Principal Bundles and Bundles with Parabolic Structure

Principal Bundles and Bundles with parabolic structure

Arijit Dey

January 22, 2013 Abstract

In this thesis we prove HN reduction for HIggs G-bundles and nonemptyness of parabolic stable bundles Contents

1 Introduction 5 1.0.1 why and how Parabolic Bundle, little bit of history . . 8 1.0.2 Balaji’swork ...... 8 1.0.3 HintofΓ-Balaji ...... 8 1.0.4 WorkDoneinshort ...... 8 1.1 Introductiontosecondproblem ...... 8 1.1.1 MainResults ...... 12

2 Preliminaries 13 2.1 Higgsbundle ...... 13 2.2 NonemptynessofParabolicModuli ...... 19 2.2.1 The category of bundles with parabolic structures . . . 19 2.2.2 The Kawamata Covering lemma ...... 21 2.2.3 The category of Γ–bundles ...... 22 2.2.4 OnlocaltypesofΓ–bundles ...... 23 2.3 Qin-Li’s proof for rank r case ...... 29

3 Harder-Narasimhan reduction for Higgs G bundle 30 3.1 ProofofexistenceofaHiggsH-Nreduction ...... 30 3.2 The Higgs structure on the adjoint bundle EG(g) ...... 35 3.3 Proof of the uniqueness of Higgs H-N reduction ...... 40

4 Nonnemptyness 45 4.0.1 Preliminaries ...... 46 4.0.2 Proof of Theorem I ...... 53 4.0.3 ProofofTheoremII ...... 54

5 holonomy 57

2 6 Ramified G-bundle 61 6.1 H-N reduction for ramified G bundles over a smooth curve . . 61 6.1.1 H-N reduction for [Γ,G,N]bundle ...... 63 6.1.2 Proof of H-N reduction for ramified G bundle . . . . . 64

3 Acknowledgments

I would like to thank Prof. V. Balaji , for suggesting me this problem, for many discussions and for his encouragement. I am very grateful to him for introducing me to the study of moduli spaces of vector bundles. I consider myself very fortunate for having found his generous help. I would like to thank Prof. D. S. Nagaraj who introduced me to algebraic geometry. I would also like to thank Shamindra Kumar Ghosh, Paramita Das and T. Muthukumar for their generous help during my course work of Ph.D program. I have also beneﬁted from discussions with R. Parthasarathi, A. Laddha, S. Nayak and many other people as well as from many seminars and courses at Institute of Mathematical Sciences and Chennai Mathematical Institute. I want to thank Institute of Mathematical Sciences for the fellowship that supported my graduate studies. Without their generous support, this thesis couldn’t have been made. Lastly I would like to thank members of IMSc for giving me a very friendly atmosphere during this work.

4 Chapter 1

Introduction

In this chapter we will explain the main results of the thesis using as little mathematical background as possible. We will always work over field of complex numbers. Also we assume that varieties are projective, i.e. there is n an embedding in PC. Moduli spaces are fundamental objects of algebraic geometry which have been at the center of interest since the beginnings of the subject. In a nut- shell, moduli spaces are spaces whose points correspond in a natural way with geometric objects that one would like to study, such as Riemann surfaces, or vector bundles on a fixed space. Understanding the geometry of moduli spaces often helps to understand the underlying geometric objects themselves. For example in [8] by exploiting moduli space of µ-stable bundles on a smooth projective surface Donaldson was able to prove some strik- ing new results on smooth classification of certain projective surfaces, for example there is only one differential structure on Rn except when n = 4, in which case there are uncountably many. Moduli are, however, not only of interest to the algebraic geometer. They also play a role in other subjects, like in theoretical Physics. In this thesis we are interested with holomorphic parabolic vector bundles (E,D,α∗,r∗) (often denoted as (E∗) if there is no scope of confusion) with normal crossing parabolic divisor D over X, which is a vector bundle E together with a weighted (α∗) flag structure (r∗) on fibres of E|D (for precise definition of parabolic bundle see chapter 2). To distinguish non-isomorphic parabolic vector bundles of fixed quasi-parabolic structure and parabolic weights, we can define certain invariants called parabolic Chern classes. 2i These are cohomology classes parci(E) ∈ H (X, R), 1 ≤ i ≤ dimC X. We have parci(E)=0if i > rank(E). Even after fixing these discrete invari-

5 ants, we can have continuous families of non-isomorphic parabolic vector bundles. I.e., to specify the isomorphism class of a parabolic vector bundle it is not enough to fix some discrete invariants, but we have to fix also some continuous parameters. For fixed parabolic structure and parabolic Chern classes we would like to define a variety called the moduli space of parabolic vector bundles. Each point of this variety will correspond to a different (at least non-isomorphic) vector bundle. In this thesis first we will study the nonempty-ness of this space for rank 2 parabolic vector bundles over any surfaces and using this we can answer nonempty-ness of higher rank Moduli space of certain type of quasi-parabolic structure. Unfortunately, in order to construct this moduli space we have to restrict our attention to parabolic stable vector bundles (this is not a very strong restriction, since it can be proved that in some sense all parabolic vector bundles can be constructed starting from parabolic stable ones). There are different notions of stability (see 2). Here we will only discuss parabolic Mumford stability (also called parabolic-slope stability). Let X be a projective variety and H an ample divisor. For a parabolic vector bundle E∗ one can define par-degree(E∗)(see chapter 2). We define the parabolic slope of E∗ with respect to H as: par-degree(E ) parµ (E )= ∗ H ∗ rank(E)

(the product is the intersection product or cup product in cohomology) E∗ ′ is called parabolic H-stable if for every parabolic subbundle E∗ of E∗ we have ′ parµH (E∗) < parµH (E∗). 0 It can be proved that there is a space PMH (r,D,α∗,pci), called the moduli space of H-stable parabolic vector bundles of rank r and parabolic Chern classes pci (if the quasi-parabolic structure is clear from the context we will drop it from the notation). In many situations it will be a quasi-projective 0 variety (maybe with singularities). In general PMH (r,D,α∗,pci) is not compact but one can compactify this variety in such a way that boundary points correponds to polystable parabolic bundles together with a 0 cycle in X. This compactiﬁcation is known as Donaldson-Uhlenbeck compacti- ﬁcation of parabolic bundles (for details see [4]). Let (X, Θ) be a smooth polarized algebraic surface, this gives rise to a underlying Riemannian manifold with metric d. There is an important relationship between gauge theory or Yang-Mills theory and stable holomorphic vector bundles; there is a bijection between Θ-stable SU(n) bundles on (X, Θ) and Hermite-Einstein or

6 Anti Self Dual (ASD)1 on a fixed differentiable SU(n) vector bundle over (X,d). This correspondence is known as Hitchin-Kobayashi correspondence. Infact more precisely both these spaces gets a canonical algebraic structure and their underlying topological spacesare homeomorphic. Note that the metric of the manifold appears in the ASD-equation. This is reflected (for bundles) that the stability condition of a bundle depends on the chosen polarization Θ. This relationship has been used, for instance, to calculate Donaldson polynomials for the study of the topology of 4-manifolds [8]. This polynomial denoted as γc2 (X) is of the form :

γc (X)(α)= µ(α) ∧··· µα 2 Z Mc2 d

2 2 | {z } where µ : H (X) → H (Mc2 and d = dim(Mc2 )/2 by exploiting the geometry of Mc2 Donaldson was able to prove some strik- ing new results on smooth classiﬁcation of projective surfaces. By exploiting algebro-geometric description of Mc2 he showed that γc2 (X) is nonzero for large c2. Hence asymptotic nonemptyness is good enough to compute Don- aldson polynomials. While studying the problem of embedding a sphere in a 4-fold Kron- heimer and Mrowka found an analogue of Donaldson polynomial for pairs (X, Σ), where Σ is a surface. They found that obstruction to embedding a sphere into a 4-fold is nonvanishing of this polynomial qk,l. They deﬁned α-twisted anti self dual connections which are actually ASD connection in α X − Σ. let Mk,l be the moduli space of equivalence class of α-twisted connections where α ∈ (0, 1/2) is the holonomy parameter, k is the usual second chern class and l denote the instanton number. In [6] they conjec- tured Hitchin-Kobayashi correspondence i.e. there is a bijection between α-twisted ASD connection and rank 2 stable parabolic bundles (E, L,α) on (X,σ). In their paper they proved this conjecture for some simple case like ruled surfaces and later it was proved in greater generality. for a suitable integer n and is nonzero for large values of c2. In this thesis we prove nonemptiness of rank-2 parabolic stable for arbitary c1 and large c2 with suitable bound on α2. The bound on c2 and α2 depends on polarisation. We beleive one should be able to prove Gieseker- Taubes kind of theorem for parabolic bundles for generic choice of parabolic datum, but due to lack of knowledge/patience we could not get such nice theorem. In future we would like to investigate this problem.

1Itoh [11] showed that for SU(n) bundles Hermite-Einstein connection is same as ASD connection

7 1.0.1 why and how Parabolic Bundle, little bit of history 1.0.2 Balaji’s work 1.0.3 Hint of Γ-Balaji 1.0.4 Work Done in short The main strategy used is the categorical correspondence of the category of Γ–bundles of fixed type τ on a certain Kawamata cover Y of the surface X with the category of parabolic bundles on X with fixed parabolic datum (see chapter ?? for definitions and terminology). The Kawamata cover Y is non- canonical and is therefore employed only as a stepping stone. Although non- canonical, the moduli problem gets defined more naturally on Y and becomes less complicated. We then go on to show that the moduli space of rank 2, µ-stable Γ–bundle of type τ is non–empty for large topological invariants, hence the moduli space of rank 2 µ–stable parabolic bundles is non–empty for large topological invariants. The proof is a generalization of the classical Cayley-Bacharach construction to the setting of orbifold bundles. Our proof of non-emptiness and existence of components with smooth points gives the same results for the Maruyama-Yokogawa’s space [12] as well in the case when X is a surface. To the best of our knowledge the non-emptiness of these moduli spaces have not been shown hitherto. In chapter ?? we present a proof of the following theorem :

s Theorem 1.0.1 The moduli space MΓ(Y ) of rank two Γ-stable bundles of type τ and arbitary c1 on a smooth projective Γ surface Y is nonempty if c2(E) ≫ 0. Hence, the moduli space of parabolic bundles on X with given full quasi–parabolic structure and with parc2(V ) ≫ 0 is non-empty.

1.1 Introduction to second problem

Let X be any smooth projective variety over C. THe concept of semistability is very important in Bundle theory and Harder-Narasimhan filtration (reduction) is some kind of measure which tells us how far a bundle is away from being a semistabe bundle. It gets more importance because of its uniqueness nature. The problem of the existence and uniqueness of Harder-Narasimhan reductions (henceforth called briefly as HN-reduction) for principal G–bundles, where G is a reductive algebraic group over C, was first introduced by A. Ramanathan (when G is reductive) and M.F. Atiyah (when G is arbitary lie group over C). was solved by Atiyah and Bott in [9]. The notion of semistability that was used in this paper was using the

8 adjoint representaion, i.e E is semistable iff E(g) is semistable as a vector bundle, where g is the lie algebra of G. In an earlier article by A.Ramanathan ([3]) this question of HN-reduction of principal G-bundle was posed as a problem intrinsically on E, and this definition is quite different from Atiyah-Bott’s definition. The good thing about Ramanathan’s definition on HN-reduction is for its applicability to the positive characteristic cases as well. The problem in this setting was solved by Behrend [1] and in positive characteristics Behrend had a conjecture which was verified by Mehta and Subramaniam [14]. Biswas and Holla gave a different approach to the solution following essentially the broad setting in Ramanathan’s paper in [16]. In this thesis we generalise the methods of Biswas and Holla to give an unified approach to the case of principal bundles with Higgs structure on smooth projective varieties as well as the case of ramified bundles on smooth curves (as defined in [17])(see chapter 2 for definitions). For the case of Higgs vector bundles on smooth projective varieties the existence and uniqueness of HN-filtrations is proved in §3 of Simpson [7]. For the case of ramified bundles we generalise the existence and uniqueness of HN-filtrations for parabolic vector bundles (proved in [18]). By a principal object in this introduction we mean a principal Higgs bundle on smooth projective varieties or a ramified bundle on smooth projective curves.(See §1 for the definition of HN-reduction’s) A.Ramanathan in [?] generalised the Narasimhan-Seshadri theorem [?] for G-bundles over curves and constructed the moduli space of G-bundles of fixed topological type, where G is a reductive group over C. In order to construct the moduli space he extended in a natural manner the notion of µ-(semi)stability to G-bundles over any dimension, which generalises the usual vector bundle µ-(semi)stability. A rational G-bundle (i.e a G-bundle over a big open set, i.e whose complement is of codimension ≥ 2) is said to be µ-(semi)stable if for any reduction EP of E |U to a maximal parabolic P , deg(EP χ)(≤) < 0 for any dominant character χ of P , where EP χ is the associated line bundle to character χ of P . In order to measure instability of a G-bundle he defined HN reduction (analogue of HNF ) for a principal G-bundle. A reduction EP to a parabolic subgroup P of G is said to be HN reduction if it satisfies following two conditions.

1. The associated Levi bundle EP ×P L is a semistable L-bundle, where L is the Levi quotient of P .

2. For any dominant character χ of P with respect to some Borel subgroup B ⊂ P of G, the associated line bundle EP χ has positive degree.

9 For the case of G = Gl(n), a parabolic reduction gives rise to a filtration of rank n bundle. It is not very difficult to see that EP is a HN reduction iff the corresponding filtration of the associated vector bundle (with respect to standard representation) coincides with its HN filtration. In [2], Ramanathan had stated (without proof) that any principal G-bundle admits an unique HN reduction. The problem of the existence and uniqueness of Harder-Narasimhan reduction for principal G-bundles, was solved by Atiyah and Bott in [9] for (more general) reductive Lie group. The notion of (semi)stability that Atiyah and Bott used in their paper was (semi)stability of adjoint bundle ad(E), i.e E is semistable iff E(g) is semistable as a vector bundle, where g is the Lie algebra of G. Since the adjoint bundle ad(E) has the self duality property, the uniqueness of HNF of ad(E) (as a vector bundle) gives a filtration of following form:

0 ⊂ E−r ⊂ E−r+1 ⊂··· E−1 ⊂ E0 ⊂ E1 ⊂···⊂ Er−1 ⊂ Er = E (1.1) where E−j is the orthogonal complement of Ej−1 with respect to the “Killing” 1 form . The middle subbundle, given by E0, is essentially given by a parabolic subalgebra p. Hence the subalgebra bundle E0 ⊂ ad(E) deter- mines a reduction of structure group of E to a parabolic subgroup P . We call this reduction to parabolic the Atiyah-Bott’s HN reduction and it is unique because of uniqueness of HNF of ad(E). At that time it was not clear whether Atiyah-Bott’s HN reduction coincides with Ramanathan’s HN reduction or not. The first proof of the existence and uniqueness of HN reduction(in Ra- manathan’s sense) was given by K. Behrend [1] for the case when X is a smooth projective curve. The proof of Behrend works in any characteristics for reductive group schemes over a curve. His argument involves usage of complementary polyhedron which is bit afar from bundle theoretic argument, but he did not compare his parabolic reduction with Atiyah-Bott’s HN reduction. In [5], I. Biswas, H. Azad and B. Anchouche proved Ramanathan’s HN reduction for the case of compact complex Kahler manifolds and they also proved that Ramanathan’s and Atiyah-Bott’s HN reduction coincide. In [16] I. Biswas and Y. Holla gave a proof of Ramanathan’s HN reduction for any projective variety over any algebraically closed field. The aim of this work is to generalise the methods of Biswas and Holla to give an unified approach to the case of principal G-bundles with Higgs structure on smooth projective varieties as well as the case of ramified bundles on smooth curves (as defined in [17]) over C. Recall that for the case

1For every x ∈ g, there exists linear map Ad(x): g −→ g. This gives rise to a bilinear map θ : g × g −→ C, given by θ(x, y) = Tr((Ad(x) ◦ Ad(y))

10 of Higgs vector bundles on smooth projective varieties the existence and uniqueness of HNF was proved by Simpson [7]. In the case of ramiﬁed bundles similarly we generalise the existence and uniqueness of HN-ﬁltrations for parabolic vector bundles (proved in [18]).

11 1.1.1 Main Results

0 1 Let E be a principal G-bundle on X and θ ∈ H (U, E(g)⊗ΩU ) with θ∧θ = 0 2 in E(g) ⊗ ΩX , where U is a big open set in X. The pair (E,θ) is called a Higgs principal G-bundle on X. Let EP be a reduction of E to maximal 0 1 parabolic P ⊂ G over an open subset U of X and θP ∈ H (U, EP (p) ⊗ ΩU ), such that following diagram commutes

θ 1 OU −→ E(g) ⊗ ΩU θP ց ↑ 1 EP (p) ⊗ ΩU ↑ 0

Then the tuple (EP ,θP ) is called Higgs reduction of (E,θ) to P . A Higgs G-bundle (E,θ) is said to be Higgs semistable if for any Higgs parabolic reduction (EP ,θP ) and for any dominant character χ of P the associated line bundle EP χ on UP has nonpositive degree. Following Ramanathan we deﬁne Higgs HN reduction as follows.

Deﬁnition 1.1.1 Let (E,θ) be a principal Higgs G-bundle on X and (EP ,θP ) be a reduction of structure group of E to a parabolic subgroup P of G, then this reduction is called Higgs HN reduction if the following two conditions hold:

1. If L is the Levi factor of P then the principal L bundle EP ×P L over U is a Higgs semistable L-bundle.

2. For any dominant character χ of P with respect to some Borel subgroup B ⊂ P of G, the associated line bundle EP χ over UP has degree > 0. In this thesis We prove the following theorem. The main theorem of this thesis is:

Theorem 1.1.2 Let E be a principal G object on X. Then there exists a canonical HN-reduction (P,σ) where P is a parabolic subgroup of G and σ : X −→ E/P is a section of the associated ﬁbre bundle E/P over X. The H-N reduction is unique in natural sense.

12 Chapter 2

Preliminaries

2.1 Higgs bundle

Throughout this paper, unless otherwise stated, we have the following no- tations :

1. We work over an algebraically closed ﬁeld k of characteristic 0.

2. G will stand for a connected reductive algebraic group (i.e, for the linear algebraic group G, unipotent radical of G, denoted by Ru(G), is trivial).

3. X is an irreducible smooth projective variety over k of dimension d ≥ 2.

4. Let X be embedded in PN for some positive integer N, which is equivalent to ﬁxing a very ample line bundle on X, we call it H which we ﬁx it throughout the paper.

Let E be a torsion-free coherent sheaf of rank r over X. Then the degree d−1 2 of E is defined as deg(E) = c1(E) · H , where c1(E) ∈ H (X, Z) is the first Chern class. Let U be an open subscheme containing all points of codimension one, (codim(X\U) ≥ 2) such that E|U is locally free. Then r ∧ (E |U ) is an invertible sheaf on U, corresponding to a divisor class D on r U. Then we have c1(E |U ) = c1(∧ E |U ) = D. From the functoriality of Chern classes, c1(E)= c1(E |U )= D. Let U be an open subset of X such that codim(X\U) ≥ 2, then i∗OU =∼ OX , further if F is a locally free sheaf on U then i∗F is a reflexive sheaf over X.

13 It follows from the above that if two bundles agree at all points of codimension one, they have the same degree. Also, we can talk about the degree of torsion-free sheaf on U(since codim(X\U) ≥ 2). Let E be a torsion- free sheaf on U, then we deﬁne deg(E) = deg(i∗(E)), where i is the inclusion i : U −→ X. Further, if 0 → F → E → G → 0 is an exact sequence of torsion-free sheaves on X, then deg(E) = deg(F ) + deg(G), also deg E∗ = − deg(E). Again from the functoriality of Chern classes, for exact sequence of locally free sheaves over U we have the same. For if, 0 → F → E → G → 0 is an exact sequence of locally free sheaves on U, then we have deg(i∗E) = deg(i∗F ) + deg(i∗G).Therefore it follows that, deg(E) = deg(F )+deg(G) on U. Now we can deﬁne for a torsion-free sheaf E on X, deg(E) µ(E)= ∈ Q rank(E) A torsion-free sheaf E is said to be µ-semistable if for all coherent subsheaves F ⊆ E we have

deg(F ) deg(E) µ(F )= rank(F ) ≤ rank(E) = µ(E). Note that it is enough to check only for subsheaves whose quotients are torsion-free. Let E be a torsion-free coherent sheaf on X and θ : E → 1 E⊗ΩX , an OX –linear homomorphism of sheaf of modules, such that θ∧θ = 0 2 in E ⊗ ΩX , then the pair (E,θ) is called Higgs sheaf on X. If E is locally free, then the pair (E,θ) is called Higgs bundle on X. Let F be a subsheaf of 1 E such that θ |F : F → F ⊗ ΩX , then we say (F,θ |F ) is a Higgs subsheaf of (E,θ). Further, (E,θ) is called Higgs semistable if for every Higgs subsheaf F ⊆ E we have, µ(F ) ≤ µ(E) . We also have the notion of Chern classes ∗ of the principal G bundle E over X; let ci(E) ∈ H (X, Z) be the Chern classes of the principal bundle E. We will say that ci(E) = 0 for any i if ci(E(W )) = 0 for every W ∈ Rep(G), where Rep(G) is the category of ﬁnite dimensional representations of G ([?] Rem.2.3,). The aim of this paper is to ﬁnd Higgs compatible H-N reduction for Higgs principal G bundle E on X. Now we recall some facts which we need in our work.

1. Let E be a principal G-bundle on X. Let Y be any quasi-projective variety on which G acts from the left, then we define by E(Y ) as the associated fiber bundle with fiber type Y which is the following object: E(Y )=(E × Y )/G for the twisted action of G on (E × Y ) given by g.(e,y)=(e.g−1,g.y) where g,e,y are in G,E,Y respectively.

14 2. Let H be a closed subgroup of G. A principal G bundle E on X is said to have a H structure or equivalently a reduction of structure group to H if there exists an open subset U of X which contains all the points of codimension one codim(X\U) ≥ 2 and a section σ : U −→ E(G/H). In other words a reduction of structure group to H means giving a principal H bundle EH over an open set U which satisﬁes codim(X\U) ≥ 2 such that EH (G) =∼ E |U . Given a reduction of structure group over an open subset U with codim(X\U) ≥ 2 there is a unique maximal open set over which the reduction extends.

3. Let H be a closed subgroup of G and U be an open subset of X. A principal G bundle E on U is said to have a H structure or equivalently a reduction of structure group to H if there exists an open subset U ′ of U which contains all the points of codimension one codim(U\U ′) ≥ 2 and a reduction σ : U ′ −→ E(G/H). In other words, a reduction of ′ structure group to H means giving a principal H bundle EH on U ∼ such that EH (G) = E |U ′ . 4. Let π : E → X be a principal G bundle, then E is called a semistable bundle if for every parabolic subgroup P of G and any reduction of structure group (P,σ) where σ : U −→ E(G/P ) is a reduction with codim(X\U) ≥ 2 and for every dominant character χ of P the associated line bundle i∗Lχ on X has degree ≤ 0 .

0 1 5. Let E be a principal G bundle on X and θ ∈ H (X,E(g) ⊗ ΩX ),( g 2 is the Lie algebra of G) with θ ∧ θ =0 in E(g) ⊗ ΩX . Then the pair (E,θ) is called a Higgs principal G bundle on X.

Deﬁnition 2.1.1 Let (E,θ) be a Higgs principal G bundle on X. Let (H,σ) be a reduction of E to H over an open subset U of X with codim(X\U) ≥ 2 0 1 where H is a connected closed subgroup of G and θH ∈ H (U, EH (h) ⊗ ΩU such that the following diagram commutes.

θ 1 OU −→ E(g) ⊗ ΩU θH ց ↑ 1 EH (h) ⊗ ΩU ↑ 0

Then the quadruple (H,σ,θH ,U) is called Higgs reduction of E to H .

15 Remark 2.1.2 Let (E,θ) be a Higgs principal G bundle over U, where U is an open subset of X with codim(X\U) ≥ 2 . We can deﬁne the Higgs reduction of E to a subgroup H in the same way as above.

Deﬁnition 2.1.3 Let (E,θ) be a Higgs principal G bundle over X, it is said to be Higgs semistable if for any Higgs reduction (P,σ,θP ,UP ) to a parabolic and for any dominant character χ of P the line bundle Lχ on UP has non-positive degree which is same as saying the degree of the reﬂexive sheaf i∗Lχ over X has non positive degree.

Deﬁnition 2.1.4 Let (E,θ) be a Higgs principal G bundle over U, where U is an open subset with codim(X\U) ≥ 2. Then it is said to be Higgs semistable if for any Higgs reduction (P,σ,θP ,UP ) to a parabolic, where codim(U\UP ) ≥ 2 and for any dominant character χ of P the line bundle Lχ on UP has non-positive degree which is same as saying the degree of the reﬂexive sheaf i∗Lχ over X has non positive degree.

Lemma 2.1.5 Let (E,θ) be a Higgs principal G bundle over X. Then (E,θ) is Higgs semistable if and only if for any maximal parabolic subgroup P of G ∗ and for any Higgs reduction (P,σ,θP ,UP ) we have deg i∗σ (TG/P ) ≥ 0 where TG/P is the relative tangent bundle for the projection E |U → E |U (G/P ) .and i is the inclusion U ֒→ X

Proof: Let P be a maximal parabolic subgroup of G. Then consider the following exact sequence of vector bundles on UP ; 0 → EP (p) → EP (g) → EP (g/p) → 0 given by the exact sequence 0 → p → g → g/p → 0. Since G is reductive there is a non-degenerate bilinear form on g invariant under G; therefore we see that deg EP (g) = 0. Therefore, deg EP (p) ≤ 0 iﬀ deg EP (g/p) ≥ 0. So the ’only if’ part is trivial. The other way can be proved following the proof of lemma 2.1. in [?]

Lemma 2.1.6 When G is GL(n,k), the Higgs semistability of Higgs principal G bundle E on X is equivalent to the Higgs semistability of the associated Higgs vector bundle E(kn) by the standard representation.

Proof: Let P be a maximal parabolic subgroup of G. Let (P,σP ,θP ,UP ) be a Higgs reduction of E to P , where P is parabolic and let EP be the corresponding P bundle. Let us consider the ﬂag 0 ⊂ kr ⊂ kn corresponding n r to the parabolic subgroup P . Let V = EP (k ),W = EP (k ); then W is a Higgs subbundle of V on UP (with Higgs structure induced by θP ). Note ∗ that EP (g/p) = V/W ⊗ W and Higgs structures on both sides are same.

16 (since they are coming from θ and θP ). Also µ(W ) ≤ µ(V ) is equivalent to µ(W ) ≤ µ(V/W ) on UP .

deg((V/W ) ⊗ W ∗)= − deg W.rank(V/W )+ rankW. deg(V/W ) =(µ(V/W ) − µ(W )).rank(V/W ).rank(W ) From the above equation it is easy to see that µ(W ) ≤ µ(V ) is equivalent ∗ ∗ to deg((V/W ) ⊗ W ) ≥ 0. Since EP (g/p)=(V/W ) ⊗ W we conclude that deg(EP (g/p)) ≥ 0 iff µ(W ) ≤ µ(V ). Suppose E is Higgs semistable. then we have, deg iP ∗(EP (g/p)) ≥ 0 on X for any Higgs reduction of (P,σP ,θP ,UP ) to a maximal parabolic subgroup P where iP is the inclusion of UP to X. Let W be a Higgs torsion- free sheaf of V on X. We can choose an open subset U with codim(X\U) ≥ 2 r such that W is a Higgs subbundle of V on U. Then W = EP (k ) for some r n Higgs reduction (P,σP ,θP ) corresponding to a flag 0 ⊂ k ⊂ k on U. Then deg iP ∗(EP (g/p)) ≥ 0 implies that µ(W ) ≤ µ(V ) on U. Thus V is Higgs semistable. Converse is also true. Given a torsion-free coherent sheaf E over X, there is a unique filtration of subsheaves

0= E0 ⊂ E1 ⊂ E2 ⊂·········⊂ El−1 ⊂ El = E which is characterised by the two conditions that all the sheaves Ei ,i ∈ Ei−1 [1,l], are semistable torsion-free sheaves and the µ Ei ’s are strictly de- Ei−1 creasing as i increases. This filtration is known asHarder-Narasimhan filtration for E The following is the definition of H-N reduction for principal bundles in the literature(cf. [5]).

Deﬁnition 2.1.7 Let E be a principal G bundle on X and (P,σP ,UP ) be a reduction of structure group of E to a parabolic subgroup P of G, then this reduction is called H-N reduction if the following two conditions hold:

1. If L is the Levi factor of P then the principal L bundle EP ×P L over UP is a semistable L bundle. 2. For any dominant character χ of P with respect to some Borel subgroup BP of G, the associated line bundle Lχ over UP has degree > 0.

For G = GL(n,k) a reduction EP gives a ﬁltration of the rank n vector bundle associated to the standard representation. It is easy to see that EP is

17 canonical in the above sense iff the corresponding filtration of the associated vector bundle filtration coincides with its Harder Narasimhan filtration.

Lemma 2.1.8 Let ρ : G −→ H be the homomorphism of connected reductive algebraic groups and let (E,θ) be the Higgs G bundle on X. Then the associated bundle EH := E(H) gets a Higgs structure. Proof: The representation ρ induces a morphism ρ′ : E(g) −→ E(h) of vector bundles on X. where g and h are Lie algebras of G and H respectively. ′ So we deﬁne Higgs structure on E(H) denoted by θH := (ρ ⊗ id) ◦ θ where ′ 1 1 1 ρ ⊗ id : E(g) ⊗ ΩX −→ E(h) ⊗ ΩX and θ : OX −→ E(g) ⊗ ΩX . Now we give a deﬁnition of a Higgs compatible H-N reduction for Higgs principal bundles.

Deﬁnition 2.1.9 Let (E,θ) be a Higgs principal G bundle on X. Then a Higgs reduction (EP ,σP ,θP ,UP ) (Deﬁnition 2.1.1) is called a Higgs compatible H-N reduction if the following conditions hold:

1. The Higgs bundle EL is Higgs semistable on UP , where L is the Levi factor of P .

2. For all dominant character χ of P with respect to some Borel subgroup B ⊂ P ⊂ G the associated line bundle Lχ has positive degree on UP . From now onwards we call a Higgs compatible H-N reduction as a Higgs H-N reduction. Suppose H1, H2 are closed connected subgroups of G. Let σ1, σ2 be reductions of structure group of E to H1 and H2 respectively. To ∗ this data we can associate subgroup schemes σi E(Hi) of E(G) and their ∗ Lie algebras σi E(hi), which are sub-bundles of E(g). For details we refer to [16]. We recall some facts which we use in our proof of Higgs H-N reduction for Higgs principal bundles:

1. Let (E1,θ1) and (E2,θ2) be two Higgs semistable vector bundles on smooth projective variety X. Then their tensor product (E1 ⊗E2,θ1 ⊗ id + θ2 ⊗ id) is also Higgs semistable.(cf.[?])

2. For two torsion-free torsion-free coherent sheaves E1 and E2 over X, the equality µmin(E1 ⊗ E2)= µmin(E1)+ µmin(E2) is valid. Similarly, we have µmax(E1⊗E2)= µmax(E1)+µmax(E2).(cf.[?])

18 3. Let E1 and E2 be two torsion-free coherent sheaves over X.(cf.[?])

0 • If µmin(E1) >µmax(E2), then H (X,HomOX (E1,E2)) = 0.

• If there is a surjective OX linear homomorphism φ : E1 → E2 , then µmin(E1) ≤ µmin(E2).

2.2 Nonemptyness of Parabolic Moduli

2.2.1 The category of bundles with parabolic structures We rely heavily on the correspondence between the category of parabolic bundles on X and the category of Γ–bundles on a suitable Kawamata cover. This strategy has been employed in many papers (for example [?]) but since we need its intricate properties, most of which are scattered in a few papers of Biswas and Seshadri, we recall them briefly. We stress only on those points which are relevant to our purpose. Let D be an effective divisor on X and Θ1 be an very ample divisor in X. All ouc degree calculations of a sheaf on X are with respect to this Θ1. For a coherent sheaf E on X the image of E O (−D) in E will be denoted OX X by E(−D). The following definition ofN parabolic sheaf was introduced in [12].

Deﬁnition 2.2.1 Let E be a torsion-free OX –coherent sheaf on X. A quasi–parabolic structure on E over D is a ﬁltration by OX –coherent subsheaves

E = F1(E) ⊃ F2(E) ⊃ ··· ⊃ Fl(E) ⊃ Fl+1(E) = E(−D)

The integer l is called the length of the ﬁltration. A parabolic structure is a quasi–parabolic structure, as above, together with a system of weights {α1, ··· ,αl} such that

0 ≤ α1 < α2 < ··· < αl−1 < αl < 1 where the weight αi corresponds to the subsheaf Fi(E).

In other words a quasi-parabolic structure on a sheaf E can also be deﬁned by giving ﬁltration by subsheaves on the restriction E|D of the sheaf E to each component of the parabolic divisor:

1 2 l l+1 E|D = FD(E) ⊃ FD(E) ⊃ ... ⊃ FD(E) ⊃ FD (E) = 0

19 A parabolic sheaf is a sheaf together with a quasi–parabolic structure and a system of weights

0 ≤ α1 < α2 < ··· < αl−1 < αl < 1

i where each αi corresponds to F |D(E). We shall denote the parabolic sheaf defined above by (E, F∗,α∗). When there is no scope of confusion it will be denoted by E∗. For a parabolic sheaf (E, F∗,α∗) define the following filtration {Et}t∈R of coherent sheaves on X parameterized by R:

Et := Fi(E)(−[t]D) (2.1) where [t] is the integral part of t and αi−1

Remark 2.2.2 The notion of parabolic degree of a parabolic bundle E∗ of rank r is deﬁned as:

1 pardeg(E∗) := deg(Et)dt + r.deg(D) (2.2) Z0

Similarly one may deﬁne parµ(E∗) := pardeg(E∗)/r. There is a natural notion of parabolic subsheaf and given any subsheaf of E there is a canonical parabolic structure that can be given to this subsheaf. (cf [12] [?] for details)

Deﬁnition 2.2.3 A parabolic sheaf E∗ is called parabolic semistable (resp parabolic stable) if for every parabolic subsheaf V∗ of E∗ with 0 < rank(V∗) < rank(E∗), the following holds:

parµ(V∗) ≤ parµ(E∗) (resp.parµ(V∗) < parµ(E∗)) (2.3)

Some assumptions The class of parabolic vector bundles that are dealt with in this thesis satisfy certain constraints which we will explained now. In a remark below, (see

20 Remark 2.2.4), we observe that these constraints are not stringent in so far as the problem of moduli spaces is concerned.

1. The first condition is that all parabolic divisors are assumed to be divisors with normal crossings. In other words, any parabolic divisor is assumed to be reduced, its each irreducible component is smooth, and furthermore the irreducible components intersect transversally. 2. The second condition is that all the parabolic weights are rational numbers. 3. The third and final condition states that on each component of the parabolic divisor the filtration is given by subbundles. The precise formulation of the last condition is given in ([?], Assumptions 3.2 (1)). Henceforth, all parabolic vector bundles will be assumed to satisfy the above three conditions.

Remark 2.2.4 We remark that for the purpose of construction of the moduli space of parabolic bundles the choice of rational weights is not a serious constraint and we refer the reader to [18, Remark 2.10] for more comments on this.

Let PVect(X,D) denote the category whose objects are parabolic vector bundles over X with parabolic structure over the divisor D satisfying the above three conditions, and the morphisms of the category are homomorphisms of parabolic vector bundles (which was deﬁned earlier). The direct sum of two vector bundles with parabolic structures has an obvious parabolic structure. Evidently PVect(X,D) is closed under the operation of taking direct sum. We remark that the category PVect(X,D) is an additive tensor category with the direct sum and the parabolic tensor product operation. It is straight–forward to check that PVect(X,D) is also closed under the operation of taking the parabolic dual deﬁned in [?]. For an integer N ≥ 2, let PVect(X,D,N) ⊆ PVect(X,D) denote the subcategory consisting of all parabolic vector bundles all of whose parabolic weights are multiples of 1/N. It is straight–forward to check that PVect(X,D,N) is closed under all the above operations, namely parabolic tensor product, direct sum and taking the parabolic dual.

2.2.2 The Kawamata Covering lemma The “Covering Lemma” of Y. Kawamata (Theorem 1.1.1 of [?], Theorem 17 of [?]) says that there is a connected smooth projective variety Y over C

21 and a Galois covering morphism

p : Y −→ X (2.4)

∗ such that the reduced divisor D˜ := (p D)red is a normal crossing divisor on ∗ ∗ Y and furthermore, p Di = kiN.(p Di)red, where ki, 1 ≤ i ≤ c, are positive integers. Let Γ denote the Galois group for the covering map p.

2.2.3 The category of Γ–bundles Let Γ ⊆ Aut(Y ) be a ﬁnite subgroup of the group of automorphisms of a connected smooth projective variety Y/C. The natural action of Γ on Y is encoded in a morphism

µ : Γ × Y −→ Y

Denote the projection of Γ × Y to Y by p2. The projection of Γ × Γ × Y to the i–th factor will be denoted by qi. AΓ–linearized vector bundle on Y is a vector bundle V over Y together with an isomorphism

∗ ∗ λ : p2V −→ µ V over Γ × Y such that the following diagram of vector bundles over Γ × Γ × Y is commutative: (q ,q )∗λ ∗ 2 3 - ∗ q3V (µ ◦ (q2,q3)) V @ @ ∗ @ ∗ (m × IdY ) λ @ (IdΓ × µ) λ @ @R ? ∗ (µ ◦ (m,IdY )) V where m is the multiplication operation on Γ. The above deﬁnition of Γ–linearization is equivalent to giving isomorphisms of vector bundles

g¯ : V −→ (g−1)∗V for all g ∈ Γ, satisfying the condition that gh =g ¯ ◦ h¯ for any g, h ∈ Γ. AΓ–homomorphism between two Γ–linearized vector bundles is a homomorphism between the two underlying vector bundles which commutes with the Γ–linearizations. Clearly the tensor product of two Γ–linearized vector

22 bundles admits a natural Γ–linearization; so does the dual of a Γ–linearized vector bundle. Let VectΓ(Y ) denote the additive tensor category of Γ– linearized vector bundles on Y with morphisms being Γ–homomorphisms. As before, VectΓ(Y ) denotes the category of all Γ–linearized vector bundles on Y . The isotropy group of any point y ∈ Y , for the action of Γ on Y , will be denoted by Γy.

2.2.4 On local types of Γ–bundles Recall that since the Γ–action on Y is properly discontinuous, for each y ∈ Y , if Γy is the isotropy subgroup at y, then there exists an analytic neighbourhood Uy ⊂ Y of y which is Γy-invariant and such that for each g ∈ G, g · Uy ∩ Uy 6= ∅.

Deﬁnition 2.2.5 Let ρ be a representation of Γ in GL(r, C). Then Γ–acts on the trivial bundle Y × Cr by (y,v) −→ (γy,ρ(γ)v),y ∈ Y,v ∈ Cr,γ ∈ Γ. Following [?] we call this Γ–bundle, the Γ–bundle associated to the representation ρ.

We then have the following equivariant local trivialisation lemma.

Lemma 2.2.6 Let E be a Γ–bundle on Y of rank r. Let y ∈ Y and let Γy be the isotropy subgroup of Γ at y. Then there exists a Γy-invariant analytic neighbourhood Uy of y such that the Γy–bundle E|Uy is associated to a representation Γy → GL(r) (in the sense of Def 2.2.5).

Remark 2.2.7 The above Lemma for Γ–bundles with structure group GL(r) can be found in [?, Remark 2, page 162] and [?]. Here the key property that is used is that Uy and Uy/Γy are Stein spaces. This result, for the more general setting of arbitrary compact groups K instead of Γ and for general structure groups can be found in [?, Section 11].

Γ–bundles of ﬁxed local type We make some general observations on the local structure of Γ–bundles on the Kawamata cover deﬁned in (2.2.3). D Let VectΓ (Y,N) denote the subcategory of VectΓ(Y ) consisting of all Γ– linearized vector bundles W over Y satisfying the following two conditions:

∗ 1. for a general point y of an irreducible component of (p Di)red, the isotropy subgroup Γy is cyclic of order |Γy| = ny which is a divisor of

23 N; the action of the isotropy group Γy on the ﬁber Wy is of order N, which is equivalent to the condition that for any g ∈ Γy, the action of N g on Wy is the trivial action;

2. In fact, the action is given by a representation ρy of Γy given as follows:

α1 z .I1 0  .  ρ (ζ)= (2.5) y .    0 zαl .I   l  where

• ζ is a generator of the group Γy and whose order ny divides N mj • αi = N and • Ij is the identity matrix of order rj, where rj is the multiplicity of the weight αj.

• z is an ny-th root of unity.

• We have the relation 0 ≤ m1 < m2 <...

∗ 4. For a special point y contained in (p D)red, the isotropy subgroup Γy contains the cyclic group Γn of order n determined by the irreducible component containing y. By the rigidity of representations of ﬁnite groups, the Γy–module structure on Wy (given by Lemma 2.2.6) when restricted to Γn ⊂ Γy is of type τ. 5. At special points y of the ramiﬁcation divisor for p not contained in ∗ (p D)red, the restriction of the representation to the generic isotropy is trivial.

Deﬁnition 2.2.8 Following Seshadri [?, page 161] we call the Γ–bundles E D in VectΓ (Y,N) bundles of ﬁxed local orbifold type τ.

Remark 2.2.9 The reason for calling it local type τ is that, for a Γ–bundle and a point y the generic point of a divisor as above, the structure of the representation deﬁnes the bundle EU for a Γy-invariant analytic neighbourhood

24 in Y . Seshadri denoted the collection of representations of the cyclic groups which deﬁne the local isomorphism type over an analytic neighbourhood by the letter τ; note that the Γ–bundle deﬁnes what is known as an orbifold bundle.

Remark 2.2.10 We remark that this definition of Γ–bundles of fixed local type easily extends to Γ–torsion–free sheaves since the local action is specified only at the generic points of the ramification divisor.

D We note that VectΓ (Y,N) is also an additive tensor category.

Parabolic bundles and Γ–bundles In [?] an identification between the objects of PVect(X,D,N) and the ob- D jects of VectΓ (Y,N) has been constructed. Given a Γ–homomorphism between two Γ–linearized vector bundles, there is a naturally associated homomorphisms between the corresponding vector bundles, and this identifies, in a bijective fashion, the space of all Γ–homomorphisms between two objects of D VectΓ (Y,N) and the space of all homomorphisms between the corresponding objects of PVect(X,D,N). An equivalence between the two additive tensor D categories, namely PVect(X,D,N) and VectΓ (Y,N), is obtained this way. Since the description of this identification is already given in [?], and [?], it will not be repeated here. We observe that an earlier assertion that the parabolic tensor product operation enjoys all the abstract properties of the usual tensor product operation of vector bundles, is a consequence of the fact that the above equivalence of categories indeed preserves the tensor product operation. The above equivalence of categories has the further property that it takes the parabolic dual of a parabolic vector bundle to the usual dual of the corresponding Γ–linearized vector bundle. D Let W ∈ VectΓ (Y,N) be the Γ–linearized vector bundle of rank n on Y that corresponds to the given parabolic vector bundle E∗. The fiber bundle

π : P −→ Y whose ﬁber π−1(y) is the space of all C–linear isomorphisms from Cn to the ﬁber Wy, has a the structure of a (Γ,GL(n, C))–bundle over Y .

Deﬁnition 2.2.11 A Γ-linearized vector bundle E over Y is called Γ-semistable (resp. Γ-stable) if for any proper nonzero coherent subsheaf F ⊂ E, invariant under the action of Γ and with E/F being torsionfree, the following

25 inequality is valid: µ(F ) ≤ µ(E) (resp.µ(F ) <µ(E)) (2.6) where the slope is as usual µ(E) = deg(E)/r and deg(E) is computed with respect to the Γ–linearised very ample divisor Θ on Y . The Γ-linearized vector bundle E is called Γ-polystable if it is a direct sum of Γ-stable vector bundles of same slope.

Remark 2.2.12 The above correspondence between parabolic bundles on X and Γ–bundles on Y preserves the semistable (resp. stable) objects as well, where parabolic semistability is as in (2.3). (cf [?])

Remark 2.2.13 We remark that it is not hard to check that for Γ–bundles, Γ–semistability (resp. Γ–polystability) is the same as usual semistability (resp. polystability). This can be seen from the fact that the top term of the Harder-Narasimhan ﬁltration (resp. the socle) are canonical and hence invariant under the action of Γ. But we note that a Γ–stable bundle need not be Γ–stable, as can be seen by taking a direct sum of Γ-translates of a line bundle.

Remark 2.2.14 We make some key observations in this remark where we also note the essential nature of assumptions of characteristic zero base fields. 1. The notion of Γ–cohomology for Γ–sheaves on Y has been constructed and dealt with in great detail in [?]. These can be realised as higher derived functors of the Γ–fixed points–sub-functor (H0)Γ of the section functor H0. (We use this notation to avoid ΓΓ, because we have denoted the finite group by the letter Γ!). We note immediately that since we work over fields of characteristic zero, the sub-functor (H0)Γ ⊂ H0 is in fact a direct summand (by av- eraging operation). Hence, we see immediately that the higher derived functors of the functor(H0)Γ are all sub objects of the derived functors of H0. 2. When we work with a Kawamata cover as in our case, then we have the following relation between the Γ–cohomology and the usual cohomology on Y/Γ= X:

i i Γ HΓ(Y, F)= H (X,p∗ (F)) ∀i.

26 Γ–bundles and orbifold bundles We make a few general remarks on the advantages of working with a Kawa- mata cover Y and Γ–bundles on Y over working with orbifold bundles or V –bundles over V –manifolds. Locally, these two notions can be completely identified but for any global construction such as the one which we intend doing, namely a moduli construction, working with a Kawamata cover al- beit non-canonical, has obvious advantages since it immediately allows us to work with a certain “Quot” scheme over Y . To recover the moduli of parabolic bundles with fixed quasi parabolic structure, we then simply use the functorial equivalence of parabolic bundles and Γ–bundles of fixed local type.

Γ–line bundles and parabolic line bundles A Γ line bundle on Y is a line bundle L on Y together with a lift of action Γ . The Γ line bundle gives a Γ invariant line bundle LΓ on X. Let D be a d divisor of normal crossing on X. Let D = i=1 Di be a decomposition into irreducible components. A parabolic line bundleP on (X,D) is a pair of the form (M,β1,...,βi,...,βd) where M is a holomorphic line bundle on X and 0 ≤ βi < 1 is a real number. When we start from a Γ line bundle on Y we Γ get a pair (L ,β1,...,βi,...,βd) where βi is a rational number and it can be ∗ written as βi = mi/N. Let D˜i = (p Di)red. Then by following [?, Section ∗ Γ d ˜ 2b] we have L = p (L ) ⊗OY ( i=1 kimiDi) P Remark 2.2.15 In our situation, by choice we work with a single weight when we consider Γ–line bundles of ﬁxed local type τ although this may not be absolutely essential.

Serre duality for Γ–line bundles of fixed local type Definition 2.2.16 By a line bundle L of fixed local type τ we mean a parabolic line bundle (L,α1,α2,...,αd), where αi = α∀i. In other words, locally, the generic isotropy on the irreducible components of the inverse image of the parabolic divisor acts by a single character namely α. We will write L(α) to specify the character.

Let L = L(α) be a Γ line bundle on Y of type τ. Then by 2.2.4, one ∗ Γ ˜ knows that L = p (p∗ (L))⊗OY ( kimiDi) where all the mi can be assumed to be equal to m since we haveP a single weight α. Then if M = M (α) is another Γ–line bundle with the same local character type we have:

27 Γ ∗ Γ ∗ Γ (p∗ (L ⊗ M))=(p∗ (L) ⊗ (p∗ (M)) (2.7)

(α) Consider the canonical bundles KX of X and deﬁne the Γ–bundle KY as follows: (α) ∗ ˜ KY = p (KX ) ⊗OY ( kiDi)m) (2.8) X Γ (α) Then, we see as above that p∗ (KY ) = KX . We then have the following duality for Γ–line bundles of type τ:

Lemma 2.2.17 For Γ–line bundles L of type τ, with local character α, (α) the Γ–line bundle KY is the dualising sheaf. In other words, we have a canonical isomorphism:

i ∗ (α) n−i ∗ HΓ(Y,L ⊗ KY ) ≃ HΓ (Y,L) for all i. We have made this statement for Γ–varieties Y of any dimension.

Proof: The proof is straightforward, but we give it for the sake of completeness. Recall the relationship between the Γ–cohomology on Y and the usual cohomology on X (Remark 2.2.14). We have the following isomorphism (using 2.7):

i ∗ (α) i Γ ∗ (α) i Γ ∗ Γ (α) HΓ(Y,L ⊗ KY ) ≃ H (X,p∗ (L ⊗ KY ) ≃ H (X,p∗ (L) ⊗ (p∗ (KY ))

Γ (α) Using p∗ (KY )= KX we then conclude from the following isomorphism: i Γ ∗ n−i Γ ∗ n−i ∗ ≃ H (X,p∗ (L) ⊗ KX ) ≃ H (X,p∗ (L)) ≃ HΓ (Y,L) where we use the usual Serre duality on X. 2

Before we give a short sketch of our proof we would like to recall two lemmas which proves nonemptiness of rank 2 stable bundles over any surface. Let X be a smooth projective surface and H be an ample divisor in X. We can certainly ask the following question: Given two line bundle L, M and a reduced complete intersection subvariety Z inside X; does there exist a stable vector bundle (locally free sheaf) E which is an extension of the form:

0 −→ L −→ E −→ M ⊗IZ −→ 0

28 To answer this question one breaks up the above question into two parts:

1. When E is locally free?

2. When it is µ-stable with respect to some very ample divisor H?

To answer the ﬁrst question one uses Cayley-Bachrach property. This idea came up in [13]. Following two lemmas give answers to above two questions respectively.

Lemma 2.2.18 There exists an extension

0 −→ L −→ E −→ M ⊗IZ −→ 0

∗ such that E is locally free iﬀ the pair (L ⊗ M ⊗ KX ,Z) has the Cayley- Bachrach(CB) property: (CB) If Z′ ⊂ Z is a subscheme with l(Z′) = l(Z) − 1 and s ∈ H0(X,L∗ ⊗ M ⊗ KX ) with s |Z′ = 0, then s |Z = 0

Lemma 2.2.19 Let Q ∈ Pic(X) be a line bundle. Then there is a constant c0 such that for all c ≥ c0 there exists a H-stable rank 2 vector bundle E which comes as extension 2.2.4 with det(E) ≃ Q and c2(E)= c.

2.3 Qin-Li’s proof for rank r case

Lemma 2.3.1 Let L, Mi’s, i = 1 to r − 1 are r line bundles over X and Zi’s, i = 1 to r − 1 are r − 1 reduced complete intersection subvarieties of codimension 2 such that Zi ∩ Zj = Ø, for i 6= j. We also assume that that ∗ each pair (L ⊗ Mi ⊗ KX ,Zi) satisﬁes CB property. Then there exists a locally free extension of the following form

r−1 0 −→ L −→ E −→ i=1 Mi ⊗IZi −→ 0 L Theorem 2.3.2 [19] For any ample divisor H on X, there exists a constant α depending on X,r,c1 and L such that for any c2 ≥ α, there exists a H- stable rank r bundle with chern classes c1 and c2 respectively.

Remark 2.3.3 Note that the above theorem is not a full generalisation of Gieseker-Taubes version ([10],

29 Chapter 3

Harder-Narasimhan reduction for Higgs G bundle

3.1 Proof of existence of a Higgs H-N reduction

For a Higgs G bundle (E,θ) on X (following [16]) we deﬁne an integer dE as follows:

∗ dE = min{deg iP ∗σ E(g/p):(P,σ,θP ,UP ) is a Higgs reduction }

∗ where, iP : UP →X is the inclusion. Since iP ∗σ E(g/p) is a quotient of a fixed bundle E(g) on X, the integer dE is well defined. Here we use the fact that the degrees of quotients of a fixed torsion-free sheaf is bounded from below.

Proposition 3.1.1 Let (P,σ,θP ,UP ) be a Higgs reduction of structure group of E to P such that the following conditions hold:

∗ 1. deg(σ E(g/p)) = dE 2. P is a maximal among all parabolic subgroups P ′ satisfying the condi- ′ ′ ′ tion that there is a Higgs reduction (P ,σ ,θP ′ ) of E to P such that ′∗ degree(σ E(g/p)) = dE.

Then the Higgs reduction (P,σ,θP ,UP ) is a Higgs H-N reduction.

Proof: Let (P,σ,θP ,UP ) be a Higgs reduction of E to P satisfying the properties stated in the proposition, i.e we have a Higgs reduction (P,σ,θP )

30 of E to P on the open subset UP of X. Let U be the unipotent part of P . First note that EL as a L bundle on UP gets a Higgs structure induced by the surjective map P −→ L −→ 0. We have, ρ : EP (p) −→ EP (l) induced by the above map, where p and l are Lie algebras of P and L respectively. So we deﬁne θ := (ρ ⊗ id) ◦ θ where ρ ⊗ id : E (p) ⊗ Ω1 −→ E (l) ⊗ Ω1 L P P UP P UP and θ : O −→ E (p) ⊗ Ω1 . P UP P UP We ﬁrst show that the associated Levi bundle EL is Higgs semistable on UP . Suppose that EL is not a Higgs semistable L bundle on UP . Therefore, there ′ is a Higgs reduction (Q,τQ,θQ,U ) of EL to a parabolic subgroup Q of L such that,

∗ ∗ deg(τ TEL/Q) = deg τ EL(l/q) < 0 (3.1)

′ ′ where, U is the open subset of X with codim(UP \U ) ≥ 2 which implies that codim(X\U ′) ≥ 2. It is easy to see that the inverse image of Q under the projection P −→ L −→ 0 which will be denoted as P1, is a parabolic subgroup of G. (Since, G/P1 −→ G/P is a fiber bundle with fiber P/P1 =∼ L/Q, hence P/P1 is complete. This implies that G/P1 is complete.) Since (Q,τQ) is a reduction of structure group of EL to Q given by the section ′ ′ τQ : U −→ EP (L/Q) we have a section σ1 : U −→ EP (P/P1). So we have ′ (P1,σ1) a reduction of structure group of EP to P1 on U open subset of UP , of X with codim(X\U ′) ≥ 2. From now onwards we fix this open subset U ′ of X. Note that we have two short exact sequences of P and P1 modules respectively.

0 −→ u −→ p −→ l −→ 0 (3.2)

0 −→ u −→ p1 −→ q −→ 0 (3.3)

′ where u, p, p1, q are Lie algebras of U,P ,P1 and Q respectively. On U we have, 0 −→ EP1 (p1) −→ EP (p) and 0 −→ EP1 (q) −→ EP (l). But we know that EP (l)= EL(l) and EP (u)= EP1 (u) The above exact sequences therefore gives the following commutative diagram of exact sequences.

0 0 0 ↓ ↓ ↓

0 −→ EP1 (u) −→ EP1 (p1) −→ EP1 (q) −→ 0 ↓ ↓ ↓ 0 −→ EP (u) −→ EP (p) −→ EP (l) −→ 0

31 This results in the following commutative diagram of exact sequences.

g 0 0 1 0 1 g 0 1 1 1 1 → H (U, EP1 (u) ⊗ Ω ) → H (U, EP1 (p1) ⊗ Ω ) → H (U, EP1 (q) ⊗ Ω ) → H (U, EP1 (u) ⊗ Ω ) ↓ ↓ k i j k 0 1 0 1 f 0 1 f1 1 1 0 → H (U, EP (u) ⊗ Ω ) → H (U, EP (p) ⊗ Ω ) → H (U, EP (l) ⊗ Ω ) → H (U, EP (u) ⊗ Ω )

0 ′ From the above diagram it follows that there exists θP1 ∈ H (U ,EP1 (p1) ⊗ 1 ΩU ′ ) such that i(θP1 )= θP .

Therefore we have got a Higgs structure (P1,σ1,θP1 ) induced by Higgs ′ structure (Q,τQ,θQ) on U . But by the above diagram it is clear that this structure is compatible with the Higgs structure induced by (P,σ,θP ) i.e, we have the commutative diagram;

θP 1 OU ′ −→ EP (p) ⊗ ΩU ′ θP1 ց ↑ 1 EP1 (p1) ⊗ ΩU ′ ↑ 0

Let p1, p, l denote the Lie algebras of P1,P,L respectively. We have the following exact sequence of P1 modules

0 −→ p/p1 −→ g/p1 −→ g/p −→ 0 (3.4)

SinceP/P1 =∼ L/Q we get p/p1 =∼ l/q as P1 modules.

Since EP1 is a Higgs reduction of E to P1, we consider the vector bundles associated to EP1 using the P1 modules in (3.4). We have the following exact sequence of vector bundles on U ′.

∗ ∗ ∗ 0 −→ τ TEL/Q −→ σ TE/P1 −→ σ TE/P −→ 0 (3.5)

Using (3.1) we conclude that

∗ ∗ ∗ ∗ deg(σ1TE/P1 ) = deg(σ TE/P ) + deg(τ TEL/Q) < deg(σ TE/P )= dE (3.6)

This contradicts the assumption on dE. Therefore, EL must be Higgs semistable. Now we need to check the second condition in the definition of the Higgs H-N reduction. Let B be a Borel subgroup of G contained in P and T ⊂ B be the maximal torus. Let ∆ be the system of simple roots. Let I denote the set of simple roots defining the parabolic subgroup P . So I defines the

32 roots of the Levi factor L of P . Take any dominant character χ of P whose restriction to T is expressed as

χ |T = cαα + cββ (3.7) α∈X∆−I Xβ∈I

with cα,cβ ≥ 0. Let Z0(L) ⊂ T be the connected component of the center of L. The characters β ∈ I of T have the property that β |Z0(L) is trivial. Hence we see that if χ is a nontrivial character of P of the above form then cα > 0 for some α ∈ ∆ − I. Also, some positive multiple of a character of Z0(L) extends to character of L. Hence there are positive integers nα for each α ∈ ∆ − I such that nαα |Z0(L) extends to a character χα on L. Now we see that χα |T can be written as nαα + β∈I nβ,αβ for some integers nβ,α. Let N = α∈∆P−I nα, then we have the following equality. Q

′ ′ Nχ |T = Ncαα + Ncββ = cαχα |T + cββ (3.8) α∈X∆−I Xβ∈I α∈X∆−I Xβ∈I

′ ′ ′ for some integer cβ,β ∈ I and cα where α ∈ ∆ − I such that cα is ′ a positive multiple of cα. Hence Nχ − α∈∆−I cαχα is a character of L 0 ′ whose restriction to Z (L) is trivial. Thus,P Nχ = α∈∆−I cαχα. Hence it is enough to prove the second condition for the charactersP of the form χα, with α ∈ ∆ − I. Fix an element α ∈ ∆ − I. Let P2 ⊃ P be the parabolic subgroup of G ′ defined by the subset I2 = {α}∪ I of ∆. Let L2 be its Levi factor and P be the maximal parabolic subgroup of L2 defined by the image of P in L2. Consider the group of all characters of P ′ which are trivial on the center ′ of L2. This group is generated by a single dominant character ω of P with respect to the root system of L2 defined by its maximal torus T . ′ Now χα defines a character on P which is trivial on center of L. Hence the restriction of χα to T can be written as χα |T = mω |T for some integer m. This enables us to write

mω |T = nαα + nβ,αβ. Xβ∈I

33 A dominant weight is always a nonnegative rational linear combination of simple roots. The fact that nα > 0 implies that m> 0 and nβ,α ≥ 0. Hence χα is a positive multiple of ω. To verify the second condition of Higgs H-N reduction, we consider the i surjective map P2 −→ L2 −→ 0 and the injective map P ֒→ P2. We note that ∼ ′ P2/P = L2/P . Let EP2 be an extension of structure group of EP to P2, that is EP (P2)= EP2 on UP . So EP is a reduction of structure group of EP2 to

P on UP . It is easy to see that EP (G)= EP2 (G) and P2 ֒→ G gives rise to ρ′ ((a reduction (P2,σ2)of E on UP . Also we have EP (p) ֒→ EP2 (p2)=(EP (p2 induced by i. ′ Now we deﬁne θP2 : =(ρ ⊗ id) ◦ θP . where (ρ′ ⊗ id): E (p) ⊗ Ω1 ֒→ E (p ) ⊗ Ω1 P UP P 2 UP Since the following diagram commutes.

θP O −→2 E (p ) ⊗ Ω1 U P 2 UP θP ց ↑ E (p) ⊗ Ω1 P UP ↑ 0 we have a Higgs structure θP2 on EP2 and hence Higgs reduction (EP2 ,σ2,θP2 ) on UP . As in the ﬁrst part of the proposition EL2 = EP2 (L2) is a Higgs L2 bundle. Using the following exact sequence of P modules,

0 −→ p2/p −→ g/p −→ g/p2 −→ 0 (3.9) we have the following exact sequence of vector bundles on UP .

′∗ ∗ ∗ 0 −→ σ T ′ −→ σ T −→ σ2 T −→ 0 (3.10) EL2 /P E/P E/P2

′ ′ where σ is the Higgs reduction of the structure group of EL2 to P which ′ ′ ′ is EP (P ) for the obvious projection of P to P . Here EP (P ) gets Higgs structure in the following way. ′ ′ Let f : EP (p) −→ EP (p ) be induced by the projection P → P . Deﬁne θP ′ : =(f ⊗ id) ◦ θP . The section θP ′ is the required Higgs structure on ′ EP (P ). Also note that the following diagram commutes.

34 θL O −→2 E (l ) ⊗ Ω1 UP L2 2 UP ′ θP ց ↑ E (p′) ⊗ Ω1 P UP ↑ 0 From the above exact sequence we have

′∗ ∗ ∗ deg(σ T ′ ) = deg(σ T ) − deg(σ2 T ). (3.11) EL2 /P E/P E/P2

∗ ′∗ Now the assumption that deg(σ T )= dE gives the inequality deg(σ T ′ ) < E/P EL2 /P ′∗ ′ 0. Note that det(σ T ′ ) is the line bundle associated to the P -bundle EL2 /P ′ ′ EP (P ), for the character of P which is a negative multiple of ω. Hence ∗ it follows from the above observation that some positive powers of Lχα and ′∗ det(σ T ′ ) coincide, proving that deg Lχ > 0. Therefore, the second EL2 /P α condition of the deﬁnition of the canonical Higgs reduction holds. This completes the proof of the proposition . This proposition establishes the existence of a Higgs canonical reduction.

3.2 The Higgs structure on the adjoint bundle EG(g)

Let (E,θ) be a principal Higgs G-bundle on X, let EG(g) denote its associated vector bundle for the adjoint representation Ad : G −→ Gl(g). That gives rise to a morphism of Lie algebras ad : g −→ End(g) = g ⊗ g∗. ∗ So we get a morphism of vector bundles φ : EG(g) −→ EG(g) ⊗ EG(g) . ∗ 1 We deﬁne ad(θ)=(φ ⊗ id) ◦ θ : OX −→ EG(g) ⊗ EG(g) ⊗ ΩX which gives Higgs structure on EG(g), note that ad(θ) induces a homomorphism ′ 1 θ : EG(g) −→ EG(g) ⊗ ΩX of vector bundles on X. Here we prove the following proposition only for semiharmonic Higgs principal G bundles i.e. for Higgs principal bundles whose all chern classes are vanishing. Therefore, the uniqueness of the H-N reduction holds only for semiharmonic Higgs principal bundles which we prove in the following section (c.f.§5)

Proposition 3.2.1 A Higgs principal G bundle (E,θ) on X is Higgs semistable iﬀ the associated Higgs vector bundle (EG(g),ad(θ)) is Higgs semistable.

Proof: The proof of this proposition goes along the same lines as that of [?]. First assuming that (EG(g),ad(θ)) is a Higgs semistable vector bundle,

35 we need to prove that (E,θ) is a Higgs semistable principal G bundle. Sup- pose not, then there exists an open subset U, with codim(X\U) ≥ 2 and a maximal parabolic subgroup P of G and a Higgs reduction (P,σ,θP ,U) ∗ such that deg(σ (TE/P )) < 0 (by Lemma 2.1.5), where TE/P is the relative tangent bundle for the projection E −→ E/P . Let E0 = EP (p) ⊂ EG(g) be the subbundle given by the adjoint bundle of the P bundle EP given by the ∗ reduction σ then σ TE/P = EG(g)/E0. Also (EP (p),ad(θP )) is a Higgs subbundle of (EG(g),ad(θ)). Since G is reductive, using a bilinear form which ∗ is non degenerate on g, we have the identiﬁcation EG(g) = EG(g) which implies that deg(EG(g))=0. From the exact sequence of vector bundles on U

∗ 0 −→ E0 −→ EG(g) −→ σ TE/P −→ 0 (3.12)

∗ we conclude, deg(EG(g)) = deg(E0) + deg(σ TE/P ). This implies that ∗ deg(E0) > 0 (since deg(σ TE/P ) < 0, µ(E0) > µ(EG(g)) which contradicts the fact that (EG(g),ad(θ)) is Higgs semistable. Now we assume that (E,θ) is Higgs semistable then we will prove that (EG(g),ad(θ)) is a Higgs semistable vector bundle. Suppose that EG(g) is not a Higgs semistable vector bundle on X and let

0= E0 ⊂ E1 ⊂ E2 ···⊂ Ek−1 ⊂ Ek = EG(g) (3.13) be the Higgs H-N filtration of the Higgs vector bundle EG(g) (lemma 3.1. [7]). For the sake of notational convenience, we denote EG(g) by V and we have the Higgs structure θ′ induced by ad(θ) on V . Note that we can choose an open subset U of X such that it contains all points of codimension one (since codim(X\U) ≥ 2) and every sheaf Ei is locally free on U. We fix this ⊥ U. For any x ∈ U we consider Ej,x = {v ∈ Vx| < v,Ej,x >= 0}, where <,> ∼ ⊥ is the non-degenerate bilinear form invariant under G on Vx = g. Let Ej be the kernel of the surjection V −→ V ∗ (defined by the form) followed by ∗ ∗ canonical map V −→ Ej . ⊥ ∼ ∗ ⊥ We claim that Ej = (V/Ej) over U. Let v ∈ Ej,x, then define f : ′ ′ Vx −→ k by f(v ) =< v,v >. Since f(w) =< v,w >= 0 for all w ∈ Ej,x. ¯ ⊥ ∗ So f induces a map f : Vx/Ej,x −→ k. Thus Ej,x ֒→ (Vx/Ej,x) , but ⊥ ∗ ⊥ ∼ ∗ dim(Ej,x) = dim(Vx/Ej,x) , therefore Ej = (V/Ej) . ⊥ ∗ ′ 1 We define Wj := Ek−j =(V/Ek−j) . Since θ |Ej maps Ej −→ Ej ⊗ΩX , ¯′ 1 ′ 1 we can define θj :(V/Ej) −→ (V/Ej) ⊗ ΩX induced by θ : V −→ V ⊗ ΩX . ∗ ¯′ ∗ 1 1 ∗ So ((V/Ej) , θj) is a Higgs subbundle of (V ,θ ) where θ : EG(g) −→

36 ∗ 1 ′ 1 EG(g) ⊗ ΩX induced by ad(θ)(since both θ and θ induced by ad(θ) and ∗ ′ (V/Ej) have Higgs structures induced by θ ). Now we claim that Wi/Wi−1 is a Higgs semistable vector bundle and µ(Wi/Wi−1) decreases as i increases. To prove this claim, observe that (E,θ) is Higgs semistable vector bundle if and only if (E∗,θ) is Higgs semistable vector bundle. Now we observe that,

∗ (V/Ek−i) ∗ Wi/Wi−1 = ∗ =(Ek−(i−1)/Ek−i) (3.14) (V/Ek−(i−1))

For, we have the diagrams: 0 ↓ Ek−(i−1)/Ek−i ↓ 0 −→ Ek−i −→ V −→ V/Ek−i −→ 0 k ↓ 0 −→ Ek−(i−1) −→ V −→ V/Ek−(i−1) −→ 0 ↓ 0

∗ ∗ ∗ 0 ←− (Ek−(i−1)/Ek−i) ←− (V/Ek−i) ←− (V/Ek−(i−1)) ←− 0 (3.15)

∗ (V/Ek−i) ∗ ∗ and hence, ∗ = (E /Ek−i) . Thus, Wi/Wi−1 =(E /Ek−i) . (V/Ek−(i−1)) k−(i−1) k−(i−1) But we know that (Ek−(i−1)/Ek−i) is Higgs semistable. Therefore Wi/Wi−1 is Higgs semistable. Note that µ(Wi/Wi−1)= −µ(Ek−(i−1)/Ek−i), µ(Wi+1/Wi)= −µ(Ek−i/Ek−(i+1)). Hence, µ(Wi/Wi−1) decreases as i increases. This completes the proof of the claim. Therefore we obtain a Higgs H-N ﬁltration of V ∗ over U.

∗ 0= W0 ⊂ W1 ⊂ W2...... ⊂ Wk−1 ⊂ Wk = V (3.16) such that each Wi is a Higgs subbundle of V and the quotient is Higgs semistable; further, µ(Wi/Wi−1) decreases as i increases. Now since V =∼ V ∗, we conclude that the filtration of V ∗ over U by the Wj’s coincides with the Higgs H-N filtration of V (by the uniqueness of ⊥ filtration). In other words, we have Ej = Ek−j on U for all 0 ≤ j ≤ k. Therefore, the above filtration is of the following form:

0= E−l−1 ⊂ E−l ⊂ E−l+1 ⊂ .... ⊂ E−1 ⊂ E0 ⊂ E1 ⊂ ...El−1 ⊂ El = V(3.17)

37 where Ej is the orthogonal complement of E−j−1 for the G-invariant form ¡,¿. Let φ : E0 ⊗E0 −→ V/E0, be the composition of the Lie bracket operation with the natural projection V −→ E0. By the Proposition 2.9 in [?] we have, µmin(E0 ⊗ E0) = 2µmin(E0) = 2µ(E0/E−1) and µmax(V/E0)= µ(E1/E0). From the fact that E−1 is the orthogonal part of E0, the form ¡,¿ induces a non-degenerate quadratic form on E0/E−1. Consequently, we have ∗ E0/E−1 =∼ (E0/E−1) which implies that µ(E0/E−1) = 0. We have µmin(E0⊗ E0) = 2µ(E0/E−1) = 0 > µ(E1/E0) = µmax(V/E0). Therefore , it follows 0 that H (X,HomHiggs(E0 ⊗ E0,V/E0)) = 0; in particular, we have φ = 0. In other words, E0 is closed under the Lie bracket. Consider

φj : E−j ⊗ E−1 −→ V/E−j−1 where j ≥ 0 deﬁned using the Lie bracket operation and the projection V −→ V/E−j−1. Repeating the above argument and using the property that µ(Ei/Ei−1) > µ(Ei+1/Ei) we deduce that φj = 0. In other words, we have [E−j,E−1] ⊂ E−j−1 for any j ≥ 0. Using the above inclusion we conclude that E−1 is a nilpotent Lie subalgebra bundle of E0, and E−1,x is also an ideal of E0,x for any x ∈ X. Now by Lemma 2.11 of [?], we see that over the open set U of X, the sub-algebra bundle E0 is a bundle of parabolic sub-algebras, and it gives a reduction σ : U −→ EG/P of the structure group of EG to a parabolic subgroup P of G. Using the above lemma we have a reduction (P,σ). Let us denote the principal P bundle over U obtained in the above lemma by EP and E0 = ′ EP (p). Since E0 is a Higgs subbundle of EG(g) we have, θ |E0 : EP (p) −→ 1 ′ 1 EP (p)⊗ΩU where θ : E(g) → E(g)⊗ΩX induced by ad(θ): OX → EG(g)⊗ ∗ 1 EG(g) ⊗ ΩX . 1 The section θ factors through EP (p) ⊗ ΩU , call it θP . Therefore we conclude that (EP ,σP ,θP ) is a Higgs reduction of E to P on U. Let χ0 be the character on P associated to the action of P on its Lie algebra p. Then it is clear that χ0 is a dominant character of P with respect to the Borel subgroup B. Hence from the Higgs semistability of EG we deduce that ∗ deg(σ EG(χ0)) = deg(EP (p)) = deg(E0) ≤ 0 which contradicts the fact that µ(E0) >µ(E) = 0. This completes the proof of the proposition. Now we will prove the lemma that for every ﬁnite dimensional linear representation of the group G the associated bundle is also Higgs semistable with some constraints. The proof of the lemma goes along the same lines as that of the Lemma 5 of [5]

38 Lemma 3.2.2 A principal Higgs G-bundle (EG,θ) over X is Higgs semistable if and only if for every ﬁnite dimensional linear representation ρ : G → Aut(V ), such that ρ(Z0(G)) is contained in the center Z(Aut(V )), the associated Higgs vector bundle EG(V ) is Higgs semistable. Proof: Take the adjoint representation of G on its Lie algebra g. Now by assumption the associated adjoint bundle is Higgs semistable, therefore by the previous proposition (EG,θ) is Higgs semistable. Conversely, suppose that (EG,θ) is Higgs semistable. By our earlier dis- n cussion EG(V ) gets Higgs section canonically; call it θV . Let V = i=1 Vi be the decomposition of the G module V into irreducible submodules.L top Consider the character Hom(Vi,Vj) of G given by the top exterior top power. By assumption ZV0(G) acts trivially on the line Hom(Vi,Vj). Since G is connected, G is a quotient(semi-direct product)V of Z0(G)×[G,G], where [G,G] denotes commutator subgroup of G. Any character of [G,G] is top trivial. So the action of G on Hom(Vi,Vj) is trivial. Let us denote the Higgs vectorV bundle EG(Vi) by (Wi,θi) associated to EG for the G module Vi. So we have

n n Wi = W, θi = θV (3.18) Mi=1 Mi=1

top By the earlier observation, the line bundle Hom(Wi, Wj), which is as- top sociated to EG for the G module HomV(Vi,Vj) is trivial. Therefore, we have µ(Wi)= µ(Wj). This meansV that W is Higgs semistable iff each Wi is Higgs semistable. So, our lemma is reduced to proving this only for irreducible representa- ∗ tions ρ : G −→ Aut(V ). Let W = EG(V ), we know that End(W)= W⊗W and W = EGL(V )(V ) , then by Lemma 2.1.6, W is Higgs semistable iff EGL(V ) is Higgs semistable. Again by proposition 3.2.1 EGL(V ) is Higgs ∗ ∗ semistable iff EGL(V )(Mn(k)) = EGL(V )(V ⊗ V ) = W⊗W = End(W) is Higgs semistable. Therefore, it is enough to prove that End(W) is Higgs semistable. Since the G module V is irreducible, from Schur’s lemma it follows that the action of the center Z(G) of G on End(V ) is trivial. We note that the adjoint representation of G on its Lie algebra g gives a faithful representation of G/Z(G). The group G/Z(G) is also reductive. Therefore, the ⊗j G/Z(G) module End(V ) is a submodule of the G/Z(G) module j∈S g , where S is a finite collection of nonnegative integers possibly with repetitions.L Therefore, the Higgs vector bundle End(W) is a Higgs subbundle of ⊗j j∈S(EG(g)) . L 39 ⊗j Since (EG(g)) is Higgs semistable of degree zero we have (EG(g)) is Higgs ⊗j semistable of degree zero for every j. Thus, j∈S (EG(g)) is Higgs semistable of degree zero. But End(W) being a HiggsL subbundle of degree zero, is also Higgs semistable. This completes the proof.

Lemma 3.2.3 Suppose (E,θ) is a Higgs principal G-bundle on X. Let (EP ,σP ,θP ,UP ) be a reduction, then EL is Higgs semistable on UP , where L is a Levi factor of P , if and only if i∗EL(l) is a Higgs semistable torsion free sheaf over X.

Proof: Note that EL is Higgs semistable on UP if and only if EL(l) is Higgs semistable on UP (proposition 3.2.1). But the semistability of i∗EL(l) on X is equivalent of the semistability of i∗EL(l)|UP = EL(l) on UP . In conclusion, we have the following theorem.

Theorem 3.2.4 Let EG be the Higgs semistable G bundle. Let ρ : G −→ H be a homomorphism of connected reductive groups such that ρ(Z0(G)) is contained in the connected component of the center of H containing the identity element. Let EH := EG(H) be the Higgs principal H bundle induced by ρ. Then the H bundle EH is also Higgs semistable.

Proof: Let φ : H −→ Aut(V ) be a representation such that φ(Z0(H)) ⊆ Z(Aut(V )), where Z0(H) is the connected component of the center of H containing the identity. By using Lemma 3.2.2 it is enough to prove that EH (V ) is Higgs semistable. consider the composition φ ◦ ρ : G −→ Aut(V ). Then φ ◦ ρ(Z0(G) ⊆ Z(Aut(V )). Since EG is Higgs semistable, we have EG(V ) is Higgs semistable. But EG(V )= EH (V ). This proves the theorem.

3.3 Proof of the uniqueness of Higgs H-N reduction

Theorem 3.3.1 Any Higgs G bundle admits a unique H-N Higgs reduction. In other words, if we ﬁx a Borel subgroup B of G, then there is an unique H-N Higgs reduction to a parabolic subgroup P containing B.

Proof: The existence of H-N Higgs reduction is proved in §2. So, it is enough to prove the uniqueness.

Let (P1,σP1 ,θP1 ,UP1 ) and (P2,σP2 ,θP2 ,UP2 ) (where P1 and P2 are two parabolic subgroups containing B) be two H-N Higgs reductions of the Higgs principal G bundle (E,θ). Note that we can choose open subset U ⊆ X with

40 codim(X\U) ≥ 2 such that, the above two Higgs reductions exist on this U. That is we have two H-N Higgs reductions of E on U. We will prove that the two Higgs subbundles EP1 (p1) andEP2 (p2) of EG(g) are the same on U. Equivalently we show i∗(EP1 (p1)) = i∗(EP2 (p2)) on X, where i is the inclusion of U in X. And these are reﬂexive sheaves occurs in the H-N. ﬁltration of E(g). For proving the above stated result we need the following lemma.

Lemma 3.3.2 Let (E,θ) be a principal G bundle on a curve and (H1,σH1 ), (H2,σH2 ) be two reductions of E. Suppose the two subbundles EH1 (h1) and EH2 (h2) of −1 E(g) coincide. Then there exists an element g ∈ G such that H1 = gH2g . Moreover, if the normalizer of H1 in G is H1 itself then the two reductions σ∗ E and σ∗ E are the same. H1 H2

The above lemma is easily seen to hold for nonsingular projective varieties. By using this lemma we conclude that (P1,σP1 )=(P2,σP2 )(since B is ﬁxed and normalizer of a parabolic subgroup is itself). Then by the deﬁni- tions of θP1 ,θP2 we arrive at the fact that (P1,σP1 ,θP1 )=(P2,σP2 ,θP2 ). So now our aim is to prove that the two Higgs subbundles EP1 (p1) and EP2 (p2) of EG(g) are the same. To prove this fact we need to prove one Lemma.

1 2 Lemma 3.3.3 Let (V ,θ1) and (V ,θ2) be two Higgs torsion-free coherent sheaves on X with Higgs ﬁltrations

1 0 = V0 ⊂ V1 ⊂ V2 ⊂·········⊂ Vk−1 ⊂ Vk = V (3.19) ′ ′ ′ ′ 2 0 = V0 ⊂ V1 ⊂ V2 ⊂·········⊂ Vl−1 ⊂ Vl = V (3.20)

′ ′ where Vi/Vi−1 and Vj /Vj−1 are Higgs semistable with deg(Vi/Vi−1) ≥ 0 and ′ ′ 1 2 deg(Vj /Vj−1) < 0 for 0 ≤ i ≤ k and 0 ≤ j ≤ l. Then HomHiggs(V ,V )= 0.

Proof: Let φ := V 1 −→ V 2 be a homomorphism of Higgs torsion-free sheaves on X with the above conditions. We need to prove that φ = 0. We choose an open set U of X with codim(X\U) ≥ 2 such that all the terms in the above two ﬁltrations are locally free on U. So it is enough to prove that 1 2 HomHiggs(V ,V ) = 0 of Higgs vector bundles on U. To prove this fact we use the induction on the number of terms in the ﬁltrations i.e. on k and l. 1 2 Let k = l = 1. Then (V ,θ1), (V ,θ2) are Higgs semistable. So we have, µ(V 1) ≤ µ(φ(V 1)) ≤ µ(V 2). But deg(V 1) ≥ 0 and deg(V 2) < 0 by hypothesis, so, µ(V 1) >µ(V 2). This leads to a contradiction.

41 We will prove the lemma for any k and l = 1. Assume the lemma for k′

i π 0 −→ Vk−1 −→ Vk −→ Vk/Vk−1 −→ 0

φ|Vk−1 ց ↓ φ ւ φ1 V 2

So, φ ◦ i = φ |Vk−1 = 0. (by the induction on k) Therefore we have 2 φ1 : Vk/Vk−1 −→ V induced by φ such that φ = φ1 ◦ π. Again by induction on k we have φ1 = 0. This forces φ to be zero. So we have the lemma for k and l = 1. Now assume the lemma for all k and l′ < l. Then we consider the following,

′ ′ ′ i ′ π ′ ′ 0 −→ Vl−1 −→ Vl −→ Vl /Vl−1 −→ 0 ↑ φ ր π′ ◦ φ V 1

′ ′ we get π ◦ φ = 0, that is im(φ) ⊂ Vl−1. 1 ′ So, φ : V −→ Vl−1, this implies that φ = 0. Thus, we have proved the lemma for all k and all l. Let (P,σP ,θP ,UP ) be a H-N Higgs reduction of structure group of E to P , we denote this bundle by EP . The adjoint action of P on p preserves the sub-algebra u. The Higgs L bundle EP (L), will be denoted by EL. By the definition of H-N Higgs reduction, EL is Higgs semistable. The Higgs vector bundle associated to EP for the adjoint representation of P on p and g/p are naturally identified with adjoint bundle of EP and ∗ σ TE/P respectively. Consider the filtrations of P modules

0 = V0 ⊂ V1 ⊂ V2 ⊂·········⊂ Vk−1 ⊂ Vk = u (3.21) ′ ′ ′ ′ ′ 0 = V0 ⊂ V1 ⊂ V2 ⊂·········⊂ Vm−1 ⊂ Vm = g/p (3.22)

′ ′ ′ such that the quotients Wi : = Vi/Vi−1 and Wj : = Vj /Vj−1 are all irreducible P modules. ′ It follows that the action of U on Wi( respWj) is trivial. Therefore, the ′ ′ action of P on Wi(respWj) factors through the quotient L. Let Vj, (respVj) denote the Higgs vector bundle over X associated to the EP for the P -module ′ ′ Vj(respVj ). Since Wi and Wj are all irreducible L-modules and EL is Higgs ′ ′ semistable, Wi : = EL(Wi) and Wi : = EL(Wi ) are Higgs semistable. This result follows by Lemma 3.2.2. Let B ⊂ P and T a maximal torus in B. Let ∆ denote the set of simple roots for G. Let I ⊂ ∆ denote the set of simple

42 roots deﬁning the parabolic subgroup P . The weights of T on g/p are of the form

γ = cαα (3.23) αX∈∆ with cα ≤ 0 and cα < 0 for at least one α ∈ ∆ − I. The weights of T on u are of the form −γ, where γ is a weight on g/p. From this it follows that the character of P deﬁned by the determinant ′ of the representation of P on Wj (resp, Wj) is non-trivial and is a nonnegative (resp. nonpositive) linear combination of roots in ∆. Now by the second condition in the deﬁnition of a Higgs H-N reduction we see that

′ deg(Wj) > 0, deg(Wj) < 0 (3.24)

The adjoint action of P on p/u factors through L and this is precisely the adjoint representation of L. In other words, the Higgs vector bundle associated to EP for the P -module p/u is EL(l). The Higgs semistability of EL implies that EL(l) is Higgs semistable. Note that the degree of EL(l) is zero. Consider the exact sequence of Higgs vector bundles,

∗ 0 −→ EP (p) −→ EG(g) −→ σP TE/P −→ 0 (3.25) corresponding to the following exact sequence of P -modules

0 −→ p −→ g −→ g/p −→ 0 (3.26)

∗ From the above observations it follows that the Higgs vector bundle σP TE/P ∗ has the Higgs ﬁltration of σP TE/P

′ ′ ′ ′ ′ ∗ 0= V0 ⊂V1 ⊂V2 ⊂·········⊂Vm−1 ⊂Vm = σP TE/P (3.27)

′ such that each quotient Wi is Higgs semistable of negative degree. In the same way we have

0= V0 ⊂V1 ⊂V2 ⊂·········⊂Vk−1 ⊂Vk ⊂ EP (p) (3.28) with each quotient Wi is Higgs semistable of positive degree and EP (p)/Vm which is identiﬁed with EL(l), Higgs semistable of degree zero. So, we have the Higgs ﬁltration of EP (p) such that each quotient is of nonnegative degree.

43 Now we consider the exact sequences of Higgs vector bundles on X.

∗ 0 −→ EP1 (p1) −→ EG(g) −→ σP TE/P1 −→ 0 (3.29)

∗ 0 −→ EP 2(p2) −→ EG(g) −→ σP TE/P2 −→ 0 (3.30)

From (3.29) and (3.30) we have,

∗ EP1 (p1) −→ σ TE/P2 (3.31) ∗ EP2 (p2) −→ σ TE/P1 (3.32) the homomorphisms of Higgs vector bundles. Now by using Lemma 3.3.3 we conclude that the above two homomorphisms are zero. This implies that EP1 (p1) = EP2 (p2). So, ﬁnally we have proved the uniqueness of the H-N Higgs reduction.

44 Chapter 4

Nonnemptyness

In this chapter our aim is to prove the following theorem:

s Theorem I. With the notation of chapter 2 the moduli space MΓ(Y, 2, Q,c2) of rank 2 Γ-stable bundles of certain type τ and fixed determinant Q, on a smooth projective Γ–surface Y is nonempty if c2(E) ≫ 0. Hence, the moduli space of rank 2 stable parabolic bundles on X with certain parabolic structure is non-empty if parc2 ≫ 0 After that we use the above theorem to prove the similiar theorem for higher rank case: s Theorem II. The moduli space MΓ(Y,r, Q,c2) of rank r Γ-stable bundles and of certain type τ and fixed determinant Q, on a smooth projective Γ– 2 surface Y is nonempty if c (E) ≫ 0 and if the highest weight α < Θ1 . 2 r P Di·Θ1 Hence, the moduli space of parabolic bundles on X of rank two with full quasi–parabolic structure and with (parc2) ≫ 0 is non-empty Remark 4.0.4 Note that by certain type of Γ–bundle (parabolic bundle) we mean full quasi–parabolic structure and a condition on highest weight i.e Θ 2 αmax < 1 , if for some choices of (X,D) the right hand side is greater P Di·Θ1 than 1 then we actually get parabolic stable bundle of full flag and arbitary weights.

Remark 4.0.5 Let X be any scheme and Z be a 0 dimensional topological subspace of Sp(X), then Z gets a canonical reduced subscheme structure induced from X. So we can talk about the ideal sheaf IZ .

Outline of the proof of theorem I Let L(α1) and M (α2) are two Γ-line bundle whose Γ structures are given by two characters α1 and α2. If Z

45 is a reduced codimension 2 subvariety away from ramiﬁcation locus D then (α2) (α2) M ⊗IZ gets the same Γ structure as M where IZ is the ideal sheaf (α2) (α1) associated to Z. Note that any extension of M ⊗IZ by L gets a canonical Γ–structure given by

exp2πiα1 0 0 exp2πiα2 Our whole idea is to get a non-split locally free Γ-extension E of the form

(α1) (α2) 0 −→ L −→ E −→ M ⊗IZ −→ 0 such that E is stable with respect to a chosen polarization Θ1. Our ﬁrst 1 (α2) (α1) step is to ensure that ExtΓ(M ⊗IZ ,L ) is nonzero, for that we need 0 (α1)∗ (α2) α1−α2 l(Z) > hΓ(L ⊗ M ⊗ KY ( 4.0.6). After that we deﬁne the notion of Orbifold Cayley-Bachrach (OCB) for a Γ–triple (L(α1),M (α2),Z) 0 ∗ (α2) (α2−α1) (4.0.7). For any such OCB tuple with l(Z) > hΓ(L ⊗ M ⊗ KY , lemma 4.0.9 tells us that if l(Z) ≫ 0 there exists an extension which is actually Γ–locally free. Note that this lemma is the reason for large c2. After that we prove any such locally free extension is infact Θ–stable for certain choice of τ.

Outline of the proof of theorem II For the proof of Theorem II we follow to the classical proof (vector bundle case) which is due to Qin and Li [19]. We give a very short sketch of their proof in 2. Our proof is just following them with little bit of more care to handle Γ–case.

4.0.1 Preliminaries In this section we will give few new deﬁnitions and prove few propositions which will be usefull in proving Theorem I and Theorem II. Let Y be a smooth projective Γ–surface and p : Y −→ X be a morphism where X := ˜ Y/Γ arising from the Kawamata covering lemma. Let DY/X = D ={y ∈ Y : Γy 6= e} be the ramiﬁcation locus in Y , then lemma 2.8 of [?] tells that D˜ is a normal crossing divisor. Let R be a codimension two subvariety of X away from p(D˜) and Z = p∗(R). Then we term the cycle Z in Y a good Γ–cycle.

Remark 4.0.6 Consider Γ– line bundles L(α1), M (α2) on Y and let P = (α2) (α1)∗ (α1−α2) M ⊗ L ⊗ KY . By tensoring the exact sequence

0 −→ IZ −→ OY −→ OZ −→ 0

46 with P we get the following long exact sequence (see remark 2.2.14): 0 0 0 0 −→ HΓ(P ⊗IZ ) −→ HΓ(P ) −→ HΓ(P ⊗OZ ) −→ (4.1) 1 1 1 HΓ(P ⊗IZ ) −→ HΓ(P ) −→ HΓ(P ⊗OZ ) = 0 0 Let dimHΓ(P ) = l1, then by choosing a reduced 2 dimensional subvariety ∗ 1 Z = p (R) such that l(Z) > l1 it is easily seen that HΓ(P ⊗IZ ) 6= 0. This implies that there is at least one Γ–torsion free sheaf E on Y which is a (α2) (α1) non-split extension of M ⊗IZ by L .

(α ) (α ) Deﬁnition 4.0.7 Let 0 ≤ α1 <α2 < 1, we say that the Γ–triple (L 1 ,M 2 ,Z), satisﬁes the Orbifold Cayley Bacharach (OCB) property if the following 0 (α2) (α1)∗ (α2−α1) holds: for any section s ∈ HΓ(M ⊗ L ⊗ KY ) if the restriction ′ ′ of s to a good Γ–cycle Z ⊂ Z is zero implies that s|Z = 0, where Z ⊂ Z is a good Γ–cycle such that l(Z′)= l(Z) − d and d is the order of the group Γ.

Let Z′ ⊂ Z be good Γ-cycles and B = Z−Z′. Consider the exact sequence of ideal sheaves:

0 −→ IZ −→ IZ′ −→ OB −→ 0. (4.2) By applying the composite functor Hom( ,L(α1)) ◦ ( M (α2)) to 4.2 we get a map N

1 (α2) (α1) 1 (α2) (α1) ψZ′ : ExtΓ(M ⊗IZ′ ,L ) −→ ExtΓ(M ⊗IZ ,L ) of Γ–extensions.

Lemma 4.0.8 Let (L(α1),M (α2),Z) be a Γ–triple which satisﬁes OCB. Then we have: 1 (α2) (α1) ∪Z′⊂Z Image(ψZ′ ) 6= ExtΓ(M ⊗IZ ,L ) for all good Γ–cycles Z′ ⊂ Z with l(Z′)= l(Z) − d.

Proof. By tensoring the exact sequence 4.2 with P we get the following long exact sequence (see 2.2.14): 0 0 0 0 −→ HΓ(P ⊗IZ ) −→ HΓ(P ⊗IZ′ ) −→ HΓ(P ⊗OB) −→ (4.3) 1 1 1 HΓ(P ⊗IZ ) −→ HΓ(P ⊗IZ′ ) −→ HΓ(P ⊗OB) = 0 Here we note that the assumption that the triple (L(α1),M (α2),Z) satis- 0 ∼ 0 ﬁes OCB implies that HΓ(P ⊗IZ ) = HΓ(P ⊗IZ′ ). Therefore by dualizing we have: 1 ∗ 1 ∗ 0 −→ HΓ(P ⊗IZ′ ) −→ HΓ(P ⊗IZ ) −→ V −→ 0

47 where V is the complex vector space invariant under Γ which is precisely the 0 dual of the space of sections of the torsion sheaf HΓ(P ⊗OB). Note that V is independent of Z′ ⊂ Z and depends only on l(Z′). This in particular 1 ∗ 1 ∗ implies that HΓ(P ⊗IZ′ ) 6= HΓ(P ⊗IZ ) . Since the finite union of proper subspaces of finite dimensional vector spaces is not equal to the vector space (we are over an infinite field!) we 1 ∗ 1 ∗ have ∪HΓ(P ⊗IZ′ ) 6= HΓ(P ⊗IZ ) . The lemma now follows by Γ–Serre 1 (α2) (α1) duality (2.2.4), which gives the identification ExtΓ(M ⊗ IZ ,L ) ≃ 1 ∗ 2 HΓ(P ⊗IZ ) .

Lemma 4.0.9 Let (L(α1),M (α2),Z) be a Γ–triple which satisﬁes OCB. Then for l(Z) ≫ 0, there exists a Γ–extension

(α1) (α2) 0 −→ L −→ E −→ M ⊗IZ −→ 0 with E locally free.

Proof. Suppose now that E is not locally free. This implies that the set Sing(E) namely the singular locus of E where E fails to be locally free is a 0 dimensional Γ–invariant subvariety A ⊂ Z. Let a ∈ A then p−1(p(a)) =

γ∈|G γ · a = B ⊂ A. Let TA and TB denote the torsion sheaf associated toP A and B respectively. Note that we have an inclusion of torsion sheaves TB ⊂ TA. Therefore we get the following commutative diagram of Γ torsion free sheaves on Y .

∗∗ 0 / E / E / TA / 0 O O (4.4)

′ 0 / E / E / TB / 0

′ ∗∗ (α1) where E is the inverse image of TB in E . Note that since L is locally free the saturation of L(α1) in E′ is L(α1) itself. ′ (α2) (α1) We therefore obtain an extension E of M ⊗IZ′ by L using the above commutative diagram where Z′ is the Γ cycle corresponding to the good cycle R′ ⊂ R induced by the set A − B and l(Z′)= l(Z) − d where d is the order of the group Γ. Also we have the following commutative diagram of Γ–sheaves on Y given by two Γ–sheaves E and E′.

48 0 0 (4.5)

(α ) (α2) 0 / L 1 / E / M ⊗IZ / 0

(α ) ′ (α2) 0 / L 1 / E / M ⊗IZ′ / 0

TB TB

0 0

′ It is clear from the above two diagrams ψZ′ (E )= E. By Lemma 4.0.8 it follows immediately that there exists locally free sheaves which can be realised as extensions as desired. 2 Now we will construct a rank two Γ–stable vector bundles as a extension (α2) of M ⊗IZ by OY .

Remark 4.0.10 Let L(α1) be a Γ–line bundle on Y and let Z be a good Γ–cycle. Γ (α1) Γ (α1) p∗ (L ⊗IZ ) ≃ p∗ (L ) ⊗IR

Existence of Cayley-Bacharach triple In this section we show that for any line bundle Q on X there exists a reduced codimension 2 subvariety R in X of suitable length such that (OX , Q, R) (β) (α) satisfies strong CB property. Using this we show that (OY , Q ,Z) satisfies (α) ∗ (α) ∗ OCB property, where Q = p (Q) ⊗OY and Z = p (R). 0 Lemma 4.0.11 Let l ≥ h (X,Q ⊗ KX ). Then for a generic 0–cycle R in Hilbl+1(X) we have the usual Cayley-Bacharach property for the triple (OX , Q, R). Proof. For the sake of completeness we briefly indicate a proof. We first l 0 observe that for generic choice of T ∈ Hilb (X), l ≥ h (X,Q ⊗ KX ) implies 0 l h (X,Q ⊗ KX ⊗IT ) = 0. Let Vl ⊂ Hilb (X) consist of reduced 0–cycles and

0 Ul = {T ∈ Vl|h (X,Q ⊗ KX ⊗IT ) = 0}

49 an open dense subset of Vl. Let T be the universal family in Vl+1 × X, i.e T = {(T,x) ∈ Vl+1 × X|x ∈ Supp(T )} and consider the surjection f : T → Vl, f(T,x)= T − x and the ﬁrst projection p : T → Vl+1. Observe −1 that p(T − f (Ul)) ⊂ Vl+1 is a proper closed subset. Choose R ∈ Vl+1 − −1 −1 −1 p(T − f (Ul)) implying p (R) ⊂ f (Ul) i.e ∀x ∈ Supp(R), (R − x) ∈ Ul, 0 hence h (X,Q ⊗ KX ⊗IR−x) = 0, ∀x ∈ Supp(R). 2

Remark 4.0.12 In fact, we observe that this choice of l forces something 0 stronger, namely H (Q ⊗ KX ⊗IR) = 0. Moreover, for any x ∈ Supp(R) 0 we even have H (Q ⊗ KX ⊗IR−x) = 0 which implies the Cayley-Bacharach property. So if both these vanishings hold, we term the triple (OX , Q, R) to have the stronger Cayley-Bacharach property.

∗ Lemma 4.0.13 There exists a good Γ–cycle Z1 = p (R1) in Y with l(R1) ≥ 2 (α1) 4Θ1 having the following property; If L is a Γ-line bundle on Y such that 0 (α1) (α1) 2 ∗ hΓ(Y,L ⊗IZ1 )) > 0, then c1(L ) · Θ ≥ 2Θ . where Θ= p (Θ1) is the ﬁxed ample line bundle on Y .

Proof. Let C1 and C2 be two smooth curves in |Θ1| in X. Choose reduced 0-cycles R1 ⊂ C1 − C2 and R2 ⊂ C2 − C1 away from D the ramification 2 divisor in X such that cardinality of each Ri is more than 2H1 . ∗ ∗ Let R = R1 ∪ R2 and Z = p (R), Zi = p (Ri). Suppose that we 0 (α1) have hΓ(L ⊗ IZ1 ) > 0. Then from the above Remark 4.0.10 we get 0 Γ (α1) h (p∗ (L ) ⊗IR1 ) > 0. Γ (α1) ′ (α1) ′ Let p∗ (L )= L . Observe that L and L are both effective. By an abuse of notation, we will continue to denote by L(α1) and L′ divisors in the linear equivalence of the line bundles. ′ Suppose that the effective divisor L contains C1 and C2 as its components. Then ′ 2 c1(L ) · H1 ≥ 2H1 . ′ If L does not have Ci for some i = 1, 2 then we have

′ ′ 2 c1(L ) · H1 = L ∩ Ci ≥ l(Si) = 2H1 .

′ 2 Therefore c1(L ) · H1 ≥ 2H1 . Now

(α1) (α1) Γ (α1) ′ 2 2 c1(L )·H = degY (L )=(pardeg(p∗ (L )) |Γ|≥ degX (L ) |Γ|≥ 2H1 |Γ| = 2H . 2

50 Choice of Q and degree bounds Remark 4.0.14 Let Q ∈ Pic(Y ) be a Γ–line bundle obtained as follows: Let L be a line bundle on X. Let Q = L ⊗OX (2nΘ1), we choose n in such a way that

2 c1(Q) · Θ1 < 3Θ1 (4.6) This choice of n is made to ensure the resultant rank two bundle (to be con- ∗ (α2) structed) is Θ-stable and has determinant Q, where Q = p (Q)⊗OY . Note (α2) ∗ that Q has same Γ-structure as OY (since p (Q) has trivial Γ–structure), (α2) where by OY we mean the trivial line bundle OY with a Γ–structure of type τ given by multiplication by the character corresponding to α2 (see (2.2.4) for notation). Let 0 ≤ α1 <α2 < 1. Then we claim that for any choice of L on X, there ∗ (α2) exists a rank 2Γ–stable bundle whose determinant is L = p (L)⊗OY . For (α1) that we twist it by a suitable n (4.6) and then produce a triple (OY , Q,Z) which satisﬁes the orbifold Cayley Bacharach property, hence there exists a α1 rank 2 bundle which comes as extension of OY by Q⊗IZ , infact we prove such a bundle is actually Θ–stable. After that we twist it by OY (−nΘ) to get a Γ–stable bundle whose c1 is c1(L). So we need to choose a cycle R such that (OX , Q, R) has stronger Cayley-Bacharach property which we get by (4.0.11) and (4.0.12). This will involve the choice of generic R with l(R) ≫ 0 since we need to avoid the ramiﬁcation locus. We choose R and Q with the bounds given by 4.0.6 which clearly does the job.

Let γ = α2 · degX (Di), where Di’s are the irreducible components of the parabolic divisor.P We assume that 2 γ < Θ1 (4.7)

This imposes a condition on the weight α2 which we therefore have as hypothesis in Theorem I. Let d = |Γ|. Then we see that by comparing degrees, we have:

Γ c1(Q) · Θ= degY (Q)=(pardeg(p∗ (Q)) d = {degX (Q)+ γ} d Γ The non-trivial contribution of γ occurs since p∗ (Q) is a parabolic line bundle with underlying line bundle Q but with non-trivial parabolic structure. Again, by (4.6) and (4.7) we have; c (Q) · Θ 1 < 2Θ2 (4.8) 2

51 ∗ αi+1 Let Qi = p (Q)⊗OY ,i = 1 ··· r−1; where each αi’s satisﬁes (4.7) and are in increasing order. Let Z1 ··· ,Zr−1 are r − 1 Γ–cycles of same length, ∗ such that Zi ∩ Zj = Ø for i 6= j; where each Zi = p (Ri) and Ri ∩D = Ø. r−1 W Let Z = ∪iZi, W = i=1 (Qi ⊗IZi ) and Wi = It is known (from (Qi⊗IZi ) L 1 α1 previous section) that there exists an extension ei ∈ ExtΓ(Qi, OY ) whose corresponding exact sequence

α1 0 −→ OY −→ Ei −→ Qi −→ 0

α1 gives a bundle Ei if and only if (Qi, OY ,Zi) satisﬁes orbifold Cayley-Bacharach property. Note that

r−1 1 α1 1 α1 ExtΓ(W, OY ) ≃ ExtΓ(Qi, OY ) (4.9) Mi=1 Lemma 4.0.15 There exists a reduced Γ, 0-cycle Z = p∗(R) with l(R) ≥ 2 2 4(r − 1) · Θ1 having the following property: 0 if Q is a Γ–line bundle on Y such that hΓ(Y, Q⊗IZ ) > 0, then we have 2 c1(Q) · Θ ≥ 2(r − 1) · Θ .

Proof. Let Ci’s are r distinct smooth curves in the linear system Θ1. For each i 2 choose a set Ri of 2(r−1)Θ1 many distinct points away from the ramiﬁcation divisor D in the open subset

Ci − j6=i Cj) S ∗ 0 of Ci. Let R = ∪iRi and Z = p (R). Suppose that hΓ(Y, Q⊗ IZ ) ≥ 0, then 0 ∗ Γ from (4.0.10) we get h (X,pΓ(Q) ⊗ IR) > 0. Let p∗ (Q) = Q, observe that both c1(Q) and c1(Q) are eﬀective. If c1(Q) contains all the curves Ci’s as its irreducible components, then we have 2 Q · L ≥ 2(r − 1)Θ1

If c1(Q) does not have atleast one Ci as its irreducible component for some i, then; 2 c1(Q) · Θ1 ≥ 2(r − 1)Θ1 Therefore

c1(Q) · Θ= degY (Q)=(pardeg(Q))|Γ| =(degX (Q)+ γ)d 2 2 ≥ 2(r − 1)Θ1kΓ| = 2(r − 1)Θ note γ ≥ 0(which we need in the above inequality), since parabolic divisor is eﬀective. 2

52 4.0.2 Proof of Theorem I

∗ (α2) Lemma 4.0.16 Let Q = p (Q) ⊗OY where Q ∈ Pic(X) any line bundle and α2 satisﬁes (4.7). Then there is a rank 2, Γ–stable vector bundle E of type τ with weights (α1,α2), with det(E) =∼ Q and c2(E)= c ≫ 0.

2 Proof. Without loss of generality we can assume degX (Q) < 3Θ1 (see (α1) 4.0.14). We start with a a triple (OY , Q,Z2) which satisﬁes the orbifold Cayley-Bacharach property. This exist by what we have already seen (4.0.1). We in fact choose a 0–cycle R2 in X to satisfy the stronger property as in ∗ (4.0.14) and (4.0.12) and let Z2 = p (R2). ′ This gives us a Γ locally free extension E of Q⊗IZ2 by OY (4.0.9). ∗ 2 Now we choose a good Γ–cycle Z1 = p (R1) such that l(R1) ≥ 4Θ1 as in Lemma 4.0.13 and let

Z = Z1 ∪ Z2,R = R1 ∪ R2

Then we observe that the triple (OY , Q,Z) also satisﬁes a orbifold Cayley Bacharach property. This can be seen as follows: if Z = p∗(R), then by (4.0.14), its enough to see that (OX , Q, R) has the usual Cayley-Bacharach property. This is immediate, for if x ∈ Supp(R) = Supp(R1) ∪ Supp(R2), 0 then it is easy to see that H (Q⊗KX ⊗IR−x) = 0 since we have assumed the stronger Cayley-Bacharach property for R2 and moreover, IR−x ⊂IR2−x or

IR−x ⊂IR2 depending on whether x ∈ Supp(R2) or not. Therefore we get a new Γ–locally free extension E:

(α1) (α2) 0 −→ OY −→ E −→ Q ⊗IZ −→ 0 We now claim that any such E is Γ–stable. To see this, consider any Γ–line subbundle L of E. If L is non-trivial, then composing the inclusion L֒→ E with the map E −→ Q⊗IZ we get a nontrivial Γ–map f : L −→Q⊗IZ . This gives a non-zero Γ–section

0 ∗ s ∈ HΓ(Q⊗ L ⊗IZ ).

0 ∗ 0 ∗ In particular, hΓ(Q⊗ L ⊗IZ ) > 0 and as a result hΓ(Q⊗ L ⊗IZ1 ) > 0. Therefore by Lemma 4.0.13 we conclude that

2 (c1(Q) − c1(L)) · Θ ≥ 2Θ .

2 Hence, µ(L) = c1(L) · Θ ≤ (c1(Q) · Θ − 2Θ ). But we know that µ(E) = (c1(Q)·Θ 2 . By (4.8) we thus have:

53 (c (Q) · Θ µ(L) ≤ c (Q) · Θ − 2Θ2 < 1 = µ(E). 1 2 Hence E is Γ–stable and clearly of determinant Q. Regarding the type of the Γ–stable bundle E of rank two constructed above, we observe that we work with a zero cycle Z coming from the complement of ramification divisor. So the action of Γ on Z is a free action. So it does not affect the type of the extension bundle we constructed. Regarding any first chern class, we want to point out that that upto now Γ 2 we got a Γ–stable rank 2 bundle such that degX (p∗ E) < 3Θ1. Now twist ′ back the vector bundle E by OX (−nΘ ), where this n is same as (4.6) and the bundle still remains Γ–stable. Note that since Θ is pull back some very ample in down, tensoring by OY (−nΘ) does not change the type τ. In this way we get a rank 2, Γ-stable bundle whose c1 satisfies 4.6, but by remark 4.0.1 we get any c1. 2 Corollary 4.0.17 There exists (Γ,µ)–stable on Y with vanishing obstruction space. Proof. To see this we make a few easy observations: 1. The obstruction space of a Γ–bundle on Y can be easily seen to be the 2 space ExtΓ(E,E)0, where the subscript stands for the trace zero part.

2. Now we compute the ExtΓ using the construction of E as a Γ–extension. The argument is exactly as in Huybrechts and we only use the Remark 2.2.14 to get the vanishing when we make degree and length large. 2

4.0.3 Proof of Theorem II

1 α1 Lemma 4.0.18 There exists an extension e ∈ ExtΓ(W, OY ) whose corresponding exact sequence gives a Γ vector bundle if and only if for each i = 1, ··· , (r − 1) the 0-cycle Zi satisﬁes OCB property.

1 Proof. Put e = (e1, ··· ,er−1) where ei ∈ ExtΓ(Y, Qi ⊗IZi ). Let Ei be the −1 subsheaf φ (Qi ⊗IZi ) of E. Then Ei is given by the extension ei

α1 0 −→ OY −→ Ei −→ Qi ⊗IZi −→ 0 Note that E is locally free outside the zero cycle Z and sits in an exact sequence

0 −→ Ei −→ E −→ Wi −→ 0

54 Since Wi is locally free at the points in Zi, we see that E is locally free at the α1 points Zi if and only if Vi is locally free at Zi. But by choice (Qi, OY ,Zi) satisﬁes OCB, hence our result follows. 2

Proposition 4.0.19 The rank r Γ-bundle (constructed above) is Θ-stable.

Proof. Let E be a rank r–vector bundle as before. Let V be a proper sub-bundle of E such that E/V is torsion free. Let U2 be the image of V in W , and V1 be the kernel of the surjection V −→ U2 −→ 0. Then we have a following commutative diagram of morphisms:

/ (α1) / / / 0 OY E W 0 O O O

0 / U1 / V / U2 / 0 O O O

0 0 0

1. U1 6= 0 From above diagram we have: c1(U1) = −E1, where E1 is ∗∗ ∗∗ = an eﬀective divisor. Let rk(U2)=r2 ≤ r − 2, we have U2 ֒→ W ≥ r−1 Q ; ∧r2 (U ∗∗) ֒→ Q , where S = {β | β =(i , ··· ,i ); 1 i=1 i 2 β∈S β k1 kr1 Lk1 < ··· 0.

r r = ( U1 = 0 V ֒→ W ; let r1 = rank(V ), hence, ∧ 1 (V ) ֒→ ∧ 1 (W .2

⊕β∈S(Qβ ⊗IZβ ) 0 r1 ∗ where, Zβ = ∪j∈βZj; Hence, HΓ(Y, ⊕β(Qβ ⊗ (∧ V ) ⊗IZβ ) 6= 0, 0 r1 r1 ∗ so HΓ(Y, (Q ⊗ (∧ V ) ⊗IZβ ) 6= 0 for some β. 0 r1 ∗ 0 r1 ∗ ( Observe HΓ(Y, (Qβ ⊗ (∧ V ) ⊗IZβ ) ֒→ HΓ(Y, (Qβ ⊗ (∧ V ) ⊗IZi 0 r1 ∗ for all i ∈ β. So HΓ(Y, (Qβ ⊗ (∧ V ) ⊗IZi ) 6= 0 for some i. 2 We choose Zi such that l(Zi) ≥ 4(r − 1)Θ . Now by lemma we have r1 ∗ 2 c1(Qβ ⊗ (∧ V ) ) · Θ ≥ 2(r − 1)Θ . 2 i.e r1 · c1(Q) · Θ − c1(V ) · Θ ≥ 2(r − 1) · Θ . 2 c1(V )·Θ) ≤ (r−1)c1(Q)·Θ) + c1(Q)·Θ) − 2(r−1)Θ . r1 r r r1 2 Now by choice c (Q) · Θ < 4Θ2 , since 4Θ2 < 2r(r−1)Θ for any r ≥ 3, 1 r1 2 r1

55 2

56 Chapter 5 holonomy

Let H be an arbitrary semisimple algebraic group over C. The aim of this Γ,s section is to prove that the moduli space MH (c) of (Γ,H)–stable bundles on a smooth projective Γ-surface Y for large characteristic classes c is non– empty. Note that over surface when H is arbitary and E is a Γ-stable bundle over Y , there is no analogue of parabolic bundle on X. For curve case there is such analogue which is known as ramiﬁed G-bundle, for details see [17]. But still we can talk about its nonemptyness. When H = Sl(r)orGl(r) this will give us nonemptyness of parabolic moduli space over Y/Γ := X. Let E be a Γ–stable Sl(2) bundle of type τ with c2 ≫ 0 over Y , this corresponds to a degree 0 parabolic stable bundle on X. let C be a general high degree Γ–invariant curve such that E|C is Γ–stable. The quotient C/Γ =: T is also smooth projective curve in Y/Γ := X. By choosing C suﬃciently general, one can make sure that the action of Γ on C is faithful and we can realize Γ as a quotient Γ1/Γ2 where Γ2 = π1(C) and Γ1 acts properly discontinuously on the simply connected cover C˜ of C and T = C/˜ Γ1. Then by Seshadri’s proposition 10 of [15] one knows that there exists an ˜ 2 irreducible representation ρ :Γ1 −→ SU(2) such that E|C ≃ C ×ρ C .

Let M(E|C )= Im(ρ) be the Zariski closure of Im(ρ) in Sl(2). It is quite well known that M(E|C ) is reductive. Ingact it is the smallest reductive subgroup of Sl(2) such that E has a reduction to M(E|C ). Since one knows the set of all ﬁnite subgroups of Sl(2), the only possibilities for M(E|C ) are the following

1. Finite cyclic groups Cn and the dihedral groups Dn.

2. The alternating groups A4 and A5 and the permutation group S5. 3. The whole of SL(2) or its maximal torus.

57 Of these, since E|C is stable, we can omit the maximal torus and the cyclic

groups Cn. So M(E|C ) can either be the alternating groups, S5, or the dihedral groups apart from the whole of SL(2). We wish to estimate the set ZC of representations of Γ1 in SL(2) which lie entirely in these families of ﬁnite groups upto conjugacy by the diagonal action of SL(2). So we have an exact sequence,

0 −→ π1(C) −→ Γ1 −→ Γ −→ 0

Since π1(C) has ﬁnitely many generators and Γ is a ﬁnite subgroup the set ZC is at most countable. This implies the following lemma.

Lemma 5.0.20 The locus of points in the moduli space of Γ–stable vector bundles MC,SL(2) of rank 2 and trivial determinant on the curve C whose monodromy is among the set of ﬁnite subgroups listed above has countable cardinality.

Now we consider the restriction map r : M Γ,s (c) −→ M Γ,s . C SL(2) C,SL(2) Lemma 5.0.21 If the Γ–invariant curve C is chosen suﬃciently general and high degree k then the map rC is an immersion. Proof. 2 Now using the above lemma we prove th following proposition:

Proposition 5.0.22 There exists a rank 2, Γ–stable bundle E with c2(E) ≫ 0 and trivial determinant on the surface Y such that the restriction E|C to a high degree G–invariant general curve C ⊂ Y has monodromy subgroup to be the whole of SL(2) itself.

Γ s Proof. For c = c2(E) ≫ 0, it is known that the moduli space MX (2, O,c2) = M Γ (c)s is non–empty and has non-zero dimension d(c). By lemma 5.0.21 SL(2)

it is clear that Im(rC ) is not completely contained in the subset ZC of Γ- stable bundle with ﬁnite monodromy groups. Therefore there exists at least

one Γ–stable bundle E on Y such that M(E|C )= Sl(2) Deﬁnition 5.0.23 Let K be a maximal compact subgroup of H. A subset A ⊂ K is said to be irreducible if

{Y ∈ H|ad x(Y )= Y, ∀ x ∈ A} = centre of H.

A (homomorphism) representation δ : Γ −→ K is said to be irreducible if the image δ(Γ) is irreducible.

58 Note that in our case since H is semisimple centre of H is trivial. We have the following equivalences and we quote:

Proposition 5.0.24 ([3, Prop 2.1]) Let K be a maximal compact subgroup of H and let A ⊂ K be a subgroup. Then the following are equivalent

1. A is irreducible.

2. A leaves no parabolic subalgebra of H invariant.

3. For any parabolic subgroup P ⊂ H, A acts without ﬁxed points on H/P .

Lemma 5.0.25 If E is a Γ–stable bundle on Y such that for a high degree

Γ–invariant general curve C in Y such that E|C is Γ–stable and M(E|C )= SL(2) then E|C (φ) is a Γ–stable H–bundle on C by an extension of structure group via the principal homomorphism φ.

Proof. Since φ is irreducible, Im(φ) is not completely contained in any parabolic subgroup of H by proposition 5.0.24 we have the following:

{Y ∈ H|ad x(Y )= Y, ∀ x ∈ sl(2)} = centre of H = trivial.

In other words, we can say that Im(φ) is an irreducible subset in H (or equivalently, by the “unitarian trick”, Im(φ|su(2)) is an irreducible subset of K, where as above, K ⊂ H is a maximal compact subgroup). By [3, Prop 2.2], to show that V (φ) is G–stable as an H–bundle, we need only show that the representation

η =(φ ◦ ρ): π1(C) −→ K

is irreducible. If Y ∈ H is such that ad x(Y ) = Y, ∀ x ∈ Lie(Im(η)), then by the density of Im(η) in Im(φ) = sl(2) and hence by continuity, we see that for such an element Y , one has adx(Y ) = Y, ∀ x ∈ sl(2) and hence Y ∈ centre of H. This shows that Im(η) is an irreducible set in K ⊂ H and we are done. 2 We can now conclude the following:

s Theorem 5.0.26 The moduli space MH (c) of Γ–stable principal H–bundles on a smooth projective surface X is non–empty for c ≫ 0.

59 Proof: We claim that if φ : SL(2) −→ H is as above a principal SL(2) in H then any E as in Prop 5.0.22 has the property that E(φ) is a G–stable principal H–bundle. By the converse to the Mehta–Ramanathan theorem (Lemma ??), we see that it is enough to prove that E(φ)|C is stable. Since M(E|C )= SL(2) this is immediate by Lemma 5.0.25. The largeness of the characteristic classes of the associated H–bundle is determined by the largeness of the c2 of the rank 2 bundle E. This can determined by the general methods of Borel and Hirzebruch (see the recent preprint Beauville [?] for this).

60 Chapter 6

Ramiﬁed G-bundle

6.1 H-N reduction for ramiﬁed G bundles over a smooth curve

We assume from here onwards that X is a smooth projective curve of genus c g, and D = i=1 Di is a normal crossing reduced divisor. Unless stated otherwise, allP the groups considered will be reductive linear algebraic groups over C and N ≥ 2 be a ﬁxed integer.

Definition 6.1.1 A ramified G bundle of type N on X with ramification over D is a smooth variety Q with an action of G and a morphism φ : Q −→ X satisfying the following properties:

1. The action G on Q is proper.

2. The triple (Q,φ,X) is a geometric quotient.

3. In the complement of the divisor D ⊂ X the morphism φ : Q −→ X is a principal G bundle.

4. At the ﬁnitely many orbits on which the isotropy is nontrivial, the isotropy is a cyclic subgroup whose order divides N.

Given a homomorphism ρ : G −→ H and a ramiﬁed G-bundle Q of type N on X, the quotient space Q ×G H has a natural structure of a ramiﬁed H-bundle of type N on X. This construction is called the extension of the structure group of Q to H. Let us denote the quotient space by Q(H)(c.f. [?]).

61 ′ Definition 6.1.2 Let φ1 : Q −→ X and φ2 : Q −→ X be two ramified G-bundles of type N over X. Let h : Q −→ Q′ be a morphism of varieties. We say h is a morphism of ramified G-bundles if

1. φ2 ◦ h = φ1 2. The isotropy for the image of the orbits(whose isotropy is nontrivial) is a cyclic subgroup whose order divides N

We recall some of the deﬁnitions stated in [17] and [?].

1. Let H be a closed subgroup of G. A reduction of structure group of a ramified G-bundle Q to H is given by a section σ : X −→ Q/H. Let QH be the inverse image of the natural projection of Q to Q/H, of the subset of Q/H defined by the image of σ. This QH is a ramified H-bundle on X. It is easy to see that QH ×H G =∼ Q.

2. Let QH be a reduction of a ramified G-bundle Q. If W is a finite H dimensional H-module, then the associated construction QH × W , is a parabolic vector bundle over X. This is denoted by Q(W )∗. 3. A ramified G-bundle Q is called semistable if for every reduction of structure group (P,σ) of Q to any parabolic subgroup P of G where σ : X −→ Q/P is a section, and any dominant character χ of P , the parabolic line bundle Q(χ)∗ (= Lχ, as in our first section) has non-positive parabolic degree.

4. Let Q be a ramified G bundle of type N over X with ramification divisor D. A Kawamata cover of type (N,D) (or simply N) is a Galois cover p : Y −→Γ X where Y is a connected smooth projective variety and Γ is a finite subgroup of Aut(Y ). A (Γ,G)-bundle is defined to be a principal G-bundle E together with a lift of the action of Γ on E which commutes with the right action of G on E.

5. The ramiﬁed G-bundle Q over X with ramiﬁcation divisor D gives rise ∗ to a (Γ,G)-bundle E over Y such that for every y ∈ (p Di)red has a cyclic subgroup Γy ⊂ Γ of order kiN as the isotropy group. 6. Let [Γ,G,N] denote the collection of (Γ,G)-bundles on Y satisfying the following two conditions:

∗ (a) For a general point y of an irreducible component of (p Di)red, the action of Γy on Ey(ﬁbre over y) is of order N.

62 (b) For a general point y of an irreducible component of a ramiﬁcation ∗ divisor for p not contained in (p D)red, the action of Γy on Ey is trivial.

7. Let E ∈ [Γ,G,N] and E′ = E/Γ. The variety E′ is smooth; moreover φ : E′ −→ X is a ramiﬁed G-bundle.

′ 8. E is a normalisation of the ﬁbre product Y ×X E . In other words, the G-bundle E on Y can be constructed back from E′.

9. A ramified G-bundle Q canonically defines a parabolic G-functor FQ : Rep(G) −→ PV ect(X,D) taking values in a category PV ect(X,D,N) for some N. Conversely, given a parabolic G-functor F there exists a ramified G-bundle Q, unique up to isomorphism of ramified G-bundles, such that FQ = F . 10. A ramified G-bundle Q is semistable if and only if the corresponding functor FQ is semistable and hence if and only if the induced (Γ,G)- bundle is semistable. A ramified G-bundle Q is stable if and only if the induced (Γ,G)-bundle is stable.

Definition 6.1.3 Let Q be a ramified G-bundle over X. Let (P,σ) be a reduction of structure group of Q to a parabolic subgroup P of G. Then the reduction (P,σ) is called a H-N reduction for the ramified G-bundle Q on X if the following two conditions hold:

1. Let L be the Levi factor of P . Then the ramiﬁed L-bundle QP (L) is semistable.

2. For any dominant character χ of P with respect to some Borel subgroup B ⊂ P of G the associated parabolic line bundle QP (χ)∗ has parabolic degree > 0.

6.1.1 H-N reduction for [Γ,G,N] bundle We deﬁne the H-N reduction for a [Γ,G,N] bundle over any connected smooth projective variety over C(c.f.[17]) and we prove that it exists and is unique.

Deﬁnition 6.1.4 A (Γ,G,N) bundle E over Y with a Γ-reduction of structure group (P,σ) over some open subset U with codim(X\U) ≥ 2 is said to

63 be canonical or Γ equivariant H-N reduction if the following two conditions hold:

1. The associated Levi bundle EP (L), is Γ-semistable. 2. For every dominant character χ of P with respect to some Borel subgroup contained in P , the line bundle EP (χ) on UP has positive degree. Proposition 6.1.5 For any (Γ,G)-bundle over Y , Γ-equivariant H-N reduction exists and it is unique with respect to some Borel subgroup B contained in P .

Proof: We will use the fact that for any G-bundle the canonical reduction (P,σ) exists and the usual semistability is same as Γ-semistability([17], Proposition 4.1). So our main job is to show that the usual H-N reduction (P,σ) is Γ-saturated. Let γ ∈ Γ(⊂ Aut(Y )), we note that γ induces ∗ ˜γ ˜γ γ˜: E −→ E. We denote σ E by EP and γ˜(EP ) by EP . Clearly EP is a P ˜γ −1 −1 bundle. [We deﬁne P action “⋆” on EP by p⋆(γ(e)):=γ(e) · p = γ(e · p ) , where · denotes the action of G on E, clearly γ(p⋆e) = p ⋆ (γ(e)). This ˜γ means the free action of “P” on EP commutes with the action of γ. Hence ˜γ EP is a P -bundle.] ∼ ˜γ ˜γ Note that EP = EP , by uniqueness of H-N reduction EP = EP . This implies that EP is Γ saturated (since γ is arbitrary). So EP is a (Γ, P ) ˜γ reduction. Now we use the fact that that usual semistability of EP (L) is same ˜γ as Γ semistability of EP (L)(c.f. Theorem 4.3, [?]). The second condition for Γ equivariant H-N reduction is satisﬁed automatically. So EP is a Γ equivariant H-N reduction, and it is unique.

6.1.2 Proof of H-N reduction for ramified G bundle We fix a Borel subgroup B. We would like to prove here that for any ramified G-bundle there exists unique H-N reduction. We already know that the existence of the H-N reduction for the (Γ,G,N) bundles. Let φ : Q −→ X be a ramified G-bundle on X and let p : Y −→ X be a Kawamata cover and E ∈ [Γ,G,N] such that E/Γ = Q. Let (P,σ) be a H-N reduction of E to a parabolic subgroup P of G where σ : Y −→ E(G/P ) is the Γ-equivariant section. We will prove that the corresponding reduction τ : X −→ Q/P of Q to P (where τ is induced by the Γ-equivariant section σ)is the H-N reduction for the ramified G-bundle Q. Note that the semistablity of EP (L) is equivalent to the semistablity of the ramified L-bundle (EP /Γ)(L)= EP (L)/Γ. Let

64 χ be the dominant character of P with respect to the Borel subgroup B ⊂ P . It is proved in [17] that deg(EP (χ)) ≥ 0 if and only if deg((E/Γ)(χ)∗) ≥ 0. Therefore, by the above discussion it is clear that the reduction (EP /Γ,τ) is a H-N reduction for the ramiﬁed G-bundle Q = E/Γ . The uniqueness of the H-N reduction follows from the uniqueness of the H-N reduction of the (Γ,G)-bundle E.

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