semistable principal bundles—i (characteristic zero)balaji/publications/balaji... · v. balaji,...

Journal of Algebra 258 (2002) 321–347

www.elsevier.com/locate/jalgebra

Semistable principal bundles—I(characteristic zero)

V. Balaji a,b,∗,1 and C.S. Seshadria

a Chennai Mathematical Institute, 92 G.N. Chetty Road, T. Nagar, 600 017 Madras, Indiab Institute of Mathematical Sciences, Taramani, 600 113 Chennai, India

Received 19 November 2001

Dedicated to Claudio Procesi on the occasion of his 60th birthday

Introduction

The moduli space of principalG-bundles (for a reductive algebraic groupG)on a smooth projective curveX was constructed by A. Ramanathan over fields ofcharacteristic zero (cf. [R1,R2]). His method using Geometric Invariant Theoryfollowed the basic lines of the construction for the case of vector bundles(cf. [Ses]). The properness (and hence the projectivity) of the moduli space is anend product of this method of construction. One knows that this property (for thecase of vector bundles) could be proved a priori, before constructing the modulispaces and is referred to as thesemistable reduction theorem(cf. Langton [L]).

The principal aim in this article is to prove this semistable reduction theoremfor principalG-bundles overX in characteristic zero (cf. Theorem 7.1); that is,the moduli functor associated to semistable principalG-bundles is proper. Theconstruction of these moduli spaces follows as an easy consequence from thecase of vector bundles (cf. Section 8).

Our approach could be termedTannakianin the sense that aG-bundle can beviewed as atensor functorand can be studied in terms of its associated vectorbundles. This arose out of an attempt to understand C. Simpson’s proof of results

* Corresponding author.E-mail addresses:[email protected] (V. Balaji), [email protected] (C.S. Seshadri).

1 The research of the author was partially supported by the DST project no DST/MS/I-73/97, andthe IFCPAR Project No 1601-2/Geometry.

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0021-8693(02)00502-1

322 V. Balaji, C.S. Seshadri / Journal of Algebra 258 (2002) 321–347

similar to Lemma 8.1, and Lemma 8.2 which he proves in the context of Higgsbundles by usingTannakian arguments(cf. [Sim2,BBN]).

The techniques developed in the proof of Theorem 7.1 are general and havemany applications. For instance, we prove the semistable reduction theorem forfamilies of semistable principal Higgs bundles over smooth projective varieties(cf. Theorem 9.3). In particular, we get a different proof of Theorem 9.15in [Sim2].

The most important application is that the methods of this paper generalizesuitably to fields of positive characteristic as well and this appears in a sequel tothis paper (cf. [BP]).

There is also a proof, due to G. Faltings, of the semistable reduction theoremfor principalG-bundles in char 0 (cf. [F]). In an earlier article of ours [Rem], theproof of this theorem had a serious error which was pointed out by G. Faltings.

Since the proof of the semistable reduction theorem is technically involvedwe outline the broad strategy so as to highlight the main difference between thepresent approach and the existing ones. This would enable the reader to appreciatehow this method is amenable for generalization to positive characteristics(see [BP]).

0.1. Outline of proof of the semistable reduction theorem

The notations are as in Section1, whereA is a dvr with residue fieldk, whichis algebraically closed and the function field ofA is K. We are given a familyof semistable principalHK -bundles onXK . The problem is to extend this asa semistableHA-bundle toXA.

We choose a faithful representationH →G, whereG= SL(n). Extending thestructure group ofPK to GK overXK we call thisGK -bundle asEK . Then, byusing the GIT construction of the moduli space of vector bundles, we extend thisto aGA-bundleEA on XA with the added property that the limiting bundle ispolystable. (For this we may need to go to a finite cover ofA.)

We now view the entire data given above as follows: we are givenEA,aGA-bundle onXA, together with a reduction of structure group toHK overXK .The reduction gives a section

sK :XK →EK(GK/HK).

The point is that, if this holds, the semistable reduction theorem follows. Oneof the crucial technical results, namely, Proposition 2.8 is that, if this sectionsK extends along any pointx ∈ X i.e. alongxA = x × Spec(A), to a section ofEA(GA/HA)|xA, then the semistability of the familyEA enables us to prove thatsK extends tosA.

The difficulty is thatsK need not extend along anyx ∈ X. One attempts toget around this as follows: we fix a base pointx ∈X; we also fix a non-canonicalA-section ofEA|xA. Given this, the reduction sectionsK alongxK can be thoughtof as giving acoset representativeθK.HK in GK/HK , which is not in generalextendable to acosetθA.HA.

V. Balaji, C.S. Seshadri / Journal of Algebra 258 (2002) 321–347 323

One modifies the group scheme to a conjugate group schemeH ′K = θK.HK.θ

−1K

and viewssK as a section ofEK(GK/H′K). Then it is a simple observation that,

sK restricted toxK is indeed theidentity coseteK.H ′K in GK/H

′K . Going to the

flat closureH ′A of H ′

K in GA, this extends as theidentity cosetof GA/H′A. Ob-

serve that the group schemeH ′A need not be semisimple.

Viewed thus, the sectionsK extends along the base sectionxA to a sectionof EA(GA/H

′A), whereH ′

A is the flat closure ofH ′K in GA. Thus,the gain of

extending the reduction section alongxA forces the choice of the flat closure in thecategory of non-semisimple group schemes. The key points of the proof thereafterare the following:

1. Extend the sectionsK to a sectionsA of EA(GA/H′A) over the whole ofXA.

In other words, reduce the structure group of theGA-bundleEA to the flatclosureH ′

A (cf. Section 5, Proposition 5.1).2. Using Bruhat–Tits theory, relate the group schemesH ′

A andHA so as toobtain the requiredHA-bundle (cf. Section 6, Proposition 6.2).

It is probably appropriate at this juncture to observe the basic differencebetween this proof and Langton’s proof in the case of families of vector bundles.

In his proof, Langton first extends the family of semistable vector bundles(or equivalently principalGLn-bundles) to aGLn-bundle in the limit althoughnon-semistable. In other words, the structure group of the limiting bundle re-mainsGLn. Then by a sequence ofHecke modificationshe reaches the semistablelimit without changing the isomorphism class of the bundle over the generic fiber.

Instead, we extend the family of semistableHK -bundles to anH ′A-bundle with

the limiting bundle remaining semistable, but the structure group is non-reductivein the limit. In other words one loses the reductivity of the structure group scheme.Then, by using Bruhat–Tits theory (cf. Definition 3.2), we relate the group schemeH ′A to the reductive group schemeHA without changing the isomorphism class of

the bundle over the generic fiber as well as the semistability of the limiting bundle.The layout of the paper is as follows: Section 2 contains some preliminary

results on principal bundles which are crucial for what follows. Section 3 to Sec-tion 7 is devoted to the proof of the semistable reduction theorem; Section 8 givesthe construction of the moduli space of semistable principal bundles. In Section 9we indicate briefly how the methods in Sections 3–7 extend to the case of princi-pal Higgs bundles.

1. Notations and conventions

Throughout this paper, unless otherwise stated, we have the followingnotations and assumptions:


(a) We work over an algebraically closed fieldk of characteristic zero andwithout loss of generality we can takek to be the field of complex numbersC.

(b) H is asemisimplealgebraic group, andG, unless otherwise stated will alwaysstand for the general linear groupGL(n). Their representations are finitedimensional and rational.

(c) X is a smooth projective curve of genusg 2.(d) A is a discrete valuation ring (which could be assumed to be complete) with

residue fieldk, and quotient fieldK.(e) Let E be a principalG-bundle onX × T whereT is SpecA. Let x ∈ X

be a closed point which we fix throughout. Then throughout this article weshall denote byEx,A or Ex,T (respectivelyEx,K ) the restriction ofE to thesubschemex ×SpecA or x × T (respectivelyx ×SpecK). Similarly,p ∈ Twill denote the closed point ofT and the restriction ofE to X × p will bedenoted byEp .

(f) In the case ofG=GL(n), when we speak of a principalG-bundle we identifyit often with the associated vector bundle (and can therefore talk of the degreeof the principalG-bundle).

(g) We denote byEK (respectivelyEA) the principal bundleE on X ×SpecK (respectivelyX × SpecA) when viewed as a principalHK -bundle(respectivelyHA-bundle). HereHK and GK (respectivelyHA and GA)are the product group schemeH × SpecK andG × SpecK (respectivelyH × SpecA andG×SpecA).

(h) If HA is anA-group scheme, then byHA(A) (respectivelyHK(K)) we meanits A (respectivelyK)-valued points. WhenHA = H × SpecA, then wesimply write H(A) for its A-valued points. We denote the closed fiber ofthe group scheme byHk.

(i) Let Y be anyG-module and letE be aG-principal bundle. For exampleYcould be aG-module. Then we denote byE(Y ) the associated bundle withfiber typeY which is the following object:E(Y )= (E×Y )/G for the twistedaction ofG onE × Y given byg.(e, y)= (e.g, g−1.y).

(j) If we have a group schemeHA (respectivelyHK ) over SpecA (respectivelySpecK) anHA-moduleYA and a principalHA-bundleEA. Then we shalldenote the associated bundle with fiber typeYA byEA(YA).

(k) By a family of H -bundles onX parameterized byT we mean a principalH -bundle onX× T , which we also denote byEt t∈T .

2. Preliminaries

Remark 2.1. Recall that ifH ⊂G

(a) a principalG-bundleE onX is said to have anH -structure or equivalentlya reduction of structure group toH if we are given a sectionσ :X →E(G/H), whereE(G/H)E ×G G/H ;


(b) further, there is a natural action of the group AutGE, of automorphisms ofthe principalG-bundleE, on Γ (X,E(G/H)) and the orbits correspond totheH -reductions which are isomorphic as principalH -bundles.

Lemma 2.2. LetV andW be semistable vector bundles onX of degree zero. ThenV ⊗W is semistable of degree zero.

Proof. Any semistable bundle onX of degree zero has a filtration such that itsassociated graded is a direct sum of stable bundles of degree zero. Hence thetensor productV ⊗W gets a filtration such that its associated graded is a directsum of tensor products of stable bundles of degree zero. We see easily that thisreduces to proving the lemma whenV andW are stable of degree zero. Then bythe Narasimhan–Seshadri theorem,V ⊗W is defined by a unitary representationof the fundamental group (namely the tensor product of the irreducible unitaryrepresentations which defineV andW , respectively), which implies thatV ⊗W

is semistable (cf. [Ses]).

Proposition 2.3. Let E be a principalH -bundle onX. Then the following areequivalent:

(a) There exists a faithful representationH → GL(V ) such that the inducedbundleE(V )=E ×H V is semistable(of degree zero).

(b) For every representationH → GL(W), the bundleE(W) is semistable(of degree zero).

Proof. (b)⇒ (a) is obvious.(a)⇒ (b). SinceH is semisimple; the vector bundleE(V ) is semistable of

degree 0. Consider the natural tensor representationT a,b(V )=⊗aV ⊗⊗b

V ∗.Then by Lemma 2.2, the bundleE(T a,b(V )) = ⊗a

E(V ) ⊗ ⊗bE(V )∗ is

semistable of degree 0.It is well-known that anyH -moduleW is a subquotient of a suitableT a,b(V )

and henceE(W) is a subquotient ofE(T a,b(V )) of degree zero. ThereforeE(W)

is also semistable.

Definition 2.4. An H -bundleE is said to besemistableif it satisfies the equivalentconditions in Proposition 2.3.

Definition 2.5. LetH ′ be an affine algebraic group not necessarily reductive. LetP be a principalH ′-bundle onX. We defineP to besemistableif it is flat (in thesense that it comes from the representation of the fundamental group ofX) andthere exists a faithful representation

ρ :H ′ →GL(V )

such that the associated vector bundleP(V ) is semistable of degree zero.


Proposition 2.6. Let H ′ be an affine algebraic group not necessarily reductiveas above and letP be a semistable principalH ′-bundle. Letf :H ′ → H bea morphism fromH ′ to a semisimple groupH . Then the associated principalH -bundleP(H) is also semistable.

Proof. By Proposition 2.3 we need only check that ifψ :H → GL(W) is anyrepresentation ofH then the associated bundleP(H)(W) is semistable. In otherwords, we need to check that ifγ :H ′ → GL(W) (e.g., γ = ψ f ) is anyrepresentation ofH ′, not necessarily faithful, then the bundleP(W) is semistable.

Observe that by Definition 2.5 we have a faithful representationGL(V ) ofH ′ such thatP(V ) is semistable of degree zero. Further, we are over a field ofcharacteristic zero and so theH ′-moduleW can be realized as a subquotient ofa direct sums of someT a,b(V ) (cf., for example, [Sim0, p. 86]). Hence the vectorbundleP(W) is a subquotient of

⊕T a,b(P (V )). SinceP(V ) is semistable of

degree zero, so isT a,b(P (V )).SinceP is flat the associated vector bundles of all the subquotients of the

tensor representations are ofdegree zero. Again subquotients of semistable vectorbundles of degree zero are semistable. HenceP(W) is semistable of degree zero,which proves the proposition.

Remark 2.7. See also Definition 8.7 for the intrinsic definition of semistability ofprincipal bundles due to A. Ramanathan.

Let G beGL(n) and letH be a semisimple algebraic group,H ⊂G. Let

FG : (Schemes)→ (Sets)

be the functor given by

FG(T )=

isomorphism classes of semistableG-bundles of degree 0onX parameterised byT

.

One may similarly define the functorFH (note that sinceH is semisimple, fora principalH -bundle the associated vector bundles have degree zero).

Let x ∈X be a marked point and letFH,G,x be the functor

FH,G,x(T )=

isomorphism classes of pairs(E,σx): E = Et t∈T isa family of semistable principalG-bundles of degree 0andσx :T →E(G/H)x is a section

.

(Recall thatE(G/H)x denotes the restriction ofE(G/H) to x × T ≈ T .)Notice that the functorFH is in fact realizable as the following functor (by

Remark 2.1(a)):

FH,G(T )=

isomorphism classes of pairs(E, s): E = Et t∈T isa family ofG-bundles of degree 0 ands = st t∈T is a section ofE(G/H) onx × T

or what we may call a family of sections ofE(G/H)tt∈T

.


In what follows, we shall identify the functorsFH with FH,G. With thesedefinitions we have the following proposition.

Proposition 2.8. Letαx be the morphism induced by “evaluation of section” atx:

αx :FH → FH,G,x.

Thenαx is a proper morphism of functors(cf. [DM]) .

Proof. Let T be an affine smooth curve and letp ∈ T . Then by the valuationcriterion for properness, we need to show the following:

If E is a family of semistable principalG-bundles onX × T together witha sectionσx :T → E(G/H)x such that fort ∈ T − p, we are given a familyof H -reductions, i.e. a family of sectionssT−p = st t∈T−p , wherest :X→E(G/H)t has the property that, atx, st (x)= σx(t) ∀t ∈ T −p;

then we need to extend the familysT−p to a sectionsT of E(G/H) onX× T sothatsp(x)= σx(p) as well.

Observe that sinceG/H is affine, there exists aG-moduleW such that

G/Hi→W is aclosedG-embedding. Thus we get a closed embedding

E(G/H) →E(W)

and a family of semistable vector bundlesE(W)t t∈T together with a family ofsectionssT−p and evaluationsσx(t)t∈T such thatst (x)= σx(t), t = p.

For the sectionsT−p , viewed as a section ofE(W)T−p we have twopossibilities:

(a) it extends as a regular sectionsT ;(b) it has a pole alongX× p.

Observe that if (a) holds, then we have

sT(X× (T − p)

)⊂ E(G/H)⊂E(W),

sinceE(G/H) is closed inE(W), it follows that sT (X × p) ⊂ E(G/H). Thussp(X) ⊂ E(G/H)p. Further, by continuity,sp(x) = σx(p) as well, and thisproves the proposition.

To complete the proof, we need to check that the possibility (b) cannot hold.Suppose it does hold. For our purposes, we could take the local ringA of T

at p, which is a discrete valuation ring with a uniformizerπ . Let K be itsquotient field. Then the sectionsT−p = sK is a section ofE(W)K ; in other words,a rational sectionof E(W) with a pole along the divisorX×p ⊂X× T of orderk 1. Thus, by multiplyingsT−p by πk we get a regular sections′T of E(W) onX× T . If s′T = s′t t∈T , then we have:


(i) s′t = λ(t) · st , t ∈ T − p, whereλ :T → C is a function given byπk, havingzeros of orderk atp.

(ii) s′p is a non-zero section ofE(W)p . Notice that by (ii), sinces′p is a sectionof E(W)p andE(W)p is a semistable bundleof degree 0, a non-vanishingsection is nowhere vanishing, i.e.,

s′p(y) = 0 ∀y ∈X. (∗)By assumption,st (x)= σx(t), t ∈ T − p, hence

s′t (x)= λ(t) · σx(t), t ∈ T − p.

Therefore, by continuity, sinceσx(p) is well-defined, we see thatλ(t) · σx(t)tends toλ(p) · σx(p)= 0 ast→ p.

Also s′t (x)→ s′p(x) ast→ p. Hence, by continuity, it follows thats′p(x)= 0,which contradicts(∗).

Thus the possibility (b) does not occur and we are done.

Remark 2.9. For a different proof of this proposition see [BP, Proposition 3.12].

We isolate the above proof for future use in the following lemma.

Lemma 2.10. LetT = SpecA and letET be a family of semistable vector bundlesof degree zero onX × T . Let sK be a section of the familyEK restricted toX×SpecK, with the property that for a base pointx ∈X, the sectionsK extendsalong x × T to give a section ofEx×T . Then the section extends to the wholeX× T .

3. Towards the flat closure

Fix a faithful representationH → G defined overC. Consider the extensionof structure group of the bundlePK via the inducedK-inclusionHK →GK . Wedenote the associatedGK -bundlePK(G) byEK .

Then, sinceG=GL(n), by the properness of the moduli space of semistablevector bundles, there exists asemistable extensionof PK(G) = EK to aGA-bundle onX×SpecA, which we denote byEA. Call the restriction ofEA toX × p (identified withX) the limiting bundleof EA and denote it byEp (as inSection 1). One has in fact slightly more, which is what we need.

Lemma 3.1. LetEK denote a family of semistableGK -bundles of degree zero onX× SpecK (or equivalently a family of semistable vector bundles of rankn anddegree zero onX× (T − p)). Then,(by going to a finite coverS of T if need be)the principal bundleEK extends toEA with the property that the limiting bundleEp is in fact polystable, i.e., a direct sum of stable bundles of degree zero.


Proof. This lemma is quite standard but we shall prove it in Section 4.

3.1. The flat closure

We observe the following:

• Note that giving theHK -bundle PK is giving a reduction of structuregroup of theGK -bundleEK which is equivalent to giving a sectionsK ofEK(GK/HK) overXK .

• We fix a base pointx ∈X and denote byxA = x×SpecA, the induced sectionof the family (which we call thebase section):

XA→ SpecA.

• Let Ex,A (respectivelyEx,K ) be as in Section 1, the restriction ofEA toxA (respectivelyxK ). Thus,sK(x) is a section ofEK(GK/HK)x which wedenote byEx(GK/HK).

• SinceEx,A is a principalG-bundle on SpecA and therefore trivial, it can beidentified with the group schemeGA itself. For the rest of the article we fixone such identification, namely:

ξA :Ex,A→GA.

• Since we have fixedξA, we have a canonical identification

Ex(GK/HK)GK/HK,

which therefore carries a naturalidentity sectioneK (i.e. the coset id.HK ). Us-ing this identification we can viewsK(x) as an element in the homogeneousspaceGK/HK .

• Let θK ∈G(K) be such thatθ−1K · sK(x)= eK (for this we may have to go to

a finite extension ofK). Then we observe that, the isotropy subgroup schemein GK of the sectionsK(x) is θK.HK.θ

−1K .

• On the other hand, one can realizesK(x) as the identity coset ofθK.HK.θ−1K

by using the following identification:

GK/θK.HK.θ−1K

∼→GK/HK, gK(θK.HK.θ

−1K

) → gKθK.HK.

Definition 3.2. LetH ′K be the subgroup scheme ofGK defined as

H ′K := θK.HK.θ

−1K .

UsingξA we can have a canonical identification

Ex

(GK/H

′K

)GK/H′K.

Then, we observe that, using the above identification we get a sections′K ofEK(GK/H

′K), with the property that,s′K(x) is theidentity sectionand moreover,


since we have conjugated by an elementθK ∈GA(K)(=G(K)), the isomorphismclass of theHK -bundlePK given bysK does not change by going tos′K .

Thus, in conclusion, theGA-bundleEA has a reduction toH ′K given by

a sections′K of EK(GK/H′K), with the property that, at the given base section

xA = x × SpecA, we have an equalitys′K(xA)= e′K with the identity elementofGK/H

′K (namely the coset id.H ′

K ).

Definition 3.3. Let H ′A be theflat-closureof H ′

K in GA.

We then have a canonical identification viaξA:

Ex

(GA/H

′A

)GA/H′A.

By definition, sinceH ′K is reduced,H ′

A is the scheme theoretic closure ofH ′K

in GA with the canonicalreducedstructure. One can easily check thatH ′A is

indeed a subgroup scheme ofGA since it contains the identity section ofGA,and moreover, it is faithfully flat overA. However, notice thatH ′

A need notbe areductivegroup scheme; that is, the special fiberHk over p need not bereductive.

Observe further that,s′K(x) extends in a trivial fashion to a sections′A(x);namely, theidentity coset sectione′A of Ex(GA/H

′A) identified withGA/H

′A.

Remark 3.4. If H ′A is reductivethen the semistable reduction theorem follows

quite easily. For, firstly, by therigidity of reductive group schemes over SpecA

[SGA3, Expose III, Corollary 2.6, p. 117], by going to a finite cover, we mayassume thatH ′

A =H × SpecA. Then we have aclosedG-immersionof G/H inaG-moduleW , and one may viewsK as a section ofEK(WK)← EK(GK/H

′K).

By choice, alongxA, the sectionsK(x) extends regularly to a section ofEA(GA/H

′A)⊂EA(WA). Hence, by Proposition 2.8,sK extends to a sectionsA,

which gives the required reduction overX× SpecA.

4. Chevalley embedding of GA/H ′A

As we have noted,H ′A need not be reductive and the rest of the proof is to

get around this difficulty. Our first aim is to prove that the structure group of thebundleEA(GA) can be reduced toH ′

A, which is the statement of Theorem 5.1.We need to prove the following generalization of a well-known result of

Chevalley.

Lemma 4.1. There exists a finite dimensionalGA-module WA such thatGA/H

′A → WA is aGA-immersion.


Proof. We follow Chevalley’s proof. LetIK be the ideal defining the subgroupschemeH ′

K in K(G) (note thatGA (respectivelyGK ) is an affine group schemeand we denote byA(G) (respectivelyK(G)) its coordinate ring).

SetIA = IK ∩ A(G). Then it is easy to see that since we are over a discretevaluation ring,IA is in fact the ideal inA(G) defining the flat closureH ′

A. Observealso thatIA is aprimitiveA submodule ofA(G), that is,A(G)/IA is torsion free;further,IA ⊗ k = Ik is the defining ideal ink(G) of H ′

k in Gk andIA ⊗K is IK .We may now choose a finite generating setfi of IK , such that modulok, theirimagesfi,k generateIk .

As in the classical proof of Chevalley, one has a finite dimensionalGK -submodule,VK , containing thefi. Now setVA = VK ∩A(G) andM = VA∩IA.Observe thatIA, VA and henceM are allGA-submodules ofA(G). This can beseen by keeping track of the comodule operations. Then clearlyVA is primitivein A(G) andM is also primitive inA(G) and in particular, primitive inVA. If weset

Mk =M ⊗ k and Vk = VA ⊗ k,

we see that the inclusionM → VA induces an inclusionMk → Vk . Observe that

fi ∈M, fi,k ∈Mk, and M ⊂ IA,

Mk ⊂ Ik and Mk = Vk ∩ Ik.We claim that, forg ∈GA(k), one has

g ·Mk ⊂Mk ⇐⇒ g ∈H ′k.

Obviously, ifg ∈H ′k , theng ·Mk ⊂Mk , sinceVk isG-stable andIk isH ′

k-stable.Thus it suffices to show that

fi,k(g)= 0 for all i;that is,fi,k vanish ong. Sincefi,k ∈Mk , it suffices to show that

F(g)= 0 forF ∈Mk.

But F(g) = (g−1 · F)(id), whereg−1 · F is the action ofG on functions onG.Now, by hypothesis,(g−1 · F) ∈Mk . SinceMk ⊂ Ik , and id∈ H ′

k , we see that(g−1 · F)(id)= 0. This proves the above claim.

Similarly, if we set

MF =M ⊗A L and VF = VA ⊗A F,

whereF is any field containingA, we see that forg ∈G(F)g ·MF ⊂MF ⇐⇒ g ∈H ′

A(F ).

Let L denote the primitive rank-oneA-submodule∧d

M →∧dV =WA, and

[L] the A-valued point ofP(WA) defined byL. Here, P(WA) is defined by


the functor associated to rank-one direct summands ofWA. Then, the abovediscussion means that, we can recoverH ′

A as the isotropy subgroup scheme at[L] for the GA-action onP(WA).

Recall that, for any fieldF , the isotropy subgroup ofGA(F), at the point ofPWA(F) represented by the base change ofL by F , isH ′

A(F ).Fix a generatorl ∈ L so thatl is a primitive element inWA and consider the

isotropy subgroup schemeH ′′A at l for theGA-action onWA. We claim that,H ′′

A

coincides withH ′A. To see this, observe that,H ′′

A is the subgroup scheme ofGA

which leaves the closed subscheme (= Spec(A)) determined byl invariant (withthe corresponding automorphism on this subscheme being identity). We see thenthat,H ′′

A is a closedsubgroup scheme ofGA. Further, we see thatH ′′A → H ′

A.SinceH ′

K is semi-simple, it has no characters and therefore the isotropy subgroupscheme at(l⊗K) ∈ (WA⊗K) is preciselyH ′

K . This means thatH ′′K =H ′

K . Now,H ′K is open (dense) inH ′

A (sinceH ′A is the flat closure ofH ′

K in GA) so thatH ′K is also dense inH ′′

A. This implies thatH ′A andH ′′

A coincide set-theoretically.Observe also thatH ′

A is reducedby the definition of flat closure. Thus, it followsthatH ′

A = H ′′A. This implies thatGA/H

′A → WA is a GA-immersion, and the

above lemma follows.

Remark 4.2. Regarding the Lemma 4.1 proved above, we note that usually thesubgroup schemeH ′

A can be realized only as the isotropy subgroup scheme ofa line in aGA-module. But here, since the generic fiber ofH ′

A is semisimple, oneis able to realizeH ′

A as the isotropy subgroup scheme of a primitive element inaGA-module and the limiting group also as an isotropy subgroup scheme for anelement in aGk-module.

5. Extension to flat closure and local constancy

Recall that the sections′K(x) extends along the base sectionxA to gives′A(x)=wA. The aim of this section is to prove the following key theorem.

Theorem 5.1. The sections′K extends, in fact, to a sections′A of EA(GA/H′A).

In other words, the structure group ofEA can be reduced toH ′A; in particular,

if H ′k denotes the closed fiber ofH ′

A then the structure group ofEk can be reducedtoH ′

k.

For the proof we need the following key result.

Proposition 5.2. LetE be a polystable principalG-bundle onX of degree zero(hereG= SL(n) or GL(n)). LetW be aG-module andY a G-subscheme ofWof the formG/H ′ whereH ′ = StabG(w) for somew ∈W .


Let s be a section ofE(W) such that for somex ∈ X, s(x) = w in the fiberof E(Y ) at x ∈ X. Then the entire image ofs lies inE(Y ). In particular,E hasa reduction of structure group to the subgroupH ′. Furthermore, this reductionof structure group toH ′ is flat, in the sense that theH ′-bundle comes from therepresentation of the fundamental groupπ1(X) ofX. In other words, the reducedH ′-bundles issemistable of degree zeroin the sense of Proposition2.6.

Proof. The bundleE being polystable, it is defined by a unitary representation

χ :π1(X)→G

which maps into the unitary subgroup ofG. This implies that if the universalcoveringj :Z→X is considered as a principal fiber space with structure groupπ1(X), then the principalG-bundleE is the associated bundle throughχ .

Let ρ :G→ GL(W) be the representation defining theG-moduleW . ThenE(W) can be considered as the bundle associated to the principal bundlej :Z→X through the representation

ρ χ :π1(X)→GL(W),

which maps into the unitary subgroup ofGL(W).By generalities on principal bundles and associated constructions, since

E(W)Z×π1(X) W,

a section ofE(W) can be viewed as aπ1(X)-map s1 :Z→ W . Now, sinceZis the universal cover of the curveX ands is a section ofE(W), therefore oneknows (cf. [NS]) that there exists aπ1(X)-invariant elementw ∈W such thats isdefined by a maps1 :Z→W , given bys1(x)=w, ∀x ∈X, i.e. “the constant mapsending everything tow.”

Sincew ∈ W is aπ1(X)-invariant vector and the action ofπ1(X) is via therepresentationχ , we see thatχ factors via

χ1 :π1(X)→H ′,sinceH ′ = StabG(w).

In particular, we get theH ′-bundle from the representationχ1 and clearly thisH ′-bundle is the reduction of structure group of theG-bundleE given by thesections.

By the very construction of the reduction, the inducedH ′-bundle isflat andalso semistable since it comes as the reduction of structure group of the polystablebundleE (by Definition 2.5). This proves the Proposition 5.2.5.1. Completion of proof of Theorem 5.1

By Lemma 4.1, we have

EA

(GA/H

′A

)→EA(WA).


The given sections′K of EK(GK/H′K) therefore gives a sectionuK of E(WK).

Further,uK(x), the restriction ofuK to x × (T − p), extends to give a sectionuA(x) of Ex(WA) (restriction ofEA(WA) to x×T ). Thus, by Proposition 2.8 andsemistability ofEp(WA), the sectionuK extends to give a sectionuA of E(WA)

overX× T .Now, to prove the Theorem 5.1, we need to make sure that

The image of this extended sectionuA actually lands inEA(GA/H′A). (∗)

This would then defines′A.To prove(∗), it suffices to show thatuA(X × p) lies in EA(GA/H

′A)p (the

restriction ofEA(GA/H′A) toX× p).

Observe thatuA(x × p) lies in EA(GA/H′A)p sinceuA(x) = s′A(x) = wA.

Observe further that, ifEp denotes the principalG-bundle onX, which isthe restriction of theGA-bundleEA on X × T to X × l, thenEA(WA)p =EA(WA)|X× p, and we also have

EA(GA/H′A)p

Ep(Gk/H′k)

EA(WA)p Ep(W),

where the vertical maps are inclusions:

EA

(GA/H

′A

)p→EA(WA)p, Ep

(Gk/H

′k

)→Ep(W),

whereEp(W)= Ep ×H ′k W with fiber as theG-moduleW =WA ⊗ k. Note thatG/H ′

k is aG-subschemeY of W .Recall thatEp is polystable of degree zero. Then, from the foregoing

discussion, the assertion thatuA(X × p) lies in EA(GA/H′A) is a consequence

of Proposition 5.2 applied toEp. (Note that the groupH ′k = StabGk(wk) satisfies

the hypothesis of Proposition 5.2.)Thus we obtain a sections′A of EA(GA/H

′A) on X × T , which extends the

sections′K of EA(GA/H′A) onX × (T − p). This gives a reduction of structure

group of theGA-bundleEA on X × T to the subgroup schemeH ′A and this

extends the given bundleEK to the subgroup schemeH ′A.

In summary, we have extended the originalHK -bundle up to isomorphism toanH ′

A-bundle. The extendedH ′A-bundle has the property that the limiting bundle

E′p, which is anH ′k-bundle, comes with a reduction of structure group to the

fundamental group ofX and issemistablein the sense of Proposition 2.6.Remark 5.3. By Lemma 5.2, since the limiting bundleE′p is polystable, we canconclude that themonodromy subgroupM ′ of E′p , i.e. the minimal subgroupto which the structure group ofE′p can be reduced, isreductive, being theZariski closure of the representation of the fundamental group ofX defining thepolystableE′p.


Now recall the followingrigidity theorem(cf. [SGA3, Corollary 2.8, III]),namely: sinceM ′ is reductive, the given inclusionM ′ → H ′ can be lifted to aninclusion of group schemesM ′

A M ′ ×SpecA →H ′A (possibly by going to aB

which is integral overA). It follows then thatM ′ can be embedded as a subgroupof H (recall that over the generic point of SpecA we haveH ′

K H × SpecK).Using this embedding we can thus extend structure group ofE′p toH !

It seems therefore that we have proved the semistable reduction theorem,for we have shown that the structure groupG of Ep can be reduced to thesubgroupH →G. However, there is one crucial point to be proved, namely thatall reductions varycontinuously, in other words they fit together to give anHA-bundle overX × T . This is carried out in the next few sections with the aid ofBruhat–Tits theory.

6. Potential good reduction

To summarise, we have extended the originalHK -bundle up to isomorphismto anH ′

A-bundle. To complete the proof of the Theorem 7.1, we need to extendtheHK -bundle to anHA-bundle.

Remark 6.1. We note that in general the group schemeH ′A obtained above need

not be a smooth group scheme overA. But in our case since the characteristic ofthe base field is zero and sinceH ′

A is flat, it is also smooth overA.

Recall thatHA denotes the reductive group schemeH ×SpecA overA.

Proposition 6.2. There exists a finite extensionL/K with the following property:if B is the integral closure ofA in L, and ifH ′

B are the pull-back group schemes,then we have a morphism ofB-group schemes

H ′B→HB,

which extends the isomorphismH ′L∼=HL.

Proof. Observe first that thelatticeH ′A(A) is abounded subgroupof HA(K), in

the sense of the Bruhat–Tits theory [BT]. Here, we make the identifications:

H ′K∼=HK asK-group schemes.

Hence,

H ′A(A)⊂H ′

K(K)∼=HK(K)=HA(K).

Then we use the following crucial fact:

There exists a finite extensionL/K and an elementg ∈H ′A(L)

such thatg.H ′A(A).g

−1 →HA(B).(∗)


This assertion is a consequence of the following result from ([Se, Proposition 8,p. 546]) (cf. also [Gi, Lemma I.1.3.2], or [La, Lemma 2.4]):

(Serre) There exists a totally ramified extensionL/K having the followingproperty: for every bounded subgroupM of H(K), there existsg ∈ H(K)

such thatg.M.g−1 hasgood reductionin H(L) (i.e.h.M.h−1⊂H(B), whereB is the integral closure ofA in L).

Larsen [La, (2.7), p. 619], concludes from(∗), in thel-adic case, the statementof Proposition 6.2. However, we give a complete proof.

For the sake of clarity we gather all the identifications of the subgroups underconsideration:

H ′A(K)=H ′

K(K), H ′A(L)=H ′

B(L)=H ′L(L),

H ′A(A)⊂H ′

B(B), HA(B)=HB(B).

Thus, we see that the isomorphismψL :H ′L→ HL, given byconjugation byg,

induces a mapψL(B) :H ′A(A)→ HB(B). The crucial property to note is the

following one:

Given a rational pointξk ∈H ′k(k), there exists a pointξA ∈H ′

A(A), and hencein H ′

B(B), which extendsξk , sinceH ′A is smooth overA andk is algebraically

closed.

The proposition will follow by the following lemma. LetA, B, etc., be asabove.

Lemma 6.3. LetA be a complete discrete valuation ring with quotient fieldK. LetZA andYA beA-schemes withZA smooth. LetψL :ZL→ YL be aL-morphismsuch thatψL(B) :ZA(A) → YB(B). Then, theL-morphismψK extends toa B-morphismψB : UB → YB , whereUB is an open dense subscheme ofZB

which intersects all the irreducible components of the closed fiberZk.In particular, ifZA andYA are smooth and separated group schemes and ifψL

is a morphism ofL-group schemes then there exists an extensionψB : ZB → YBas a morphism ofB-group schemes.

Proof. Consider the graph ofψL and denote its schematic closure inZB ×B YBby ΓB . Let p :ΓB → ZB be the first projection. Thenp is an isomorphism ongeneric fibers. So, it is enough if we prove thatp is invertible on an open denseB-subschemeUB of ZB , which intersects all the componentsC, of the closedfiberZk.

We claim that the mappk :Γk→ Zk is surjective onto the subset ofk-rationalpoints of each components, and this will imply thatpk is surjective sincek is


algebraically closed. Note thatZA is assumed to be smooth and so, the closedfiber is reduced and alsok is algebraically closed. Thus, eachzk ∈ Zk(k) lifts toa pointz ∈ ZA(A)⊂ ZB(B), A, being a complete discrete valuation ring. SinceψL(B) mapsZA(A)→ YB(B), we see that, there exists ay ∈ YB(B) such that(z, y) ∈ ΓB(B). Thus,zk lies in the image ofpk . This proves the claim.

In particular, the generic points,α’s, of all the componentsC, of Zk (byChevalley), lie in the image ofpk . Letpk(ξ)= α. Consider the local ringsOΓB,ξ

andOZB,α . Then, by the above claim, the local ringOΓB,ξ dominatesOZB,α .SinceZB is smooth and hence normal, for everyα the local ringsOZB,α areall discrete valuation rings. Further, sinceΓB is the schematic closure ofΓL, itimplies thatΓB is B-flat andΓL is open and dense inΓB . Moreover, sincep isan isomorphism on generic fibers, both local rings have the same quotient rings.Finally, sinceOZB,α is a discrete valuation ring, we have an isomorphism of localrings. Thus, the schemes being of finite type overB, we have open subsetsVi,BandUi,B for each component ofZk, which we index byi, such thatp inducesan isomorphism betweenVi,B andUi,B . This gives an extension ofψ to opensubsetsUi,B for everyi, with the property that these maps agree on the genericfiber. SinceYB is separated, these extensions glue together to give an extensionψB on an open subset, which we denote byUB ; this open subset will of courseintersect all the components of the closed fibers ofZk .

The second part of the lemma follows immediately ifYA is affine (whichis our case). More generally, we appeal to the general theorem of A. Weil onmorphisms into group schemes, which says that if a rational mapψB is definedin codimension 1 and if the target space is a group scheme then it extends toa global morphism. (cf., for example, [BLR, p. 109]). As we have checked above,this holds in our case and implies that as a morphism of schemesψL extends togiveψB :ZB→ YB .

Further, by assumption,ψL is already a morphism ofL-group schemes, andhence it is easy to see that the extensionψB is also a morphism ofB-groupschemes. This concludes the proof of the lemma.

7. Semistable reduction theorem

The aim of this section is to prove the following theorem.

Theorem 7.1. Let PK be a family of semistable principalH -bundles onX ×SpecK, or equivalently, ifHK denotes the group schemeH × SpecK, a semi-stableHK -bundlePK onXK . Then there exists a finite extensionL/K, with theintegral closureB of A in L such thatPK , after base change toSpecB, extendsto a semistableHB -bundlePB onXB .


Remark 7.2. LetH ⊂G, whereG is a linear group. In the notation of Section 2,let FH andFG stand for the functors associated to families of semistable bundlesof degree zero (cf. Proposition 2.8). The inclusion ofH in G induces a morphismof functorsFH → FG. We remark that the semistable reduction theorem forprincipal H -bundlesneed notimply that the induced morphismFH → FG isa proper morphism of functors. Indeed, this does not seem to be the case.However, it does imply that the associated morphism at the level of moduli spacesis indeed proper (cf. Theorem 8.5).

7.1. Completion of the proof of the Theorem 7.1

Thus, in conclusion, first by Proposition 5.1 we have anH ′A-bundle which

extends theHK -bundle up to isomorphism. Then, by Proposition 6.2, going to theextensionL/K, we have a morphism ofB-group schemesψB :H ′

B→HB , whichis an isomorphism overL. Therefore, one can extend the structure group of thebundleE′B to obtain anHB -bundleEB which extends theHK -bundleEK .

Moreover, the fiber ofEB over the closed point is indeedsemistable.To see this, observe first that it comes as the extension of structure group of

E′p by the mapψk :H ′k→Hk . Recall (Proposition 5.2) thatE′p is thesemistable

H ′k-bundle obtained as the reduction of structure group of the polystable vector

bundleE(VA)p and so remains semistable by any associated construction (cf.Proposition 2.6).

This completes the proof of the Theorem 7.1.

8. Construction of the moduli space

For the present purpose, we takeG = SL(n,C) andH ⊂ G a semisimplesubgroup.

We recall very briefly the Grothendieck Quot scheme used in the constructionof the moduli space of vector bundles (cf. [Ses]).

Let F be a coherent sheaf onX and letF(m) beF ⊗OX(m) (following theusual notations). Choose an integerm0 = m0(n, d) (n is a rank,d is a degree)such that for anymm0 and any semistable bundleV of rankn and degreed onX and we havehi(V (m))= 0 andV (m) is generated by its global sections.

Let χ = h0(V (m)) and consider the Quot schemeQ consisting of coherentsheavesF on X which are quotients ofCχ ⊗C OX with a fixed HilbertpolynomialP . The groupG = GL(χ,C) canonically acts onQ and hence onX ×Q (trivial action onX) and lifts to an action on the universal sheafE onX×Q.

Let R denote theG-invariant open subset ofQ defined by


R = q ∈Q ∣∣ Eq = E |X×q is locally free such that the canonical map

Cχ →H 0(Eq) is an isomorphism, detEq OX

.

We denote byQss theG-invariant open subset ofR consisting of semistablebundles and letE continue to denote the restriction ofE toX×Qss .

Henceforth, ‘by abuse of notation,’ we shall writeQ for Qss .

Proof of Lemma 3.1. Note that the moduli space in question, namely, ofG-semi-stable bundles, is a GIT quotientQ→M by G, and the familyEA(G) is givenby a morphismT →M. Lift the K-valued point, namely,rK , given by the familyEK , to Q and consider theG-orbit R0 of rK in Q. Let R0 be its closure inQ.Since theK-valued pointrK is in fact anA-valued point ofM, the GIT quotientof R0 is indeed the curveT . Also, observe that the closure intersects the closedfiber. Consider the morphismψ :R0→ T . Since the base is a curveT , one hasa multi-sectionfor the morphismψ , and one obtains the curveS. The generalfiber has been modified only in the orbit, therefore, the isomorphism class of thebundles remains unchanged.8.1. The construction of the moduli space for principal bundles

Fix a base pointx ∈ X (cf. Remark 2.3). Letq ′′ : (Sch)→ (Sets) be thefollowing functor:

q ′′(T )= (Vt , st )

∣∣ Vt is a family of semistable principalG-bundles

parameterised byT andst ∈ Γ (X,V (G/H)t) ∀ t ∈ T,

i.e. q ′′(T ) consists pairs of rank-n vector bundles (or equivalently principalG-bundles) together with a reduction of structure group toH .

By appealing to the general theory of Hilbert schemes, one can show withoutmuch difficulty (cf. [R1, Lemma 3.8.1]) thatq ′′ is representable by aQ-scheme,which we denote byQ′′.

The universal sheafE on X ×Q is in fact a vector bundle. Denoting by thesameE the associated principalG-bundle, setQ′ = (E/H)x . Then in our notationQ′ = E(G/H)x ; i.e., we take the bundle overX ×Q associated toE with fiberG/H and take its restriction tox ×Q≈Q. Let f :Q′ →Q be the natural map.Then, sinceH is reductive,f is anaffine morphism.

Observe thatQ′ parameterizes semistable vector bundles together withinitialvalues atx of possible reductions toH .

Define the “evaluation map” ofQ-schemes as follows:

φx :Q′′ →Q′, (V , s) −→ (V, s(x)

).

Lemma 8.1. The evaluation mapφx :Q′′ →Q′ is proper.

Proof. The lemma follows easily from the proof of Proposition 2.8.


Lemma 8.2. The evaluation mapφx is injective.

Proof. Let G/H →W be as in Proposition 2.8 and let(E, s) and(E′, s′) ∈Q′′such thatφx(E, s) = φx(E

′, s′) in Q′, i.e., (E, s(x)) = (E′, s′(x)). So we mayassume thatE E′ and thats ands′ are two different sections ofE(G/H) withs(x)= s′(x).

Using G/H → W , we may considers and s′ as sections inΓ (X,E(W)).Observe that, by definition,E being semistable of degree 0, so isE(W).

Recall the following fact:

If E and F are semistable vector bundles withµ(E) = µ(F), then theevaluation map

φx : Hom(E,F )→Hom(Ex,Fx) (∗)is injective.

In our situation,s, ands′ ∈Hom(OX,E(W)) and hence by(∗), sinceφx(s)=φx(s

′), we gets = s′, proving injectivity. Remark 8.3. It is immediate that theG-action onQ lifts to an action onQ′′.

Recall the commutative diagram

Q′′ φx

ψ

Q′

f

Q

By Lemmas 8.1 and 8.2,φx is a proper injection and hence affine. One knows thatf is affine (with fibersG/H ). Henceψ is aG equivariant affine morphism.

Lemma 8.4. Let (E, s) and (E′, s′) be in the sameG-orbit of Q′′. Then we haveE E′. IdentifyingE′ withE, we see thats ands′ lie in the same orbit ofAutGE

onΓ (X,E(G/H)). Then using Remark2.1(b), we see that the reductionss ands′ give isomorphicH -bundles.

Conversely, if(E, s) and (E′, s′) are such thatE E′ and the reductionss,s′ give isomorphicH -bundles, again using Remark2.1(b), we see that(E, s) and(E′, s′) lie in the sameG-orbit.

Consider theG-action onQ′′ with the linearization induced by theaffineG-morphismQ′′ → Q. It is seen without much difficulty that, since a goodquotient ofQ by G exists and sinceQ′′ → Q is an affineG-equivariant map,


a good quotientQ′′/G exists (cf. [R1, Lemma 4.1]). Moreover, by the universalproperty of categorical quotients, the canonical morphism

ψ :Q′′//G→Q//G

is alsoaffine.

Theorem 8.5. Let MX(H) denote the schemeQ′′//G. Then this scheme is thecoarse moduli scheme of semistableH -bundles. Further,MX(H) is projectiveand if H → GL(V ) is a faithful representation, the canonical morphismψ :MX(H)→MX(GL(V )) is finite.

Proof. We need only check the last statement. By Theorem 3.1 one sees easilythat the moduli spaceMX(H) is projective, and thereforeψ is proper. By theremarks aboveψ is alsoaffine, therefore it follows thatψ is finite. Remark 8.6. We have supposed thatH is semisimple; however, it is not difficultto treat the more general case whenH is reductive. LetH be reductive andH =H

mod centre, its adjoint group. LetP be a principalH -bundle andP theH -bundle,obtained by extension of structure groups.We defineP to be semistable ifP issemistable. If we fix a topological isomorphism classc for principalH -bundles,this fixes a topological isomorphismc for principalH -bundles. Then the modulispaceMX(H)c is “essentially”MX(H)c× (product of Jacobians). This can bemade rigorous and it leads to the construction ofMX(H)c.

8.2. Points of the moduli space

In this subsection we will briefly describe thek-valued points of the modulispaceMX(H). The general functorial description ofMX(H) as a coarse modulischeme follows by the usual process.

Recall the following definitions from [R1].

Definition 8.7 (A. Ramanathan).E is semistableif for any parabolic subgroupP of H , any reductionσP :X→ E(H/P) and any dominant characterχ of P ,the bundleσ ∗P (Lχ)) has degree 0 (cf. [R1]). We note that in this convention,a dominant characterχ of P induces a negative ample line bundle onG/P .

Note that this definition makes sense for reductive groups as well.

Definition 8.8. A reduction of structure group ofE to a parabolic subgroupPis calledadmissibleif for any characterχ on P , which is trivial on the centerof H , the line bundle associated to theP -bundleEP , obtained by the reductionof structure group, has degree zero.


Definition 8.9. An H -bundleE is said to bepolystableif it has a reduction ofstructure group to a Levi subgroupR of a parabolicP such that theR-bundleER ,obtained by the reduction, is stable and the extendedP -bundleER(P) is anadmissible reduction of structure group forE.

Proposition 8.10. The “points” of MX(H) are given by isomorphism classes ofpolystable principalH -bundles.

We first remark that, since the quotientq :Q′′ →MX(H) obtained above isa good quotient, it follows that each fiberq−1(E) for E ∈MX(H) has the uniqueclosedG-orbit. Let us denote the orbitG ·E byO(E). The proposition will followfrom the following lemma.

Lemma 8.11. If O(E) is closed thenE is polystable.

Proof. Recall the definition of a polystable bundle (Definition 8.9) and thedefinition of admissible reductions(Definition 8.8). If E has no admissiblereduction of structure group to a parabolic subgroup then it is polystable, andthere is nothing to prove.

Suppose then thatE has an admissible reductionEP to P ⊂H . Recall by thegeneral theory of parabolic subgroups that there exists a 1-PSξ :Gm→H suchthatP = P(ξ). LetL(ξ) andU(ξ) be its canonical Levi subgroup and unipotentsubgroup, respectively. The Levi subgroup will be the centralizer of this 1-PSξ

and one knowsP(ξ) = L(ξ) · U(ξ) = U(ξ) · L(ξ). In particular, ifh ∈ P thenlim ξ(t) · h · ξ(t)−1 exists. From these considerations one can show that there isa morphism

f :P(ξ)×A1→ P(ξ)

such thatf (h,0)=m · u, whereh ∈ P andh=m · u, m ∈L andu ∈U (see [R1,Lemma 3.5.12]).

Consider theP -bundle EP . Then, using the natural projectionP → L

whereL = L(ξ), we obtain anL-bundleEP (L). Again, using the inclusionL → P → H , we obtain a newH -bundleEP (L)(H). Let us denote thisH -bundle byEP (L,H). It follows from the definition of admissible reductionsand polystability thatEP (L,H) is polystable.

Further, from the family of mapsf defined above, composing them with theinclusionP(ξ) →H , we obtain a family ofH -bundlesEP (ft ) for t = 0; all thesebundle are isomorphic to the given bundleE. Following [R1, Proposition 3.5,p. 313], one can prove that the bundleEP (L,H) is the limit of EP (ft ). Itfollows thatEP (L,H) is in theG-orbit O(E) becauseO(E) is closed. Now,by Lemma 8.4,E EP (L,H), implying thatE is polystable.


9. Semistable reduction for principal Higgs bundles

The aim of this section is to extend the methods of Section 3 and to provethe analogue of Theorem 7.1 for the case of principal Higgs bundles (cf.Theorem 9.3).

9.1. Higgs vector bundles

We recall briefly the usual category ofsemistableHiggs bundles with vanishingChern classes or what are called by Simpsonsemiharmonic bundles(for details,cf. [Sim0, p. 49]).

Suppose thatX is a smooth projective variety overk = C and we havefix a polarization to enable us to definedegreeof bundles. AHiggs bundleisa holomorphic vector bundleE together with a holomorphic mapθ :E→E⊗Ω1

X

such thatθ ∧ θ = 0 in End(E)⊗Ω2X . Define a Higgs bundleE to besemistable

(respectivelystable) if for every non-zero subsheafV ⊂E preserved byθ ,

degV

rkV degE

rkE(respectively<),

where we choose a hyperplane classh and define the degree asc1(E).[h]n−1.Let us say that a Higgs bundleE is polystableif it is the direct sum of stableHiggs bundles of the same slope (where slope is defined as usually as deg/ rk).Following Simpson, we call a semistable Higgs bundleof semiharmonic typeifall its Chern classes are zero.

If E is a Higgs bundle, define thespace of Higgs sectionsorH 0dol(X,E) to be

the space of holomorphic sectionss such thatθ.s = 0.

• Then, one has the basic theorem on Higgs bundles which says that thereis an equivalence of categories between the category of polystable Higgsbundles of rankn which aresemiharmonicand the category of semisimplerepresentations ofπ1(X)→GL(n).

• From this one can deduce as in Section 2, thetensor product theoremforsemistable Higgs bundles. This in particular implies that, ifρ : GL(n)→GL(W) is a finite dimensional representation and if(E, θ) isa semistable Higgs bundle with vanishing Chern classes, then the associatedbundleE(W) is also semistable with the induced Higgs structure (with van-ishing Chern classes).

Let (E, θ) be a Higgs bundle. Then we consider the characteristic polynomialof Higgs structureθ with its coefficients as points in the space⊕

1≤i≤nH 0(X,Symi Ω1

X

).


We define this element in the above direct sum as thecharacteristic tuple of(E, θ). Then one has the following basic theorem due to N. Hitchin, N. Nitsure,and C. Simpson.

Theorem 9.1. There is a quasi projective varietyMHiggs whose points parameter-ize polystable Higgs bundles onX with vanishing Chern classes. The map fromMHiggs to the space of polynomials with coefficients in the symmetric powers ofthe cotangent bundle, taking any(E, θ) to its characteristic tuple is proper.

We have the following proposition similar to Lemma 3.1.

Proposition 9.2. Let T = Spec(A) whereA is a discrete valuation ring withresidue fieldk = C and quotient fieldK. Let EK be a family of semistableHiggs bundles onXK = X × SpecK with vanishing Chern classes and fixedcharacteristic tuple. Then, by going to a finite cover if necessary, there existsa familyEA which extendsEK to XA so that the fiberEp over the closed pointp ∈ SpecA is polystable with vanishing Chern classes.

9.2. Semistable principal Higgs bundles

Following Simpson [Sim2, Section 9], we define aprincipal Higgs bundleonX for a reductive algebraic groupH with Lie algebrah, is a principalH -bundleE→ X together with a sectionθ of (E ×H h) ⊗ Ω1

X such thatθ ∧ θ = 0 in(E×H h)⊗Ω2

X. Given such an object and a representationH →GL(V ), we geta Higgs bundleE(V )=E ×H V .

Say thatE is semistableif the Chern classes ofE are all zero and if fora faithful representationH →GL(V ) the associated bundleE(V ) is a semistableHiggs bundle. As has been noted above, by the tensor product theorem, this isindependent of the choice of the representation. Note that this definition can benaturally relativized for a variety overT . With this definition our aim is then toprove the following theorem.

Theorem 9.3. Suppose that we are given a family of semistable principalHiggsH -bundle(EK, θK) on X × SpecK, or equivalently, ifHK denotes thegroup schemeH × SpecK, we are given a semistable HiggsHK -bundleEK

on XK . Suppose further that, for the associated Higgs vector bundle family(E(V )K, θ(V )K), the characteristic tuple is fixed. Then there exists a finiteextensionL/K with the integral closureB of A in L such thatEK , when pulledback toSpecB, extends to a semistable HiggsHB -bundleEB onXB .


9.3. Higgs section

We have the following lemma which is necessary to prove the resultcorresponding to Proposition 2.8 or Lemma 2.10. (For a related result, cf. [Sim2,Theorem 9.6].)

Proposition 9.4. Let T = SpecA and let (ET , θT ) be family of Higgs bundleswith vanishing Chern classes and fixed characteristic. LetsK :OXK → EK bea family of Higgs sections such that for a base pointx ∈ X the section extendsalongx × T . Then the familysK extends tosT :OXT →ET , to wholeT .

Proof. The proof of this proposition is much the same as the proof ofProposition 2.8. The only new ingredient needed to complete it is the followingfact about Higgs bundles.

Lemma 9.5. Let V andW be semistable Higgs bundles with vanishing Chernclasses. Letx ∈X be a base point. The we have an injection

HomHiggs(V ,W) →Hom(Vx,Wx),

or equivalently, a non-zero Higgs section is nowhere zero.

Proof. (Cf. [Sim1, Lemma 4.9].) 9.4. Monodromy subgroups, polystability and local constancy

Following [Sim2, Theorem 9.8] we definethe monodromy subgroupof thepolystable Higgs bundleE as a subgroupM which is minimal among allsubgroups ofG with the property that the structure group ofE can be reducedto M and such that the reduced bundleEM is semiharmonic(note that as definedM is not unique as we have not fixed a base point in the fiber ofE at x ∈ X asSimpson [Sim0, p. 29]).

We need the followinglocal constancyproperty which we isolate in a propo-sition (cf. also [Sim0, Lemma 2.10]).

Proposition 9.6. Let (E, θ) be apolystableprincipal HiggsG-bundle onX withvanishing Chern classes(or semiharmonic) (hereG= SL(n) or GL(n)). LetW beaG-module andY aG-subscheme ofW of the formG/H ′, whereH ′ = StabG(w)for somew ∈W .

Then ifs is a Higgs section ofE(W) such that for somex ∈X, s(x)=w is inthe fiber ofE(Y ) at x ∈X, then the entire image ofs lies inE(Y ). In particular,the principalG-bundle has a reduction of structure group toH ′. Furthermore,the reducedH ′-bundle is also semiharmonic in the sense of[Sim2].


Proof. The proof follows the proof of Proposition 5.2. The sections being a non-zero Higgs section, is nowhere zero by Lemma 9.5. By [Sim2, Theorem 9.8],sinceE is polystable,E(W) is also polystable. Therefore the sections of E(W)

gives a splitting ofE(W) as

E ⊕

Vi ⊕OX.

This gives a reduction of the structure group ofE(W) to a group which is theLevi subgroupL of a maximal parabolic subgroup ofGL(W) corresponding tothe extension ofE(W) asOX by

⊕Vi , Vi being stable Higgs bundles.

Thus, the bundleE(W) has twosemiharmonicreductions of structure group,namely, to the subgroupsG andL of GL(W). Therefore, by the definition of themonodromy subgroup ofE(W), we haveM → G ∩ L. Let EM be the reducedM-bundle. The sections which gives the copy ofOX in E(W) can therefore bethought of as a section ofEM(W) obtained by thetrivial characteronM.

Since the value of the section is given atx ∈X, namelys(x)=w, the sectionof E(W) = EM(W) can be seen as obtained by theconstant mapEM → W

which maps the whole ofEM to anM-invariant vectorw ∈ W (cf. proof ofProposition 5.2).

Exactly as in the proof of Proposition 5.2, we see that the inclusionM → G

factors via an inclusionM →H ′ sinceH ′ = StabG(w).Now, takingE′H = EM(H ′), we get the required reduction of structure group

of E toH ′.Again, sinceEM , by the definition of monodromy subgroup, issemiharmonic,

it follows that the inducedH ′-bundle is alsosemiharmonicproving the proposi-tion. Remark 9.7. In the proof of Proposition 5.2 instead of the monodromy reductionwe realize the bundle as extension of structure group from the principalπ1(X)-bundlej :Z→X, the universal covering space ofX. Notice that the monodromysubgroup of(E, θ) can be identified with the Zariski closure of the monodromyrepresentation giving the polystable Higgs bundle(E, θ).

9.5. Extension to the flat closure and potential good reduction

Once this proposition is proven, then we follow the strategy of the proof ofTheorem 7.1. We can extend the family(EK, θK) to a family of Higgs bundleswith structure group schemeH ′

A, the flat closureof HK in GL(VA). Here wecan define the notion of a Higgs bundle for a non-reductive group simply asa principal bundle which becomes Higgs semistable for a faithful representation,etc. (in fact, Simpson does not assume his group is reductive to define the notionof a semistable principal Higgs bundle).

The rest of the proof is verbatim from Section 3, Proposition 6.2, and we haveTheorem 9.3.


Remark 9.8. One can proceed as in Section 4 and obtain a construction of themoduli spaceMHiggs(H) of semistable HiggsH -bundles with vanishing Chernclasses, and, as a consequence, in much the same way one can prove, as inTheorem 8.5, that the natural morphismMHiggs(H)→MHiggs(G), induced froma representationH →G, is finite (cf. [Sim2, Section 9]).

Acknowledgments

We thank Gopal Prasad and M.S. Raghunathan for helpful discussions.

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