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.
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