Deformations of Bundles Over the Riemann Sphere

BY STEPHEN J.H. XEW.

DEPARTMENT OF >~THE.M.%TICS A S D ST.~TISTICS.

M ~ G I L L VXIVERSITY. MONTREAL

-4 thesis submitted to the faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of Ph.D.

@ Stephen J.H. New, 1997

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Abstract

Ué study holomorphic bundles over Pl x U with structure group G. where G is

a complex reductive Lie group. Pl denotes the Riemann sphere, and U is an open

set in Cm. We show that such a bundle may be given locally by a (non-unique)

normal form transition function which takes values in a Borel subgroup of G. Ln

the case in which G = GL(n. C). we use the normal form transition function to

construct matrices S(r) whose kernels may be identified with the spaces of global

holomorphic sections of various associated vector bundles. We use the matrices

S ( x ) to calculate many integrd invariants of these bundles. and also to study the

divisor of jumping Lines. We also study spaces of framed bundles oyer Pl xC. which

form the building blocks in a stratification of instanton moduli spaces.

Résumé

Xous étudions les fibrés holomorphes de groupe de structure G sur Pi xC-. où

G est u n groupe de Lie réductif complexe. Pi la sphère de Riemann. et L- un

ouvert de Cm. Sous montrons qu'un tel fibré est localement déterminé. mais non

uniquement. par une fonction de transition à mleurs dans un sous-groupe de Borel

de G. Quand G = Gl(no C), nous utilisons cette forme normale de la fonction de

transition pour contruire des matrices S(x) dont les noyaux peuvent être identifiés

avec les différents espaces de sections holomorphes des fibrés vectoriels associés. Ces

matrices sont ensuite utlisées pour construire des invariants entiers de ces fibrés et

pour étudier le diviseur des droites de saut. Nous étudions ensuite les espaces

de paires (fibrés sur Pi x U, trivialisation sur {O} x U), qui forment les éléments

constitutifs d'une stratification de l'espace des instantons.

Acknowledgments

During the preparation of this thesis. 1 received funding fiom WSERC. FCAR

and ISM. M y supervisor. Dr. Jacques Hurtubise. also provided intaluable financial

support. but of far greater d u e were his insightful suggestions, his elucidative

explmations and his affable nature. In fact many of the techniques used in this

thesis were originated by Dr. Hurtubise in his own ground-breaking works. His

influence cannot be underestimated, and 1 am pleased to have this opportunity

to offer him my deepest gratitude. I a m grateful also to other members of the

mathematics department, notably Dr. Georg Schmidt and Dr. Niky Kamran. and

to the staff of the departmentai office for their fiequent and ready assistance. From

amongst my feilow students with whom 1 have shared usefd discussions. 1 would

like to single out Christopher Anand. Djordje Çubrit and Thomas Mattman who

have proffered their friendship. The typesetting of the thesis was performed using

the A*,@-Qhacro package. Finally. 1 wish to thank my wife and my children.

who bring me great pleasure in Life.

Table of Contents

Introduction

1. Lie Groups and Lie Algebras 1.1 Root Systems and Representations 1.2 Embeddings of the Simple Lie Algebras in ~ [ ( n , C) 1.3 The Campbell-Hausdorff Formula

2. Holomorphic Bundles Over Pl 2.1 Holomorphic Principal BundIes Over Pl 2.2 Holomorphic Vector Bundles 2.3 -4ssociated Vector Bundes 1

2.4 The Semicontinuity Theorem

3. Deformations of Bundles Over Pl 3.1 The Transition Function Normal Fonn 3.2 On the Uniqueness of the Xormal Form 3.3 The Transition Matrix of the Endomorphism Bundle

4. The Section Matrices 4.1 The Construction of the Section Matrices 4.2 some Applications 3.3 A Reformulation of the Section Matrices

5. Some Invariants 3.1 Formal Xeighbourhoods 5.2 The Graph and the Muitiplicity 5.3 The Divisor of Jumping Lines

6. The Cascade of Bundles 6.1 The Cascade of Bundles 6.2 An Application to SU(2)-instantons

7. fiamed Jumps 7.1 Framed Bundles Over Pz and F'ramed Jurnps 7.2 Framed Bundles Over Pi x U 7.3 Isomorphic Framed Bundles 7.4 Deformations of Framed Jumps 7.5 Bundles of Minimal Type

Conclusion

References

Introduction

The study of various moduli spaces of holomorphic bundles has had a rich and

interesthg history. In 1957, Grothendieck [Gr] classified the set of holomorphic

principal bundles over the Riemann sphere; they may be considered as elements of

the integral lattice of the structure group, which is a discrete set. Versions of this

theorem go back to Birkhoff [Bi] in 1913. Soon after this, the space of holomorphic

vector bundles over an elliptic c w e was described by Atiyah [At 11. He found that

the space of indecomposable bundles of given rank and degree could be identified

with the elliptic curve itseif. The problem of classiS.ing bundles over Riemann

surfaces of genus g 3 2 proved more dificult, and contributions were made bom

many sources, beginning with the classical paper of Weil [We].

One difficulty with the case g 2 2 is that the space of al1 bundles is not a

Hausdorff space. One way to get around this problem is to consider the space of

stable bundles: a concept introduced by Mumford. Narasimhan and Seshadri [NaSe]

made the usehl discovery that vector bundles are stable if and oniy if they arise

from irreducible unitary represeatations of certain discrete groups which are closely

related to the fundamental group of the Riemann surface, and this was generalised

in 1975 to principal bundles by Ramanathan [Ra]. Eventually it was possible to

give an explicit formula for the Betti numbers of the moduli spaces of stable bundIes

(see [DeRa]). A comprehensive account of the study of holomorphic bundles over

Riemann surfaces was given by Atiyah and Bott [AtBo] in 1982.

By this t h e , a new interest in the subject had been aroused by the discovery

(see [At HiSi] ) of a correspondence between solut ions to the Yang- Mills equat ions

over the 4sphere and certain holomorphic bundles over the complex projective space

P3. Furthermore, a monad description of these bundles led to a complete set of

fairly explicit solutions to the self-duality equations for the compact classical groups

(see [AtHiDrMa] or [At 21). Atiyah and Jones [AtJo] studied the global topology

of instanton moduli spaces and conjectured that the inclusion of the space A& of

based instantonç of charge k (that is framed anti self-dual connections on a principal

bundle P over S* with c2(P) = k) into the space Bk of all kamed connections on

P should induce the homotopy equivalence xq(Nk) "- rq(Bk) for sd5ciently large

values of k. In 1984, Donaldson [Do] simpmed things by showing that the instanton

bundles over Pg were determined by their restrictions to P2, and the study of based

K-instantons over S4 was reduced to the study of fkamed G-bundles over Pa, where

G is the complexification of the compact classical group K. This simplification

was exploited by Boyer, Hurtubise, Mann, and Milgram ([Hur] and [BoHuMaMi])

to prove the Atiyah-Jones conjecture for SU(2)-instantons, and later by Youliang

Sian [Ti 1,2] to extend this to the compact classical groups. Hurtubise described

the ra&-2 instanton bundles in terms of framed jumps, while Youliang Tian used

Donaldson's monad description of the instanton bundles. W e the monads were

easier to work with for higher rank bundles, the framed jump approach yielded more

detailed information about the topology of the moduli space.

This brings us to the present thesis. Our goal is to extend some of the tech-

niques and results developed in [Hur] and [BoHdvlaMi] to the case of higher rank

bundies and, when possible, to the case of principal bundles. It is hoped that t hese

techniques will prove useful in the study of instanton moduli spaces, and other mod-

uli spaces (such as uniton moduli spaces [An]) which may be described in terms of

holomorphic bundles.

Our primary objects of study will be deformations of principal G-bundles over

the Riemann sphere Pl, where G is a complex reductive group, or in other words

G-bundles over Pl x U where U is a small open set in Cm. In particular, we shall

study framed jumps (that is, germs of bundles over Pi x U , where U is a complex

disc centred a t the origin, which are trivial over Pl x {x) when O # x E U and

which are equipped with a harning over {z = O)). The reason for studying framed

jumps is that they may be glued together to form kamed bundles over various two-

dimensional complex manifolds, such as ruled surfaces (see [HuMi] for a proof of

the Atiyah-Jones conjecture for SU(2)-instantons over ruled surfaces).

The first chapter provides a review of some of the theory of Lie groups and Lie

algebras and their representations which will be of use tu us in the study of princi-

pal G-bundles. We shall describe explicit representations of the simple algebras as

matrix algebras. and some of these wili appear intermittently in examples. In par-

ticular, we shail be using a variant matrix representation of the speciai orthogonal

and symplectic algebras in which the Borel subalgebra consists of upper triangu-

lar matrices. We shall also describe an offshoot of the Campbell-Hausdorf formula

which relates the conjugation of elements in a Lie group with the Lie bracket of

elements in the Lie algebra.

Before discussing bunàles over Pl x Li we must fmt deal with bundies over Pi.

The second chapter provides a leisurely review of the subject of bundles over Pl.

The fundamental result is Grothendieck's theorem [Gr] which states that. for a

reductive group Go the space of principal G-bundles over Pl is the discrete set

-1 n where .l denotes the integral lattice of G and denotes the closure of the

fundamental Weyl chamber. In the case of rank-n vector bundles. the splitting type

X E :i n W of a bundle F over Pi determines, and may be determined frorn. the

dimensions of the cech cohomology spaces Hl ( P l , E(2)) where E(1) = E O(I).

We shall also discuss a natural partial ordering on the set of G-bundles over Pl and

we s h d use this ordering to reword the sernicontinuity theorem so that it applies

to principal G-bundles over Pi x 5.

, h y G-bundle E over Pl x C'. where C: is a polydisc in Cm. may be described

bu a single transition function o : C* x C + G. The main result of the third chapter

states that if the polydisc Ir is sufficiently s m d then we ma>- take O to be in normal

form. that is of the form c i = exp(X in r) exp(P) for anq. X 2 Eo in :\ n YV and for

some P : G + H1(Pl . AdX) where AdA denotes the (sheaf of sections of the) vector

bundle over P associated to the G-bundle X under the adjoint representation. This

result was proven in the special case that G = SL(2 , C) by Hurtubise in [Hur]. In

the case in which Eo = A, our result arnounts to a concrete realization of a versal

deformation of Eo whose existence was stipulated by Ramanathan in [Ra]. One

consequence is that the structure group of E may be reduced to a Borel subgroup

Given a vector bundle E over Pl xU and given a vector bundle X over Pl

with X 3 Eo, there exist iinear maps Si(z) : HO(Pl , X ( 1 ) ) + H1(Pi . X(1)) with the

property that HO(Pi, E=( l ) ) may be identified with the kernel of Si(z). These maps

dl be the subject of the fourth chapter. We shall show how to construct these maps

explicitly from a normal form transition matrix of the bundle E. Our construction

of these section matrices S l ( z ) was motivated by the construction of the section

matrix S-i (1) which appears in [Hur] in the case that G = SL(2 , C). The section

matrices may be used to calculate explicit bases for the spaces H o (Pl . E x ( [ ) ) . or

they may be used to compute the splitting type of E, for any x E C .

In the fifth chapter we shail discuss some invariants of bundles over Pl x C. The

section matrices play a ubiquitous role in the calculation of these invaxiants from a

normal form transition matrix of the bundle. For a vector bundle E over Pl x C wit h

CT c C. we let E(" denote the restriction of E to the kth formal neighbourhood

of the line Pl x { O } in Pl x C:. The dimensions h o ( P i . E ( [ ) ( ~ ) ) are then integral

inm.riants of the bundle E. If E, is trivial whenever x # O then for large values

of k we fhd that hO(p i . E ( - I ) ( ~ ) ) = h l ( P l x c E(-1)) and we c d this integral

invariant the multiplicity of E. .Zn invariant of a different nature is the divisor of

jumping lines J of a bundle over Pi x LT where C C Cm. For a vector bundle E with

q ( E ) = 0. J is defined to be the divisor of zeroes of the determinant of the section

matrix S-l (E). We also define the divisor of jumping lines of a G-bundle E over

Pa, and n-e use methods similar to those of Barth [Ba] to show that the degree of

J is equal to the second Chern claçs of E (Barth proved this in the case of rank-2

vector bundles).

Ln the sixth chapter we s h d discover some new invariants for a vector bundle

E over Pl x l' where u' c C. CVe shall introduce the cascade of bundles below

E. which is a collection of bundles EI over Pl x17 each of which is deterrnined up

to isomorphism from the bundle E. From these bundles we obtain the additional

integal invariants h O ( P l , E ~ ( I ) ( ~ ) ) of the bundle E . We s h d show that these

inva.riants detexmine and may be deterrnined by the splitting types Ar of the bundles

Er restricted to Pi x (O).

In the final chapter we shall consider framed jumps, that is bundles over Pl x U

where C/' c C which are equipped with a fiaming over {O) xU and which are trival

over P 1 x {x) when x # O. A h i t e collection of framed jumps Ei with multiplicities

m, rnay be glued together dong their fiamings to obtain a fiamed bundle E over

Pz with c z ( E ) = Cmi. This means that framed jumps may be used to study

the moduli space Mt of fiamed bundles E over Pz with cZ( E) = k (which is

diffeomorphic to the moduli space .t; in the case of the classical groups). tVe s h d

show t hat a fiamed jump E may be @en by a transition function in the normal form

o = exp(Xln;)exp(P) whereA = Eo in Anwandwhere P : I F -r H1(Pl.AdA(-1))

with P(0) = O. We s h d consider the question of which normal form transition

functions yield isomorphic fiamed jumps and we shall conclude that the transition

function of a hamed jump may be truncated to finite order. We s h d also study

deformations of framed jumps. We s h d see that the space of linear deformations

of a given framed jump E is parametrized by the space Hl (Pl x G. AdE(-1)). and

we shall determine how to calculate whkh deformations presert-e the inmriants

h O ( P i . AdE( l ) (k ) ) of E. Finallp we s h d present a detailed malysis of the space of

fiarned jumps E of multiplicity k for which Eo is of minimal type. l é s h d find

that this space is a smooth complex manifold of dimension hl (PI x C: AdE(-1)) - k.

L é believe that this space is open and dense in the space of al1 framed jumps of

multiplicity k. and if it is dense. then the techniques of [BoHublaMi] may be used

to show that H9(.Mr) i H9(.Uk+,) for k 3 l ( q + 1).

6

1. Lie Groups and Lie Algebras

This chapter contains material which will often be used in the sequel. but little

of the material is new. The first section provides a review of the basic theory of

root systems and representations. It is included primarily because it gives us a

chance to establish the notation which will be used later. In the next section. we

describe explicit representations of the simple Lie algebras as matrix algebras. R e

shail be implicitlÿ using some of these representations in numerous examples. It is

perhaps worth emphasizing that uTe shall not be using the standard representations

of the special orthogonal algebra and the symplectic algebra. In the final section we

review the Campbell-Hausdorff formula. which describes the relationship between

the multiplicative structure of a Lie group and the structure of its Lie algebra. and

we shail derive a similar but simpler formula which describes conjugation in the

group in terms of the Lie bradet of the Lie algebra.

1.1 Root Systems and Representations

In this section we provide a bnef review of the theory of root systems and

representations. All of the material presented here may be found in [Bolo [Hum] or

[BrDi].

Suppose that G is a complex semisimple Lie group with Lie algebra g. Let H

be a fked Cartan subgoup of G with Lie algebra B. Then the Weyl group W of G

is defhed to be W- = - V ( H ) / H where : V ( H ) is the normalizer of H in G. The Weyl

group acts on H by conjugation (hence W acts also on 0) and the induced map

M.- + Xut(H) is injective. The Killing form defines a W-invariant inner product on

which may be used to identify with its d u d b*. The roots of G are defined to be those a f b* such that the root space

gP = {x E gl [h, z] = a ( h ) t for al1 h E H)

is not empty. We denote the set of al1 mots by R. These root spaces are all

one-dimensional, and the Lie algebra g decomposes into orthogonal root spaces as

To each root a E $' there corresponds

identification t) S t)' we have

an inverse root ô- E t). and under the

For any two roots a. 3 E R we have a(3) E Z and a - a($)3 E R.

ive shall fix a b a i s A = {a,} c R of simple roots of C. This means that each

root a c m be uniquely expressed as a = C n, a, where either all the n, are positive

in which case o is called a positive root or ail the ni are negatit-e in which case o is

called negative. We denote the set of positive roots by R+ and the set of negative

roots by R-. The Borel subalgebra b of g is defmed by

and by exponentiating a-e obtain the Borel subgroup B of G.

It is possible to find a basis e, for each root space g, so that the Lie bracket

on g is deterrnined bÿ the relations

1 c , je ,+j if a + 3 is a root.

[ e , . e3 ] = â i f c r + 3 = 0 .

O otherwise. and

[h . e ,] = a ( h ) e , .

with c , ~ = - c-,-a f 2. If G is semisimple then the roots wiU span and the

inverse roots a-iii span b' . In this case, the basis {â. a E A: e,. a E R} is cailed the

Chedley basis of g.

Let bR denote the span over R of the inverse roots of G. The ba is A determines

for us the so called fundamental Weyl chamber

W = { h E f i ~ J a ( h ) > O for a l l a E A}

which is the open convex set whose waiis are the hyperplanes ker(a) c 4 ~ . It turns

out that the Weyl group W is the group generated by the reflections in the waiis

of W. that the W-orbit of any point in hR m e t s the closure of W in exactly one

point, and that the Weyl group permutes the inverse roots of G.

LVe introduce several lattices in [IR- Let ï denote the inverse root lattice. let ."\

denote the integral lattice. and let C denote the central lattice:

ï = the integral span of the inverse roots . 1

-1 = -ker(exp: -, H) . and 2 AZ

X = { h E b l o ( h ) E Z for all a E R } .

W e have r c ;\ c S. If G is simply connected then = .\. More generally. if G

denotes the universal cover of G and if Z ( G ) denotes the centre of G. we have

(G) = A/r. Z(G) = C/.l . and

2(G) = s/r.

Dually, we have the root lattice ZRo the lattice of integral forms Il. and the weight

lat tice R. which aii lie in the real span of the roots of G:

ZR = the integral span of the roots.

Il = {a E f)'/w(A) E Z for di X E -1) . and

Q = {d E t f lw (6 ) E Z for ali cr E R ) .

If G is simply connected then II = R. The simple roots {a i } make up a Z-basis for

ZR. and they also determine a Z-basis {éi) for R. The e; are cailed the fundamental

dominant weights. and they are defined by e i ( â j ) = hi j . -4 weight u E R is called

dominant if w(hi) 2 O for al1 i.

A finite dimensional representation p : G + End(V) of the Lie group gives rise

to a representation p : g + End(V) of the Lie algebra g. If we restrict this to b then

a e obtain the decomposition V = @ V, where R ( p ) c II is the set of weights of wf Qb)

p and the V, are the weight spaces

V, = { v E Vjp(h)v = w(h)v for a l l h E f ~ ) .

The dimension of the weight space Vu is called the multiplicity of the weight d.

The set of weights of p contains a unique weight w, with the property that u, - i

is a sum of positive mots for ail other E f l ( p ) . This weight w, is a dominant

weight. and it is called the highest weight of the representation p. Conversely.

given any dominant weight w E II there is a unique finite dimensional irreducible

representation p with w, = u. The highest weight has multiplicity 1. and the

mu1 t iplici t ies of the ot her weights may be calculated recursively using Freudent halas

formula (see also [BrMoPa] for tables of weight multiplicities ). The dimension of the

representation. which is the dimension of V , is giwn by the s u m of the multiplicities

of the weights ~z E f l ( p ) .

1.2 Embeddings of the Simple Lie Algebras in gl(n. C)

In this section we shall describe explicit embeddings of the simple Lie algebras

into the Lie algebra gl(n. C) of n x n matrices. in each case the embedding is given

by the faithful irreducible representation p of minimal dimension. -And in each case

we may take the Cartan subalgebra to be contained in the set of diagonal matrices.

and the Borel subalgebra to be contained in the set of upper triangular matrices.

This will prove useful later on when working with holomorphic bundles.

The embeddings are all obtained by the foilow-ing process. W e begin with the

abstract description of the root system found in [Bo] or [Hum]. W e find the set of

weights Q ( p ) of the minimal dimensional representation p. These weights are all of

the form w = w, - C nia i , where the ni are nonnegative integers. w, is the highest

weight. and the a, are the simple roots. We d e h e the height of the weight to

be the sum - C ni. and we choose an ordering of the set of weights by height. say

R(p) = {di,. . . , w, } with <*.1 = w,. Wé have chosen this ordering to ensure that

the Borel subalgebra wiU be sent to the set of upper triangular matrices. Once we

know the weights, we know how p acts on the Cartan subalgebra:

In particular, we know how p acts on the inverse rwts â, and this allows us to

consider the inverse mots as diagonal matrices. These matrices are not unduly

complicated (the entries are all O. f-1, or f 2) and it is not difficult to find matrices

eo which. together with the simple inverse roots. form a Chevalley basis for the Lie

algebra. The full details of this process will only be revealed for the Lie algebra G2

as a detailed exposition in all cases would be cumbersome.

In the examples below. Ej denotes the standard basis element for gl(n. C) and

E, denotes the diagonal element E: .

1.2.1 The Special Linear Group: Foilowing the procedure outlined above

yields the standard embedding sl(n, C) c gl( n. C ). FVe review some information

about its root system.

SL(n . C) = {.-I E G L ( n . C)I det .-î = l}

d(n. C ) = {.A E gI(n.C)ltraceA = O}

Simpleinverseroots: ai = Ei -Ei+t ' aith 15 i < n

Positive inverse roots: E, - E,, with i c j

Root spaces: if a = E, - E, then e, = E; and e-, = e,' = E/

Fundament al Weyl chamber:

W = {diag(tl.. . . . tn) l t i E R. ti > - . . > tn. C ti = O }

Integrai lattice:

-1 = {diag(tl,. . . . i ,)lti E 2, C t , = O }

1.2.2 The Special Orthogonal Group: The embedding so(n. C) C d(n. C)

ahich one obtains on applying our embedding procedure is not quite the standard

one. but it is better suited to o u purposes, so we s h d be using an altemate

description of the special orthogonal group. Let J denote the matrix

Given an n x t? matrix -4 we let -4% denote the matrix

Definition: We define the special orthogonal g o u p SO(n,C) to be the set of n x n

matrices which preserve the symmetric form with matrix J .

Our definition for SO(n. C) differs £iom the standard one by conjugating by an

invertible matrix L with Lt J L = I. such as

where the terms in brackets occur only when n is odd. What makes our defini-

tion more convenient is that we may take the Cartan subgroup to be the diagonal

subgroup and the Borel subgroup to be the upper triangular subgroup.

To describe the Cheialley basis for so(n. C) we consider the two cases in w-hich

n is even and n is odd separately.

so(2n + 1, C):

Simple inverse roots:

hi = (Ei - Ei+i + E2n-i+1 - Ezn-;+2), with 1 5 i 5 n - 1

â n = 2 (En - En+î)

Positi~*e inverse roots:

(Et - El + EZn-j+2 - E2n-i+2). ~ith 1 5 i < j < n (Ei + E, - Ezn-l+2 - Ezn-n+2). ~ i t h 1 < i < j 5 n

?(Et - E2n-t+2). mith 1 5 i 5 R

Root spaces:

2n-]+2 if & = ( E , - Ej + E2n-j+2 - E2n-i+Z) then e, = E; - E2n-i+2

if Q = (Et + El - E2n-j+~ - & n - i + ~ ) then e , = E;,-,+, - E;,-,+,

if ci = 2(Ei - Ez,-,+2) then e, = EA+I - E,"+' ,n-:+2

and e,, == e,'

Fundamental iVeyl chamber:

W = {diag(tl , . . . . t n . O . - tn , . . . , - t l ) l t i E R. t l > . - > tn > O }

1.2.3 The Symplectic Group: -4s with the special orthogonal group. our defi-

nition of the symplectic group and its Lie algebra may be slightly unfamiliar.

Definition: The symplectic group Spf3n.C) is defined to be the set of 272 x 2n

matrices whicb preserve the skew-symmetric bilinear form K =

Simple inverse roots:

âi = ( E , -Ei+l + Ezn-, - E2n- i+i ) , with 1 < i 5 n - 1 Gn = (En - En+l)

Positive inverse roots:

(Ei - Ej + E2n-j+l - E2n-i+l) l with 1 5 i < j 5 n

(Ei + Ej - E ~ n - ~ + i - Ezn- i+t ) , with 1 _< i < j 5 n (Ei - & n - i + l ) , with 1 5 2 5 n

Root spaces 2n-j+l if a = ( E ; - Ej + EZ, - ,+~ - EÎn-i+i) then e, = E; - Ezn-i+i

if â = (Ei + Ej - Ea,-j+l - Ezn-i+1) then e, = Ein-j+l + EJ 2n-j+i

if â = (Ei - E2n-i+L) then e , = Ein-i+l

and e-, = e, t

Fundament al Weyl chamber:

W = {diag(ti.. . . , t , . - t , . . . . , - t i ) l t i E R . t l > - - - > ta > 0 )

1.2.4 Gz : Sext, we tum our attention to the exceptional Lie algebra G2. LVe

shall describe an embedding G2 c so(7. C) which is similar to the one given in

[Hum]. -4s mentioned earlier. this is the only example for a-hich we supply the full

det ails of the embedding procedure.

Abstractly. G2 has a root system with basis al = (1.-1.0). a2 = (-2.1.1) E R3.

The set of positive roots is then

The fait hful irreducible represent ation of smallest dimension is the representat ion

given by the fundamental dominant weight €1 = 2ai + a2 = (0. -1,l). W e tabulate

a list of al1 the weights c. of this representation together with the dalues of &(&,).

These weights all occur with multiplicity 1 so the representation is of dimension

7. and the columns w ( â i ) reveal the images of ai in gl(7. C), for example âl is

mapped to diag(1, -1,2,0, -2,1, -1). This enables us to consider the inverse roots

and hence also the roots as diagonal matrices in gt(7, C), and the root spaces can

ensily be found by triai and error:

e,, = Ei +AE:-&E,S-E:

e,, = E: - E;

e,, = E: - E:

and e -,, is the transpose of e,, . Thus the Lie algebra G2 consists of matrices of 6 6

the form A = a& + bâ2 + C C i e p i + .x c - , ~ - ~ , or i= 1 :=I

1.2.5 F4: The Lie algebra FI has a root system with basis ai = (0.1.-1.0).

a2 = (0.0.1. -1). a3 = (0.0.0.1). and a4 = ($. -', -1 -'). F4 mas be embedded 2 2 ' 2

into sl(ll6. C) by means of the representation given by the fundamental dominant

weight €4 = ( 1.0.0.0). which is also the highest short mot. The weights of this

representation are the 24 short roots. each appearing with multiplicity 1. together

with the trivial weight. which has multiplicity 2. By ordering these weights z, by

height (this ordering is not unique) and by caiculating a(&, ) we can determine the

images of the inverse roots in st(26. C):

By trial and error. we can find mot spaces which. together with the inverse roots

ai. generate F4 c ~ ( ( 2 6 , C). 4 n arbitrary element of FI is of the following form.

where the diagonal is a linear combination of the inverse roots.

1.2.6 E, : The Lie algebra E6 may be embedded into ~ ( ( 2 7 , C) by means of the

representation corresponding to the fundamental dominant weight e, . It 'urns out

that the positive inverse roots are all of the form

(1) â = E i i + . . . + E i 6 - E i ; - . . . - Eh2

for some collection of distinct i, with il < - - - < i, and i, < - - - < i12 and i, < ij+,

(not d such combinations occur). For s u c h a root, we may take

(2) ea 17 1 a 112 = ~ ( 1 & ~ ( 2 f . . . f ~ ( 6

and, for the root spaces of the negative roots. we take e-, to be the transpose of e,.

The signs wiii uniquely determined once we have chosen the signs for the 6 simple

roots.

W e summarize the matrix representation E6 c ~ ( ( 2 5 . C) by displaying the the - -

matrix ke,. . where a l . - --- . a,, are the positive mots of E6 (ordered by height ). k=I

From this matrix. it is easy to read off any root space eOk (s imp- find all occurences

of k in the matrix) and hence it is easy to h d any inverse root (ûk and eQk are

related by (1) and (2) above).

1.2.7 E, : The Lie algebra E7 may be considered as a subalgebra of ~ ~ ( 5 6 , C )

through the representation with highest weight e,. It tu rns out that the inverse

roots are ail of the form

and the corresponding root space may be given the basis

For the negative roots we may take e-, = e,'. Ué summarize the embedding

E7 C ~ ~ ( 5 6 . C ) by dispiaying (below and on the following page) a portion of the 6 3

56 x 56 matnx C k e O k . The rest of the matrix may be recovered by the symmetry k=l

of the symplectic aigebra.

18

(The following is to be afaued to the matrix on the previous page)

1.2 -8 E, : The minimal dimensional representation of E8 is the adjoint representa-

tion which has highest weight c8 which is also the highest root. This representation

embeds E8 into sf(248, C ) .

1.3 The Campbell-Hausdorff Formula

The exponential map exp : g + G provides a holomorphism between a neigh-

bourhood of O in g and a neighbourhood of 1 in G. Given -4. B E 0 with exp(C) =

exp(;l) exp(B). the Campbell-Hausdofi formula is a formula for C in terms of -4

and B and the Lie bracket in g. Thus the multiplicative structure of G is determined

locally by the Lie algebra g.

To obtain the formula (see [Ja] ). one may a-ork in the universal enveloping alge-

bra of g where one can define exp(.-L) and log(1 + -4) to be given by their standard

expansions. One formally expnds the expression C = log( 1 + (exp( A) exp( B) - 1 ))

and verifies that C is a Lie element. The Specht-Wet-er criterion for a Lie el-

ement (see [Ja].page 169) states that C will not be changed - replacing each

monomial of the form -4"s Bbl A a 2 ~~2 --- . Aan B6n in the expansion of C by the

expression r- [ 4 B - * - 4 B n where in the latter expression we have

used the iterated Lie bracket notation: for example [A] = A. [BI = B. and

[-4B3 --l'BI = [[[[[[A B] B]B].-L]A] BI. The resulting explicit formula for C is known

as the Campbell-Hausdorff formda.

1.3.1 Theorem: (The Campbell-Hausdorff Formula) Lf -4.B E g and if C is

given b~ exp C = exp -4 e'rp B then

where [(a) is the length of the mdti-index a.

This formula describes the relationship between multiplication of elements in

the group G and the Lie bracket of elements in g. We would like to use these same

ideas to find a simpler formula describing conjugation in G in terms of the Lie

bracket in g.

1.3.2 Proposition: (The Conjugation Formula) If A. B E g and if C is given

where [B .-Ln] means [- - - [[B rl].4] - - . -41.

Proof: tVorking in the universal enveloping algebra. we have

( - 1 ) ' 3 C ( n + l ) i ! ~ ! [A'B AJ] . by the Specht-Wever cntenon n = O i+j=n

W7e would like to make note of one consequence of the conjugation formula a-hich

wiil often be used in the sequel.

1.3.3 Coroiiary: Let P E g and let A E .\. where :\ denotes theintegral lattice

of G . Suppose that Q is @en by

a-here ln z denotes the multivalued logarithm. Then, if w e write P = Ph + C Pde,

and Q = Qh + C Q,e,, a-e have aER

Q, = p ( * ) p , and Qh = Ph.

ProoE Since [Ph A] = 0. and since [e, A] = +A). the conjugation formula gives a0

Q, = C *n[~h An] lnn z = Ph , and n=O

2 1

2. Holomorphic Bundles Over Pl

Holomorphic bundles over the Riemann Sphere P, are weil understood. The

foundational result is Grothendieck's theorem which states that the set of holomor-

phic principal G-bundles is the set A VV where .\ is the integrai lattice of G and

W is the fundamental Weyl chamber. There is a natural partial order on this set

which arises in the study of loop groups. These things s h d be discussed in the fist

section. In the se5ond section aie will specialize to holomorphic vector bundles. for

which Grothendieck's theorem states that al1 vector bundles split into a direct sum

of line bundles. Here we aiso study the spaces of holomorphic sections of bundles

in relation to the ordering on the set of bundles. In the following section we follow

the passage from a principal G-bundle to a vector bundle through a representation

of G. And in the £inal section w e give a restatement of the semicontinuity theorem

which applies to principal bundles.

2.1 Holomorphic Principal Bundles

Let G be a connected reductive complex Lie group. -4 holornorphic principal

G-bundle E over P may be git-en by a single transition function o : Co f l + G .

where = Pl \ {oc} and Lwx = PI \ {O}. A change of triiialization over t o

or L, will alter the transition function O but not the isomorphism class of the

bundle E. and so the set of isomorphism classes of G-bundles over Pl is given by

the cohomology set Hl (P l . O ( G ) ) . A theorem of Grothendieck [Gr] asserts that

where H is the Cartan subgroup, and W- is the Weyl group. Thus we can take the

image of o to lie in H and we are still £re to reduce further by the action of W.

We have a natural isornorphism C* Bz A + H given by ( z @ A ) ++ exp(X ln z)

where A is the integral lat tice of G and in z denotes the multivalued logarithm. This

gi ves

These identifications preserve the action of the Weyl group W, and the point A E A

corresponds to the bundle E with transition function d = exp ( A h r) : C' + H. t -

B y acting on A by the Weyl group we may assume that X E A n W , where W is the

fundament al Weyl chamber. The point X E -1 i l W is caiied the (holomorphic ) type

of the bundle E.

The topological type of a G-bundle E over Pl is determined by its first char-

acteristic class c , ( E ) E H2(Pl. rl(G)) = n1 (G) (see [Ra]): if G is the universal

covering group of G then the exact sequence 1 -. a,(G) + G -r G -+ 1 gives rise to

the coboundary map Hl (P, , Q(G) ) -% HZ (Pl, nl (g)) which sends the transition

function o : Ca -r G to the corresponding element in r1 (G). Gnder the identifica-

tions H1(P1? O(G) ) = A n W and xl(G) = *\/I' (recali that l? denotes the integrai

span of the inverse mots of G) the coboundary map -1 n W + :1/r is given b~

X - X + r. Thus the set of holomorphic G-bundes of a given topological type

X + r E a,(G) is the set ( A + I') n W c .\. Let us summarize.

2 .l. 1 Theorern: (Grothendieck) The set of isomorphism darses of holomorphic

G-bundles over P, is given by

The point X E A fi LV corresponds to the bundie E given by the transition function

The topological type of tti is buncüe is

There is a partial ordering on the set .\ n W which is dual to the ordering on

weights used in representation theory.

2.1.2 Definition: We put a partial order on the set A n W by defining A c p

whenever p - X is a sum of positive inverse mots.

This ordering also arises in the ceii decomposi tion of loop groups (see [Pr Se] ).

For a compact group K. the loop group R K may be decomposed into finite codîmen-

sional strata CA indexed by the set :ln with C, C FA if and only if A < p.That

this ordering is also natural in the context of holomorphic bundles can best be seen

through the correspondence beta-een holomorphic bundles and loop groups [PrSe]:

if is a compact group whose complexïfication is G. then a point in the loop group

RK may be identified with a pair (E. r ) where E is a holomorphic G-bundle over

P , and T is a trivialization of E over C,. The topological type of the bundle E

is determined by the connected component of Rh- in which (E. r ) lies. and the

holomorphic type of E is determined by the straturn X, containing ( E . T).

Let us make a few remarks about this partial ordering. Recall that the set

11 n W is partitioned by topological type into components ( A + î) n W labelled by

1 / The partial ordering does not compare element s fiom distinct components.

When restricted to any one of the components. the ordering may be given several

alternate descriptions.

2.1.3 Proposition: Let A, p E .l n W with X + ï = p + ï. Then the following

are equivalent:

(4) (r. A) < ( r . p) for aU r in the fundamental Weyl chamber W .

( 5 ) ci(X) 5 e i ( p ) for aü i .

Here R+ denotes the set ofpositive roots, Conv(WA) denotes the convex h d of the

CV-orbit of A, and the ei are the fmdamental dominant a-eights.

Proof: The proof is not difficult and may be found in [BrDi] or in [AtBo]. O

kioreover, each set ( A + r) n W becomes a lattice in the sense of having meet

and join operations: if a, is a basis of simple roots, then the meet operation tl is

and the join operation u is given by taking rnax(ni. mi). It can be seen that the

convex hull of W(X n p ) is the intersection of the convex hulls of W A and W p . Each

lattice has a unique minimum element A,. and every element of (A, + ï) f~ VV is

then uniquely expressible as X = X, + C nial with the ni necessarily positive. The

sum C nt dl be c d e d the height of A.

2.1.4 Examples: One can draw a picture of the lattice of S L ( n . C)-bundles

(or the lattices of G L ( n . C)-bundles) by noting that X = diag(t,. . . . . t,) aill lie

immediately below p = diag(sl . . . . s, ) provided that eit her

( 1 ) si 2 s i+ , + 2 and X = p - cri . or

( 3 ) 5 , - 1 = = ... = sj-i = s - - 1 and X = p - (a i + ai+, + - - - + CI,-,).

1

ahere the ai are the basic roots.

Here is a picture of the lat tice of S L ( 4 . C) bundles beneat h X = diag(4.0.0. -4).

Energy

The lat t ice of Sp(2n, C)-bundles is the sublat tice of self-dual bundles ( that is

bundles of type X = diag(t,. . t z n ) with t2n-i+1 = - f i for all i ) in the lattice of

SL(2n. C)-bundles. Here is a picture of the lattice of Sp(4. C )-bundles:

Sow SO(n. C) is not simply connected. in fact a,(SO(n. C)) = 2,. and so the

set of SO(n.C)-bunciles breaks into two components. For example. here are the

two lattices of SO(4, C)-bundles:

2.2 Holornorphic Vector Bundles

More concretely. we may use a representation G + G L ( n . C) to convert a

principal bundle E into a holomorphic vector bundle over Pl . CVe shall on occasion

use the embeddings of the classical groups described earlier to think of G-bundles as

vector bundles. We shall also use the adjoint representation to produce the bundle

of g - d u e d endomorphisms of E. which we shall denote by AdE. For this reaçon it

is instructive to consider the case G = G L ( n . C ) in more detail.

The integral lattice of G L ( n . C) is the set D(n. 2) of integer-dued diagonal

matrices and the fundamental Weyl chamber is the set V of r e d d u e d diagonal

matries with entries in decreasing order. So the set D( n. 2) n Ü is the set of all n x n

integer-valued diagonal matrices X = diag(t . . . . . t , ) wit h t 2 - - - 2 t,,- and the

). vector bundle of type X may be given by the transit ion mat rix O = dia& z t . . . . . - Thus Grothendieck's theorern tells us that a r d n holomorphic vector bundle E

splits holomorphically into a direct sum of line bundles

where O([) denotes the line bundle with transition matrix t = (z-' ).

The first characteristic class cl(E) of a vector bundle is called the f i s t Chern

class: for the bundle of type X as above it is given by

\Vhen working with a vector bundle E a-e may also study its s h e d of holo-

rnorphic sections (which a-e shall also denote by E). Let us compute the cech

cohomology Hq(P,, E ) using the open cover Co. Cm of Pl . In terms of the transi-

tion function @ : uo n U- + GL(n. C) a global section of E is a map u : Go + Cn

holornorphic in z such that 4 u is holomorphic in z-' . If E = O ( [ ) then 4 u = z-'u

and in order for this to be holomorphic in z - l . either u = O or we must have

1 2 O and u = u, + --• + u l z L . Now an element of H1(Pl,E) is given by a map

v : LTo nU, + Cn and we are free to add u, and d - l u , for any maps uo : Uo + Cn

and u x : LI, + Cn without altering the class of c in H1(P1. E). If E = O(I) then

0 - l ~ ~ = i iuri . So if 1 >_ -1 then v may be identified with O and if Z 5 -2 then v

ma'- be put into the normal form v = c - , z-' + c - , z - ~ = - + L . ( + ~ Z ' + ' . Thus we

have

~~(~,,C3(l))=span{~.~.----.~') f o r 1 1 0 and

and in general. for the bundle E of type X = diag( t , ---- - . t n ) we have n

H0(p1. E) = @ H o ( P , . O ( - t , ) ) and i= 1

-4s an irnmediate consequence. we obtain the follon-ing formulas in\-olving the

dimensions of the cohomologv groups. E(2) denotes the bundle E 2 O([) which is

of the type A ( / ) = diag( t , - 1. - - -- . t , - 1 ) and E* denotes the dud bundle which is

of type A' = diag(-t , . - - - , -t, 1.

2.2.1 Proposition: Let E be the GL(n. Cl-bundle of type X = diag(t,. - - - : t , ) .

Then

Serre duality: h O ( p I . ~ ( 2 ) ) = h l ( p l . E*(-1 - 2)) . and

Riemann-Roch: h O ( p l . ~ ( 1 ) ) - h l ( p , . E(1)) = n(1 + 1) + cl ( E ) .

W e have just seen that the splitting t-ype of a bundle determines the dimensions

of al1 of the cohomology spaces. On the other hand. if we know the dimensions

h O ( P l o E(1)) for all 2 , then we can recover the splitting type of the bundle:

2.2.2 Proposition: Let E be the GL(n'C)-bunde of type X = diag(t, , . . , t , ) .

Then for any integer 1, the number of indices i s u d that ti = 1 is given by

Proof: We have

= C ( Z + l - t i ) - 2 C (1- t i )+ C (Z -1 - t i ) i3ti<l+l i3t; 51 i 3 t , g - 1

= (Cl + C, + C,) - 2(C, + C , ) + C, ,

where Cl = C Z + 1 - t i , C , = 1 + 1 - t i , C , = 1 + 1 - ti' and i3tiSl-1 i3ti=l i 3 t i = l f 1

C,= C 1 - t i , C , = C 1 - t i , a n d C 6 = C 1 - 1 - t i . Butwehave i3t;Sl-1 i3t,=l i3ti si-1

C , = C , = 0. and C, - 2C, + C, = 0, and C, = #{i 3 ti = 1 ) . This proves

the first equality, and the second equality follows directly from the first using the

Riemann-Roch formula. O

Now let us explore the ordering on the set of vector bundles over P,. The set

A nW is partitioned into disjoint open sets labeiled by the Chern class. The partial

order makes each of these sets into a lattice (with meet and join operations), and

does not compare elements with distinct Chern class. One may draw a picture of

each of these lattices as was done for SL(n, C)-bundles. For a given rank n and

Chern class k: the minimum element in the lattice is X o = (r + 1 , . . . . r + 1. r. . . . . r )

where r is the largest integer with r 5 kln. And we may give an alternative

description of the ordering in terrns of the cohomology groups.

2.2.3 Proposition: Let E and F be two GL(n, C)-bundles over Pl of types

X = diag(t,, - , t,) and p = dia&, , - , s,) respectively with cl ( E ) = cl ( F ) . Let

A* and p* be the types of the dual bundles E* and F*. Then the following are

eq uivalent .

(3) t l + - - + t , p s i + - - + s , for1 5 1 < n .

(4) h0(p,,E(1)) 5 h o ( ~ , , F ( l ) ) foralll .

( 5 ) h ' ( ~ , , E(1)) 5 h1(p1, F(1)) for d l 1 .

(6) A* S p * .

Proofi From proposition 2.1.3 we know that X 5 p if and only if €,(A) 5 e i ( p ) for

each of the fundamental dominant weights c i . For G L ( n . C). as for SL(n. C). the n-i - i fundamental dominant weights are given by c, = diag ( 7. - - - - - , .,,! - - - - O n 5)

n-t where 7 is repeated i times. So

1 ~ ~ ( ~ ) + - c ~ ( E ) = t , + - - - + t ~ n and

1 ~ ~ ( p ) + - c l ( F ) = s , + - - ++

n

Since w e have assurned that c, (E) = cl ( F ) n-e see that ( 1 ) is equivalent to (2) .

The equit-alence of (2 ) and (3) follows easily from the identities

t 1 + - - - + t l = c l ( E ) - ( t l+ , + ... + t , ) and

s1 + - - - + s l = c , ( F ) - ( s l f l + a - - + s n ) .

Sext we shall show that (2) is equivalent to (5). Suppose that (2) is false. and

choose k such that t , + - - - + tl 5 sl + . - . + s i whenever 1 < k but t l + S . - + t k >

s1 + .. - + s,. Then a-e must have tk > sk. and so { i 3 t , > sk} > ( 1 . - - - . k} w-hile

{i 3 si > s k } C (1. - - . . k - l}. Thus

So we have h l ( P l . E(sk-1) ) > h l ( P l , F(sk - 1 ) ) and ( 5 ) fails to be true. Conversely.

suppose that (5) is false. Choose Z so that h l ( P , , E(Z + 1 ) ) 5 h l ( P , F(1 + 1 ) ) but

h l ( P , . E(1)) > h l ( P , ? F(1)) . Gsing the identities

~ ' ( P , . E ( I ) ) = C ( t i - 1 - 1 ) = # { i 3 t i ~ l + 2 ) + C ( t i - Z - 2 ) i 3 t ; 11+2 i3ti>l+2

= ~ ' ( P , , E ( z + 1 ) ) + #{i 3 ti 1 1 + 2 ) and

we see that {i 3 s i 2 2 + 2} c {i 3 t i 2 1 + 2) . So we have

Hence C t i > C s i . showing that (2 ) is false. i3t; >1+2 i3t; >1+2

The e q u i ~ d e n c e of (4) and (a) foilows immediately from the Riemann-Roch

formula. and the equidence of (5) and (6) follows from Serre duality. CI

2.3 Associated Vector Bundles

Let G be a semisimple group. Given a G-bundle E over PI a-ith transition

function O : ri C, -+ G and given a represent ation p : G -+ G L(n. C ) the

associated vector bundle p ( E ) is the G L ( n . C)-bundle given by the transition matrix

p o. The infinitesimal version of the representation is a map p : g -+ pl( n . C ). I f *-e

choose a basis for Cn consisting of weight vectors then p( h ) = &ag(;, ( h ) . . - - . d ( h ) )

for al1 h E 4. where the di are the weights of p which ali lie in the lattice of

integral forms II. In particular p sends the integral lattice of G to the integral

lattice D(n.2) of G L ( n . C ) . I f the G-bundle E is given by the transition function

o = exp(X ln =) then the associated bundle p( E) is given bÿ the transition matrix

p O = exp(p(X) ln z).

2.3.1 Proposition: I f E isaholomorphicpnncipaiG-bundleoi-erP, of type

X E -1 n and if p : G + GL(n . C) is a representation of G then the associated

vector bundle p(E) is given by

a-here R(p) is the set of weights of p.

In particular, when p is the adjoint representation, we obtain

AdE = @ O(-&(A)) @ O(0)GGm H. a E R

It should perhaps be mentioned that, even if p is an embedding of G into

GL(n, C), then although the map p : A + D(n, C) will be injective, the rnap

p : A n W + D(n, 2) n v sending a G-bundle to its associated vector bun-

dle need not be injective; for example, the two SO(2n, C)-bundles of types A =

diag(l,.. . , 1, 1, -1, - l y . . . , -1) and p = diag(l,.. . ,l, -ly 1, -1,. . . -1) are not

isomorphic, but they become isomorphic when treated as GL(2n, C)-bundles.

Nonetheless we can use associated bundles to destinguish betnreen non-isomor-

phic G-bundles (see [AtBo]). Let cl , - - - , c, be the fundamental dominant weights

of G and for each i let pi be the representation of highest weight nié, where ni is

the smallest positive integer such that niei lies in the lattice II of integral forms.

For the vector bundle E = O(-t,) @ - - - @ O(-t,) over P l , with tl 2 - . - 2 t,,? we

shall define IE[ = t , .

2.3.2 Proposition: Let 3 and F be two G-bundles over PI. Then

(1) E = F if and ody if I P i ( ~ ) I = I p i ( ~ ) I for al1 i , and

Proof: Since niei is the highest weight of pi we must have Ipi(E)I = niei(X)

and Ipi(F)I = niei(p) where X and p are the respective types of E and F. So

Ipi(E)I = Ipi(F)I if and only if e i ( X ) = ei(p), and this is true if and only if A = p

since the ei span b*. And Ipi(E) 1 5 Ipi(F)I if and only if ei(X) 5 ci(p) which is true

if and only if X 5 p by proposition 2.1.3. O

2.4 The Sernicontinuity Theorem

We r e c d the semicontinuity theorem for vector bundles E over Pl x U , where

U is a polydisc in Cm. For x E U, let E, denote the restriction of E to Pl x {x}.

2.4.1 Theorem: (The Semicontinuity Theorem for Vector Bundies) Let E be a

vector bundle over Pl x U . Then for any k E Z the sets {z E U 1 hq(Pli E,(I)) 2 k }

are closed analytic su bspaces of U for q = 0 , l and for al. 1 E 2.

After we have described the section matrices of a bundle in section 4.1 we s h d

provide an elementary proof of this well known theorem. But at the moment we

would like to give a restatement of the semicontinuity theorem which applies to

principal bundles.

2.4.2 Theorem: (The Semicontinuity Theorem for G-bundles) Let E be a G-

bundle over PlxU. Then for any G-bundle F over Pl the set {s E UIE, 2 F} is

a closed anaiytic su bset of U.

Proof: Recall that (by 2.3.2) we have E, 2 F if and only if Ip,(E),I 2 Ipi(F)I for

al1 i. Let k, = Ipi(F)I. Note that Ip,(E)=I 2 ki if and only if Ipi(E),(ki - 2)1 2 3

a-hich is true if and o d y if hl(Pl ,p i (E) , (k i - 2 ) ) 2 1. So

which is a closed analytic variety by the semicontinuity theorem for vector bundles.

-4lthough the proof is now complete, we would like to show that this theorem

not only follows fiom theorem 2.4.1 but also impiies it thus showing that 2.4.3

is a restatement of 2.4.1 in the case that G = GL(n7 C). So we suppose that

{x E U 1 E, > F) is a closed analytic subset for any GL(n, C)-bundle F over P , . In the lattice D(n, 2) f l V of GL(n, C)-bundles, if A lies directly above p then either

h O ( P l . A) = h O ( P l , p ) or hO(P,, A) = 1 + hO(P,+ ), In particular every bundle X

with h O ( P l , A) > k lies above some bundle p with ho ( P l , p ) = k (provided that

k 2 h0(P17 A , ) where A, is the minimum element of the lattice of bundles with first

Chern class equal to cl(X)). So we let .F be the set of bundles F with cl (F) = cl (E,)

such that ho(Pl , F(2)) = k . Then 3 is a finite (possiblp empty) set. and

(if 3 is empty then we take the union to be the whole set Li) which is a closed

analytic set. O

33

3. Deformations of

Throughout this chapter we let G be a

and a-e let E be a G-bundle over Pl xC'

Bundles Over

connected reductive complex Lie group

where C is a polydisc in Cm. Let r

denote the standard affine coordinate on Pl and let I = ( rl . - - - . x m ) denote the

coordinates in C. The bundle E may be given by a single transition function

o : L-o n CF, -r G. u~here Co = { z # x} and L-, = ( z # O}. For x E Ci. the

restriction of the bundle E to Pl x {r } will be denoted by E,. and the isomorphism

class of Ez aill be denoted by t'pe(E,).

It is known (see [Nase] or [Ra]) that Hl (Pl . AdA) is a local versal deformation

space for the G-bundle of type X E .! n VV over P l . This means that there is a

bundle E (called the versal deformation) over P, xH1(Pl. AdX) which is universal

in the following sense: if E is any bundle over P, xC with Eo of type A. then for

some 1- c C the restriction of E to Pl x I d v is the pullback of E via some holomorphic

map P : 1,' -, H L ( P , . AdX). Our primary goal in this section w-ill be to explicitly

describe such a versal deformation E by exhibiting its transition h c t i o n . This

\vas done in [Hur] for the case in which G = SL(2 . C). and the transition function

proved to be an induable tool in the study of deformations of SL(2 . C)-bundles.

11-e exhibit the transition function of E in the first section. do this by

showing that a G-bundle over Pl x C may be given locaily by a transition function

of a particular form. which we c d the normal form. The next section probes

the extent to which the normal form ma)- be considered unique. And in the final

section we give an expression for the transition matrix of the bundle AdE of 0-dued

endomorphisms of E.

3.1 The Transition Function Normal Form

Recall that, by proposition 2.3.1, if X E A n W is a G-bundle over PI then AdA

is the GL (0)-bundle over Pl given

AdX = @ O(-a(X)) @ 6(0)@ dim . a E R

An element u E H1(Pl.AdX) = @ H1(Pl.C3(-a(X))) can be represented by a oER+

g-valued function (in fact a n-valued h c t i o n , w-here n is the nilpotent subalgebra

n = @ go) u : C* + g of the form a E R +

3.1.1 Deflnition: Let E be the holornorphic G-bundle over Pl x H1(P , . AdX)

Sotice that cP takes values in the Borel subgroup of G.

The buk of this section dl be devoted to proving that E is a versal deformation

for the bundle over Pl of type A. and indeed any bundle of type less than A:

3.1.2 Theorem: Let E be a G-bundle orer Pl xC aith type(Eo) 5 A. Then

for some polydisc V c l - . the restriction of E to P l xCr may be obtained by p u h g

back £ via some holornorphic map P : C' + H 1 ( P l . .AdA). Moreoi-er a-e will have

type(E,) = A if and O&- if P(0) = 0.

The bundle E over P l x 1:- pulled back from E via P : V + H L (P l . AdX) is

given by the transition function

.A transition function of this form will be said to be in A-normal form. Thus

another way of stating the above theorem ~ ~ o u l d be to say that the bundle E may

be given l o c d y by a transition function in A-normal form.

3.1.3 Examples: According to this theorem. the transition function O of a

GL(n , C)-bundle may be put in the form b = t e P , where

This bundle uill be an SL( n. c )-bundle if in addition a-e have C t, = O. In the

case of SL(2 . C) t his agrees with the normal form described in [Hur].

For an Sp(n. C)-bundle (with n even). the normal form is as above but P and

t must also satisfy the additional requirernents that ta = -t,-,+, . Pt = - P,":::: 2 +a

f o r 1 5 a < b < 2 . a n d P b 2 = ~ ~ ~ ~ ~ \ f o r l < a . b < ~ - - 2 -

For an SO(n. C)-bundle. the normal form is again as above. but P and t must

be such that P," = - P:I:,"=: and t a = - tn- ,+, . and if n is even then we need ody

have t l 2 - - - 3 tn j2 - l 2 Itnjz 1 (we do not necessarily have tn,Z 2 0).

3.1.4 Remark: Sotice that if a-e rewrite eP as e P = (1 + p) then the entries p:

of the matrix p will involve the same orders in z as the entries Pa of the matrix P: - 1

we have pi = C p i l z J . j = l - t a + t b

Before launching into the proof of the theorem. let us smooth the waj- with a

pair of lemmas.

3.1.5 Lemma: Let F be the G-bundle over P l given by the transition function

O = exp(Ah z)exp(P(z)) . a-here P E H 1 ( P 1 . AdA)? and let E be the G-bundle over

P, x C @en by c = exp( A ln z ) exp(Q(x, r ))' a-here Q = 1 Q,ea is defined by

Then E,, is of type A. and E, is isomorphic to F for al1 r # O.

Proof: It is clear that E, is of type X so we consider x # O. Let p be equal to

haif the sum of the positive inverse mots of G. Then (see [Hum]) a ( p ) = height(a)

and so [e, p] = -height(o)e,. This, together with the conjugation formula 1.3.2.

implies that for an. t # O we have

where lux is any branch of the Iogarithm. Thus we have an isomorphism Q 2 tC.l

away £rom x = 0. O

3.1.6 Lemma: I f A E . inWaadifo is anyroot. then the bundleoi-erP, given

by the transition function O = exp((A + à) ln z ) exp( r-' ea ) is of type A .

Proof: B y changing trivialkation over L-, we may rnultiply on the left by a =

exp( -= -"(*)-' ). W e ~ = o u l d Like to show that n-e can then change trivialization

over to get a ob = t. where t = exp(X ln z ) . For us to be able to do this we must

have t - ' a o holomorphic in z . l e have

AI1 the terms in this latter expression lie in the subgroup of G whose Lie dgebra

is generated by e-,. e, and â. This subgroup is isomorphic to either SL(2. C) or

SO( 3. C ). lf it is a copy of S L ( 2 . C) then we may arite the above expression as

rvhich is indeed holomorphic in 2. Similarly w e find t hat t - l a O is holomorphic in

z in the case in which it t&es d u e s in SO(3 .C) . (7

3.1.7 Proof of Theorem 3.1.2: To prove the theorem we must show that the

transition function ma- be put in the form O = e-xp( X ln z ) exp( P) where

The proof involves passing back and forth between the Lie group and the Lie alge-

bra. \t-e use the fact that the exponential map defines a holomorphism between a

neighbourhood of O in g and a neighbowhood of 1 in G.

ié begin the proof by working over Pl x {O}. where a-e have assumed that O E I.-.

Let us write O, = O ( = . O). Shen we know from Grothendieck's theorem that oo may

be put in the form oo = exp(p ln z) where p is the type of Eo. Let us suppose. for

the purpose of an inductive argument. that for any p with type(Eo) 5 p < X the

transition function 4o may be put in the form

for some Po E H1(P1. Adp) .

Sext. let us write o(r. z) = o,02, with O, as above. Then O,, : C * X C + G - with O,, (r. O ) = 1 E G. Let h- be any annulus in Cm. and choose a polydisc

-

1' c L- small enough that the image O , ~ ( ~ X C * ) lies inside SO we may write &

O z 1 = exp(Q) and we have

u-here Q : h* x t' -+ g.

By changing trivialization over C o . we may multiply O on the right by a function

of the form exp -4. where -4 is holomorphic in r and z . Suppose that -4 is given

bu a forma1 power series -4 = C --I~X'. .Ir = C -4,jz'. "th .4(r. O ) = 0. and let I J 20

R = loglexp(Q) exp(,4)). By the Campbell-Hausdorff formula we have

From this we can see that -4 ma!: be chosen (in a unique way) so that the power

series e.qansion of R contains no positik-e powers of 2. -4 must be given by

and this ma' be solved for .-II (having first solved for al1 -4, with 1 JI 5 111 ) since -4

and Q contain no constant term in x. -4ssuming. for the moment. that the potver

series -4 converges. we have shown that E may be given locally by a transition

function of the form

O = e.xp(pIn t)exp(P,)exp(R).

where R involves only strictly negative powers of +.

Next we s h d change trivialization over C, by multiplying on the left by

exp(8) with B holomorphic in r and z-'. But first, let us rewrite O . If RI

is given by exp(R') = exp(Po) exp(R) exp(-Po) and if S is given by exp(S) =

exp(p ln Y ) exp (R t ) e x p ( - ~ ln z) then we may rewrite 4 as

Notice that the power series for R', like that of R, contains no positive powers of z ,

and notice also that, as in corollary 1.3.3, Sb = Rk and S, = ra(') R' Q - Now suppose that B is given by a fonnal power series B = B ~ X ' with

I B(r , O) = 0, and let T = log(exp(l3) exp(S)). Then we may choose B, in a unique

way, so as to ensure that T contains no negative powers of z. If B converges then

Ive have put the transition function into the form 4 = exp(T)exp(rhz)exp(P,)

where

T = C Taea with Ta = C Tai;'. a E R + i= 1

Again rewriting 4, this time as

where exp(P) = exp(-p ln z ) exp(T) exp(p ln z ) exp(P,), we have succeeded in put-

ting + in p-normd form; we have

P = C P,e, with Pa = C PQiz i . Q € R + i=-a(X)+l

Recall that we began the proof with the inductive hypothesis that Eo could be

given by the transition function 4, = exp(p In z ) exp(P,,) with Po E HL (P, , Adp).

Our final inductive step in the proof will be to show that E, could just as well be

given by a transition function of the form = exp((p + â) ln r ) exp(Qo ). where

o is a root with p < p + & 5 A, and where Qo E H1(P1,Ad(p + â)). Let us

define R : C -+ H1(Pl , Adp) by Ra = xheight(") (Po), as in lemma 3.1.5, so that

the transition function exp(p ln z ) exp(R) defines a bundle F over Pl x C with the

property that F, 5 E,, whenever x # O. By an application of lemma 3.1.6 we

find that this bundle F could equally well be given by a transition function of

the fonn exp((p + â) ln z ) exp(S) with So E H l (P l , Ad(p + â)). As we have just

shown in the preceding paragraphs: by changing trivializations over Uo and U ,

we may assume that S : W + H1(Pl, Ad(p + â)) for some polydisc W. Finally,

by restricting the bundle F to Pl x {x} for any x E W, we obtain the transition

function ?ho = exp((p + â) ln z ) exp(S,) as desired.

The proof would now be complete but for the question of the convergence of

-4 (and 8). Recall that -4 was uniquely determined by the requirement that R =

log(exp Q exp -4) contained no positive powers of 2. The question of convergence is

more easily dealt a-ith by working. not in the Lie algebra g. but in the Lie group

it self which we regard locally as a group of matrices Si .G C G L( n. C ). So we

shall let a = exp -4. q = e.xp Q. Mé suppose that a is given bu a forma1 (i-e. not

necessarily convergent ) power series a = C a ,r '. a = C a ,[ zf . with a( z . O ) = 1. I r>o

The power series a is uniquely determined by demanding that the product ga must

contain o d y strictly negative pou-ers of 2: the power series for qa is

so we are forced to define

where the subscript 2 0 indicates that we take only the positive powers of z .

Wé shail check that this power series a converges by finding bounds on its

coefficients. We know that q converges in h-xC' for some annulus iï and polydisc

Let L be any annulus L = {r 5 1 - 1 5 R} which lies inside h-. Choose p with

O < p < 1 so that the annulus L p = {rp 5 121 5 R / p ) also lies in K. And for

O < o < 1 a-e suppose that the polydisc M- = { l x , 1 5 S,} is srnail enough that

Ri = {Ili 1 5 S i l o } lies in L'. We define t j = C I q l f Ir' z i . which also converges. and

we let Mo = mmau{[iq~lr'zl'lll.(i.r) E L,xw'). where the norm J I II of a rnatrix

is the largest of the absolute d u e s of the matrix entries. Then on L x U ' we have

for all 1.

where . M = . b f o z ? and i is the degree of the index K. To find bounds on the

coefficients of a, we shall also need to estimate the number #(i, K) of indices I of

degree i which appear in the sum C pIaJ. Since the total number of indices 1 of I+ J=K

10

degree i (including those which do not appear in the sum) is the binomial coefficient

("+,'-' ) . we have

-1nd now we can find bounds on the coefficients of a.

one. aK is defined by aK = (-qK)<,. so we have IlaKx K - make the inductive assumption that

For indices

Il 5 114'h-sKl

K of degree

1 5 MU. né

for all indices J of degree J < k (recall that rn is the dimension of the polydisc and

n is the size of the matrices). Then for an index of degree k we have

x Sotice that C ("+,'-')(2" j-' = ( 1 - 2-m) - 1. and it is not difficult to ~ e n f y that

a= 1

O < (1 - 2-m)-m - 1 < 1 for al1 m 2 1. So we have shown that

l which we can arrange Thus the power series a converges provided that a < m. by taking the polydisc W to be sufficiently srnail. Consequently. the power series

for -4 also converges. And so. similal>-. does the series for B.

One final remark is that converges not only in K xW but also in C'x W

because 6 is polynomial in zdl.

We shall close this section by recording a couple of immediate consequenees of

t his theorem.

3.1.8 Corollary: A G-bundie over Pl x U may be given locally by a transition

function which takes d u e s in the Borel su bgroup B of G.

3.1.9 CoroUary: If p and X Lie in A f l W then the foflowing are equivôlent:

(1) P I A,

( 2 ) There exists P E R1(Pl , Ad)<) such that Ep is of type p,

( 3 ) There exists a G-buode E over Pl x C with E, of type X and El of type p

when x # 0.

Proof: That (1) implies (2) follows fiom the theorem: the transition function

e x p ( ~ l n z ) defines a bundle E over Pl x C with E, of type p for ail x E C ,

and this bundle may be given locdy by another transition function, say 4 =

exp(X ln r ) exp(P(x)), in A-normal form; then P(O) defines the required element

of Hl (Pl AdX). Actually. the proof of the theorem contains an algorithm for con-

structing such an element P, and we shall illustrate this procedure in example 3.1.10

below .

(2) implies (3) by lemma 3.1.5, and (3) implies (1) by the sernicontinuity theo-

rem. R

3.1.10 Example: The purpose of this example is to illustrate how, gib-en A, 11 E

A n W with v 5 X one can h d an element P E H' ( P l . -4dX) such that Ep is of

type v.

We s h d take G = SO(4,C) with simple roots a, = idiag(1,-l,1, -1) and

a2 -- fdiag(i, 1, -1, -1); and we s h d let v = a, = diag(l , l , -1, -1) and X =

Y + â, + û2 = diag(3,1, -1, -3). Since A - v is a sum of two roots, we shall make

use of lemma 3.1.6 twice.

Using lemma 3.1.6 we see that the transition function

defines a bundle over P, of type v. Then, applying lemma 3.1.5, we obtain a bundle

over Pl x C with transition function

which is of type p := u + hl = diag(2.0.0. -2) over z = O and of t-ype u elsewhere.

-4nother application of lemma 3.1.6 shows t hat the transition function

defines a bundle over Pl of type p. But this time Ive need not just the transition

function 8 but also the tx-iiializations a and 6 a-hich give the isomorphism a8b =

exp(pLnz). and these we obtain from the proof of lernma 3.1.6: we must choose

a = e x p ( z - a 2 ( ~ ) - l e-== ) and we have

Finally a e use these trivializations a and 6 to compute

Like 6. this defines a bundle of type p above x = O and of type v elsewhere. So we

obtain our element P E H1(P1. AdX)) by ei-aluating at x = 1 (any other z # O

couId serve just as weU):

3.2 On the Uniqueness of the Normal Form

The normal form for the transition function of a G-bundle is certainly not

unique. However. we may say the following.

3.2.1 Proposition: ff a G-bundle E over P, x t ' is considered as a B-bundle

(a-here B c G is the Borel subgroup) then the normal form transition function

exp( X ln z ) exp( P ) is unique up to the adjoint action of the group of invertible holo-

morphic maps fkom L* to X (a-here H c G is the Cartan subgroup).

Proof: We consider G as a subgroup of G L ( n . C) with H contained in the set of

diagonal matrices and B contained in the set of upper triangular matrices. Suppose

that n-e can change trivializations to put O into a new normal form. s a - aob = c or

with a holomorphic in x. z-' and b holomorphic in z. z . Concentrat,ing on the

diagonal part. we have

a c Ztibc = = J i . t t

If s, were greater than t , then n-e a-ould have a : ( x ) = zss-t l b : ( t ) - ' which is holo-

morphic in z but not z-' . Sirnilady t t cannot be greater than s, . Instead. we m u s

have sg = t , and a: and b: must be constant in z with a : ( r ) = b : ( x ) - ' .

Lp to the adjoint action of the group of int-ertible holomorphic maps fiom C1 to

H. we may assume that a: = bi = 1. Then the above-diagonal part of the equation

gives

,ti+1 + z t i 1 ~ f + i + a:+,- b1+1=d+,

Comparing coefficients we see that ai+, = b:+, = O for all i. The proof may be

complet ed using induction on successively higher diagonals. O

3.3 The Transition Matrk of the Endomorphisrn BundIe

Suppose that a G-bundle E over Pl x C . L- C Cm is given by the normal form

transition function o = t exp P where t = exp(X In z ) . We shall show how to com-

pute the transition matrix c. of the associated vector bundle AdE of g-valued en-

domorphisms.

.A section of AdE is a map -4 : L-,, + g holomorphic in x and z such that Ad, -4

is holomorphic in 2-' . so Ive must have

where s and Q are defmed by

s.4 = Ad,.4 and Q.4 = [P. -41.

To express Q and s as a matrices we must introduce an ordered basis for the

Lie algebra g. Let A be any basis for 9. For o E R we let e, be a non-zero elernent

of the root space g,, and for a E A we let e, = a. Then {e, Io E R u Li} is a basis

for g (the union R U S is the disjoint union). W e shall order t his basis in such a way

that the positive roots corne before the elernents of A which. in turn. corne before

the negative roots. and we aiso require. for roots a and 3. that if & ( A ) > 3 ( X ) then

a cornes before J.

3.3.1 Proposition: Lf a G-bunde E is given by the nomal form transition

function 4 = exp(X ln z)exp(P), where P = C P,e,, then AdE is given by the oER+

normal form transition matrix 1L = s eQ where s is a diagonal matrix and Q is an

upper triangdar matrix &th

-a(3)Po i i a ~ R+ and 3~

b,,P-, if a E and 3 c R-

a-here ao3 and baJ are constants.

Proof: The form for the matrix s has already been computed in 2.3.1. To

determine the matrix Q we must compute [P. e3] = C P, [e , . ej]. u-hich may r E R+

-. do as follows: if 3 E R and y + 3 E R then [ e , . ~ ~ ] = c , ~ ' ~ + ~ -. aa3ep where

a = 7 + 3: if 3 E A then with a = u-e have [e,. e 3 ] = [ep. 31 = -a (3)ea: and if

3.3.2 Examples: Let E be a GL(2.C)-bundle given by the transition matris

lvith p = 2 pirr. Then AdE (which in this case is also denoted by EndE) 1=-(2-s ) t l

is given by the transition matrix

Let E be an SL(2, C)-bundle given by the transition function

If E is an SO(3. C)-bundle then -\dE is isomorphic to E. since so(3. C) is the image

of st(2. C ) under the adjoint representation. If E is an SO(4. C)-bundle git-en by

given by

4. The Section Matrices

Let E be a GL(n.C)-bundle over P l x C with G c Cm. let E, denote the

restriction of E to P l x {r}. let O = exp! X ln i) e.xp(P) be a A-normal form transition

matrix for E with X = diag(t l . . . + . t , ). and let E(1) denote the bundle given bu the

transition matrix =-'o. W e u-ould like to be able to describe the cohomology spaces

Hq(P, xL-. E(1 ) ) in terms of the transition matrix. We s h d do this by studying

maps S l ( x ) : Ho(P, . A ( [ ) ) -. H 1 ( P l . X ( 1 ) ) u-hich have the property that there are

~ ~ ( ~ ~ . ~ , ( l ) ) = k e r S ~ ( x ) . and

H ~ ( P , . ~ ~ ( 1 ) ) = H ' ( P , . W ) ) / h S I W .

Of particu1a.r importance d l be the section matrix S-1(2). For vector bundles

with lanishing first Chem class the matrix S-, (r ) will be a square matrix. and E,

will be trivial precisely when H O ( P , . E, ( - 1) ) = O. that is a h e n ker S-, ( r ) = O or

when det( S-, (2)) # 0.

In the first section u-e shall construct the maps S l ( r ) in matrix form. Our

construction is motivated by the construction given in [Hur] for the section matrix

S - , ( I ) for an S L ( 2 . C)-bundle.

These matrices. which tve cal1 the section matrices of E (u-ith respect to the

transit ion matrix O ) have numerous applications. -4 few irnmediate applications -are

provided in section two. For example. we shall shoa- hoa- to use the section matrices

in conjunction with proposition 7 - 2 2 to compute the splitting type of E,. -41.~0.

we use the existence of the section matrices to provide an elementary proof of the

semicontinuity theorem for vector bundles over P l x C. Fust her applications will

appear in later chapters.

The section matrices of course depend on a choice of basis for HO(P,.A(I))

and H l ( P , . X ( 1 ) ) . In the third section we discover an alternate choice of basis for

H 1 ( P , . A([)) with respect to which the section matrices exhibit greater symmetry

(see proposition 4.3.7) and may be given a more concise description.

4.1 The Construction of the Section Matrices

IVe try now to compute Ho(Pl . E,(l) j from the normal form transition matnx

O of a vector bundie E over P, xL-. For convenience. rat,her than wrîting o in the - t n ) form O = t e P we shail wite o in the form O = ( I + p)t where t = diag( zt l . . . - . -

~ 5 t h t l - 2 t n and where p is a strictly upper t r k g ~ l a r matrix with pp = t a - t b - 1

pf,:' (note that p is polynomial in z rather than in r - ' ) . t = 1

For a point x E l-. a global section of E r ( [ ) is an n x i column vector c(t )

holomorphic in { r # r) such that =- 'oc is holomorphic in {t # O}. If -ive let

We require that

we must have

ua be holomorphic in z-' for al1 a. If ta - 1 > 1 then for ail j > ta

and for 1 < j < ta we must have

if ta - 1 = 1 then va, is given by (1) for all j 2 0. Findy. if ta - 1 < 1 then condition

(1) must hold for j > 1 and u-e have no further conditions on cf for O 5 j 5 1 - ta.

B - treating those caj with t , - 1 < 1 and O 5 j 5 1 - t a as independent variables.

we are led to the follow-ing definition.

4.1.1 Definition: Consider the vaj with t , - 1 < 1 and O 5 j 5 ! - t a as

independent variables. The condition (1) alows us to wnte a.ll the other v l as linear

combinations of the independent variables, and the condition (2) then becomes a

set of linear equations in these variables. The matrix corresponding to these linear

equations is called the section matrix of E(1) and will be denoted by Sl(x).

By their definition. these matrices S, (x ) satisfy

H O (P,. E x ( [ ) ) 2 ker S,(z).

L é also note that the number of roa-s of S l ( x ) is given by the number of equations

in ('2 ) which is

and the number of columns is the number of independent variables which is

In particular. if the fkst Chem class of E vanishes. then S-, is a square matrix.

and in this case E, is trivial if and only if h O ( P , . E,(Z)) = O which holds if and only

if det S - , ( x ) # O.

13-e can also compute H 1 ( P I . E,(Z)) from the section matrix O = (1 + p)t. The

space H1 ( Pl . E , ( l ) ) is the space of ( n x 1 ) column vectors IL*(=) holomorphic in

{ z # O. r } quotiented by two spaces: the space of vectors u ( z ) holomorphic in 2-'

and the space of veztors of the form : - 'O r nhere r is holomorphic in z . ( i v e are

working in the set { z # O} rather than the set { z # r } ) .

The vector u ( s ) may be used to eliminate al1 the negative powers of = in the

vector u.( z ) . so w e need only worry about w Z l . Likewise. the entries va( z ) of the

vector c ( z ) may be used to ehminate cca,ra-l . so w-e need only worry about rxas -

when 1 $ i < ta - 1. W e rremark that the space of wa, with 1 5 i < ta - Z i s precisely

the space H1(P1. A(/)).

LI-e are still left with the freedom to quotient the space H1(P , . A([)) by the

space of vectors of the form =-'O o where va, is arbitrary for O < j 5 1 - ta. and

to ensure that w ~ ? , ~ - , remains zero. we must choose va,, j > 1 - t a to satisfy the

condition (1) above. The space of vectors z-'Q v with v of this form is precisely the

image of S l ( x ) . Thus we have

Our next task is to give an explicit description of the section matrices. Let us

fis Z and wnte S = S,. It is naturd to u-rite the matrix S in block form: for each a

suchthat t a - [ > 1 andeachbsuchthat t b - I < 1 a-ehavea( ta -Z- l )x ( l - tb+l )

matrix Sa which forms one block of the matrix S. We noa- examine the block Sp

more closely. The first row of the matrix Sz is determined by the condition (2) in

Since al1 the c: involved in this s u m are in the set of independent variables. we see

that the first row of the block Sp is

and the ( j + 1)" rom- of (SP.. . . . ST-) is given by ( C c c ) l + j + l = O . or c 3 t . < t a - 1

Xotice that the surn on the left looks like the expression giving the jth row-but

with the coefficient of each pz shifted up by one. As for the sum on the right, the

expression inside the parentheses looks like the condition giving the jCh rom* but

with the superscript raised from a to c. Thus we have the follow-ing lernma.

4.1.2 Lemma: The block Sz of the section matrix Sl of the bundle E is con-

structed fiom the transition matrix 4 = ( 1 + p ) t of E in the foUowing manner. The

first row of S,O is

The kth entry of the j th row aiU be a finite sum of the form

The entries of the ( j + 1)" roa- can be obtained from those of the jth row according

a-here shift( S:: ) is obtained fkom S:; by replacing each jo b~ j, + 1. and raisec( SZ 1

is ob tained from S l i by dianging each a into a c .

This lemma is perhaps intimidating enough that it should be softened by some

reassuring examples.

4.1.3 Example: Let E betheGL(2.C)-bundlegi\-enby

1 2 - t 1 - 1 wherep= C pi:'. Then thesection matrices S l ( s ) are the Toeplitz matrices

i= 1

In the case that t , = -tl and 1 = -1, this agrees with the section matrix for an

S L ( 2 . C )-bundle dehed by Hurtubise in [Hur].

4.1.4 Example: Suppose that the bundle E is given by the transition matris

ta - t b -1 a i th each p: = C p t i z t . Then the section matrices are

4.2 Some Applications

Our first application is to use the section matrices for a GL( n. C)-bundle E over

P, x L - to compute the dimensions h o ( P 1 . E x ( [ ) ) and to use these in turn. together

with proposition 2.2.2. to determine the spiitting t'pe of Er .

4.2.1 Example: Let us consider the vector bundle given by the transition matrix

This \vas the transition matrix computed for the SO(4. C bbundle in example 3.1.10.

which we now consider as a GL(4. C)-bundle. Its section matrices are

We can then calculate the dimension of the kernel of each of the matrices S , ( x ) :

Then, since dim ker Sl (x) = h0(Pl7 E,(I)), we see £iom proposition 2.2.2 (as we

already saw in example 3.1.10) that x defines a GL(4, C)-bundle E over Pl x C

with E, of type p = diag(2,0,0, -2) and with E, of type u = diag(l,l, -1, -1)

when x # 0.

We c m also calculate the type of E, in the more general case that E is a

principal G-bundle over Pl x U . This may be done by studying the vector bundes

p ( E ) associated to one or more representations p of G. For example, if G is one of the

groups SL(n, C), Sp(n, C), or G2, then the type of the G-bundle E, is completely

determined by the splitting type of the vector bundle p(E,) where p : G -+ GL(nl C)

is the embedding descrîbed in section 1.2. Ln general, the type of El may always

be determined by the splitting types of pi(E,), where the pi are the representations

of highest weight niei as in proposition 2.3.2.

4.2.2 Example: We return yet again to the bundle E given by the A-normal fonn

transition matrix X. R e c d that SO(4, C) has inverse roots âl = diag(1, - 1,1, - 1)

and&* = diag(l,l,-1,-1) and that we had v = â2, p = â l + â 2 and A = &,+2â2 .

In the previous example we showed that the GL(4, C)-bundle E, was of splitting

type p for x = O and of type û2 for x # O. The calculations given there were not

in themselves s&cient to guarantee that E, would still be of type â2 if E were

considered as an SO(4,C)-bundle (although we know this to be the case from

example 3.1.10) as E, could be of type âl . 1 - The fundamental dominant weights of SO(4, C) are E , = iâl and E, = 7a2.

The four dimensional representation which we have been using is the representation

of highest weight el + E ~ , but in this example we consider instead the representa-

tions p, and p, with highest weights Sa, = al and 2e2 = ct2 respectively. These

representations are both three dimensional and we have

while pi(hj) = O = pi(eaj) for i # j. The transition matrices 4, and #2 of the

GL(3.C)-bundles p , ( E ) and p,(E) axe given by

Csing the section matrices for these bundles as we did in the previous example. we

find that p l ( E,) is of type diag(2.0. -3) if z = O and of type O if x # O ahile p 2 ( E , )

is of type diag(3.0. -2) for all x. In particular. if r = O we have Ip,(E,)I = 2 and

Ip2! Ez ) I = 2 implying that the SO(4. C)-bundle E, is of type â, + c i 2 = p. and if

r # O w e have [p l (E,)I = O and jp2(E, )I = 2 ixnplying that E, is of type a2.

-4 GL(n. C)-bundle E, over Pl. which is given by a A-normal form transition

matrix o. could be of any spkitting type p = diag(s, . . . . . s, ) with p 5 A. and

we have seen how one may use the section matrices to determine p. We c m go

a little further and h d actual tritializations u and c such that uov = s wbere

s = diag(r91 ' . . . . z9- ). We shall need to be able to do this later when we compute

the cascade of bundles. so let us e-xplain how these trivializations may be found.

Since E is of type p. we know that there eas t sections ui of E(s , ) such that

every section of E ( 1 ) is of the form C a' ui a-here ai is a polynomial in : of degree 131>s;

Z - si. These sections vi can be computed a i th the help of the section matrices:

one begins by finding a section zYn of E(s,). then. having found c,. . . . one looks for a section ck of E ( s , ) which is independent of the set of sections 2.5 ivith k + 1 5 j $ n and0 5 i s s k - s Thematrix r~ nithcolumns c i will then

I - be invertible and holomorphic in z . and the matrix u is defined by u = s v-'O-'.

Here is an example.

4.2.3 Example: W e know from any one of the examples 3.1.10. 4.2.1. or 4-22

that the bundle E given by

is of type v = diag(1, 1, -1. - 1). Let us use the section matrices computed above in

example 4.2.1 to find trivializations u and v such that u ~ ( l ) v = diag(z, z, z- ' , z-').

The column vectors (O O 1 O)* and (O O O l ) t span the kernel of S-I and they

determine sections v, = (O 1 O z)t and v, = (-1 z z z2)t of E(-1). The kernel of

S, is spanned by

; and

.l

which correspond to the sections v3, z v, , zZv3; u4, z v4 , iAv4 : v I ; and v2 of E(1)

ivhere v, = (O O O l)t and v, = (O 1 - 1 0)t. So we have found v' and then u is

We remark that u and v do not take values in SO(4, C), but from example 3.1.10

or 4 - 2 2 we h o w that it is possible to fmd other trivializations which do. In fact

For our final application of this section we use the existence of the section matri-

ces to provide an elementary proof of the semicontinuity theorem for the cohornology

spaces of vector bundles E over Pl x U . Recall that it states that for any k, Z E Z

and for q = 0 , l the set {x E Ulhq(P, , E r ( [ ) ) 3 k} is a closed analytic subspace of

0'.

4.2.4 Proof of the semicontinuity theorem: Setting r = ho(Pl, E,( l ) ) - k,

we have

{. E u 1 h0(~,,E,(1)) 2 k} = {x E U 1 rank(S1(x)) l r }

a-hich is the intersection of the closed analytic subsets cut out by the determinants

of the ( r + 1) x ( r + 1) minors of the section matrix Sl(x). The same semicontinuity

property holds for h l ( P l . Er([)) since

bu the Riemmz-Roch formula.

4.3 A Reforrnulation of the Section Matrices

Our description of the maps S l ( r ) : Ho(Pl. A([)) + X1 (P,. X ( 1 ) j in matrix form

depended on a choice of basis for H o (P l . X ( 1 ) ) and H L (PI. A ( ( ) ) . In this section

ive shall proride an alternate description of the section matrices. and a-e shall ser

that there is a more natural choice of basis for Hl (Pl . A(1) ) . Csing this alternate

basis we find that the section matrices have a sl-mmetric property which is related

to Serre duality.

Let E be a GL(n. C)-bundle over Pl xLV aven b - the A-normal form transition

matrix O = ( 1 + p ) t with t = diag(ztl . . z t - ). Fix the integer 1 and let r be any

integer such that t , 2 1 + i and t,+l 5 1 + 1. Let E+ denote the subbundle of E(1)

whose transition matrix O+ is the upper left hand r x r submatrix of --'o. and let

O- be the quotient bundle E- = E(l)/E+ (whose transition matrix O- is the lower

right hand ( n - r) x ( n - r ) submatrix of : - 'O ). Then the exact sequence

of bundles gives rise to the exact sequence

of cohornology spaces. The map Sl(z) : H"(Pl, X(1)) + HL (PI X(1)) of section 4.1

may be recovered by making the identifications Ho ( P l , E;) = HO(P,, A([)) and

H1(Pl, E z ) = H1(Pl, A(1)) . We would like to discuss these identifications in more

detail. as we shall discover a more natural choice of basis for Hl (Pl , A([)) than the

basis which was implicitly used in the construction of the matrix S,(x) .

We begin with the identification HO(Pl. Ep ) = HO ( P, . A([ ) ) . A global section

of E; is an ( n - r ) x 1 colurnn vector u holomorphic in s such that O-u is holomorphic

in t-'. We have

To make this holomorphic in z-' we may choose u a j as a-e wish provided that

t a - Z 5 O and O 5 j 4 Z - t a . and then for d j > Z - t a tve must have

Thus u-e obtain a basis {c; Ita -1 5 0. O 5 i 5 1-ta} of sections for HO(Pl . E T ) which

may be identified with the standard basis for HO(PI.X(l)) . It was this standard

basis for HO(Pl . X(2)) which --as used irnplicitly in section 4.1 when we constructed

the section matrix S [ ( x ) .

Som- we turn to the identification H1(Pl . E f ) = H1(P1 . A ( [ ) ) . .An elernent of

H l ( P l . E:). in the C o trivialization. is given by an r x 1 column vector u. holo-

morphic in {r # O. x} and we are fiee to add aqv vector u holomorphic in z and

any vector of the form O+-' v where v is holomorphic in i-' wirhout changing

u. E H 1 ( P l . E:). LI-e may use a vector u to elirninate rca, for j 2 0. and it is

not hard to check. using the definition of oC. that a-e may use a vector of -' c to

eliminate uqa for j 5 I - ta. Thus we obtain a basis {w: l t= - 1 1 2. Z - t , < j < O} j

for H' (P, . E,f ) &ich may be identified with the standard basis for Hl (PI . X ( 1 ) )

in the L;, trivialization. ln the construction of the section matrix Sl(x). however.

w e worked in the C, trii-ialization. In the C , trivialization. the above basis for

HI ( P l . EI+ ) becomes {o+wy ] whereas in section 4.1 we implicitly used the stan-

. et us now recalculate dard basis {t.:lt, - I >_ 2,O < j < t a - 1) for H1(P, , X ( 1 ) ) L

the section matrices using the basis {d+wg} for Hl ( P t , A([)). To distinguish nota-

tionaly between the section matrices calculated with respect to different bases for

H1(P, , A([)) we shall denote the new section matrix by Ti(+

4.3.1 Proposition: The section matriv T l ( z ) is made up of ( ta - 2 - 1) x (1-t,+l)

blocks Tt whicb may be constructed fkom the )<-normal form transition mat&

a-here the subscript < j denotes truncation to order j - 1 in r and the subscript

1 - t, + j - k + 1 indicates that a-e pi& out tbe coefficient of Z ' - ~ ~ + J - ~ + ' .

The proof will make use of tw-O lemmas.

4.3.2 Lemma: The r o m sa' . s a 2 , . . - . - 1 - 1 of the section matrùr Si( x ) are

related tc the rows ral. - - - . ratta-'-' of Ti(x) by the row operations

Proof: In our calculation of the section matrïx Sl(r ) we irnplicitly used the basis

{ r : l t , - 1 2 2.0 < j < ta - l } in the l-, trivialization for H X ( P l . A(!)). where

ru is the column vector (v;) ' = b a C z J . Ive non* wish to use the basis { u : ) . where 1

u: = O+ v C 1 * to calculate the section matrix Tl ( z ) . i é have

-'pz i f a < c .

(u : )" = if a = c ? and

So n-e may express each u: in terms of the barsis u,D as

This means that Sl(x) = RTi(r) where R is the hl(P,, A ( 1 ) ) x hl(P,, A([)) matrix

10 otherwise .

This implies that S,(z) and T,(z) are related by the required row operatiom. O

4.3.3 Lemma: The first row of the blodr T,O of the matzix Tl is gii-en by

The kth entry of the jth rom- WU be a finite sum of the form

The entries of the ( j + 1)" row are obtained kom those of the jth row according to

the rule

This is almost identical to the description of the section matrix of lemma 4.1.2.

but notice that here the sum is taken over al1 c > a while there it was taken over

only rhose c > a with tc 5 1 + j .

Proof: Let sa'. s a 2 . . . . . denote the roWs of S&I) and recail from lemma

4.1.2 that

Let ra' ..... ravta-'-' denote the rows of the matrix constructed as outlined in the

lemma. so that

To show that these are indeed the rows of Tl (r ) we must verifj- that they are related

to the rows of Sl(x) by the relations of lemrna 4.3.2: we claim that

To verify this claim. suppose inductively that (3) holds for a given a and j. Then

substituting sa' as given by equation (3) into the equation (1) we obtain

where

From equation (2) we see that shift(raJ) - C, - Cs = ra~j+l , and we notice also

that C, vanishes since i 5 t d - 1 - 1 and j 2 tc - 2 imply that j - i > t , - td

so = O. We can rewrite C, as CS = C pzlrc', so we have C 2 + C, = c 3 t e - l > j

E ~ P C , ~ - , + ~ ~ " . Thuswehave c 3 t , - l > j i = i

sa ,j+l = (shift(raj) - C, - C , ) + (Pl + C2 + Cs)

So formula (3) holds for al1 a and j. O

4.3.4 Proof of Proposition 4.3.1: One shows, by induction on the rows of T;,

that the sum in lemma 4.3.3 is taken over a.ll muti-indices 1, J satisfying a = 2, 5

i, 5 - - . - < i m = b, and jo+ j l+ - - -+jmd1 < j , and j o + j l + - - - + j , = 2 - t b + j - k + l .

O

4.3.5 Example: If E is the SL(2,C)-bundle given by

then (see example 3.3.2) AdE is given by the transition matrix

and £rom proposition 4.3.1 we have

For example if t = 2 and if p = p , ; + + p,z3 then we have

It can be shown that the pattern which is apparent in the above matrix. a-here

t = 2. persists when t > 2:

ié make several remarks about this matrix T-, . One remark is that the matrix

T-, is skew-symmetric in the sense that

-4nother is that it is equal to the section matrix dehed earlier:

A third remark is that if we treat E as a G L ( 2 . C)-bundle then we h d that the

associated G L ( 4 . C)-bundle EndE has the same section matris as the GL(3. C ) -

4.3.6 Example: Let us return to example 4.1.4. Let O be the transition matrix

defined there. Then

a d T, = So and Tl = S, . If one compares these matrices aith the section matrices

from example 4.1.4. one sees immediately that the matrices SI may be obtained

from the matrices Tl by the row operations of lemma 4.3.3.

Xow we corne to a skew-symmetric property of the matrices T, (x ) which is

related to Serre duality.

4.3.7 Proposition: ff a G L ( n . C)-bundle over Pl xi7 is given by the normal

form transition function o. then its dual bundle may be given by c = O-" ti-hich

is also in normal form. and ti-e bave

Proof: The dual bundle has transition matrix O-' ' which is lower triangular. To

convert this to upper triangular normal form. we muhipl? on the right and left by

the matrix J to get = J Q-' J = O- ' ' .

Let us write O = (1 + p ) t and v = (1 + q ) s with

Recall that each block Tl(@): is a ( t a - I - 1) x ( 1 - tb + 1) matrix. and note that

T-,-,(lit): is an (s, + Z + 1) x ( - s d - [ - 1) matrix. So in order to prove that

T,(o) = - ~ , - ~ ( i 1 > ) " we must show that

for d a. 6 with t a - 1 > 1 and t , - 1 < -1. And indeed

This proposition has an interest ing consequence for SO( n. C )- bundles. If O

takes d u e s in SO(n.C) then c = O-'" = o. so the proposition states that

T-, (O) = -T-,( ( O ) % . But such skew symmetric matrices have even dimensional

rank. so we obtain the following corollary.

4.3.8 Corollary: f l an SO(n. Cl-bundle over Pl x C is considered as a G L ( n . C)-

bundle. t6en for all x. y é C u-e have

In fact it is not difficult to check that c l ( E , ) m h O ( P l . Er(-1)) (mod 2) where

cl ( E x ) is the first characteristic class (the first Stiefel-Whitney class) of Er.

lies in al(SO(n. C) = :\/r = Z,.

-4nother a-ay to see that SO(n. C )-bundles corne in these two types is to refer to

[PrSe] in which the correspondence between bundles and loop groups is discussed. A

holomorphic SO(n. C)-bundle over PI x U toget her with a trivialization over { r # O}

corresponds to a holomorphic map fiom U into the loop group f2 Soin . R). and this

loop group has two comected components. For those bundles E corresponding to

maps whose images lie in the identity component of the loop group, the space

H o (Pl . E,(- 1)) is even dimensional for ail 2.

5 . Some Invariants

In the first two sections we shall concentrate pximarily, though not exclusively,

on one-parameter deformations of vector bundles over P,, that is vector bundles E

over Pl xU where U is a disc in C, and we shaii discuss some invariants of such

bundles. These invariants are all local invariants, by which we mean that they are

determined by the restriction of E to Pl x V for any disc V C LI.

The splitting type of the restriction Eo of the bundle E to Pl x {O) is one

such invariant, and we have seen that it may be determined from the integral

ivariants hO(P1, Eo(l)) which may be calculated using the section matrices of E.

In the first section we s h d use the section matrices to compute a larger collection

of invariants: we s h d calculate the dimensions hO(P1, E ( Z ) ( ~ ) ) of the cohomology

spaces HO(Pl, E(z) (* ) ) where E(z)(*) denotes the restriction of E(1) to the kth forma1

neighbourhood of Pl x {O} in P, x U. We h d it convenient to work also with the

spaces wf(~) of sections of E(l)(O) which extend to E(I)( ' ) . These spaces are of

dimension w f ( E ) = hO(P,, E ( [ ) ( ~ ) ) - hO(P1, E(Z)(~-')) .

In the next section we define the graph G(E) = { w * _ , ) and the multiplicity

m ( E ) which, roughly speaking, measure how quickly the bundle E, "jumps dom"

to the bundle E, for x near O. In the case that El is trivial for x # 0, the multiplicity

may be caiculated as the order of vanishing of the determinant of the section matrix

S-, (z), and it has the property that if the bundle E is extended to a bundle È over

P, xP, with Ê, trivial whenever r # O then c,(Ê) = n(E).

In the final section we shall discuss the divisor of jumping lines for a principal

G-bundle over Pl xU where U c Cm and also for a G-bundle over P,. Barth

showed in [Ba] that the degree of the divisor of jumping lines of a bundle E over

P, is equal to c2 (E) . Although Barth stated his results only for vector bundles of

r a d 2, we s h d see that this result also hoids for principal G-bundles.

5.1 Formal Neighbourhoods

We restrict our attention for the moment to one parameter deformations of

vector bundles over the Riemann sphere. Let E be a GL(n, C)-bundle over Pl x U

where C' is a disc in C. In order to study the behaviour of E near O E G we s h d

consider the restriction of E to formal neighbourhoods of the line L = P, x {O} in

P, xl'.

Let (3 denote the shed of holomorphic functions on P l x l'. let 1 denote the ideal

shed of functions vanishing on L. and consider the quotient shed O( = 0/1~+' .

The kCh formal neighbourhood L(') of L is defined (see [Ha]) to be the closed

subscheme ( L. O( '1) of (P, x L'. (3). The bundle E is identified a i th its shed of

holomorphic sections (also denoted by E). and the restriction E( of the bundle E

to L(') is defined to be the shed

For any k 2 O. 1 > O. n E Z we have the exact sequence

where 5 n - 1 denotes truncation to order n - 1 in x-

If the bundle E is descnbed bu a transition mat+ o. then the sheaf ~ ( 1 ) " ) can

be more elementarily desribed. For an open set I7 C L. a section of E(z ) ( " ) over

C- is given by an n x 1 column vect.or u(z. 2). holomorphic in { z # x} c 1.- and

polpornial of degree k in z. such that ( z - 'O u ) < ~ is holomorphic in { z # O} C V. -

5.1.1 Definition: W e shall denote the space of sections of E(l)(O) that extend

to E ( l ) ( k ) by cV. and the dimension of this space by w f . !\-e dso define = x W;. and w r = dirnCV;C.

k=O

From the long exact sequence associated to

we see that for 1 > O

W,' = H'(P , , E ( Z ) ( ~ ) ) / Z H'(P* E(l)( '- ' )) , and

tu: = hO(P, , E(I)( ')) - h 0 ( p l , E(1) ( k - 1 ) )

= h 1 ( p 1 , E ( Z ] ( ~ ) ) - h 1 ( p 1 , E ( z ) ( ~ - * ) ) .

66

There are several remarks which may be made about these spaces.

(1) Wf = Ho(Pl.E,(i)). where E,,(l) denotes the rest.riction of E ( i ) to L.

(2) We have WC' c Mi:. so decreases as k increases.

(3) tVe have 2 CF-: C C V ~ ~ . 50 t ~ . : increases with I .

Suppose now that E is given by a normal form transition matrix. iVe s h d de-

scribe how one may use the section matrices to determine the spaces Ho ( Pl . E(Z)( ') )

and H i k . R e d that each column vector ~ ( x ) E ker Sl(r ) determines a unique sec-

tion of E ( f ) .

To f b d an explicit basis for ker(Sl(t)) we c m perform a sequence of roa oper-

ations on the matrix S l ( r ) . in order to convert it into a row reduced matrix R , ( z )

(1) For some m l : the fkst ml rows of R , ( x ) are non-zero and the rest of the

rows are zero.

(2 ) Each non-zero row R' contains an entry R:(i, of the form x " ~ . ~ for some

nlTZ 2 0- (3) Al1 other entries of the row Ri vanish to order nlsZ in z.

(1) Ali entries beneath R:(t, in the colurnn RI( , ) vanish.

(5) Ive have nl., 5 nlSz 5 - - - < - n r e m , .-

To construct R, ( z ) we begin by hding. from arnongst the non-zero entries of

SI( x ). an entry. say Si. of minimum order of vanishing in r. If S; = xnm f ( r ) where

f ( O ) # O. then we multiply the row St by f - ' ( x ) and we interchange the row Sa

with the &st row SI. Then we eliminate all 10%-er entries of the column S, in the

obvious way. and we proceed to hunt for the next entry (on some other row) of

minimal order of vanishing.

W e remark that f -' (x) need not be defhed on all of C, so the row reduced

matrix R l ( x ) may only be defined in some disc V C L'. We also remark that

the row reduced matrix RI is not uniquely determined £rom the section matrix Si.

but the integers nlti are, and in tact we shall see that they depend only on the

isomorphism type of the bundle E.

From the rnatrix R l ( z ) we may read off a basis for Ho(P, x V, E(Z)). Let J

denote the set of integers j with 1 5 j 5 ho(P1, A ( 1 ) ) such that j = j(i) for some

i. Then for each o 4 J there is a unique column vector va(x) with v' = 6: for d

j # J such that v a E ker Sl. We have

Just as a vector v E ker Sl determines a section of E(l ) , so too is a section of

E(z)(*) determined by a vector v which is a polynomial of degree k in z and satisfies

where 5 k denotes truncation in x. And so the matrix Rl(x) may also be used

to obtain a basis for H O ( P l , E ( [ ) ( ~ ) ) . Let J~ denote the set of integers j with

1 5 j 5 h0(P17 A([)) such that j = j ( i ) for some i and nlPi 5 k . Then for each

a $ J' there is a unique vector v:(s) which is polynomial in x of degree k? with

( u t ) j = 6: for al1 j $ which satisfies ( ~ ~ ( x ) v ~ ( z ) ) ~ ~ = O. We have

We also notice that for any k 2 nl,,, we have

where the subscript O denotes evaluation at x = O. This last remark also follows

from Grot hendiekis t heorem on formal functions (see [Ha]).

Ln a similar fashioqwe can use the section matrices to determine the spaces

X 1 (P, , E ( z ) ( ~ ) ) . To do this, we carry out column operations to convert Sl(x) into

a column reduced matrix C l ( z ) , say with c:") = I " ' . ~ , defined in sorne disc V C W .

The integers nlPi appearing in this matrix will be the same as those occuring in the

row reduced matrices Rl(x). We have

ahere CJ denotes the jth column of the matrix C l ( r ) . The space

m. be identified with the space of h l (P l . X(1) ) x 1 column vectors w(r) with

uj is a polynomid of degree n,,i - 1 if j = j ( i ) for some i . and

w j is holomorphic in z if j # j ( i ) for an? i.

The space

H'(P,. E ( z ) ' ~ ) ) = H ' ( P ~ . A ( / ) ) .Z O(')( c ) / ( I ~ s ~ ) ~ ~

may be identified with the space of cab vectors u: &th

upJ is a polynomial of degree nl,, - 1 if j = j(i) and nl 5 k, and

urJ is a polynomial of degree k otherwise.

Some immediate consequences of these computations are gathered together in

the following theorem.

5.1.2 Theorem: The inlariants tc: and nl-, possess the foUoa-ing properties:

( 3 ) = hO(Pl xC: E(1)) = hO(Pl. E,(Z)) for any x # O

( 7 ) (Serre Duality) hl (P,, E * ( - 2 ) ( * ) ) = h 0 ( p 1 y ~ ( ' 1 ) 30

(9) h l ( P , xV: E(1)) = ( w f - c , ( X ) - ( 1 + 1)n) (this sum may be uifinite)

Serre duaEty follows from theorem 4.3.7: since T,(E) = -T-[-,(E*)%, we have

R,(E) = C-l-,(E*)t which implies that the integers nl,, ( E ) : - --- , nl,ml (E) are iden-

tical to the integers n-l-2,1 (Ee), - - - nWlhZ ( E ) Serre duality then follows

69

from (5 ) and (6). N7e also note that

5.1.3 Example: Let E be the G L ( 4 . C)-bundle over P. x C git-en by the transi-

tion matrix

Then referring to example 4.1.4 we have

From these u-e obtain the row reduced matrices

and R, = So and RI = SI. Rom the matrices RI we can read off the dimensions

IL.: of the spaces CV::

ive would also like to record a stronger version of corollaq 4.3.8. R e c d

that if E is an SO(n,C)-bundle then the section matrix of E is skew-symmetric:

T- , ( x ) = -Tdl(s))". To calculate the integers n-l ,g we rnay use both row and

column operations on the matrix T-, (x) in such a way that the reduced matrix is

also ske~l--symmetric. and so the set {zln-, -, 4 k}. which appears in part (3) of the

above theorern. will dways have an even number of elements.

5.1.4 Proposition: Let E be an SO(n.C)-bundle over Pl xC- aith C c C .

Then the invariants IL.L.~,(E) are either ail even or all odd. (The>- are dl even in the

case that cl(E) = O E 2,).

Return now to the more general case of rn-parameter deformations of vector

bundles. If E is a vector bundle over Pl x C with C C Cm and if h- is a multi-

index = ( k l . - - . km ) then we can use the section matrices of E to calculate

XO(P, . E ( z ) ~ - ) : a &bal section of ~ ( 1 ) ~ is given by a column vector u - n-hich is

holomorphic in z and polynomial in s of degree ki in x,. such that ( ; - ' du ) , , is - holomorphic in z-' , where the subscript 5 denotes truncation to order hi in x, . \\è demonstrate how this may be done in the follouring example.

5.1.5 Example: Let E be the GL(2, C)-bundle over P, x c 2 given by the tran-

si t ion mat rix

where (x, r ) are the coordinates in C2. We s h d use the section matrix

to help us find a basis of sections for H O ( P 1 . E ( -1 ) (2*1 ) ) . W e must find a column

vector u = (:) where a and b are polynomids of degree 2 in x and 1 in e such that

( S - , ( x . c ) U ( X . c ) ) < ( ~ , ~ ) = O- W e have -

Sett,ing the coeficients which appear in the latter matrix all equal to zero we find

- that a,, = boO = a l , - sol = O and a l , + blo + bol = O . S o the kernel of S - , ( x . e )

modulo r2 and is spanned by

Each of these determines a section of E( -1 ) (2 -1 ) and we obtain the basis

for HO(P, . E ( - 1 ) ( 2 - 1 ) ) .

5.2 The Graph and the Multiplicity

Let E be a GL(n. C)-bundle over P l x C Bwhere C is a disc in C.

5.2.1 Definition: The graph of the bundle E is the sequence

The length of the graph is

which. by theorern 5.1.2 (4). is the smallest integer k such that w ! , ( E ) = w : ( E ) .

The multiplicity of E is defined to be

The multiplicity is most interesting in the case of a bundle E for which E, is

trivial whenever z # O. In this case we have wml ( E ) = O and, using theorern 5.1.2,

we can give several equivalent characterizations of the multiplicity:

5.2.2 Theorem: Let E beaGL(n,C)-bundleoverP,xC a-hereGisadisc in

C such that E, is trivial wheo z # 0 . and let Ê be an extension of the bunde E to

Pl xPl aith Ê, trivial for z # O. Then

= the order of kanjshing of det S-, (z )

= c * ( Ë ) . and

(Ê(-1) denotes the bundle E ;̂ p*0(-1) û q8C3(-1) where q and p are the two

projections from Pl x Pl down to Pl).

Proofi The e s t three equalities foilow immediately from t heorem 5.1 .S. The fourth

equality is proven in lemma 5.2.3 below. The equality c , ( Ê ) = hl(Pl x P I , Ê(-1))

follows fiom the Riemann-Roch-Hirzebnich formula ( a-hich may be found in [At Si] )

using the fact that HO(Pl xPl.Ê(-1)) = O since E, is trivial for r # 0. and hence

also H2 ( Pl x P l . Ê(-1) ) = O by Serre duality. The final equahty foUoWs from the

fact that c , ( ~ n d Ê ) = 2nc2(Ë) for any bundle Ê over P lxP1 with cl(Ê) = O (in

order to show this. one may assume that É is a direct sum of line bundles. by the

split t ing principle ).

5.2.3 Lemma: Let x i be r distinct points in C . let C; be s m d (non-overlapping)

discs about xi- let E, be GL(n , C)-bundles over Pl xCi with (Ei ), trivial for z # L.,'

and let Ê be a bunde over Pl x P t with ÊIPi xL7i = E, su& that Êz is trivial when

x # x i . Then ..

Proofi Let n : P, xPl + P, be the projection to the second copy of P l . There

is a Leray spectral sequence ( s e [GrHa]) which converges to H*(Pl xP1, Ê(- 1))

with E, term given by

E;9q = H P ( P , , 3 1 q ) ,

where Hq is the presheaf given by W ( U ) = Hq(x- 'C . Ê(-l)ln-IL.) for any open

set l: c Pl . Notice that 'HO(C) = O for aii C'. and that Hl is supported at the

points r, . The only non-zero term of E:". then. is the term

5.3 The Divisor of Jumping Lines

Suppose non- that E is a G L ( n . C)-bundle over P, x C with L- C Cm such that

Ex is trivial for some x E L-. Then the &st Chern class of E must xanish. and the

set J of points x E C for which E, is nontrivial is given by

l é have seen that for such bundles the section matrix S-, (x) is a square matrix

a i th h O ( P 1 . Er( -1 ) ) = dim(ker S - , ( x ) ) . and so we have

J = {z E FI det S&) = O } .

So the set J is a hypersurface in Lw. We can also view J as the divisor of zeroes

of det S - , ( x ) . and then J is called the divisor of jumping lines of E. This aas

studied in the case of r a d - 2 bundles by Barth in [Ba].

Let us show that the multiplicities of the irreducible components of J are in-

dependant of the choice of transition function for E. Our argument is es sent id^ t hat used by Barth in [Ba] for rank-2 bundles. Write det S-, ( x ) = n pi ( z . where

the p, are the irreducible factors with multiplicities k i . For each i. choose a smooth

point zi of J with p , ( x , ) = O. let LG be a smail one-parameter disc with x i E LI

transverse to the hypersurface of zeroes of p , ( x ) , let E, denote the restriction of E

to P, xL-, . and let Èt be a bundle over P, xP, which restricts to E, over P, x l ;

such that the restriction of Êi to Pl x ( x ) is trivial for al1 x # x i .

Then k, is the order of vanishing at xi of det S-,(x) restricted to Ci, which is the

multiplicity of E,:

So the multiplicities

We would like t O

are indeed invariants of the bundle E.

repeat the above argument in the case in which E is a principal

G-bundle where G is a simple Lie group, but first we shall digress for a moment to

discuss the second characteristic class of a principal bunde- A similar discussion

may be found in (AtHiSi]. The second characteristic class c2(E) of a G-bundle E

may be defined using the Chern-Weil theorem in terms of a second degree invariant

polynomial on g (see [Ch]). For the simple groups, there is only one such polynomial

up to scale (it is given by the Killing form) and so the second characteristic class is

uniquely determined up to a scaling factor. It is this scaling factor which we would

like to fix.

We shall be dealing with G-bundles E over M = Pl xPl which are trivial

over L = Pl x {oo} U {O) xP1 and also G-bundles E over M = P2 which are

trivial over a line L C P,. Such a bundle may be described by a single transition

function 4 : Uo n U1 -+ G where LI, is a tubular neighbourhood of L and where

U, = ll1 \ L = c2. Since LTO 17 U1 is homotopic to the 3-sphere? the transition

function defines an element of x, (G)? and the principal G-bundle E is deterrnined

topologically by $J E 7r3 (G) .

Now for any simply connected simple Lie group G: there is an embedding

i : SL(2, C ) G which is obtained by sending the positive root of S L ( 2 : C ) to

the highest long root of G. The embedding i induces an isomorphism 7r3(G) =

7r3(SL(2, C)) = Z. This means that, as a topological bundle, the G-bundle E may

be reduced to an SL(2, C)-bundle. We would me, then, to normalize the second

characteristic clam in such a way that for any SL(2, C)-bundle k, the second charac-

teristic class c2 ( 1 ( ~ ) ) of the G-bundle i ( ~ ) associated to F through the embedding

i should be equal to the second Chern class c,(F) of fi. We do this in the following

way.

5.3.1 Definition: Let F be the SL(2, C)-bundle over P, x C which is given by

4 = (' =zi ) , and let F be an extension of F to PlxP1 which is trivial over

P, x {oo} and {O} x Pl (note that c , (F) = m ( F ) = 1). For any irreducible repre-

sentation p of a simple group G we d e h e the scaling factor

We d e h e the second characteristic class of a G-bundle E (over any base man-

ifold) to be

c2(E) = C ~ ( P ( ~ ) / C ( P ) 7

where c2(p(E)) is the Chern class of the vector bundle p(E).

For the embeddings p of the simple Lie algebras d(n, C), so(n, C), sp(n, C),

G2, F4? E6, E7 and E, which were described in section 1.2, the scaling factors c ( p )

are equal to 1, 2, 1, 2, 6, 12 and 60 respectively. Later on (in section 7.5) we shall

also calculate the scaiing factors c(Ad) for the adjoint representation.

Although this choice of scaling is well suited to our purposes, we remark that

some G-bundles over Pl x P or P2 (which are not trivial over any lines) d l have

fractional second characteristic class. Here we shall end our digression and retum

to our study of the divisor of jumping lines.

Suppose now that E is a principal G-bundle over Pl x U with C' c Cm with Er

trivial for some z E U. Then c l ( E ) will be trivial, which means that we may assume

that G is simply connected. The set of points x E U at which E, is nontrivial may

be given by

J = {x E U I det S&(E))(x) = O ) ,

where p is a faithful representation of G. If we mite det S-, ( p

In particular, the multiplicities ki(p) are d integral multiples of c(p), so we may

make the following definition:

5.3.2 Definition: The divisor of jumping lines of a G-bundle E over P, x C

where C c Cm is defined to be the dirisor of zeroes of ( det S - , ( p ( E ) ) ) I / c ( P ) for

any irreducible represent at ion of G.

5.3.3 Example: The section matrix of an SO(n . C)-bundle (or a G2-bundle)

is skew-symmetric. and the divisor of jumping lines is the divisor of zeroes of the

Pfaffian of the section matrix (which is the square root of the determinant of the

section matrix).

Finally. we wish to discuss the divisor of jumping lines of a G-bundle over Pz.

The set of lines 1 C P, are parametrized by the dual space P,'. The flag manifold

F. whosepoints are the pairs (r.1) with x E P2 and 2 E P,*such that x E 1 c P,.

is a P, fibre bundle over both P, and Pz*.

For small discs

5.3.4 Definition: Let E be a G-bundle over Pz which is trivial over some line

L c P,. The divisor of jumping lines of E is the divisor in P2* which. over a srnail

disc C c P,'. is defined by the divisor of E l q - , c , which denotes the restriction of

E to q- 'C = P, xLÏ .

in [Ba] Barth showed that. for a rank-2 vector bundle. the degree of the divisor

J is equal to the second Chern class of the bundle. Let us show that this result holds

also for principal G-bundles. The pencil of lines through a fixed point z E L C P,

is parametrized by a single line D in Pzo which pases through the point L E Pz'.

The divisor of jumping Iines will then intersect the line D in a finite number of points

li with deg J = C mi, where mi is the intersection multiplicity mi = multli (D fI J).

Blolving up P, at the point r a*e obtain PZ = p-lD c F together with the maps

p : pz + P1 and g : P~ + D. The bundIe E pulL badr via p to a G-bundle Ë

over PZ and we have c 2 ( Ë ) = c 2 ( E ) . Surrounding each point Z, E D by a s m d disc

L-z we obtain bundles E, over Plxt; by restricting Ë to q-lL; = P,X(-~. Then

the intersection multiplicities mi may be calculated as the order of Mnishing of

( det S&(E,)) ) l 'c(p' which is equal to m(p( E,))/c(p). so we have

where the third equality follows from lemma 5.2.3 and the fourth follows £rom the

Riemann-Roch-Birzebruch t heorem [At Si]. Thus ne obtain the folowing generaliza-

t ion of Barth's t heorem.

5.3.5 Theorem: (Barth) I f E is a G-bundle over P2 whicb is trivial over some

fine L c P2. and if J is the divisor of jumping lines of E . then deg J = c 2 ( E ) .

6. The Cascade of Bundles

In the first section of this chapter we s h d introduce the cascade of bundes

below a vector bundle E over Pl x L- where C is a complex disc. The cascade is a

collection of bundles EI each of which is canonically determined by the bundle E-

The normal form transition matrices for the bundles E, may be computed and used.

in turn. to calculate the dimensions h O ( P , , ~ ~ ( 1 ) ' k, ) or. equivalently, the dimensions

W . In this way we obtain many new invwiants for the bundle E.

In the next section w e shdl discuss the special case of rank-2 vector bundles

E with cl(E) = O. \17e s h d see that in this case al1 of the invariants u ~ ( E [ )

are completelp determined from the graph of E. so the cascade provides no new

invaxiants. W e shall find the cascade useful. however. as we shall use it to show

that the graph of E detexmines the graph of the endomorphism bundle EndE.

and this result deepens our understanding of the topology of the moduli space of

SL'( 2)-instantons.

6.1 The Cascade of Bundles

Ln the last chapter we showed how to calculate the imuiants IL':(E) h m the

normal form transition matrix of a G L ( n . C)-bundle E over Pl x L - . where C is a

complex disc. Yow we shall show how to construct a cascade of GL(n . C)-bundles

EI over Pl x V for some disc V c C from which we obtain a whole collection of

invariants w:(E~).

6 1 Definition: Suppose that E is given by the normal form transition matrix

o = ( 1 + p)t where t = diag(ztl , - + - , z t - ) is the transition matrix for E, ( the

restriction of the bundle E to {x = O}) . so that p(z.0) = O. Then for 1 5 i < n we

define O( i, by

if a 5 i and b > i. 9(i) = (1 + q)t , where qt =

pi , otherwise.

The bundle over Pl x U given by the transition matrix 4(i) wiU be denoted by E( i ) .

6.1.2 Proposition: U t i > t i+, thentheisomorphismclassofE~,~isdetermined

by that of E: it does not depend on the choice of transition matriu m .

Proof: Suppose that the bundle E. given by o with ~ ( r , O) = t is isomorphic to

a bundle F given by 3 with lu(z .0) = t . and say uov = ri. where u is holomorphic

in z- ' and x and II is holomorphic in z and t. If we mite uo = u ( z . 0 ) and -1 a r , = c( s. O ) then n-e have uot v, = t. which implies that (uo ) f = ~ ' a - ~ b ( v , )*.

Since uo is holomorphic in r-' and is holomorphic in s we see that

( I L , ) : = (L?~); = O whenever ta > tb .

So. provided that ti > titi we can define u( , , by

-1 a x u, . i f a s i a n d b > i o

( " ( i l )a = s u t . i f a > i a n d b s i .

u i , otherwise.

and we can define v ( ~ , in a sirnilar way. It is not hard to see that we then have

and thus E(;) is isomorphic to F( ,).

The transition matrix d = (1 + p) t which we used to define the bundle E(, , had

the property that p ( z . O ) = 0. but the transition function O,, , = (1 + p ) t need not be

such that q(z . O ) = O. We do know however, by theorem 3.1.2. that the bunde E(i ,

(suitably restricted) may be given another transition function, say = (1 + r)s

which does satisfy r(z_ O ) = O. So aie may repeat the above procedure and define

the bundle E(i) ( j , Thus we are led to the following definition:

6.1.3 Definition: If I is the null multi-index then we set EI = E, and for the

multi-index I = (il,&, . . . , i,) ivith 1 4 ij < n we define

To ensure that El is weli defined. we require that each ij is chosen as in proposition

5.2.2: if E(il . - - + , i j - l , is of type X = diag(sl.. . . .s,) over z = O then we must have

siJ > s i 3 + , . The collection of bundles EI will be called the cascade of bundles

below E.

From the cascade we immediately obtain many new integral invariants for the

bundle E. namely the inlariants UT:( E r ) . Our next results describe certain rela-

tionships amongst these invariants.

6.1.4 Lemrna: FE, is of type A = diag(t,.--- - . t , ) and i f t i + , - 1 5 15 ti - 1

Proof: Looking back to the description of the section matrix given in lemma 41.3.

one sees that exactly one of the P : : , , , ~ appearing in the sum of lemma 1.1.2 aill

have i k 5 i and ik+, > i. and so dividing each p: with a 5 i and b > i by z has

the effect of dividing the section matrices S l ( x ) by x : we have Sl(E( , , ) = x - ' S I ( E )

whenever t,+, - 1 5 2 5 t i - 1. This implies that R1(E( , , ) = r- 'R,(E) (recall that

R, denotes the row reduced matrix of section 5.1 which we used to calculate uf )

which irnplies in turn that w : ( E ( , , ) = <Ü:+'(E). O

6.1.5 Proposition: The intariants w : ( E I ) may aLi be determined fiom the

invariants u f ( E I ) .

Proof: Let X I = diag(s,:*-m. . s , ) be the splitting type of E I . If 1 2 s, - 1 then

w ~ ( E ~ ) = cl(A1) + n( l+ 1) for all k: if 2 5 s, - 1 then w f ( E I ) = O for ali k: and

ifs, - 1s 15 s, - 1 then we can choose i such that si+, - 15 1 < s i - 1 so the

proposition follows from the lemma using induction on k. O

This proposition provides us with an alternative way of describing the innriants

obtained £rom the cascade. For any fixed rnulti-index 1, the invariants w : ( E I )

determine and may be determined by the splitting type X I of the restriction FI of

the bundle EI to P, x {O}. By restricting to x = O then, the cascade of bundles EI

over Pl x V determines a cascade of bundes FI over Pl which we rnay think of as

a descending cascade X I in the lattice of splitting types. The cascade { X I } is an

invariant of the bundle E which determines all the integral invariants w:(E[ ) .

It should be pointed out that this entire cascade of bundles is cornputable in

a straightforward (albeit time-consuming) manner from the transition matnx 4 of

the initial bundle E. To renormalize the transition matrix d ( i ) to obtain a transi- -

tion matrix = (1 + r)s with r(z,O) = O one first follows the procedure outlined

in example 4.2.3 to normalize above {x = O}, then one follows the renormaliza-

tion procedure described in the proof of theorem 3.1.2. We provide an illutrative

example.

6.1.6 Example: As an example, let us compute a few of the bundles in the

cascade of bundles below the bundle given by the transition matrix

We have already computed the dimensions w f for this bundle in example 5.1.3.

From the definition of Q ( i ) we immediately obtain the transition matrices for the

bundles E( ) : E(2), and E(31, and as in example 5.1.3 we can compute the dimensions

w: for each of these:

Mlen z = 0. the matrix O , , , is diagonal. so we could easily continue by finding

E ~ ~ , w E ~ ~ , 2 p and E( .3,. U é choose instead to h d the bundles which lie below

E To do this we must first renormalize the matrix o ( ~ ) . FoUowing the procedure

outlined in exarnple 4.2.3. we begin by renormalizing over {x = O}:

This is stili not in normal form as the matrix contains several terms whose degree

in = is lower than allou-able. To elirninate these terms we must multiply this matrix

on the left by a matrix u with u( r. O ) = O which is holomorphic in =-' as we did in

the proof of theorem 3.1.2. There is a unique such change of trivialkation u which

puts the matrix into normal form. and it may be found term by term in t:

Xow that we have renormalized O(, , we can continue one stage hr ther and compute

the bundles that lie beneath E(2) : .

It is not immediately clear exactly how much new information about the bundle

E one obtains by calculating the invariants w ~ ( E ~ ) . W e have already seen that

there are some relationships amongst these inkariants. and it is conceivable that

they could all be determined by the invariants u f ( E ) (in fact this is true for rank-2

bundles. as we shall see in the next section). However. we shall provide one example

which reveals that the cascade of bundles below a bundle E can indeed furnish some

additional informat ion about that bundle.

6.1.7 Example: Consider the GL(4. C)-bundle F given by the transition matrix

'3 0 ;tt4z x 2 = - 1 + x 3 , 2 - z 0 =-2

L. = ( , - I 0 I

,-3 - If one computes the dimensions uf ( F). one h d s that they are all identical to those

computed in example 6.1.6 for the bundle E defined there. Thus the invariants uf

are insufficient to destinguish between the two bundles E and F. However. if we

compute ~ ( 1 ) and the invariants tc:( F( ). we find that

I \ k O l 2 3

1 9 8 8 8

Comparing this with the results of the previous example, we see that w:,(F(, , ) = 0

while wY2 (E( ) = 1 - We conclude that the bundles E and F are not isomorphic.

6.1.8 Remark: It is possible to give a more abstract definition of the cascade

of bundles which applies also to principal G-bundles. This may be done as follows:

if E is the G-bundle over P, xC. L: C C given by o = exp(Xhz)exp(P) with

P( z . O ) = O then a-e define the transition function O( i , for the bundle E(,) by o ( ~ ) =

exp( X ln z ) e.xp(Q) where

If ai ( X ) > O and if i is such that a-e have ni = 1 or O for all positive roots o = C n a then the isomorphisrn class of E(,, is determined by that of E. If we choose multi-

indices I = (il. - - - -2,) such that each i, satisfies the above two conditions then

Ive obtain a cascade of G-bundles EI canonically determined fiom the bundle E.

B y resricting to r = O we obtain a descending cascade X I in the lattice .\ n %V of

G-bundles over P, . This cascade is an invariant of the G-bundle E.

Cnfort~nately~ if G = G , or E, then this cascade is empty because if ri = C ni ai

is the highest long root then we have n, > 2 for ail i. Also. the cascade is ingeneral

difficult to compute. since we have not developed a renormalization algorithm whîch

generalizes the one given in example 4.2.3 to the case of arbitrary groups G.

6.2 An Application to SU(2)-instantons

In this section we s h d concentrate on the case in which G = SL(2. C). so

suppose that E is a GL(2,C)-bundle over P1xL- with c l ( E ) = O where C is a

cornplex disc. To simplify our notation. let us write E(, ) to denote the nth bundle

Ew.-- .I, in the cascade of bundles belon- E.

6.2.1 Proposition: For a G L ( 2 . C)-bundle E over Pl xL' with cl(E) = O , aii

of the invariants w:(E( , ) ) are determined by the graph G ( E ) according to

w c t n ( ~ ) + l + 1 , if 1 < W::,(E), and

21 + 2 , i f ' Z 2 w ~ ~ " ( E ) .

Proot: A GL(2, C)-bundle F over P, with Mnishing first Chern class is of splitting

type diag(h, -h) if and only if h = ho(P, , F(-1)). dpplying this to the bundles

E( ,, restricted to PI x {O] we obtain

Combining this result with lemma 7.1.4 we obtain

IC!,(E) + 1 + I . if Z < W ~ ~ ( E ) . and k-1

W : ( E ) = W ~ ( E ( , , ) = - - - = ~ ~ ( E ~ ~ ) ) , if I 2 ~ L _ , ( E ) .

and

E k+1 ul( (n)) = W I (E(n-11 ) = ... = w ~ + " ( E ) .

-4 similar result holds for GL(2. C)-bundles with nonvanishing first Chern class.

FVe should emphasize? though. that the above result is not true for bundles of higher

rank. For example. the GL(3, C)-bundles given by

have the same graphs but different collections of innriants u':.

6.2.2 Proposition: I f E is a GL(2 . C)-bundle over PI x C with c l (E) = 0, then

the graph of the endomorphism bunde EndE is related to the graph of E by

Proof: If h = hO(P l . E, ( -1 ) ) then the splitting type of E, is diag(h. -h) and so

the splitting type of EndE, is diag(2h.0,O. -2h): so we have

If E is given by the normal form transition function 6 = (1 + p)t with p( z . 0) = 0

then, from example 4.3.5 (in which we computed the section matrix of the end*

morphism bundle), we see that the section matrix S,(EndE) vanishes to order 2 in

x, and this implies that

Now consider E( ,) which is given by = (1 + $p) t . Referring again to

exarnple 4.3.5 we see that S-l(End(E(l,)) = 3s-,(EndE) which implies that

i c !T2(~nd~) = w f , ( ~ n d ( ~ ( , , ) ) . In particular

and

U . : ~ ( E ~ ~ E ) = u * \ , ( ~ n d ( ~ ( , , ) ) = w: , (~nd(~( , , ) ) = 2 w ~ , ( E ) .

Proceeding inductively down the cascade we obtain

Sotice that this is consistent with the formula m(EndE) = 2n m ( E ) .

The above proposition is of some consequence as i t slightly increases our under-

standing of the moduli space .Ut of based SC(')-instantons over S4 of charge k. In

[BoHuSlaMi]. a stratification .M = U SG of the modidi space .M was constructed

and used to prove the Atiyah-Jones conjecture. Each stratum SG is labelled by a

collection of graphs Ç = (G, . - - - . Gr) whose multiplicities add up to the charge

k. The stratum SG fibres over the space DPr(C) of r distinct unordered points

in C (which is an r-dimensional complex manifold) the fibres being the product

3ZG, x - - - x 3JG, where 3JG denotes the space of framed jumps of graph G (we

shall discuss framed jumps in section 6.1). The space 3gG is the total space of an

iterated fibration

mhere 332) denotes the space of Gamed jumps ~ ( 1 ) on the jtb forma1 neigh-

bourhood of Pl x {O) with graph G up to order j. It was shown that the fibre

('+" + 3e) over a fixed bundle E(J) E 3.72) is the smooth variety of F&

S::+, x kerp where the w j are the decreasing sequence of integers defining the

graph G and s$+, denotes the space of w j x w j Toeplitz matrices of corank wJ+'

and where p is a surjective linear map C( : Wj+'(EndE) -r H1(Pl,O(-2wj)). It

a-as also show that the Toeplitz variety s$+, is a smoot h cornplex manifold of di-

mension 2 ( w j - UJ+' ) (unless U J # O and WJ+' = O in which case it is of dimension ( i + I )

2111 - 1 ). and it follows that the dimension of the fibre of 3JG -+ 3 ~ 2 ) over

the bundle ~ ( j ) is

It was not known. however. that these fibres were al1 of the same dimension. inde-

pendent of the ehosen bundle ~ ( j ) E 3.7"'. The above proposition reveals that

indeed they are. since i c< : ' (~nd~) is determined by the graph of E'J) .

7. Framed Jumps

-4 framed jump is a bundie E over Pl x LT where C c C which is equipped with a

framing oves {O} xL7 and which is trivial when restricted to P, x {z} whenever x # O.

In the first section we shall motivate the study of fiamed jumps by describing the

manner in a-hich Hurtubise [Hur] reduced the study of instanton moduli spaces to

the study of framed jumps. He did this using Donaldson's description [Do] of the

relationship between instanton moduli spaces and framed bundles over P2 - W e shall

also review the manner in which a dimension count of the space of framed jumps

\vas used in [BoHuhla!Zli] and [Ti 1.21 to prove the .\tiyah-Jones conjecture.

In the next section we shaii briefly describe modifications that ma" be made to

the results of previous chapters in order that the framing of a bundle is taken into

account. In particular. we s h d describe the space of framed bundles over PI . and

we shall describe a normal form for the transition function of a frarned bundle over

P, xL' a-here C c Cm. For unLameci bundles. the versal deformation space was

H1(Pl. -4dX). but for framed buociles it is H1(Pl..4dA(-1)).

In the t hird section we shall ponder the question of which normal form transition

functions yield isomorphic bundles. One consequence of our results is that the

transition function of a frarned jump may be tnrncated to order Z(AdE). where

l ( r \ d E ) denotes the length of the graph of AdE. A similar result holds for unframed

jumps.

In the following section we s h d discuss deformations of framed jumps. CVe shall

see that Hl (P, x L', A U ( - 1)) parametrizes the space of infinitesimal deformations

of the framed jump E. We shall also determine which deformations preserve the

multiplicity of E and which deformations presen-e the graph of AdE.

In the final section we shall give a description of the space FU of framed jumps

of multiplicity k which are of minimal type over x = O. It is a smooth complex

manifold of dimension (c(.4d) - 1) k where c(4d) is the scaling factor, which we s h d

calculate, and it is an open subset of the space of all fiarned jumps of multiplicity k.

In the case of a GL(n, C)-bundle, the space 3.7; is a vector bundle over TP,-, O

which denotes the tangent bundle of P,-, with the zero section removed.

7.1 Framed Bundles Over P, and Ramed Jumps

X framed G-bundle over Pz is a holomorphic principal G-bundle E over PZ

which is trivial over a fixed Line L c Pz and which is equipped with a framing

e : L -+ G there. Two framed G-bundles are isomorphic if there is an isomorphism

of the unframed bundes which sends the framing of the first bundle to that of the

second. -4 fiamed bundle always has Mnishing first characteristic class c, (E) = 0.

The second Chern class c2( E) may be defined. using any faithfd representation p

of G. by c , (E) = c , ( p ( E ) ) / c ( p ) ( s e section 5.3). \Te shall denote the moduli space

of isomorphism classes of trazned G-bundles over P, ~ 5 t h c2( E) = k by iM k ( G )

-4 strong motivation for studqing the moduli spaces .Uk(G) cornes fiom their

relationship with instanton moduli spaces. In [Do], Donaldson showed that if F is

a compact classical group (F = SLW(n). SO(n). or Sp(n) ) and if G is the com-

plexification of F (G = SL(n. C ) , SO(n. C), or Sp(n,C) ) then the moduli space

. l ; (F) of based F-instantons over S4 of charge k is diffeomorphic to the moduli

space .M &3).

B y Kuranishi theory. .Mk(G) is a smooth complex manifold whose tangent

space at the bundle E E .Uk(G) is the cohomoloy space HL (PZ. AdE(-1)) which.

by the Riemann-Roch-Hirzebnich formula. is of dimension c, - ( AdE) which is equal

to c(.Ad) k. If one calculates c(Ad) for the simple groups (which we shall do in detail

in section 7.5) then one obtains the following:

G dim M, (G)

60k

These dimensions agree with the dimensions of instanton moduii spaces which were

calculated in [AtHiSi].

In [Hur], Hurtubise used framed jumps to study the topology of the moduli

spaces &tk(SL(2. C ) ) with some success. Let us take a moment to describe hoa-

framed jumps rnay be used to study framed bundles over Pz.

7.1.1 Definition: A framed G-jump is a G-bundle over Pl x L'. where C' is a disc

in C. which is trivial over Pl x {x) whenever r # O and which is equipped with a

framing e : C + G above {O}xL- c P,xC. TWo framed jumps (E + P , x G . e :

l- - G) and (F -t Pl x V. f : Y -t G) are considered to be isornorphic if there is a

local isomorphism £iom the bundle E to the bundle F above Pl x CV for some disc

Lt' c C ri b- which sen& the framing e to the framing f: if E and F are given by

the transition functions o and ti? then ( E . E ) dl be isomorphic to (F. f ) provided

that there exists a map v : {; # s} -+ G holomorphic in x and z such that O c u-'

is holomorphic in 2-' and e c, = f.

;\ G-bundle E over PZ with a kaming over L c P2 p d s back via p : P, 4 P,

to a G-bundle Ë over the blow up P, with a framing over E fI D. where Z denotes the

proper transfo- of L and D denotes the exceptional divisor of P,. The divisor of

jumping lines J of E intersects D in a finite number of points 1, # L each of which we

surround by s m d open discs C: c D. Then we obtain fiamed jumps Ei over Pl xL7,

by restricting the framed G-bundle É. and we have c 2 ( E ) = deg J = m ( E i ) . a

Conversely. the framed jumps Ei over Pl x Li may be glued toget her dong the

framings to the trivial bundle E, over CF0 = D \ {L} to recover the tramed bundle

Ë and hence the fiamed bundle E . This shows that the study of framed G-bundles

over P, may be reduced to the study of framed G-jumps:

7.1.2 Theorem: (Hurtubise) Let II : .M,(G) -. SP'(C) 2 C' be the map which

sends a bunde E E .Mk(G) to the point {(li. mi ) } E SP'(C) where the li are the

jumping lines of E (which lie in D \ {t} = C ) and where mi = m(E,). Then the

fibre of II above the point {(li, mi)) is the product 3Jml x - - x3Jmv, where 3Jm

denotes the space of isomorphism classes of framed G-jumps of mdtiplici ty m.

To further motivate the study of framed jumps we would now like to briefly

sumrnarize the manner in which the spaces 3gk are used in [BoHuMaMi] and

[Ti l,2] to prove the Atiyah- Jones conjecture [4t JO]. There is a natural inclusion

6, : i\;;(F) -+ B k ( F ) where F is a simple compact Lie goup. h\(F) is the space

of based F-instantons of charge k (that is the space of self-dual connections on the

principal F-bundle Pk over S4 of index k modulo gauge-equivaience) and B k ( F ) is

the space of all based connections on PI modulo gauge equidence. It --as shown

in [AtJo] that the latter space Bk(F) is homotopic to the space Q:(G). and it was

conjectured t hat . for any fixed q. the induced map

should be an isomorphism for sufficiently large d u e s of k.

In [BoHuMaMi] and [Ti 1) it was shown that dirnF, = (2n - 1) k in the case

that G = SL(n. C), and in [Ti21 it was shown that d i m 3 A = (2n - 5) k in the

case that G = SO(n. C). In either case we have

As we shall see. this is all that needs to be known about the spaces 3.7, to obtain the

. h i p h - Jones conjecture. In fact it suffices to know that dim 3gk 5 (c( Ad) - 1) k.

Consider the stratification

where K is the set of multi-indices h- = (k,. -'a - . kr) aith kl 2 - .'- 2 k, and

C k, = k. and .MA- is the set of framed bundles in .Mk(G) with r jumping lines

of multiplicity k,. The projection II : .bfk(G) -r SP'(C) of theorem 7.1.2 restricts

to a projection II : 'U h. + Ur where Ur is the smooth r-dimensional submanifold

of SP*(C) consisting of points ( ( l i , m i ) } E S P 4 ( c ) such that the l i are r distinct

points in C and the mi are any permutation of the ki . The fibre ,FgkL x - - . x FJkP r

is of dimension C (c(Ad) - 1) ki = (c(Ad) - 1) k, and so we obtain i= 1

dim .%f = r + (c(Ad) - 1) k .

and since dim M (G) = c(Ad) k we have

In ~articular Af ( , , ..- , 1 1 is the only stratum of codimension zero and hence it is an

open dense submanifold of .U ,(G).

Choose an)- total ordering on the index set K which has the property that

(k,. - - - . kr) < ( 1 , . - - - . I , ) whenever r > s. and define

Then each Ch. is an open dense submanifold of M k ( G ) (a small deformation of

a framed bundle dl either fi all its multipLicities or reduce some multiplicities

and add some jumping lines). In order to simplifp our exposition. we shall assume

for the moment that the spaces 3Jk. and hence also the strata .M are smooth

manifolds. Then for any coefficient ring -4. the filtration Ch- of .M C(G) may be used

to obtain a homology Leray spectral sequence E; with

which converges to @ Hq(,Uk(G); -4). Xotice the shift in homological dimension by 9

2 codim ( .MA-) which a-e know to be 2 ( k - r).

There is an inclusion i k : M k ( G ) + ,bf k+ , (G) which is obtained by glueing a

fixed framed jump of multiplicity 1 to each framed bundle in ,Uk(G). This inclusion

sends the stratum ,MA- of .Mk(G) to the stratum .McK,,, of JMk+ i (G) . and we note

that these two strata are of the same codimension. It was shown in [BoHullaMi]

that the induced map in homology

is a homology equiidence whenever q 5 4, where s is the number of indices ki in

the multi-index K with ki = 1. The proof of this result is technical, but it applies

also to other spaces of labelied particles, for example. it was used in [BoHu'uiaMi 21

in the study of the space of rational maps into flag manifolds.

The inclusion ik also induces a map ik : El + E;+ of spectral sequences, which

w e s h d now examine with the help of our knowlege of the codimensions of the strata

.M K . Consider a stratum M of (G). If K' = (kl , - - , kr) is not of the form

(K. 1) for any multi-index K. then we must have k, 2 2 for all i which implies that

r 5 2 and hence 2 codim (M K, ) 2 k. Thus Hq-, codim ( du,l ) ( M A-l : A) = O

for al1 q < k . Suppose. on the other hand, that = (K. 1) for some multi-

index h- = (k,:-. .k,) ai th say s of the ks equal to 1. Then k, 2 2 for dl

i 5 r - s which implies that r 5 y and hence 2 codim (,U K ) 2 k - S. From ( 1)

we see t hat i k : HP-* ( M n ) (.M : A) a Hq-2 codim K , , (M : A) whenever k q-3codim(,MK) 5 5 : andsince ; +2codim(MK) 2 i + ( k - S ) = k - ; 2 5 we

see that i induces a homology isomorphism whenever q 5 $. In terms of the spectral

sequence. this means that the inclusion i k induces an isomorphism at the El level

for q 5 $. Since the spectral sequence converges to H,(.Mk(G): -4) it follows that

the map

ik : Hg ( .Uk(G) : A) + H&U &+JG): A)

is an isomorphism for aii q 5 5 - 1. The h a 1 step in the proof of the Atiyah-Jones conjecture is to show- that the in-

clusions i k : .M,(G) -. M , + , ( G ) are homotopic to similar inclusions j k : .\\(F) -t

-l*+,(F) which were used by Taubes in [Ta] to prove a stable version of the con-

jecture. This is proven using intermediary maps which a-ere constructed in [BoMa]

using the [At HiDrMa] monad description of instanton bundles. This portion of the

proof makes use of Donaldson's theorem and applies only to the classical groups.

FVe return now to our assumption that the spaces 3Jk are smooth manifolds.

In fact this assumption is unnecessary. and it has not been shown to be true for any

group G . It suffices that the spaces 3gk are C W complexes. Then the spaces 3Jk

may t hemselves be st rat ified into smoot h complex manifolds

and this yields a h e r stratification of the moduli space M k(G) . Since each stratum

of .Fgk has dimension smaller than or equal to that of FA. the proof of the

Atiyah- Jones conjecture as outlined above remains valid. The only modification

which must be made is to enlarge the index set K: to include multi-indices of the

form K = (k,, - - - ,k,; j , , - - - , j , ) with j, E J k i . In [Ti 1,2] the exact nature of the

stratification of FJk was unspecified. In [BoHuMaMi] however, the space F& was

stratified by the spaces F J ~ (or simply 3&) of framed jumps E of multiplieity

m ( E ) = k with a given graph G ( E ) = G. It was shown that the space FJG is a

smooth manifold of dimension 2k + 1 where 1 is the length of the graph.

We condense our summary of the proof of the Atiyah-Jones conjecture into the

following theorem.

7.1.3 Theorem: (see [BoHuMaMi] and [Ti 11) Let G be a simple Lie group.

If'dim.Fgk 5 (c(Ad) - 1) k for al1 k then for any coefficient ring A, the inclusion

i, : M,(G) + M k + , ( G ) induces a homology isomorp~sm

for all k 2 2(q + 1).

7.1.4 Remark: It is not easy to prove that dirn.F.7'' 5 (c(Ad) - 1) k for any

simple group G, but it is easy to obtain a weaker upper bound. We have M ( k ) =

F z k x C (recall that M ( k ) c M (G) is the space of fiamed bundles of multiplicity

k with a single jumping Iine) and so

7.2 F'ramed Bundles Over P, x U

In this section we shall retrace our steps and we s h d study framed bundles over

P, x U (that is a bundle over P, x U equipped with a fiaming over {O} x U) as Ive

have studied \inframed ones.

Let us begin our study by considering framed bundles over P,.

7.2.1 Proposition: The space of h e d G-bundles over Pl of a given type

X E i\nw is the gener&ed fîag manifold G / P , where PA is the parabolic subgroup

Proof: There is a unique unframed bundle E of type A, and two fiamings e, f E G

will be isomorphic if there is an automorphism v : { z # oo} -+ G of E with

e u (0 ) = f . But every automorphism is of the form exp(A) for some endomorphism

-4. andweknowfrom2.3.1 that theseareoftheformA= A,e,+ A,P. a€ R3 4 A ) <O

where A, is a polynomial of degree - a ( X ) and each Ag is a constant.

Next we shall describe a normal forrn for transition functions of framed G-

bundes (E, e) over P, x U, where U is a polydisc in Cm. Notice that we may

always truncate the framing e to order O in x by modifying the transition mat&

by 6 ++ d e-le,, and then we may consider eo as an element of the generalized flag

manifold G/ PA. Rather than working with arbitrary bundle isomorphisms, then, we

allow only Tame-fixing bundle isomorphisms: a fiamed G-bundle may be given by

a pair ( 4 , e) with e E G / P A ; two frarned G-bundles (4, e) and (+, f ) are isomorphic

if and o d y if e = f and 4 G $ via a £rame-fixing isomorphism.

Recall £rom theorem 3.1.2 that by changing trivializations over U , and Li, we

were able to put the transition function into normal forrn. If we modie the argument

by requiring that the change of trivialization map a : U, - G preserves the haming,

that is by requiring that a ( ( z = O}) = 1, then we find that the transition matrix

may be put into a modified normal form.

7.2.2 Proposition: A framed G-bunde over Pl x U with type(E,,) $ X E

may be given by a pair (9, e ) wtiere e E G/PA and where q5 is in framed A-normal

form (see definition 7.2.3).

7.2.3 Definition: The transition function 4 of a framed G-bundle ( 6 , e) over

P, xU is in framed A-normal form if # = exp(XInr)exp(P) where X E A n W and P : U -, X1(P,, Ad(A)(-1)). We may \mite

If two framed bundles are isomorphic, then certainly the underlying unframed

bundles are also isomorphic, and so any invariant for unframed bundles is also an

invariant for fiamed bundles. The invariants of chapter 5 were calculated using the

section matrices of chapter 4 which were calculated Tom a ,\-normal from transition

matrix of a bundle. One may follow exactly the same procedure to calculate the

section matrices from a framed A-normal form transition matrix; both lemma 4.1.2

and proposition 4.3.1 hold in the case of fiamed bundes.

One further remark on the subject of invariants is that the cascade of bundles

may be modified to obtain more invaxiants for framed bundles than were obtained for

unframed ones: the proviso that t i be greater than t i+, which appears in proposition

6.1.2 (or more generally, the proviso that a( A) > O as in remark 6.1.8) is unnecessary

in the case of framed bundes.

7.3 Isomorphic Ekamed Bundles

We consider the problem of determining which framed A-normal form transition

functions yield isomorphic framed G-bundles over P, x U, where U is a complex disc.

The group of all bundle isomorphisms is knom as the gauge group, and this group

acts on the space of al1 transition functions. We shall describe the intersection of

the orbit of a given A-normal form transition function under the restricted gauge

group of fiame-fixing isomorphisms with the space of d A-normal form transition

func t ions.

Recall that an isomorphism between the two fiamed bundles given by the framed

normal form transition functions and 111 is given by a G-valued function v ( z , x)

with v(0, x) = 1 which is holomorphic in x and z such that 4 v $-' is holomorphic

in z- ' . Locdy, v may be expressed as v = sexp(A) where s = v(z, O). We shall

begin by restricting our attention to isomorphisms of the form v = exp(A) such

that -4(z, 0) = .4(0, x) = 0.

Fix a bundle E over P l x U , where U is a disc in C, with framed A-normal

form transition function t exp(P) where t = exp(A ln ;) and P ( r , O) = O. Given

two elements Q and R of H 1 ( P l , AdA(-1)) consider the two bundles given by

the framed A-normal form transition functions 11, = t exp(P + Q X " + ' ) and x =

t exp (P + RX'++') which agree with 4 up to order k in z. Given an endomorphism

A E t z H o ( P 1 7 A ~ E ( - I ) ( ' - ' ) ) , let us ask whether the fiame-fixing automorphism

v = exp(A) up to order k of the bundle E may be extended to a frame-fixing

isomorphism w = exp(A + BX'+') up to order k + 1 of the two bundles given

by w and X . For this to be so. we must have (ii> wy-' )k+l holomorphic in z-'.

Embedding G into some G L ( n . C) we compute

u t d ) k + l modulo t\d,B? with B arbitrary. is precisely the obstruction to

extending t.he endomorphism -4 from order k to an endomorphism of order k + 1.

Thus v is isomorphic to up to order k + 1 in x if and only if (Q - R) lies in the

image of the obstruction map 6 : H ~ ( P , . A ~ E ( - ~ ) ( ~ - ' ) ) + H1(Pl ,AdE(-l) (0))

in the long exact sequence associated to

a n d we have

dirn(Im6) = hO(pl . -4d~(-l)(O)) - hO(pl . A~E(-I)( ' ) ) + h0(p1 . ~ d ~ ( - l ) ( ~ - ' ) )

= h0(p1. AdX(-1)) - w l l ( . 4 d ~ ) .

7.3.1 Proposition: Let V ( k ) denote the space of Q : C' + H1(Pl . AdX(-1)) of

degree k in x with Q ( r . O) = O such that the bundle F given by t exp(Q) is isomor-

phic to the b u d e E u p to order k in z via a fiame fUcing isomorphism of the f o m

exp(;L) with A(;, O) = O. Then C'fk+l) + Vk) is the subbunde of the trivial bun-

dle I ~ ( ~ ) X Z ~ + + ' H ~ ( P , . AdX(-1)) a-hose fibre or-er Q(') is 6 H O ( P ~ . ~ d ~ ( - l ) ( ~ - ' ) )

which is of dimension hO(Pl . AdEo(-1)) - wkl(r\dE).

In the case that E is a framed jump. that is in the case that E is trivial over

Pl x {r} when r # O. we have w k l ( AdE) = O whenever k 2 l(AdE), where l(AdE)

is the length of the graph of AdE. and so for k 3 Z( AdE) we have

B y the above proposition, t his implies t hat d extensions of Q(*) to order k + 1 yield

isomorphic bundies. Consequently, the framed A-normal form transition function

(and hence any transition function) may be tmcated to order Z(AdE) without

altering the isomorphism class of the bunde.

7.3.2 Theorem: If E is the fkamed jump given by ( O . e).then d may be trun-

cated to order 1(,4dE) in z.

7.3.3 Remark: The same arguments may also be applied to untramed bundles.

One finds that the transition function of an unfiamed jump may be truncated to

order k in z. where k is the smallest integer such that W ; ( A ~ E ) = dimg. We always

have k 5 Z(AdE).

!Ve have so f a considered only fiame-Gng isomorphisms of the form exp(il)

wit h .A( z . 0 ) = O. but no= ne consider general kame-&eng isomorphisms s exp(.-l).

with s E .4utEo(-1). where AutEo(-1) denotes the space of irame-fixing automor-

phisrns of Eo (the exponential map provides a holomorphism from HO(Pl. AdX(-1))

to .4utEo(-1) ) W e must notice two things. The first is that if O i v via sexp(.4)

and if o i via s exp( B) then w will also be isomorphic to via exp(C) for some C

vrith C(r. O ) = O. The second is that ç5 can be isomorphic to Ii via one frame fixing

isornorphism of the form s exp(.-i) and another of the form exp(B) if and only if s

estends to a lame-fixing automorphism of E over Pl x C'. that is s E (;\ut E(- 1 ) ), .

Thus over and above the iterated vector bundle there is an additional orbit of

which is of dimension hO(P1. AdX(-1)) - w zl ( AdE). If E is a framed jump then

(.-lut E(-1) ) , is the trivial group.

7.3.4 Remark: We can use our description of the orbit of a bundle to obtain

a crude estirnate of the dimension of the space 3Zk- Let 73; denote the space

of fiamed jumps of multiplicity k with type(Eo) = A. Choose 2 large enough that

dl bundles in Fg; rnay have their transition functions truncated to order 1 in x

(we could choose 1 = c(.4d) k for example). The space Q of al l truncated framed 1

X-normal form transition functions is @ z* Hl (P 1, AdX(- 1)) which is of dimension i= 1

Z h l (P l Ad)<(-1)). If we denote the bundle given by an element Q E Q by Eq,

and if we write ( det(S-,(AdEQ))) = SO + S1z + - - - S k z k + - - - then we

see that the space of Q E Q which define bundes Eq of multiplicity k is the open

subvariety (defined by Sk # O) of the closed subsariety cut out by the k - 1 equations

sl = ... - - Si-, = O (S, is automatically zero since Q ( z . 0) = O). The dimension of 1-1

the orbit of Q in Q is ho(P1, AdX(-1)) + C (ho(P1? A d X ( - 1 ) ) - d1(.4dEQ)) = &=O

( Z f l )hO(P , . AdX( -1 ) ) -c(Ad) k. Finally a-e must not forget that a framed jump is

determined not only by the transition hinction (@\-en by Q) but also by the fkaming

e E G / PA. Putting all these ingredients together. a-e obtain the foilowing dimension

estimate:

=(-Ad) k - d(X) - (k - 1 ) 5 d i m ~ ~ t 5 c(;\d) k - d ( X )

where d ( X ) = h O ( p l . .4dX(-1)) - dim(G/P,) = C ( & ( A ) - 1) . Since d ( A ) 2 1 a E R 3 a ( X ) 2 2

for ail X we deduce that

dirn3& 5 c(-Ad) k - 1 ,

and we have recovered the dimension estimate which was obtained (with much less

effort) in remark 7.1.4. Yotice that the k - l equations (which single out the proper

multiplicity) do not enter into the calculation of the upper bound. It is difficdt to

determine how many of these equations are independent.

7.4 Deformations of Framed Jumps

By a deformation of a bundle E over Pl x L* we mean a bundle E over Pl x C x V.

and we shall denote the coordinates in V by E. If E cornes with a framing over

{O} x C then we require that E have a framing over {O} x Cx TV. In this section we

s h d be discussing linear deformations of framed bundles: we consider two linear

deformations E and F to be isomorphic if they are isomorphic to first order in e via

a frarne-Wng isomorphism.

Let E be a framed G-bundle over Pl x C where U is a disc in C with Eo of type

A. and suppose that E is given by the framed X-normal form transition function

GJ = t exp(P) where t = exp(X in z) and P : C + R1(P,, A~x( -1 ) ) . Since Q takes

d u e s in the Borel subgroup B c G we may also consider E as a B-bundle. If we

consider E as a B-bundle and if we allow only fiame-fixing automorphisms then,

referring to section 3.2 in which we discussed the uniqueness of the normal form,

we see that the map P is uniquely determined. Consequently, the space of aii

deformations of the B-bundle E is the space of bundles E with transition h c t i o n

9 = t exp(P)e.xp(cQ) for some map Q : C + H1(Pl.AdX(-1)). To simplifv

notation we shall embed G in some GL(n. C) and write a = o( l + EQ). Ive would iike to be able to determine the space of deformations of the G-

bundle E. To do this we must determine which deformations of the B-bundle E

are isomorphic to the trivia1 deformation E when considered as G-bundles. in other

w-ords we must determine for which Q : C + H1(P, . AdX(-1)) we have o(l + cQ)

isomorphic to ci to first order in e. W e must find changes of trivialization a (1 + cr t )

over L-= and b ( l + rB) over U,,. with b ( O o x ) = 1 and B(O. r) = O so that b(l + eB)

fixes the frarning. such that

Extracting the constant term in E we obtain o = a-ld b so the pair (a. b) determines

a frame-fixing automorphism of o (if E is a kamed jump then we necessarly have

a = 6 = 1 ). Equating t.he first order terms in c gives

IYe may interpret this condition on Q in the following waj-. Recail that the section

map S-l ( AdE) : XO(P1, E- ) 4 H1(P, . E + ) was defined using the long exact

sequence associated to the exact sequence of bundles

of section 1.3. From the definition of E+ and from the description of the normal

form transition matrix IL of AdE given in proposition 3.3.1. we see that E+ may

be given by the submatnx ++ of rzb with entries (z+); whece a,$ E R+ U (R+ is the set of positive roots and A is a basis for f ~ ) . This means that the subbundle

Ef may be taken to be the bundle associated to the B-bundle E under the adjoint

representation of B tensored by O(- 1):

The map Q : U + H1(Pl,AdX(-1)) may be considered in a natural way as

an elernent of Hl (Pl x U, E+) , so the space Hl (Pl x U, Ad,E(-1)) parametrizes

the space of deformations of the B-bundle E. The section matrix S-,(AdE)

\vas constructed in terms of the transition function + of AdE by demanding that

B = +- 'A = #-'A4 as an elernent of H1(P, xU, E+). So the condition that

Q = B - +-'A # in H1(Pl x U , E + ) for some pair (A, B) is precisely that Q must lie

in the image of the section map. Thus the space of deformations of the G-bundie

E is the space Hl (Pl x U, E+)/Im(S-l(AdE)) = H1(PI x U, AdE(-1)).

7.4.1 Proposition: The space of h e a r deformations of a h e d G-bundle E

of multiplicity k is parametrized by the space H1 (Pl x U, AdE( - 1)) which is of

dimension c(Ad) k.

Recd that the moduli space Mk(G) fibres over C* and that the fibres are

products of spaces FZm of fiamed jumps of rnultiplicity m. This suggests that it

would be interesting to determine which linear deformations E of a given fiamed

jump E preserve its multiplicity to fxst order in E . This may be determined in the

following way. Let p be any faithfd irreducible representation of G and expand

Shen the multiplici ty rn of E is the order of vanishing of S, (x) and the deformation

£ will preserve the multiplicity to first order in E provided that Sl(x) also vanishes

to order rn in x. Notice that the section matrix S-, ( p ( E ) ) and also its determinant

are both linear in Q to first order in E and so the condition that S, ( x) must vanish

to order m defines rn linear constraints on Q E Hl (Pl, AdX(-1)). If these linear

constraints were independent, then we could deduce that dim 3Jm 5 (c(Ad) - 1) m

which is exactly what we need in order to apply theorem 7.1.3. Unfortunately, as

we shall see in the following example, the constraints are not always independent.

The example shows that it is necessq to consider higher order deformations if one

hopes to obt ain m independent constraints.

7.4.2 Example: Let E be the SL(2. Cl-bundle given by c = ( z 2 *P 7 -2 ) where

2 p = x + tr. This bundle has graph G ( E ) = (2.1.1.0) and it is of multiplicity 4.

Consider an a r b i t r q linear deformation t of E given by = ( rhere

q = q - l ( x ) z - l + q,(z) + q , ( x ) z . The determinant of the section matrix of is

x2 + eq, det S-,(a) = det

and so the deformation E preserves the multiplicity to first order in E provïded that

- L i . * - 0- q -,,, = %,., and q-,.J = -qo,, -

Let us determine whether £ may be extended to a second order multiplicity-

preserving deformation given by \k = P f c q + r 2 r ) ,-2 ive -

Since r- , (z) may be chosen arbi trarily. we ttid t hat q must satisfv the one additional

constraint that qOeo2 = q- lq ,q , ,o . In all. q must satisfy the four linear constraints - -

Q - 1 . 0 - P o . * - q - l , l = O - and q 4 2 = 2 q 0 J .

In [BoHuMaMi] the spaces 3gk of framed SL(2. C)-jumps of multiphcity k a-ere

stratified as the union of smooth spaces 3gH of framed jumps with a given graph

H of multiplicity k. In the case of SL(2.C)-bundles. we have seen that the graph

of -4dE is determined from the graph of E and vice versa. So ~ir-e could generalize

this stratification of ; f J k by redefining 3JH to be the space of fÎamed G-jumps E

whose endornorphism bundle AdE has graph H. A perusal of section 7.3 supports

the idea that the graph of the endomorphism bundie is of particular importance. For

this reason we shall now try to determine which linear deformations of E preserve

the graph of the associated bundle AdE to hrst order in E . We s h d discuss two

met hods of doing t his.

The first method is computationdy simpler and may be applied to any vector

bundle E; in this paragraph we suppose that E is a GL(n, C)-bundle. The graph of

E is determined by the dimensions hO(P1, E(- 1)(')) and the graph will be preserved

provided that every element of H O ( P I , E(-1)(')) may be extended to an element

1 O 3

of HO(P,. E(-l)('el)) which. we recall. denotes the space of sections of E(-1) up to

order k in x and order 1 in e. Let us expand the section matrix of E as S-,(E) =

So + ES, + - - - (note that So is the section matrix S-, (E) ). Then given any column

vector uo in the kernel of S, to order k in x (so that uo defines a section of E ( k ) )

we rnust be able to find a column vector u, such that (So + eS,)(u, + e u l ) = O to

order k in x and 1 in E. This can be done if and only if SI u, + S o u l = O. Thus E

d l preserve the graph of E to first order in e if and only if

SI ker S, c LmS,

to order k in x for any k. Since S, is Linear in Q this defines linear constraints on

Q. The second method applies specifically to the endomorphkm bundle AdE. -4

deformation E aiil preserve the graph of AdE to first order in e provided that

every section of A ~ E ( - I ) ( ~ ) extends to a section of A ~ E ( - I ) ( ~ , ' ) . A section of

.4dE(- 1 )( '1 is given by a pair (A. B) of g -dued maps with -4 holomorphic in s-'

and B holomorphic in ; with B(0. r ) = O such that A = o B O-' to order k in

x. This section extends to Ad£(- 1)( if there exists a similar pair (-4,. B, ) such

t hat

(-4 + ~-4 ) = 4 (1 + eQ)(B + el?, )(1 - c Q ) Q-' = O

to order k in s and 1 in e. So we must have

; I l = o ( Q B + B l - ~ ~ ) o - l = O .or

[Q, BI = O-'.?, 4 - B,

to order k in x. This means that [Q. BI must lie in the image of the section map

S - , ( M E ) or, equividently, that [Q. B] = O in the space H1(P1. ~ d ~ ( - l ) ( ' ) ) . Thus

P preserves the graph of AdE if and only if [Q, BI = O E X1(P, , 4dE(-1)(')) for

all B E Ho(Pl , AdE(-1)(')) and for ail k. So far this method has been nothing

more than a special case of the previous method, but we shall continue to refine the

result using the Serre duality pairing

FVe have

[Q. BI = O for aii B E H ' ( P , . A ~ E ( - I ) ( " ) iff

( [ Q . B ] . C ) = O t o r d B . C E H ~ ( P ~ . - . ~ E ( - ~ ) ' " ) ifF

(Q. [B. Cl) = O for all B. C E H'(P,. A ~ E ( - I ) ( ~ ) ) .

7.4.3 Proposition: The deformation E given by O = o(l + cQ) preserves the

graph of the G-bundle E given by o to first order in E if and o d y if

for all k.

7.4.4 Remark: Recall that the space of all linear deformations of a bundle E

of multiplicity k is the space H' ( P l x C. AdE(- 1)) a-hich is of dimension c(Ad) k.

The above proposition describes linear constraints which must be satisfied by a

Linear deformation in order that it preserve the graph of AdE. If it could be shown

that at least k of these constraints are independent. then we could deduce that the

space 3JH. and hence also the space 33'. is of dimension less than or equal to

(c(-Ad) - 1) k. and then we would obtain the stability result of theorem 7.1.3.

7.4.5 Example: Consider again the bundk E given b>* O = ( z 2 : P 2 ) rhere

- - l + p = r2 + iz and the deformation 8 given by + = (=* P>:q ) where q = q- , . q, + q , z. I f one uses the first method described above to determine which g presen7e

the graph G ( E ) = (2.1.1,O) one finds that q must sa t i se the following constraints:

- when k = O one h d s that q-,., = q0., - q, , , = O : when k=l one h d s that in

addition q - , , , = O: when k = 2 one finds that q- l , 2 = 2qoS1 : and when k 2 3 there

are no further constraints on q.

Applying either of the two methods to determine which deformations preserve

the graph G(AdE) = (4,4,2,2,2,2,0) one h d s that the deformation E preserves

G(AdE) if and only if E preserves G ( E ) if and only if q satisfies the above five

constraints. However, when determining which E preserve G( Ad E), the constraints

manifest themselves at higher values of k (the constraints appear when k = 1, 4

and 5 ) than when we determined which ê preserved G ( E ) ( k = 0. 1 and 2). t2 r2 + rt + X E

To be more specific. consider the bundle E given by CP = *-2 C

a-hich was the subjeet of example 5.1.5. This bunde does not ireserve the graph of

E because we do not have qeiS2 = 2qoSi . This constraint appeared (using the tirst

method) when k = 2. Indeed. from example 5.1.5 it is apparent that the section -r of E(*) does not extend to a section of E(2*1). (:-J

7.5 Bundles of Minimal type

We shall focus our attention nom- on some important examples of G-bundles E

over Pl x L*. If G is a simple group. then the lattice flm (recall that ï denotes the

integral span of the inverse roots) contains a unique minimal element, namely the

highest short inverse root g. For semisimple or reductive groups. the lattice ï n E contains one minimal element for each simple summand of g. Our aim is to describe

the space 3.7: of framed G-bundles E of multiplicity k with E, of minimal type

$ E -1 ri W. Since any deformation of a bundle E will either fk or reduce the type of

E,. we see that 3.7: is an open subset of the space 3& of all frarned G-bundles of

multiplicity k. Furthermore. it is reasonable to expect that. if G is a simple group

(SO that there is a unique minimal type). then the space 33; should also be dense

in FJk. -1lthough we have no proof for this conjecture. it motivates the study of

the space 3.7;. FVe shall see that 33: is a smooth cornplex manifold of dimension

(c(Ad) - 1) k. This immediately gives us a lower bound on the dimension of 3gk

If ~ 3 ; is dense (or at least of maximal dimension) in 3gk. as we expect, then the

above inequality may be replaced by an equality and we obtain the stability result

stated in theorem 7.1.3.

Our first step is to show that a bundle E with E, of minimal type may be given

by a particularly nice transition function.

7.5.1 Proposition: Let E be an ( d a m e d ) G-bundle over Pl x U whose re-

striction to Pl x{O} is given by a minimai nonzero element # of the lattice I' m.

Then, over P, x V , for some V c U, E rnay be given by a transition function of the

f orm

Q = e u p ( ~ l n z - l ) e ~ P ( ~ k r e , )

for some positive integer k , where p is the root whose inverse is @ and e, is a bais

for the root space g, .

Proof: By going through a List of root systems (such as that in [Hum] or [Bo]) one

rnay verify that B ( F ) 2 for a.ll roots 13, and that equality holds only for the root

p whose inverse is f i . So in view of theorem 3.1.3 we know that E may have its

transition function put in the normal form 4 = t e ~ ~ ( P ( x ) z - ~ e , ) with P : V -+ C

and t = exp(fi ln r): the bundle is determined by a single holomorphic function.

A s we have seen the normal form is not unique, and we shall sharpen it by

conjugating by a map V -r X. Write P ( z ) = pkxk + Pk+IxkC1 + . . . , with P, # 0.

Then the function x-* ~ ( s ) admits a holomorphic logarithm, and we conjugate b

by the rnap exp(- $ ln(z-k~(s)) j i ) , whidi commutes with t . So we can replace the

transition function by q!~ = t exp(Q),where

By the conjugation formula 1.3.3: we have

and by the definition of a root space, we have [fi e p ] = 2 e p . So the iterated Lie

bracket becomes [ep bn] = (-2)"ep. Replacing this in the formula for Q gives

complet ing the proof. O

7.5.2 Remark: If i : SL(n, C) -t G is the homomorphism obtained by sending

the root of SL(2, C) to the root p of G, then the G-bundle of the above proposition

is the bundle i ( F ) associated to the SL(2, C)-bundle F given by 4 = (' ) . So a G-bundle of minimal type reduces to an SL(2, C)-bundle.

7.5.3 Examples: By t heorem 3.1.2, an S L(n , Cl-bundle (or an Sp(n , C)-bundle

for even values of n) over Pl xU of splitting type X = diag(l,O,. . . ,O, -1) above

O E U may be given over V c U by a transition function of the form

Unless p ( x ) = O, this is not unique. In fact, if p ( x ) # O then for some k we have

A 4 = PP' + P ~ + ~ x ~ ~ ~ + . - - with pk # O and, as in the proof of proposition

7.5.1, we can conjugate by diag(Jr"p", 1,. . . , 1 , J-) to obtain the transition

funct ion

where we may have to restrict V further in order to choose a holomorphic branch

of the square root.

For SO(n .C) , the point X = diag(l,O,. . . , O , -1) does not lie in r: it is the

minimum element of the non-identity cornponent of X n W. The unique b u -

dle of this splitting type over O E U is given by the transition function O =

The minimal splitting type for an SO(3, C)-bundle is A = diag(3,0, -2). A

bundle of this type rnay be given by

And for n 2 4, X = diag(l,l, O,. . . , O , -1, -1) is the minimal splitting t-ype for

an SO(n, C)-bundle, and a bundle E with E,, of this type may be given by

Wè should also remark that for n = 4, there are two minimal elements: the

hndle of Qpe A = diag(1, 1, - 1. - 1) is not isomorphic to the bundle of type

X = dia@' -1.1. -1).

Having determined the transition function d for the G-bundle E. we know by

proposition 3.3.1 how to compute the transition matrix ~ = s exp(Q) for the endo-

morphism bundle AdE. R e c d that s will be a diagonal matrix with s: equal either

to zO(" if û E R or to 1 if a E A. A list of root systerns reveals that [a(fi)( 5 2

Rith equality only when a = IF. so we have

The matrix Q is given b - Q j = zk i-L [ec. e $1, If we choose A to be an orthogonal

basis for with + E A. then Q takes the form

= { - rkz-' if u = p and 3 = p .

bzk;-' if a = jî and Y = -p . and

By again sifting through a list of root systems. one h d s that for anJ- root a.

û ( F ) == i if and only if a - p E R.

Let # denote the number of roots a such that a(A) = 1. Then for any G-bundle E

w-ith Eo of minimal type. the transition matrix for AdE is given by

109

Then using lemma 4.1.2 we may compute the section matrices.

From t hese. we me>- calculate the dimensions of all the spaces kt-:.

7.5.4 Proposition: Let E be a G-bundle or-er Pt xCo with Eo of minimal type

i î . Thm the iniariants ~f(..\dE) are given by

m-kzere g is the dimension of g and where # is the number of roots a sud l that

& ( f i ) = 1.

7.5.5 Proposition: For any simple goup G w e have

where c(Ad) is the scaling factor (definition 5.3.1), # is the number of roots a such

that a(jî) = 1 , h is the duai Coxeter number of G (which is equal to 1 plus the

height of Ci), c is the Casimir element of the adjoint representation (which may be

computed by the formula c = ( P + 2p)(fi) where 2p is the s u m of the positive roots

of G ) and ~ ~ d ( f i ) l ~ = a(b)2. a E R

Proof: The graph of AdE is given by the row 1 = -1, and this determines the

multiplicity

m(AdE) = (# + 4) k.

By our definition of the scaling factor c( Ad) and by remark 7 - 5 2 . this implies that

c(Ad) = # + 4. This proves the fist equality. and the other equalities are easily

verified with a list of root systems. U

Let us now discuss the spaces 3 ~ ' of framed G-jumps E of multiplicity k with

E, of minimal type 4. As remarked earlier. the space 3.7: is an open subset of the

space 3gk. and for a simple group G we expect it to be both open and dense. W e

shall give two descriptions of the space ~3:.

Our first description of the space 3.7; will make use of the results of section

7.3. A framed jump E in FJ; may be given by a pair ( 0 . e ) where e E G/PG and

O = t e x p ( P ) is a framed @-normal form transition function, so we may mite

13-e know £rom theorem 7.3.2 that P may be truncated to order l (-4dE) = Ik in

x. and in order that the bundle E have multiplicity k, one may verify that P

must satisfy the constraints that is that Pp.-l(x) vanishes to order k - 1 in x and

pp.-~.* # O. Thus the space of ail possible P is

If we quotient this space P by the restricted gauge group of frame fixing isomor-

phisms of the form exp(.-i) with -A(=. O) = O as in proposition 7.3.1 we find that

P/Ç is the total space of an iterated fibration

where the fibre of 30) + flj-l) is equal to C#+' for 1 $ j < k, C* XC#+' for

j = k , and C2 for k < j 5 2k. Findy, after quotienting by .SutE,(-1) (which, as

a cornplex manifold, is equivalent to Ro(PI , Adfi(-1)) = cQ+~) we obtain

which is a smwth complex manifold of dimension

?;ow we s h d give our second description of the space 33:. Since there is a

unique (unhamed) bundle of minimal splitting type ji and multiplicity k. we need

only describe the space of equivalence classes of framings t hat may be placed on the

bundle: recail that two framings e, f : C -r G are deemed equident if e a, = f for

some automorphisrn a of E (ao denotes the restriction of a to { z = O} ). W e can use

the transition matrix c for AdE to explicitly compute the endomorphisms of E:

t hey are given by a column vector -4 holomorphic in r with w.4 holomorphic in z- ' . If one cornputes the endomorphisms one finds that their restrictions to { z = 0) are

given by .-l(x) = A,(x)e, which meet the following requirements: P E R U ~

(1) -4, is a r b i t r q for a E R with a($) < 1 and also for a E A \ jî: (2) AG vanishes up to order k in x. as does il, for a E R with a(b) = 1: and

Let us introduce the following notation: we let qr denote the subalgebra of g given

b y

where bL c i). and w-e let Q p denote the corresponding subgroup of G (notice that

we have qp span(F) = Pf i ) -

spfitt ing type b and of multiplicity k is a vector bundle of r a d ((# + 3)k - (# + 2))

over G/QP which. in tum7 is a Ck-bundle over the generalized flagmanifold G/Pfi.

The space 3: is of dimension (# + 3) k = (=(Ad) - 1) k.

Proof: Thespace~: is thequotient of the spaceofallhamings by the restriction

of the space Ho(P, x U, AutE) to {z = 0). -4ny fkaming may be written as exp(F)g 00

with g E G and with F = C Fi+', F, E 0. The elements of H O ( P ~ XC? - ~ U ~ E Y ~ , ) ~ ~ . , i= 1

on the other hand, may be written as exp(A)b with b E Q, andwith A = C A , x ~ , i= 1

u-herefor 1s i < k w e h a ~ e . 4 ~ E q,,. andfor k 5 i c 2k we haveAi E $ g,Bb. QER\P

and for i >_ 2k we have Ai E g. Cl

7.5.7 Proposition: If G = GL(n . C) then the flag manifold G/Pi is the pro-

jectivization P(TP,-I ) of the tangent bundle of P,-l. a d C/Q, is TPn-, O which

denotes the tangent bundle of Pn-, with the zero section removed.

Proof: The last column of a matrix -4 = (a:) E G L ( n ? C) provides homoge-

nious coordinates Xt = a: for an element of the base space P,-i. We shall cover

GL(n. C) by open sets

Each set C'k may in t u m be covered by open sets c,?. 1 # k

where -4; denotes the matrix obtained by deleting the first column and the Pb row

£rom the matrix -4. Given a matnx -1 E CF there is a unique matrix B = ( b ; ) E Pb

such that the product -4B = C = (ci) satisfies the folloaing conditions

where C i ; denotes the matrix obtained by removing the first and last columns

and the k t h and Ph rows from the matrix C. To obt,ain cf, = 1 we must choose

b i = l/aL, so we see that the last column of the matrix C provides a f h e coordinates

X i = .Y'/x' = CL on the base space P,-l. In order to satisfS. conditions (1) and

(3). the (n - 1) x (n - 2) matrix Bin must be chosen to be

and this allows one to compute the entsïes b i , j > 1 in terms of the matrix A. These

in turn may be used to compute the Ph row of C in terms of A; if we write the ph

row of C as

where yk and y, have been omitted, then

- (-l)~+f+l det A: Y, - det Ai '

So the Ph row of C provides afnne coordinates, Y: = yj for j # k and 5' = -1, for

the fibre Pn-* of the fibration G/P, -r PnV1.

Now we shall consider the quotient G/Q, , . If we alter the above matrix B E P,

by multiplying the fkst column by b:/b: then we obtain a new matrix, say D, which

lies in Q,. The product C = AD stiil satisfies conditions (1) and (3) above. but

condition (2) must be replaced by

(2) c: = X 6; for some A E C* .

The lth row of the matrix C becomes

with y j as above. Since X is given by X = b : / b i , and since b: is the first entry of

the 1' colurnn of A-' we have

det A ), = (-1)'+'

an det A: '

Notice that yj/X = (- l ) ja i det Ai/ det A, which is independent of 1. So if îr :

G/Q, + P,-1 is the projection defined by the final column of A E G, and if

P,-, is covered by the sets Uk7 then we may define trivializations (bk : rr-'(Cik) +

0 - k c n - 1 \ (0) by

- X" --( det A - d e t ~ : , d e t ~ : , - - - , ( - l ) k d e t ~ : , - - - - , ( -1)"det~; ) .

omit

With respect to these trivializations, one may compute the transition matrix

Pk : r - l (uk n u') -P GL(n - 1, C). To obtain the jth colwnn of the transition

matnx mk we must express the jth entry of d L ( A ) in terms of the entries of 4r(A)-

For j # 2 we have

s o the jth column of Ok is given by

To find the Ph coluron of dk we notice that if the first column of the matrïx A is

replaced by a copy of its last column, then the resuiting matrix has two identical

columns and hence is of determinant zero:

O = C(-~) '+'X' 8 det Aj .

From this we obtain

( - 1 ) ' ~ ' det A: = C(-l)'+lxi det 4 if 1

and hence

(-1)'xk det A: 4 , ( 4 , = det A

-x; ( -1 ) 'XL det A:

i# 1 det A

So the Ph column of the matrix 4 k 1 is given b y

Notice that the transition matrix q j k coincides with the transition matnx of the

tangent bundle TF,-, , and so the proof is cornpiete. O

115

Conclusion

In t his t hesis we have developed some t 001s for s t udying deformat ions of holo-

morphic principal bundles over the Riemann sphere. t hat is bundles over P , x C

where CF c Cm. A bundle E over P, x C may be described by a single transition

function. and we have shown that Iocally. the transition function O may be put

into a (non-unique) normal fom in which o takes d u e s in a Borel subgroup of the

bundle's structure group.

For computational purposes, there are great advantages to having the transition

function in this normal form. We have developed algonthms for cdculating many

invariants of a buadle E in tenns of a normal form transition function: we can

calculate the dimensions of the spaces of holomorphic sections of various associated

bundles; we can calculate the topological type of the restriction E, of E to P, x {x)

for any t E C': and we can caiculate the divisor of jumping lines of E in (7 .

For one- parameter deformations ( bundles over P , x tr a-it h C: c C ) . we obt ained

fur t her information by studying the restriction of the bundle to formal neighbour-

hoods of the line P, x {O). In particular. Rie gave a description of the space of normal

form transition functions which yield isomorphic bundles. and we found that if E,

is trivial for x # O then the transition function of E may be truncated to h i t e

order in x. We also obtained some new invariants by constructing a collection of

bundles EI called the cascade of bundles below E. These bundles were obtained by

pex-forming a series of simple operations on any normal form transition function of

the bundle E.

ive have also attempted, with Limited success. to study the topology of spaces

of framed jumps, and hence also the topology of the moduli space .Mk(G) of h m e d

bundles E over PZ with c , (E) = k. We have given a satisfactory description of the

space 3.7: of hamed jumps of multiplicity k and of minimal type b. Mie believe

that this space is dense in the space Fgk of all kamed jumps of multiplicity k, but

a-e have no proof for this conjecture. If it is dense, then the methods of [BoHuMaMi]

may be used to deduce that H,(M,(G)) i Hq(M,, , (G)) whenever k 2 2(q + 1).

We would like to end the thesis by raising a few questions.

The space of normal form transition functions which yield bundles of a given

multiplicity is an open set in a closed variety cut out from the space of all normal

form transition functions by k equations. It is difficult to determine how many of

these k equations are independent. Might it be possible to circumvent this difficulty

by refining the normal form transition function? Perhaps it is possible to define a

different notion of "normal form" for each different multiplicity or for each different

graph type.

1s it possible to use the cascade of bundes in some way to find a aice stratifi-

cation of the space 3,7& ? For example, if G = S L ( n . C ) then we can construct a

chah of bundles Ei below E in such a way that WL_~(E,,,) = U-:'(E,) for ail i. k

(see lernma 6.1.4). By restricting the bundles E, to Pl x {O) we obtain a descending

chain A t in the lattice .2 n W . It may be able to understand the topology of the

space of framed jumps with a given descending chain A, (in fact this possibility

inspired the definition of the cascade of bundles).

The space of linear deformations of a framed jump E of rnultiplicity k is

paramet nzed by the space Hl (Pl x u, AdE( - 1)) which is of dimension c( Ad) k.

Can i t be shown, perhaps using proposition 7.4.3 (see remark 7.4.4), that the space

of linear deformations which preserve the graph of AdE is of dimension less than

or equal to (c(Ad) - 1) k ? If SO. we would obtain the desired dimension estimate

dim(F,&) 5 (c(Ad) - 1) k.

How can we prove that. for a simple group G. the space 73: is denee in the

space FJk? It suffices to prove that given any framed jump E of multiplicity k

for which E, is not of minimal type. there is at least one multiplicity-preserving

deformation of E which reduces the type of E, (or one which reduces the graph

of -4dE). .\lternatively, if 3gk were shown to be an irreducible variety then F J ~

would necessarily be dense, since it is a nonempty open subvariety.

[At 21

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