of the symplectic weighted filtration(V•,γ•) of V defined by

Vi =⊕

j≤i

V j .

Here,(Vi)⊥ =V−i−1 andγ−i = −γi . Giving a singular principalG-bundle(A ,τ) on X is equivalent

to giving a symplectic sheaf(A ,ϕA) of rankn, i.e., a torsion free sheafA onX of this rank together

with a skew-symmetric homomorphismϕA

:A ⊗A → ØX, such thatA → A ∨ is a monomorphism(namely, the homomorphism

∧2A|U → ØU amounting to the principal bundleP := P(A ,τ) on thebig open setUA whereA is locally free, then naturally extended to

∧2A → ι⋆

∧2A|U → ι⋆ØU = ØX .)

A reduction ofP to the “weighted” parabolic subgroupQG(λ ) ⊆ G amounts, by extension toX, toa saturated filtrationA• ⊆ A which is an orthogonal weighted filtration with the weightsγ•, definingA ⊥

i as ker(A → A ∨ → A ∨i ). Thus, our notion of (semi)stability becomes

t

∑i=1

(γi+1− γi)(rkAi ·P(A )− rkA ·P(Ai)

)()0,

for all orthogonal weighted filtrations(A•,γ•) of A . Since this expression is linear in the sequenceγ•, this condition can be further expressed in terms of just orthogonal filtrations of two terms 0(F ⊆ F⊥ ( A with weightsni −n andni , thus becoming the very simple condition announced in theintroduction.

The same definition of (semi)stability and corresponding moduli are obtained for orthogonalsheaves(A ,ϕ) of rank n, i.e., pairs made by a torsion free sheafA of this rank and a symmetricform ϕ :A ⊗A → ØX, which amount to singular principalG-bundles for the standard inclusionρ ofOn(k) in GLn(k). (Here, we use the fact that it suffices that the connected component of the identityelement maps to SLn(k).)

The semistable singular principalG-bundles for the natural inclusionρ of SOn(k) in SLn(k) turnout to be semistable special orthogonal sheaves(A ,ϕ

A,ψ

A), i.e., semistable orthogonal sheaves

(A ,ϕA) together with an isomorphismψ :det(A ) ∼= ØX, such thatψ2

A= det(ϕA ). (The proof is

just as in the symplectic case, but then one has to check that the extra datumψA

does not alter thedefinition of semistability.) The projectivity of the moduli spaces follows from the arguments givenin [10].

6.2 Stable algebra sheaves

LetG⊆GL(V) be the group Aut(V,ϕ) of automorphisms of a non-associative algebra structureϕ :V⊗V → V on a finite dimensional spaceV. Assume thatG lies in fact in SL(V) (for instance, ifG isconnected and the Killing form(v,w) 7→ Tr(ϕ(v, )ϕ(w, )) of V is non-degenerate), so that we canuse the embeddingρ :G → SL(V).

A one parameter subgroup of SL(V) amounts to a decomposition

V =t+1⊕

i=1

V i


into subspacesV i acted on byGm(k) with increasing integer weightsγi which are balanced, i.e.,∑γi dimV i = 0. It factors throughG= Aut(V,ϕ), if and only if ϕ(V i ,V j) is contained inVk, if thereis ak, such thatγi + γ j = γk, and otherwise is null. The corresponding parabolic subgroup QG(λ )⊆ Gis the stabilizer of the filtration ofV by the subspacesVi =

⊕j≤i V

j which we weight by the balancedsequenceγ•, i.e., ∑γi(dimVi − dimVi−1) = 0, so that it becomes a weighted algebra filtration, i.e.,ϕ(Vi ,Vj)⊆Vk wheneverγi + γ j ≤ γk.

Just as in the former example, a singular principalG-bundle onX amounts to aϕ-algebra sheaf(A ,ϕ

A) of rank dimk(V), as defined in the introduction, i.e., a torsion free sheafA of this rank

together with a homomorphismϕA

:A ⊗A → A ∨∨, defining fiberwise algebras that are isomorphicto (V,ϕ). A reduction to the “weighted” parabolic subgroupQG(λ )⊆Gamounts to a balanced algebrafiltration of A , i.e., A• ⊆ A with balanced weightsγ•, such thatϕ

A(Ai ,A j) ⊆ A ∨∨

k wheneverγi + γ j ≤ γk. The condition of (semi)stability of aϕ-algebra sheaf(A ,ϕ

A) then becomes

t

∑i=1

(γi+1− γi)(rkAi ·P(A )− rkA ·P(Ai)

)()0,

for any balanced weighted algebra filtration(A•,γ•) of A . This can be expressed more easily andequivalently in the usual terms of just algebra filtrations in the way mentioned in the introduction.

It can be applied, in particular, to the standard realizations of the exceptional groups as auto-morphism groups of some non-associative algebras. For instance,G2 is the automorphism group ofthe octonions overk. Any groupG of adjoint type (i.e., with trivial center) can be identifiedwiththe (reduced) component of identity Aut0(g,ϕ) of the group of automorphisms of its Lie algebraϕ = [ , ]:g⊗g→ g. SinceG has finite index in Aut(g,ϕ), principalρ-sheaves, for the adjoint repre-sentationρ , are just semistableϕ-algebra sheaves together with a reduction toG over the big open setwhere they are locally free, so there is a coarse moduli spacefor them.

Acknowledgments

We thank Professor Metha for suggesting this research whilepointing out, in a conversation with thelast author, that, since our previous work in characteristic 0 deals with the semistability of a tensorrather than with the semistability of a vector bundle, it wasmore suited for attacking the case ofpositive characteristic. Our thanks go also to N. Fakhruddin, J. Heinloth, Y. Holla, S. Koenig, V.Mehta, J.S. Milne, A. Premet, and S. Subramanian for discussions related to this work.

Tomas Gomez and Ignacio Sols are supported by grant numberMTM2004-07090-C03-02 fromthe Spanish “Ministerio de Educacion y Ciencia”. T.G. thanks the Tata Institute of FundamentalResearch for the hospitality during his visit in December 2004. He also thanks the Polish Academy ofScience (Warsaw) and the University of Duisburg-Essen for hospitality during his visits where partsof this work were discussed.

Adrian Langer was partially supported by a Polish KBN grant (contract number 1P03A03027).A.L. also thanks the DFG Schwerpunkt “Globale Methoden in der Komplexen Geometrie” for sup-porting his visit to the University of Duisburg-Essen.

Alexander Schmitt acknowledges support by the DFG via a Heisenberg fellowship and via theSchwerpunkt program “Globale Methoden in der Komplexen Geometrie—Global Methods in Com-plex Geometry”. A.S. was also supported by the DAAD via the “Acciones Integradas Hispano-Alemanas” program, contract number D/04/42257. In the framework of this program, the third authorvisited the Consejo Superior de Investigaciones Cientıficas (CSIC) in Madrid where some details of


the paper were discussed. During the preparation of the revised version, A.S. stayed at the Insti-tut des HautesEtudes Scientifiques where he benefitted from support of the European Commissionthrough its 6th Framework Program “Structuring the European Research Area” and the contract Nr.RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific SupportAction.

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Addresses

T.G.: Instituto de Matematica y Fısica Fundamental (IMAFF), C/Serrano 113bis, E-28006 Madrid,Spain; E-mail:[email protected].

A.L.: Institute of Mathematics, Warsaw University, Ul. Banacha 2, PL-02-097 Warszawa, Poland,and Institute of Mathematics, Polish Academy of Sciences, Ul. Sniadeckich 8, P.O. Box 21, PL-00-956 Warszawa, Poland; E-mail:[email protected].

A.H.W.S.: Universitat Duisburg-Essen, Campus Essen, Fachbereich 6: Mathematik & Informatik,D-45117 Essen, Germany; E-mail:[email protected].

I.S.: Departamento de Algebra, Facultad de Ciencias Matem´aticas, Pza. de las Ciencias, 3, Uni-versidad Complutense de Madrid, E-28040 Madrid, Spain; E-mail: [email protected].

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