ABSTRACT
Title of dissertation: PARABOLIC HIGGS BUNDLES AND
THE DELIGNE-SIMPSON PROBLEM
FOR LOXODROMIC CONJUGACY
CLASSES IN PU(n, 1)
Robert Maschal, Doctor of Philosophy, 2017
Dissertation directed by: Professor Richard Wentworth
Department of Mathematics
In this thesis we study the Deligne-Simpson problem of finding matrices Aj ∈
Cj such that A1A2 . . . Ak = I for k ≥ 3 fixed loxodromic conjugacy classes C1, . . . , Ck
in PU(n, 1). Solutions to this problem are equivalent to representations of the k
punctured sphere into PU(n, 1), where the monodromy around the punctures are in
the Cj. By Simpson’s correspondence [27], irreducible such representations corre-
spond to stable parabolic U(n, 1)-Higgs bundles of parabolic degree 0. A parabolic
U(n, 1)-Higgs bundle can be decomposed into a parabolic U(1, 1)-Higgs bundle and
a U(n − 1) bundle by quotienting out by the rank n − 1 kernel of the Higgs field.
In the case that the U(1, 1)-Higgs bundle is of loxodromic type, this construction
can be reversed, with the added consequence that the stability conditions of the
resulting U(n, 1)-Higgs bundle are determined only by the kernel of Φ, the number
of marked points, and the degree of the U(1, 1)-Higgs bundle. With this result, we
prove our main theorem, which says that when the log eigenvalues of lifts C˜j of the
Cj to U(n, 1) satisfy the inequalities in [4] for the existence of a stable parabolic
bundle, then there is a stable parabolic U(n, 1)-Higgs bundle whose monodromies
around the marked points are in C˜j. This new approach using Higgs bundle tech-
niques generalizes the result of Falbel and Wentworth in [12] for fixed loxodromic
conjugacy classes in PU(2, 1).
This new result gives sufficient, but not necessary, conditions for the existence
of an irreducible solution to the Deligne-Simpson problem for fixed loxodromic con-
jugacy classes in PU(n, 1). The stability assumption cannot be dropped from our
proof since no universal characterization of unstable bundles exists. In the last
chapter, we explore what happens when we change the weights of the stable kernel
in the special case of three fixed loxodromic conjugacy classes in PU(3, 1). Using
the techniques from [11], [12], and [25], we can show that our construction implies
the existence of many other solutions to the problem.
PARABOLIC HIGGS BUNDLES AND THE DELIGNE-SIMPSON
PROBLEM FOR LOXODROMIC CONJUGACY CLASSES IN
PU(n, 1)
by
Robert Maschal
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor Richard Wentworth, Chair/Advisor
Professor Alexander Barg
Professor William Goldman
Professor John Millson
Professor Christian Zickert
c© Copyright by
Robert Maschal
2017
Dedication
This thesis is dedicated to Darryl Ruppert and Mary Maschal.
ii
Acknowledgments
There is a long list of people who have had an impact on my success as a
student. First I would like to thank my advisor, Richard Wentworth, for his help
and support throughout my academic career. Ever since reaching candidacy I felt
more like your colleague than your student. Through our many meetings I learned
a lot of interesting mathematics. But perhaps most importantly, I learned what it
means to be a good mathematician. Thank you for being a great advisor and friend.
Second, I want to thank my lovely wife and best friend, Xia Hu. Your love and
support over the last few years has pushed me further than I could have ever imag-
ined. I will always cherish the good meals, big laughs, and thoughtful conversations
we’ve shared along the way.
I want to thank my family for all of their unconditional love and support both
before and during my graduate career. Your actions showed me what it means to
be a hardworking and productive adult, and your positive influence helped shape
me into the happy person I am today.
Life wouldn’t be complete without my good friends Sean Doged and Andrew
Schmitt. You’ve been like a second family to me, and I’m glad that we have stuck
together for so long.
I want to thank Sean, Ryan, Adam, Rebecca, Ariella, JP, Steve, Matt, James,
and all of the other friends I’ve met along the way. Mathematics is most fun when it
is done with other people, and I am grateful to have been a part of this community.
Special thanks to Ryan Hunter and Steve Balady for always being available to talk
iii
in the final two months before my defense. It helped me more than you know.
Finally, I want to Dr. Robert Warner, without whom I may have never at-
tended graduate school in the first place.
iv
Table of Contents
Dedication ii
Acknowledgements iii
1 Introduction 1
2 Background 11
2.1 The group PU(n, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Conjugacy classes in U(n, 1) . . . . . . . . . . . . . . . . . . . 12
2.2 Flat bundles, representations, and Higgs bundles . . . . . . . . . . . . 14
2.2.1 Flat bundles and representations . . . . . . . . . . . . . . . . 15
2.2.2 Higgs bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Hitchin’s equations and the Nonabelian Hodge Correspondence 16
2.2.4 U(p, q)-Higgs bundles . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Parabolic vector bundles and the Mehta-Seshadri Theorem . . . . . . 21
2.3.1 Statement of theorem . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Existence of solutions to Deligne-Simpson for G = U(n) . . . . . . . . 26
2.5 Simpson’s Correspondence . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 Filtered local systems . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.2 Parabolic Higgs bundles . . . . . . . . . . . . . . . . . . . . . 28
2.5.3 Statement of theorem . . . . . . . . . . . . . . . . . . . . . . . 29
3 Parabolic PU(n, 1)-Higgs Bundles 33
3.1 Parabolic PU(n, 1)-Higgs bundles . . . . . . . . . . . . . . . . . . . . 34
3.2 Invariant subbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Kernel of the Higgs Field . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Invariant rank-2 subbundles . . . . . . . . . . . . . . . . . . . 36
3.3 Elementary reductions . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Useful lemmas about extensions . . . . . . . . . . . . . . . . . . . . . 41
v
4 Deligne-Simpson for PU(n, 1) 47
4.1 Constructing stable parabolic U(1, 1)-Higgs bundles . . . . . . . . . . 48
4.1.1 Constructing the Higgs field . . . . . . . . . . . . . . . . . . . 50
4.1.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.3 Existence of stable U(1, 1)-Higgs bundles . . . . . . . . . . . . 54
4.2 Main Theorem for PU(n, 1) . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 The parabolic structure . . . . . . . . . . . . . . . . . . . . . 57
4.3.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Deligne-Simpson and Representations . . . . . . . . . . . . . . . . . . 59
4.5 Return to PU(2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Solutions to the Deligne-Simpson Problem for PU(3, 1) 63
5.1 The product map µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Description of reducible walls for PU(3, 1) . . . . . . . . . . . . . . . 67
5.2.1 U(1, 1)× U(1)× U(1) walls . . . . . . . . . . . . . . . . . . . 68
5.2.2 U(1, 1)× U(2) walls . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.3 U(2, 1)× U(1) walls . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Determining which chambers are full . . . . . . . . . . . . . . . . . . 77
5.3.1 Stability conditions for the parabolic U(3, 1)-Higgs bundles
with three marked points . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 Moving along U(2, 1)× U(1) walls . . . . . . . . . . . . . . . . 83
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography 92
vi
Chapter 1: Introduction
Recall Horn’s Problem: given two hermitian matrices A and B with known spectra,
what can we say about the spectrum of their sum, A + B (see for example [13],
[21])? Similarly, consider the multiplicative version of this problem: given two
unitary matrices A,B ∈ U(n), with known eigenvalues, what can we say about
the eigenvalues of their product AB? In this thesis, we consider the generalization
of this problem to other subgroups G ⊂ GL(n,C), known as the Deligne-Simpson
Problem: given conjugacy classes C1, . . . Ck in G, what are necessary and sufficient
conditions on the Ci such that there exists matrices Ai ∈ Ci such that A1 . . . Ak = I?
Solutions to this problem are often of geometric interest, such as the moduli
of polygonal linkages [18], [19], [20], [30]. For example, consider the case when
G = SU(2). We can use the three sphere S3 as a model for SU(2), where a matrix
in SU(2) corresponds to a point on the sphere. In this context, the conjugacy
class of matrix is represented by the distance between the point it represents and
the identity in S3. Solutions to the Deligne-Simpson problem then correspond to
different configurations of n-sided polygons on the sphere with given side lengths.
Necessary and sufficient conditions are then given by the triangle inequalities on S3.
More generally, the problem already has a complete solution when G = U(n).
1
This was first proved in the case G = U(2) by Biswas in [3], which he later general-
ized to G = U(n). Independent solutions were given by Belkale in [2] and Agnihotri
and Woodward in [1]. A conjugacy class in U(n) is determined by the eigenvalues of
any matrix in the conjugacy class. Necessary and sufficient conditions are given as a
set of affine inequalities involving the logarithms of the eigenvalues. For simplicity,
we state the case when n = 2:
Theorem 1.0.1. [1], [2], [4], [3]
Let S = {1, 2, . . . , k}, and denote the log eigenvalues defining a conjugacy class
Cs in U(2) by 0 ≤ αs1 < αs2 < 1. Assume Σs∈S(αs1 +αs2) is an odd (respectively even)
integer, say 2N (respectively 2N + 1). Then there is a matrix Aj ∈ Cj such that
A1 . . . Ak = I if and only if for every D ⊂ S of size 2j (resp. 2j + 1), where j is a
non-negative integer, the following inequality holds:
−N − j + Σs∈Dαs2 + Σs∈S−Dαs1 < 0.
In [28], Simpson gives necessary and sufficient conditions for existence to a
solution to the Deligne-Simpson problem in the case G = SL(n,C) (see also the work
of Crawley-Boevey on this problem in [9]). However, whereas the conditions in the
U(n) case depend directly on the eigenvalues of the individual conjugacy classes, the
eigenvalues for the SL(n,C) case don’t matter at all! Instead, conditions are given
in terms of the multiplicity of the eigenvalues and the sizes of the corresponding
Jordan blocks. For fixed conjugacy classes C1, . . . , Ck in SL(n,C), let ri denote the
minimum rank of a matrix Ai − α, where Ai ∈ Ci and α ∈ C. Then:
Theorem. [28]
2
Suppose Ck is regular (i.e. semi-simple with distinct eigenvalues). Then there
exists a solution A1 . . . Ak = I with Ai ∈ Ci if and only if Σ dim(Ci) ≥ 2n2 − 2 and
r1 + . . .+ rk−1 ≥ n− 1.
While these results are quite different, both use the following topological rein-
terpretation of the linear algebra problem. Let S = {p1, . . . , pk} be a finite collection
of points on the Riemann Sphere P1. Setting X = P1 − S, the fundamental group
pi1(X) has the following presentation:
pi1(X) =< γ1, . . . γk|γ1γ2 . . . γk = 1 >
where γi is a loop around the point pi. Then a solution to the Deligne-Simspon
Problem is equivalent to the existence of a representation ρ : pi1(X) → G with
ρ(γi) ∈ Ci.
When G = U(n), the Mehta-Seshadri Theorem [22] gives an equivalence of
categories between the category of irreducible representations pi1(X) → U(n) and
the category of stable parabolic vector bundles of rank n and parabolic degree 0 on
P1. The problem of constructing a representation ρ with ρ(γi) ∈ Ci then becomes
one of constructing a stable parabolic vector bundle whose weights at a marked
point pi are determined by the eigenvalues defining Ci. The inequalities in theorem
2.4.1 come from the stability condition on the parabolic vector bundle.
When G = SL(n,C), Simpson’s Correspondence [27] gives an equivalence of
categories between the category of stable filtered local systems of degree 0 and the
category of stable parabolic Higgs bundles of rank n and parabolic degree 0 on P1.
With the solution in the U(n) and SL(n,C) cases understood, it is natural to
3
ask what happens for other real forms, such as PU(p, q), of GL(n,C). There are
already some results in this case.
In [25], Paupert studies products of elliptic isometries in PU(2, 1). Multiplying
both sides by A−13 , the problem of finding a solution A1A2A3 = I with Ai ∈ Ci
becomes A1A2 = A
−1
3 . For fixed C1, C2, the Deligne-Simpson problem then becomes
one of studying the product map µ : C1 × C2 → PU(2, 1). Let C1, C2 be elliptic
conjugacy classes in PU(2, 1). Then Cj can be represented by a matrix Aj of the
form:
Aj =

1
e2piiθ
1
j
e2piiθ
2
j
 (1.1)
where 0 ≤ θ1j < θ2j < 1. Thus the space of elliptic conjugacy classes can be identified
with T2/S2, the two-torus modulo the action of the symmetric group S2. Then
Paupert proves the following concerning the product map µ˜:
Theorem. Let C1, C2 be two elliptic conjugacy classes in PU(2, 1), represented by
matrices Aj of the form in equation 1.1, and let
µ˜ : (C1 × C2) ∩ {elliptics} → T2/S2.
Then µ˜ is onto if and only if
θ11 − 2θ21 + θ12 − 2θ22 ≥ 1
2θ11 − θ21 + 2θ12 − θ22 ≥ 3
The key point of the proof is that the image of µ˜ is composed of “reducible
walls” and “irreducible chambers.” The reducible walls are found by computing the
4
image under µ˜ of pairs (A,B) generating a reducible subgroup of PU(2, 1). These
walls bound irreducible chambers, which are completely empty or completely full.
The theorem can be interpreted as giving conditions on C1 and C2 so that every
chamber is full.
In [12], Falbel and Wentworth consider products of loxodromic isometries.
They prove that when the Ci are loxodromic conjugacy classes in PU(n, 1), the
configurations of reducible walls are especially simple when n = 1, 2. In particular,
the complement of the reducible walls is nonempty and connected. This gives:
Theorem. Let C1, . . . , Ck, k ≥ 3 be arbitrary conjugacy classes of loxodromic ele-
ments of PU(n, 1), for n = 1, 2. Then there exists Ai ∈ Ci such that A1 . . . Ak = I.
However, the proof does not generalize to larger n. As we shall see later, the
complement of the reducibles is disconnected when n ≥ 3.
In this thesis, we consider the case of k ≥ 3 fixed loxodromic conjugacy classes
in PU(n, 1) for n ≥ 3. As we shall see later, the fact that PU(n, 1) is a rank 1 group
is important (see for example Theorem 2 in [12]). The restriction to loxodromic
conjugacy classes is reasonable for two reasons. First, the elliptic case seems to be
much more difficult. In particular, Simpson’s Correspondence between filtered local
systems and parabolic Higgs bundles cannot completely identify the conjugacy class
of an elliptic monodromy. Second, purely loxodromic representations often corre-
spond to interesting geometric structures. For example, when Σ is a closed surface
of genus g ≥ 2, Teichmueller space is identified with discrete, faithful, and purely
loxodromic representations pi1(Σ) → PSL(2,R) = SU(1, 1), up to conjugation.
5
Discrete, faithful, purely loxodromic representations pi1(Σ) → SU(2, 1) are related
to quasi-Fuchsian representations and complex hyperbolic structures, the space of
which admit a coordinate system which is a direct generalization of Fenchel-Nielsen
coordinates of Teichmueller space [24].
Similar to [28] and [4], we interpret solutions to the Deligne-Simpson problem
as representations pi1(P1 − {p1 . . . , pk})→ PU(n, 1). By Simpson’s Correspondence
(Theorem 2.5.3), irreducible such representations correspond to stable parabolic
Higgs bundles.
The use of Higgs bundles and Simpson’s Correspondence is attractive for two
reasons. First, this strategy has already proved successful when G = SL(n,C)
and G = U(n). Additionally, we expect the solution for loxodromic classes in
PU(n, 1) to somehow be a combination of these two cases. In particular, it would
be interesting to see the extent to which the eigenvalues defining a loxodromic
conjugacy class matter. Second, the topology of the PU(p, q) character variety
Hom(pi1(X), U(p, q))//U(p, q) has been well-studied in the literature using Higgs
bundle techniques. In particular, the case where X is compact and genus g ≥ 2 is
handled in [33], [34], [6], [32]. The case where X is a punctured Riemann surface of
genus g ≥ 1 was handled in [14]. Our main theorem builds on the latter result by
guaranteeing in some cases that the PU(n, 1) character variety is nonempty for a
punctured Riemann surface X of genus 0. An obvious next question then is to ask
about the topology of these spaces, for which having a Higgs bundle description of
the elements would prove useful.
Constructing irreducible representations pi1(P1 − {p1 . . . , pk}) → PU(n, 1) is
6
equivalent to constructing stable parabolic Higgs bundles (E,Φ) of the following
form:
E = E1 ⊕ E2
Φ =
 0 b
c 0
 .
Information about the conjugacy class (namely the eigenvalues) of the monodromy
around a marked point pj can be found by studying the parabolic structure and the
residue of Φ at pj. In chapter 3, we study the stability properties of Higgs bundles of
the above form. A stable parabolic Higgs bundle corresponding to a representation
with loxodromic monodromies has in particular b, c 6= 0. As a result, when n ≥ 2,
E can be decomposed into a parabolic U(1, 1)-Higgs bundle (E˜, Φ˜) and a rank n− 1
parabolic bundle S. Here S is the kernel of b, and E˜ is obtained by quotienting E by
S. Finally we show that you can reverse the process by finding an extension of E˜
by S such that Φ˜ lifts and has nice additional property. This property allows us to
directly compute the invariant subbundles of Φ, which is important for computing
the stability conditions of (E,Φ).
The above decomposition of a PU(n, 1)-Higgs bundle into a U(1, 1)-Higgs
bundle and a rank n − 1 parabolic bundle suggests the solution for loxodromic
conjugacy classes in PU(n, 1) should be related to the solution for G = U(n − 1).
The main result of this thesis is that assuming some of the data defining the fixed
loxodromic conjugacy classes satisfy the conditions in Theorem 1.0.1, then solutions
to the Deligne-Simpson problem exist.
7
To be more precise, let C1, C2, . . . , Ck be k fixed loxodromic conjugacy classes
in PU(n, 1). Lift each Ci to a conjugacy class C˜i in U(n, 1), represented by a
diagonal matrix of the following form:
Aj =

rj
r−1j
exp(2piiα1j )
. . .
exp(2piiαn−1j )

(1.2)
where r is a positive real number and 0 ≤ αi1 < . . . < αin−1 < 1. In Chapter 4, we
prove the following:
Theorem 1. (Main Theorem)
Assume the αjl ’s defining the conjugacy classes C˜j satisfy Σα
j
l ∈ Z and the
strict U(n− 1) inequalities in [4]. Then there exists a stable parabolic U(n, 1)-Higgs
bundle (E,Φ) with k ≥ 3 marked points such that:
• the filtration of Epj has a rank 2 jump at 0 and rank 1 jumps at each of
αj1, . . . , α
j
n−1;
• the residue of Φ at pj has eigenvalues ±i log(r)/4pi, and 0 with multiplicity
n− 1.
By reinterpreting these stable parabolic U(n, 1)-Higgs bundles as irreducible
representations (technically filtered local systems with necessarily trivial filtration),
we have the following corollary:
8
Corollary 2. Let Ci be given loxodromic conjugacy classes in PU(n, 1), lifted to C˜i
to U(n, 1) as in 1.2. If the αij satisfy the U(n − 1) inequalities in 2.4.1, then there
is a matrix Ai ∈ Ci such that A1 . . . Ak = 1.
In particular, our theorem generalizes the result in [12] for PU(2, 1). Further-
more, the Higgs bundle techniques we use give a new approach not used in any
of [12], [25], and [28].
Theorem 1 gives sufficient conditions for the existence of an irreducible solu-
tion. In fact, we can see in the proof in Chapter 4 that the requirement that the
kernel S of the Higgs field be stable is not necessary. However, this stability assump-
tion is required for our proof. While stable bundles vary nicely in families, this is
not the case for unstable bundles, which can fail to be stable in many interesting
ways. As a result, we lack the tools required to approach the problem from the
point of view of finding necessary conditions.
In chapter 5, we explore this issue further in the special case of three fixed
conjugacy classes in PU(3, 1). Following [11], [12], and [25], we consider the product
map µ : C1×C2 → PU(3, 1). We restrict µ to pairs (A,B) with loxodromic product,
and then project onto Clox = T/S2 × (1,∞), the space of loxodromic conjugacy
classes. The image, as in [12] and [25], decomposes into a set of reducible walls and
irreducible chambers. We first compute the set of reducible walls, and then consider
the question of which chambers are full. Starting with a fixed stable parabolic
U(3, 1)-Higgs bundle (E,Φ) coming from Theorem 1, we examine how the stability
changes as we vary the weights. As a result, we can show that many chambers are
9
full, and therefore that Theorem 1 guarantees a solution to the Deligne-Simpson
problem in many more cases.
10
Chapter 2: Background
In this chapter, we briefly outline the material which we use in the rest of this
paper. In section 2.1, we define the indefinite unitary groups PU(p, q), and give a
classification of the conjugacy classes in the group PU(n, 1) of isometries of complex
hyperbolic space.
In section 2.2, we give an overview of the Riemann-Hilbert Correspondence
between local systems and flat bundles, and the Nonabelian Hodge Correspondence
between Higgs bundles and flat bundles. We conclude the section with a discussion
about U(p, q)-Higgs bundles.
In section 2.3, we define parabolic bundles and state the Mehta-Seshadri Theo-
rem, which gives an equivalence of categories between the category of representations
of the fundamental group of the punctured surface into U(n) with the category of
stable parabolic vector bundles of parabolic degree 0. As an example, we give a
solution to the Deligne-Simpson problem in the case of three conjugacy classes in
SU(2). The section concludes with a statement of the general result in the case
when G = U(2).
Finally, in section 2.5, we define parabolic Higgs bundles and filtered local
systems, and give a statement of Simpson’s Correspondence between filtered local
11
systems and parabolic Higgs bundles, simultaneously generalizing both the Non-
abelian Hodge Correspondence and the Mehta-Seshadri Theorem. Since we are
interested in using this correspondence to construct representations, we conclude
the section with a careful discussion of the correspondence between the residues of
the Higgs field acting on the parabolic structure at a marked point and the residue
of the monodromy around the marked point.
2.1 The group PU(n, 1)
Let V p,q(C) denote the p + q = n-dimensional complex vector space together with
the Hermitian form of type (p, q):
〈z, w〉 = w∗Jz
J =
 Ip 0
0 −Iq
 .
Definition 2.1.1. Define the group
U(p, q) = {g ∈ GL(n,C) | 〈gz, gw〉 = 〈z, w〉}.
For the rest of this section, we’ll restrict our attention to the group U(n, 1).
2.1.1 Conjugacy classes in U(n, 1)
Here we’ll briefly review the different types of conjugacy classes in U(n, 1), as de-
scribed in [7].
For z ∈ V n,1, define Φ(z) = 〈z, z〉.
12
Definition 2.1.2. For z ∈ V n,1, define Φ(z) = 〈z, z〉. Then for a subspace W ⊂
V n,1, we say W is:
• hyperbolic if Φ|W is non-degenerate and indefinite
• elliptic if Φ|W is positive definite
• parabolic if Φ|W is degenerate.
Following [15], we define n-dimensional complex hyperbolic space as the set of
negative lines in Cn,1:
Hn(C) = {z |Φ(z) < 0}/C∗ = U(n, 1)/U(n)× U(1).
We can also define the boundary of Hn(C) as the set of null lines in Cn,1:
∂Hn(C) = {z |Φ(z) = 0}/C∗.
PU(n, 1) acts transitively and effectively by isometries under the Bergman
metric on Hn(C). Elements g ∈ PU(n, 1) extend to conformal transformations of
boundary ∂Hn(C). By the Brouwer fixed-point theorem, any g ∈ PU(n, 1) has at
least one fixed point in Hn(C) ∪ ∂Hn(C). This gives the following classification of
elements in PU(n, 1):
Definition 2.1.3. An isometry g ∈ PU(n, 1) of Hn(C) is:
• elliptic if g has (at least) one fixed point in Hn(C)
• parabolic if g has exactly one fixed point on the boundary of Hn(C)
• loxodromic if g has exactly two fixed points on the boundary.
13
The following alternate Hermitian forms are useful for describing elliptic and
loxodromic elements, respectively:
Je =
 In 0
0 −1
 , Jl =

In−1 0 0
0 0 1
0 1 0

Using Je, any elliptic isometry in U(n, 1) is conjugate to a matrix of the fol-
lowing form:  U 0
0 λ

where U ∈ U(n), λ ∈ U(1). The eigenvalue λ is said to be of “negative type”
and the eigenvalues of U are said to be of “positive type.” The conjugacy class of
an elliptic isometry is determined by its groups of positive type and negative type
eigenvalues.
Using Jl, any loxodromic isometry in U(n, 1) is conjugate to a matrix of the
following form: 
U 0 0
0 λr 0
0 0 λr−1

where U ∈ U(n− 1), λ ∈ U(1), and r > 1.
2.2 Flat bundles, representations, and Higgs bundles
In this section, we briefly outline the Riemann Hilbert Correspondence between flat
bundles and representations, and the Nonabelian Hodge Correspondence between
14
flat bundles and Higgs bundles, in the case of a compact Riemann surface X of
genus ≥ 2. We conclude this section with a description of U(p, q)-Higgs bundles.
2.2.1 Flat bundles and representations
Definition 2.2.1. A flat bundle on X is a vector bundle E on X together with a
connection ∇ with vanishing curvature: ∇∧∇ = 0.
Definition 2.2.2. A local system L is a vector bundle given by constant transition
functions. Fixing a base point x, a local system is determined by its monodromy
representation pi1(X, x)→ Lx.
For a flat bundle (E,∇), parallel translation of ∇ along a path γ only de-
pends on the homotopy class of γ. Thus parallel translation gives the holonomy
representation pi1(X)→ GL(n,C).
Given a representation ρ : pi1(X)→ GL(n,C), we can construct a flat bundle
(E,∇) on X as follows. Let X˜ be the universal cover of X. Then letting E =
X˜ × Cn/(x, v) ∼ (γ(x), ρ(γ)v), the differential d descends to a connection ∇ on E
which is flat.
These two constructions are inverse to one another, and give the following:
Theorem 2.2.3. (Riemann-Hilbert Correspondence)
The above construction gives an analytic homeomorphism between the charac-
ter variety Hom(pi1(X), GL(n,C))/GL(n,C) and the space of isomorphism classes
of flat bundles (E,∇).
15
2.2.2 Higgs bundles
Definition 2.2.4. A Higgs bundle on X is a pair (E,Φ) where E→ X is a holomor-
phic vector bundle, Φ is a holomorphic section of Hom(E,E ⊗ KX), and KX → X
is the canonical bundle.
Definition 2.2.5. A subbundle S ⊂ E is Φ-invariant if Φ(S) ⊂ S ⊗KX .
Definition 2.2.6. The slope µ(E) of a holomorphic vector bundle is defined to be
µ(E) = deg(E)/rank(E)
Definition 2.2.7. A Higgs bundle (E,Φ) is (semi-) stable if for every proper Φ-
invariant holomorphic subbundle S ⊂ E, µ(E) < (≤)µ(E).
2.2.3 Hitchin’s equations and the Nonabelian Hodge Correspondence
Let (E,Φ) be a Higgs bundle. If we fix an hermitian metric h on E, then ∇ =
dA + Φ + Φ
∗ defines a connection on E. This is our candidate flat connection.
However, its curvature F∇ = FA + [Φ,Φ∗] may not vanish. Thus our Higgs bundle
corresponds to a flat bundle precisely when Hitchin’s equations are satisfied:
FA + [Φ,Φ
∗] = 0 (2.1)
∂¯E(Φ) = 0 (2.2)
The obstruction to finding Higgs bundles which solve Hitchin’s equations is
exactly the stability condition for Higgs bundles defined previously, as evidenced by
the following proposition:
16
Proposition 2.2.8. If (E,Φ) is a Higgs bundle satisfying 2.1, then (E,Φ) is
polystable.
However, the converse of this also true:
Theorem 2.2.9. [17], [29] If (E,Φ) is polystable, then it admits a metric satisfying
equation 2.1.
Let (E,∇) be a flat bundle. If we again fix an hermitian metric h, then the
flat connection ∇ splits uniquely as ∇ = dA + Ψ, where dA is a metric connection
and Ψ is an hermitian 1-form. In a local frame {si}, Ψ is defined by the equation
〈Ψsi, sj〉 = 〈∇si, sj〉+ 〈si,∇sj〉 − d 〈si, sj〉 .
Ψ is hermitian, so we can write Ψ = Φ + Φ∗ for some endomorphism-valued 1-form
Φ. However, Φ may not be holomorphic. In fact, Φ is holomorphic iff d∗AΨ = 0 (we
also require dAΨ = 0, which we get for free since ∇ = dA + Ψ is assumed to be flat).
In this case, Hitchin’s equations become:
FA +
1
2
[Ψ,Ψ] = 0 (2.3)
dA(Ψ) = 0 (2.4)
d∗A(Ψ) = 0 (2.5)
Intuitively, an hermitian metric is a specification of which holomorphic frames
are unitary. As such, h is a section of the GL(n,C)/U(n) bundle naturally associated
to E. In this way, we can view an hermitian metric h as a pi1(X)-equivariant map
h : X˜ → GL(n,C)/U(n).
17
More explicitly, let h : X˜ → GL(n,C)/U(n) be such an equivariant map,
where GL(n,C)/U(n) corresponds to the set of positive definite hermitian matrices.
For a section s of E, think of s as a pi1(X)-equivariant map s : X˜ → Cn. Then
define ||s||2h : X˜ → C by
||s||2h(x) = 〈s(x), h(x)s(x)〉Cn
||s||2h is easily seen to be equivariant, and hence descends to X, giving an hermitian
metric.
In the other direction, if s1 and s2 are sections of E with corresponding equiv-
ariant maps s˜1, s˜2 : X˜ → Cn, then for x ∈ X and x˜ ∈ X˜ sitting above x, we can
define h : X˜ → GL(n,C)/U(n) by
〈s1(x), s2(x)〉h = 〈s˜1(x˜), h(x˜)s˜2(x˜)〉Cn .
Equivariance of h follows from the equivariance of s˜1, s˜2.
Define the energy density of h to be
E(h) = 1
2
∫
M
|dh|2ω.
The hermitian metric corresponding to h is said to be harmonic if h is a critical
point of E (i.e. if h is a harmonic map). We have the following lemma:
Lemma 2.2.10. E(h) = 2||Ψ||2.
From this it is easy to compute the Euler-Lagrange equations d∗AΨ = 0. We
conclude that Φ defined above is holomorphic if and only if h is a harmonic metric,
and the problem of relating flat bundles to Higgs bundles becomes one of finding
harmonic metrics.
18
Theorem 2.2.11. [8], [10] (E,∇) admits a harmonic metric iff ∇ is semisimple.
Let M0Dol be the moduli space of polystable Higgs bundles of degree 0. De-
fine MB = Hom(pi1(X), GL(n,C))//GL(n,C) to be the character variety (or Betti
moduli space). Above we defined maps from each moduli space to the other. By
Theorems 2.2.9 and 2.2.11, these maps are inverse to one another. Thus we have
the following theorem:
Theorem 2.2.12. Nonabelian Hodge Correspondence, [8], [10], [17], [29]
The correspondence above defines a homeomorphism between the Betti moduli
space M0B(X) and the Dolbeault moduli space M0D(X).
Example Let K
1
2
X be a fixed square root of the canonical bundle KX . Define
E = K
1
2
X ⊕K
− 1
2
X . Define the Higgs field Φ by
Φ =
 0 q
1 0
 ,
where
1 ∈ H0(X, (K
1
2
X)
∗ ⊗K−
1
2
X ⊗KX) = H0(X,O)
q ∈ H0(X, (K−
1
2
X )
∗ ⊗K
1
2
X ⊗KX) = H0(X,K2X).
Then the Higgs bundle (E,Φ) is stable of degree 0, and so corresponds to an ir-
reducible representation pi1(X) → GL(n,C). Actually, this family of Higgs bun-
dles corresponds to the uniformizing representations pi1(X) → SL(2,R), and the
quadratic differential q gives coordinates for Teichmueller space.
19
2.2.4 U(p, q)-Higgs bundles
We’ve seen above how stable vector bundles of degree 0 correspond to irreducible
representations of pi1(X) into U(n), and stable Higgs bundles of degree 0 corre-
spond to irreducible representations into GL(n,C). We can also consider other
non-compact real forms of GL(n,C), namely the groups U(p, q), where p + q = n
(see for example [6], [14], [31]).
The maximal compact subgroup of U(p, q) is U(p) × U(q). The Lie algebra
u(p, q) has a Cartan decomposition
u(p, q) = u(p)⊕ u(q)⊕m
where m is the set of matrices of the form 0 A
−A¯t 0

A metric in this case is a reduction of structure group h : X˜ → U(p, q)/U(p)×
U(q). Any such reduction uniquely splits the flat connection∇ = dA+Ψ, where dA is
a U(p)×U(q) metric connection and Ψ is a one-form with values in m. The splitting
of the dA gives a holomorphic splitting of the holomorphic bundle E. Writing Ψ =
Φ + Φ∗, we have the following:
Definition 2.2.13. A U(p, q)-Higgs bundle is a Higgs bundle (E,Φ) of the form
E = V⊕W
Φ =
 0 b
c 0

20
where b ∈ H0(X,Hom(W,V)) and c ∈ H0(X,Hom(V,W)).
2.3 Parabolic vector bundles and the Mehta-Seshadri Theorem
Now we begin to introduce a generalization of the Nonabelian Hodge Correspon-
dence to punctured Riemann Surfaces. For the rest of this chapter, let X be any
compact Riemann surface, and D = p1 + . . . + pk the reduced divisor consisting of
the marked points pj. The main result of this section is a correspondence between
representations pi1(X −D) → U(n) on the punctured surface and parabolic vector
bundles on the compact surface.
Let E→ X be a holomorphic vector bundle. Adopting the notation in [5], we
have:
Definition 2.3.1. A parabolic structure on E consists of weighted flags
Ep =E1(p) ⊃ E2(p) ⊃ . . . ⊃ El(p) ⊃ El+1(p) = 0
0 ≤α1(p) < α2(p) < . . . < αl(p) < 1
for each p ∈ D.
A morphism of parabolic vector bundles is a morphism of holomorphic vector
bundles which preserves the parabolic structure at every point p ∈ D. Formally, if
Φ : E1 → E2 is a holomorphic map between vector bundles, then Φ is parabolic if
whenever α1i (p) ≤ α2j (p), we have Φ(E1i (p)) ⊂ E2j+1(p), again for all p ∈ D. Denote
the set of all parabolic morphisms by H0(ParHom(E1∗,E
2
∗)).
Additionally, we say Φ is strongly parabolic if it is nilpotent with respect to
the flag at every marked point. Equivalently, Φ is strongly parabolic if whenever
21
α1i (p) < α
2
j (p), we have Φ(E
1
i (p)) ⊂ E2j+1(p). We denote the set of all strongly
parabolic morphisms by H0(SParHom(E1∗,E
2
∗)).
2.3.1 Statement of theorem
The parabolic degree and parabolic slope of a parabolic bundle are defined as follows:
pdeg(E) = deg(E) +
∑
pi
∑
j
αj(pi)rk(Ej(pi)/Ej−1(pi)
pµ(E) =
pdeg(E)
rk(E)
A parabolic bundle is called stable (resp. semistable) if for every proper sub-
bundle S ⊂ E with the induced parabolic structure satisfies pµ(S) < (≤)pµ(E).
Theorem 2.3.2. (Mehta-Seshadri Theorem) [22]
There is an equivalence of categories between the category of irreducible rep-
resentations γ : pi1(X − D) → U(n) and stable parabolic bundles E of rank n and
parabolic degree 0 on X.
Moreover, fixing the conjugacy class Cj of the monodromy around pi fixes the
parabolic structure on E. More explicitly, let Cj be represented by the following
diagonal matrix:
A =

exp(2piiαj1) 0 · · · 0
0 exp(2piiαj2)
...
...
. . .
0 · · · exp(2piiαjn)

22
where 0 ≤ αj1 ≤ αj2 ≤ . . . ≤ αjn < 1. The jumps in the parabolic structure of E
occur at the αjs, and they are of rank equal to the multiplicity of the corresponding
eigenvalue.
The correspondence between the eigenvalues defining the conjugacy class of
the monodromy at pi and the parabolic structure of E at pi is very important. For us,
this means the problem of constructing irreducible solutions to the Deligne-Simpson
problem when G = U(n) is equivalent to constructing stable parabolic bundles with
a certain parabolic structure.
Example We’ll use the Mehta-Seshadri Theorem to give necessary and sufficient
conditions for the existence of a solution to the Deligne-Simpson problem for 3 fixed
conjugacy classes in SU(2). As mentioned in the introduction, these conditions can
also be interpreted as the triangle inequalities on the three sphere S3.
Let C1, C2, and C3 be three fixed conjugacy classes in SU(2). Then Cj can
be represented by a matrix of the form e2piiαj 0
0 e2pii(1−αj)

where 0 < αj < 1− αj < 1. From the discussion in the introduction, the existence
of Aj ∈ Cj such that A1A2A3 = I is equivalent to a representation ρ : pi1(P1 −
{p1, p2, p3}) → SU(2) with ρ(γj) ∈ Cj. By the Mehta-Seshadri Theorem, this
is equivalent to the existence of a parabolic vector bundle of parabolic degree 0,
whose weights at the marked point pj are αj < 1 − αj. We’ll also require that the
representation be irreducible, which implies that the parabolic bundle be stable.
23
Notice that Σj(αj + 1 − αj) = 3. Therefore, since we want to construct a
bundle with parabolic degree 0, the degree of the underlying holomorphic bundle
must be −3.
To determine the underlying holomorphic bundle E, we note the following two
facts. First, every holomorphic line bundle on P1 is determined up to isomorphism
by its degree. If L is a line bundle on P1 of degree d, we can unambiguously write
L = O(d). Second, Grothendieck’s Lemma (see [16]) says that every vector bundle
on P1 decomposes as a direct sum of line bundles, which is unique up to permutation
of the factors. These two facts plus the requirement that E has degree −3 means
we can write E = O(d) ⊕ O(−d − 3). The stability condition can be interpreted as
all subbundles of E having strictly negative parabolic degree.
First, we must have d = −1 (or equivalently d = −2). Suppose for a contra-
diction that d ≥ 0. Then the line subbundle L = O(d) has nonnegative parabolic
degree:
pdeg(L) ≥ deg(L) = d ≥ 0.
Since the parabolic degree of E is 0, E is necessarily unstable. The argument for
d ≤ −3 is similar, where L = O(−d − 3) has strictly positive degree. We conclude
that d = −1 is the only possibility, and therefore E = O(−1)⊕ O(−2).
Now we need to determine the parabolic structure on E. First, we’ll consider
subbundles of E. The only two we need to worry about are O(−1) and O(−2). Any
other subbundle has strictly negative parabolic degree, and can be eliminated from
24
consideration. More explicitly, let L = O(d), with d ≤ −3. Then:
pdeg(L) = d+ Σ(1− αj) < d+ 3 < 0.
There is a unique inclusion O(−1) ↪→ O(−1) ⊕ O(−2). Since
H0(Hom(O(−2),O(−1))) = C2, and H0(Hom(O(−2),O(−2))) = C, an embedding
O(−2) ↪→ O(−1) ⊕ O(−2) is determined by three constants, up to scaling. These
constants are fixed by specifying the image of the inclusion at two points (assuming
the image is not entirely contained in the fiber of O(−1)).
Now we are ready to consider the parabolic structure on E. This is a choice of
a flag E1(pj) ⊂ Epj for each j = 1, 2, 3. The first observation to make is that E1(pj)
should not be contained in the fiber of O(−1) at pj for any j.
Suppose for a contradiction that E1(p1) = O(−1)p1 . Without loss of generality,
assume E1(pj) 6= O(−1)pj for j = 2, 3. Then pdeg(O(−1)) = −1+(1−α1)+α2 +α3.
Additionally, any choice of E1(p2) and E1(p3) determines an embedding O(−2) ↪→
O(−1)⊕O(−2) whose image at pj is E1(pj) for j = 2, 3. This subbundle has parabolic
degree pdeg(O(−2)) = −2 + α1 + (1− α2) + (1− α3). The stability condition on E
for this parabolic structure requires:
pµ(E)− pµ(O(−1) = α1 − α2 − α3 > 0
pµ(E)− pµ(O(−2)) = −α1 + α2 + α3 > 0
which is clearly impossible. The general case follows similarly, so we must have
E1(pj) 6= O(−1)pj for all j.
By a similar argument, the choice of flags E1(pj) must be generic, in the sense
that for any embedding O(−2) ↪→ O(−1) ⊕ O(−2), the image at pj is E1(pj) for
25
at most two of j = 1, 2, 3. This requirement on the parabolic structure gives the
following necessary and sufficient conditions for stability:
1− α1 − α2 − α3 > 0
α1 + α2 − α3 > 0
α1 − α2 + α3 > 0
−α1 + α2 + α3 > 0
where each of the above inequalities come from the requirement that the parabolic
degree of any subbundle of E must have strictly negative parabolic degree. The first
inequality comes from the uniquely embedded O(−1). The last three inequalities
come form 3 separate embeddings of O(−2) , the fibers of which correspond to the
flag E1(pj) at exactly two of the marked points pj.
2.4 Existence of solutions to Deligne-Simpson for G = U(n)
The case of k ≥ 3 conjugacy classes in U(2) was handled by Biswas in [3]. This
result was later generalized to U(n) by Biswas in [4], with alternative proofs given
by Agnihotri and Woodward in [1], and Belkale in [2]. For simplicity, we state
the result from [3] on the existence of irreducible solutions to the Deligne-Simpson
problem when G = U(2). Later, we will not directly reference the inequalities when
n > 2. Therefore, the reader need only be aware that they exist.
Let C1, . . . , Ck be fixed regular conjugacy classes in U(2). Then Cs can be
26
represented by a matrix of the form e2piiα
s
1 0
0 e2piiα
s
2

where 0 ≤ αsi < αs2 < 1. Then we have the following
Theorem 2.4.1. [4], [3], [1], [2]
Let S = {1, 2, . . . , k}, and assume Σs∈S(αs1 + αs2) is an odd (respectively even)
integer, say 2N (respectively 2N +1). Then there is a stable rank 2 parabolic bundle
with parabolic weights {αs1, αs2} at the marked point ps ∈ P1 if and only if for every
D ⊂ S of size 2j (resp. 2j + 1), where j is a nonnegative integer, the following
inequality holds:
−N − j + Σs∈Dαs2 + Σs∈S−Dαs1 < 0.
By the Mehta-Seshadri Theorem, such a stable parabolic bundle corresponds to
an irreducible representation pi1(P1−{p1, . . . , pk})→ U(n). The parabolic structure
at pj determines the eigenvalues (and hence the conjugacy class) of the monodromy
around the marked point pj. As such, we have the following Corollary from the
Introduction:
Corollary 2.4.2. Let S = {1, 2, . . . , k}, and denote the log eigenvalues defining a
conjugacy class Cs in U(2) by 0 ≤ αs1 < αs2 < 1. Assume Σs∈S(αs1 + αs2) is an odd
(respectively even) integer, say 2N (respectively 2N + 1). Then there is a matrix
Aj ∈ Cj such that A1 . . . Ak = I if and only if for every D ⊂ S of size 2j (resp.
2j + 1), where j is a non-negative integer, the following inequality holds:
−N − j + Σs∈Dαs2 + Σs∈S−Dαs1 < 0.
27
2.5 Simpson’s Correspondence
2.5.1 Filtered local systems
Definition 2.5.1. A filtered local system is a local system L on X − D together
with a filtration Lpi,β of Lx. It is assumed that the filtration Lpi,β is preserved by the
monodromy around pi.
The degree of a filtered local system is defined by:
deg(L) =
∑
pi
∑
β
βrk(Lpi,β/Lpi,β+).
A filtered local system L is said to be stable if for any subsystem M ⊂ L with
the induced filtrations, we have
deg(M)
rk(M)
<
degL)
rk(L)
An important thing to keep in mind is the relationship between the stability
of the filtered local system and the irreducibility of the associated representation.
When the local system has the trivial filtration at every point, stability is equivalent
to the irreducibility of the corresponding representation (since any subsystem in this
case has filtered degree 0, stability asserts no subsystems exist).
2.5.2 Parabolic Higgs bundles
Definition 2.5.2. A parabolic Higgs bundle is a parabolic vector bundle (E,Epi,αi)
together with a meromorphic map Φ : E → E ⊗K with poles of order at most 1 at
28
the marked points pi. Φ is regular everywhere else. The residue of Φ at a marked
point pi is assumed to preserve the given filtration Epi,αi.
Fix local coordinates z vanishing at pi and a local frame ei. Then near pi, Φ
can be written
Φ(ei) = Φijej
dz
z
+ (regular stuff)
The statement that the residue of Φ preserves the filtration means that the matrix
(Φij) preserves the filtration.
Equivalently, a parabolic Higgs bundle is a parabolic vector bundle (E,Epi,αi)
and a parabolic morphism Φ ∈ ParHom(E,E ⊗ K(D)). With this interpretation,
the residue of the Higgs field Φ is simply its value at a point pi.
The parabolic degree and parabolic slope of a parabolic Higgs bundle are
defined as follows:
pdeg(E) = deg(E) +
∑
pi
∑
αj(pi)rk(Ej(pi)/Ej−1(pi)
pµ(E) =
pdeg(E)
rk(E)
A parabolic Higgs bundle is called stable (resp. semistable) if for every Φ-
invariant subbundle S ⊂ E with the induced parabolic structure satisfies pµ(S) <
(≤)pµ(E).
2.5.3 Statement of theorem
Theorem 2.5.3. (Simpson, [27]) There is a bijective correspondence between stable
filtered local systems of degree 0 and stable parabolic Higgs bundles of degree 0.
29
Additionally, there is a correspondence between the residues of the filtered
local system and parabolic Higgs bundles at each pi, as described in [27], [28]. To
make this more explicit, let L be a filtered local system and (E,Φ) a parabolic Higgs
bundle (both stable, of degree 0). Then we define
respi(L) =
⊕
β
respi(L)β
respi(E) =
⊕
α
respi(E)α,
where respi(L)β and respi(E)α are the blocks in the associated graded of each fil-
tration at the point pi:
respi(L)β = Grβ(Lpi) = (Lpi,β/Lpi,β+)
respi(E)α = Grα(Epi) = (Epi,α/Epi.α+).
Since the monodromy Ai around pi is assumed to preserve the filtration of Lpi ,
it acts by an endomorphism, which we denote by res(Ai), on the associated graded
respi(L). Similarly, since the ”residue” of Φ is assumed to preserve the filtration of
Epi , it acts by an endomorphism respi(Φ) on the associated graded respi(E). The
residue of L at pi is defined to be the pair (respi(L), res(Ai)). Similarly, the residue
of (E,Φ) at pi is the pair (respi(E), respi(Φ)).
Moreover, res(Ai) acts on the blocks respi(L)β, and res(Φ) acts on the blocks
respi(E)α. Therefore, we can decompose the blocks respi(L)β and respi(E)α into
30
the generalized eigenspaces of res(Ai) and res(Φ). We have
respi(L)β =
⊕
λ
respi(L)β,λ (2.6)
respi(E)α =
⊕
τ
respi(E)α,τ (2.7)
where λ is a generalized eigenvalue of res(Ai) with generalized eigenspace
respi(L)β,λ ⊂ respi(L)β, and τ is a generalized eigenvalue of res(Φ) with gener-
alized eigenspace respi(E)α,τ ⊂ respi(E)α.
Finally, res(Ai) acts on respi(L)β,λ by a matrix res(Ai)β,λ. Putting res(Ai)β,λ
into Jordan normal form induces a partition P β,λpi of its generalized eigenspace
respi(L)β,λ. The collection P
β,λ
pi
forms the ”residue diagram” of (respi(L), res(Ai)).
The residue diagram P β,λpi uniquely determines the residue (res(Ai), respi(L)), up to
isomorphism.
The same process applied to res(Φ) acting on respi(E)α,τ associates to
(res(Φ), respi(E)) its partition diagram P
α,τ
pi
. Again this uniquely determines
(res(Φ), respi(E)) up to isomorphism.
In the correspondence in Theorem 2.5.3, the residue diagrams P β,λpi and P
α,τ
pi
of
(res(Ai), respi)L)) and (res(Φ), respi(E)) are the same, up to the following change
of labels:
(α, τ = b+ ci) 7→ (β = −2b, λ = exp(2piiα− 4pic) (2.8)
(β, λ) 7→
(
α =
1
2pi
log(λ), τ = −β
2
− i log|λ|
4pi
)
(2.9)
To describe the correspondence more simply, the weights and rank of each
jump in the corresponding filtrations at pi are the same, up to the change of labels
31
in 2.8. The eigenvalues of Ai and the residue of Φ at pi are the same and have the
same multiplicities, up to the change described in 2.8. The sizes of the nilpotent
parts of res(Ai) and res(Φ) acting on the associated graded objects respi(L) and
respi(E) are also the same.
Theorem 2.5.3 allows us to reinterpret our problem of constructing represen-
tations into one of constructing parabolic Higgs bundles. The requirement on the
local monodromy around a given puncture becomes a requirement on the parabolic
structure and the residue of Φ at pi. Going in the other direction, if we know every-
thing about our parabolic Higgs bundle (i.e. the parabolic structure, Jordan form
of res(Φ)), then we can use this to gain knowledge about the monodromy around
pi. In particular, we know everything about the eigenvalues of the local monodromy
Ai. However, we may not know everything. If the filtration on the local system is
nontrivial (i.e. if the eigenvalues of the residue of Φ have nonzero real part) then we
only have partial information about the nilpotent part of Ai. This could also mean
the underlying representation is reducible. Since we are only interested in semisim-
ple conjugacy classes, we do not need to worry about the nilpotent pieces. However,
we must keep in mind that a stable filtered local system may not correspond to an
irreducible representation.
32
Chapter 3: Parabolic PU(n, 1)-Higgs Bundles
In this chapter, we discuss the structure of the Higgs field that guarantee the mon-
odromies of the corresponding representation are in U(n, 1). In particular, since we
are interested in constructing representations with loxodromic monodromies, Simp-
son’s Correspondence says the residue of Φ at the marked points must have two
nonzero (in particular with nonzero imaginary part) eigenvalues. This requirement
adds further restrictions to the structure of the Higgs field.
A parabolic U(n, 1)-Higgs bundle will always have invariant subbundles when
n ≥ 2). The restrictions on the structure of the Higgs field put restrictions on the
structure of the invariant subbundles. In particular, the subbundles of interest are
the rank n − 1 kernel of Φ, and a rank 2 subbundle corresponding to the nonzero
eigenspaces of Φ.
Quotienting out by the kernel, we can break a parabolic U(n, 1)-Higgs bundle
into a parabolic U(1, 1)-Higgs bundle and a rank n − 1 parabolic vector bundle.
The main result of this chapter is that we can put these pieces back together into
a parabolic PU(n, 1)-bundle in a way that allows us to easily compute the two
invariant subbundles.
33
3.1 Parabolic PU(n, 1)-Higgs bundles
Let D = p1 + . . .+ pk be a fixed reduced divisor on X = P1. Following section 2.2.4,
we have the following:
Definition 3.1.1. A parabolic PU(n, 1)-Higgs bundle is a parabolic Higgs bundle
(E,Φ) such that:
• E = E1 ⊕ E2 where rank(E1) = 1 and rank(E2) = n
• With respect to the above splitting of E,
Φ =
 0 b
c 0
 .
where b : E2 → E1 ⊗K(D) and c : E1 → E2 ⊗K(D).
These Higgs bundles are exactly the ones corresponding to representations
pi1(X −D)→ PU(n, 1) [33], [34], [6], [14].
We are interested in constructing (irreducible) representations of pi1(P1 −
{p1, . . . , pk}) with loxodromic monodromies. From the above, we can determine if a
given (stable) Higgs bundle has loxodromic monodromies by looking at its residues
at the marked points.
Definition 3.1.2. Let (E,Φ) be a parabolic PU(n, 1)-Higgs bundle. We say Φ is of
loxodromic type if its residue at each marked point pj has exactly two eigenvalues
with nonzero imaginary parts.
34
For the rest of the paper, we will only consider parabolic PU(n, 1)-Higgs bun-
dles of loxodromic type. This assumption has important implications regarding
which types of invariant subbundles can occur.
3.2 Invariant subbundles
Let (E,Φ) be a parabolic PU(n, 1)-Higgs bundle of loxodromic type, where E and Φ
the decomposition as in Definition 3.1.1. In this section, we determine all invariant
subbundles of Φ. These correspond to the eigenspaces of Φ. The first corresponds
to the 0 eigenvectors, i.e. the kernel of Φ:
3.2.1 Kernel of the Higgs Field
Lemma 3.2.1. The kernel of Φ corresponds exactly to the kernel of b : E2 →
E1 ⊗K(D).
Proof. Since Φ is of loxodromic type, c : E1 → E2 ⊗ K(D) is necessarily nonzero.
Since E1 is a line bundle, c is generically injective. This implies ker(Φ) = ker(b).
Note that since rank(E2) = n and E1 = 1, rank(ker(Φ)) = n − 1. As such,
when n > 2, any subbundle of ker(Φ) is also necessarily invariant. This will not
be an issue, since for our main construction we will assume ker(Φ) is stable, when
guaranteed by [4]. In chapter 5, we will explore the case where ker(Φ) is actually
unstable.
35
3.2.2 Invariant rank-2 subbundles
There is one other important subbundle, corresponding to the nonzero eigenspace
of the Higgs field. If we write the Higgs bundle (E,Φ) as
E = E1 ⊕ E2
Φ =
 0 b
c 0
 .
then F = E1⊕L, where L is the saturation of Im(c). In order to check the stability
of E, we need to compute the degree of F. Since E1 is given, we only need to compute
the degree of L. This depends on the vanishing of the map c:
Lemma 3.2.2. Im(c) = E1((c))⊗K(D)∗, where (c) is the vanishing divisor of the
section c. In particular, when c is non-vanishing, Im(c) = E1 ⊗K(D)∗
Proof. E1 and c fit in the following exact sequence of coherent sheaves:
0 E1 E2 ⊗K(D) Q 0c
If c vanishes, then the quotient sheaf Q is not locally free. On a Riemann
surface, the stalk of the structure sheaf is a PID. As a consequence, any coherent
sheaf splits as a direct sum of a locally free sheaf and a torsion sheaf. Therefore we
can write Q = V⊕ T, where V is locally free and T is a torsion sheaf supported on
the vanishing divisor (c) of c.
We want to compute the saturation L of the image of E1 in E2 ⊗K(D) of E1.
The projection of V⊕ T → V onto V induces a map E2 ⊗K(D). L is the kernel of
36
this map, and sits inside the following commutative diagram:
0 E1
L
E2 ⊗K(D) V⊕ T
V
0
0
0
c
Computing determinants, we see that
det(E2 ⊗K(D)) = det(E)⊗ det(V)⊗ det(T)
= det(L)⊗ det(V)
It follows that L = E1 ⊗ O((c)). Tensoring with K(D)′, we conclude that Im(c) =
L⊗K(D)∗ = E1 ⊗ O((c))⊗K(D)∗.
3.3 Elementary reductions
We can reduce the problem of constructing U(n, 1)-Higgs bundles into constructing
a parabolic U(1, 1)-Higgs bundle and a parabolic U(n−1) bundle. The U(1, 1) piece
determines the nonzero eigenvalues of the Higgs field and the U(n− 1) piece deter-
mines the kernel. The U(n, 1)-Higgs bundle is constructed by taking an appropriate
extension.
37
Let (E,Φ) be a U(n, 1)-Higgs bundle of loxodromic type:
E = E1 ⊕ E2
Φ =
 0 b
c 0

b, c 6= 0
Then the invariant subbundle corresponding to the kernel of Φ is the kernel of b:
S = ker(b) ⊂ E2. This fits in an exact sequence:
0 S E2 Q 0
We can construct a U(1, 1)-Higgs bundle (E˜ = E1 ⊕ Q, Φ˜), where Φ˜ has the
following form:
Φ˜ =
 0 b˜
c˜ 0
 . (3.1)
The b˜ and c˜ determining Φ˜ are obtained in the following way. First c˜ is determined
by the following diagram:
0 S E2 Q 0
E1 ⊗K(D)∗
c c˜
Put slightly differently, c˜ ∈ H0(E∗1 ⊗ Q ⊗ K(D) is the image of c ∈ H0(E∗1 ⊗
E2 ⊗K(D) in the following exact sequence in cohomology:
0 H0(E∗1 ⊗ S⊗K(D)) H0(E∗1 ⊗ E2 ⊗K(D)) H0(E∗1 ⊗ Q⊗K(D))
38
b˜ is slightly harder to obtain. For this, we have the following exact sequence
in cohomology:
0 H0(Q∗ ⊗ E1 ⊗K(D)) H0(E∗2 ⊗ E1 ⊗K(D)) H0(S∗ ⊗ E1 ⊗K(D))
Recall that b ∈ H0(E∗2 ⊗ E1 ⊗ K(D)). b˜ should be the preimage of b in the group
H0(Q∗ ⊗ E1 ⊗ K(D)). This might not always be possible. Consider the image of
b in H0(S∗ ⊗ E1 ⊗ K(D)). This is the morphism L → E1 ⊗ K(D) obtained by
restricting b to the subbundle S. However, recall that S was defined to be the
kernel of b. Therefore b restricts to the zero map on S, and hence its image in
H0(S∗ ⊗ E1 ⊗K(D)) is 0. Thus b is the image of an element, which we name b˜, in
H0(Q∗ ⊗ E1 ⊗K(D)). This proves the following lemma:
Lemma 3.3.1. Let (E = E1⊕E2,Φ =
 0 b
c 0
) be a parabolic U(n, 1)-Higgs bundle
of loxodromic type. Then quotienting E2 by the kernel S of b decomposes (E,Φ) into
a parabolic U(1, 1)-Higgs bundle of loxodromic type and a parabolic U(n−1) bundle.
Now we wish to invert the above process. Given a parabolic U(n− 1) bundle
S and a parabolic U(1, 1)-Higgs bundle (E,Φ), how can we construct a parabolic
U(n, 1)-Higgs bundle (E˜, φ˜)? First, assume (E,Φ) decompose as follows:
E = E1 ⊕ Q
Φ =
 0 b
c 0

b, c 6= 0
39
Together with S, we will show how to construct a parabolic U(n, 1)-Higgs bundle
(E˜, φ˜) of the following form:
E˜ = E1 ⊕ E2
Φ˜ =
 0 b˜
c˜ 0

b˜, c˜ 6= 0
E2 is given by taking some extension of S by Q:
0 S E2 Q 0
For now, we do not fix the extension class β ∈ H1(Q∗⊗L). The existence and
uniqueness of the extension b˜ of b follows immediately from the following lemma:
Lemma 3.3.2. Given an extension E of a bundle Q by a bundle S and a map of
bundles b : Q→ F, there exists a unqiue extension b˜ : E→ F of b.
Proof. In the long exact sequence of cohomology, we have in particular
0 H0(Q∗ ⊗ F) H0(E∗ ⊗ F)α
The (necessarily unique) extension b˜ of b is the image of b under the injective map
α.
On the other hand, c is slightly more difficult to obtain. Consider the following
long exact sequence:
40
H0(E∗1 ⊗ E2 ⊗K(D)) H0(E∗1 ⊗ Q⊗K(D)) H1(E∗1 ⊗ S⊗K(D))
The section c ∈ H0(E∗1 ⊗ Q⊗K(D)), and we want to pull it back to a section
c˜ ∈ H0(E∗1⊗E2⊗K(D)). This is only possible if the image of c in H1(E∗1⊗L⊗K(D))
is 0 (see Lemma 3.4.1).
To further complicate matters, just finding an extension for which c lifts is
not enough. By Lemma 3.2.2, the vanishing of the lift c˜ has a direct effect on the
stability of E˜. As such, we must find an extension for which c lifts to a nonvanishing
element c˜ ∈ H0(E∗1 ⊗ E2 ⊗K(D)). We’ll tackle this issue in the next section.
3.4 Useful lemmas about extensions
To construct a U(n, 1)-Higgs bundle out of a U(n − 1) bundle and a U(1, 1)-Higgs
bundle, we need to identify for which extensions H1(Q∗⊗S) the map c˜ can be lifted.
The following lemma due to Narasimhan and Ramanan in [23] does just that.
Lemma 3.4.1. (Narasimhan-Ramanan) Let 0→ S→ E→ Q→ 0 be a short exact
sequence of holomorphic vectors bundles, and c : F → Q a map. Then the space of
extensions of S by Q for which c lifts to a map F → E corresponds to the kernel of
the map H1(Q ∗ ⊗S)→ H1(F∗ ⊗ S) induced by c.
Proof. Precomposition by c induces the following diagram:
0 Q∗ ⊗ S Q∗ ⊗ E Q∗ ⊗ Q 0
0 F∗ ⊗ S F∗ ⊗ E F∗ ⊗ Q 0
Taking cohomology, we get the following:
41
. . . H0(Q∗ ⊗ Q) H1(Q∗ ⊗ S) H1(Q∗ ⊗ E) . . .
. . . H0(F∗ ⊗ Q) H1(F∗ ⊗ S) H1(F∗ ⊗ E) . . .δ
c∗
A necessary and sufficient condition for a lift of c to exist is that δc be 0. But
c is the image of the identity under c∗. By commutativity of the diagram, δ(c) is
the image of the extension class β ∈ H1(Q∗ ⊗ S). Thus c can be lifted if and only if
β is in the kernel of c∗ : H1(Q∗ ⊗ S)→ H1(F∗ ⊗ Q).
Lemma 3.4.2. If F and Q above are line bundles and c : F → Q is a nonzero
morphism, then the map H1(Q∗ ⊗ S)→ H1(F∗ ⊗ S) is surjective.
Proof. F and Q fit into the following short exact sequence, where T is a torsion
sheaf:
0 F Q T 0
c
Dualizing and tensoring with S, we have
0 Q∗ ⊗ S F∗ ⊗ S T′ ⊗ S 0
c∗
where T′, and hence T
′ ⊗ S, is torsion. In the long exact sequence, this induces
H1(Q∗ ⊗ S) H1(F∗ ⊗ S) H1(T′ ⊗ S) 0c∗
T
′⊗S is torsion, and thusH1(T′⊗S) = 0. Therefore, the map c∗ : H1(Q∗⊗S)→
H1(F∗ ⊗ S) induced by c is surjective.
Lemma 3.4.3. Let p ∈ P1 be a point at which c ∈ H0(F∗ ⊗ Q) vanishes. Then
42
• The extensions E of Q by S for which c lifts to a c˜ ∈ H0(F∗⊗E) which vanishes
at p are in the kernel of the map H1(Q ∗ ⊗S)→ H1(F(p)∗ ⊗ S) induced by c.
• There is a natural inclusion of ker(H1(Q ∗ ⊗S) → H1(F(p)∗ ⊗ S)) into
ker(H1(Q ∗ ⊗S)→ H1(F∗ ⊗ S)).
Proof. Suppose c lifts to a map c˜ which vanishes at p. Then c and c˜ factor through
the inclusion F → F(p), giving the following commutative diagram:
0 S E Q 0
F
F(p)
c˜
c
c
′c˜
′
Thus a lift c˜ of c which vanishes at p is equivalent to a lift c˜
′
of c
′
. By Lemma
3.4.1, these extensions are contained in ker(H1(Q ∗ ⊗S)→ H1(F(p)∗ ⊗ S)).
Next we show that ker(H1(Q ∗⊗S)→ H1(F(p)∗⊗ S)) is naturally included in
ker(H1(Q ∗ ⊗S)→ H1(F∗ ⊗ S)). Consider the following short exact sequence:
0 F F(p) O|p 0i
where F is included in F(p) as the subsheaf of sections vanishing at p. Dualizing
and tensoring with S, we have
0 F(p)∗ ⊗ S F∗ ⊗ S S|p 0
i∗
Taking cohomology, we have by Lemma 3.4.2 the following commutative diagram:
43
H1(F(p)∗ ⊗ S) H1(F∗ ⊗ S) 0
H1(Q∗ ⊗ S) H1(Q∗ ⊗ S)
i∗
=
The containment of ker(H1(Q ∗ ⊗S) → H1(F(p)∗ ⊗ S)) into ker(H1(Q ∗ ⊗S) →
H1(F∗ ⊗ S)) follows from a simple diagram chase, proving the lemma.
When ker(H1(Q∗⊗S)→ H1(F(p)∗⊗(S))) is strictly contained in ker(H1(Q∗⊗
S) → H1(F∗ ⊗ S)), then we can find extensions E of Q by S such that the map
c : F → Q has a lift c˜ : F → E which is nonvanishing at p. If the containment is not
strict, the existence of a nonvanishing lift is measured by H0(F∗ ⊗ S). By adding
an element in the image of H0(F∗⊗ S) to the lift c˜, we can find new lift which does
not vanish.
Proposition 3.4.4. Given a vector bundle S, line bundles Q and F, and a nonzero
morphism c : F → Q, there exists an extension of Q by S such that c has a lift c˜ that
is nonvanishing.
Proof. The lift c˜ of c can only vanish along the divisor of c. First, we’ll pick an
extension. We have the following commutative diagram, with exact top row:
H0(F∗ ⊗ S) S|p H1(F(p)∗ ⊗ S) H1(F∗ ⊗ S) 0
H1(Q∗ ⊗ S) H1(Q∗ ⊗ S)
γ δ i∗
c′∗ c∗
id
Note that the maps γ and δ only depend on F and S, and not the extension class
β ∈ H1(Q∗ ⊗ S). Let p be a point in the divisor of c. Then there are two cases:
44
Case 1: The map γ is not surjective
If γ is not surjective, then there is an element α ∈ Sp whose image δ(α) ∈ H1(F(p)∗⊗
S) is not zero. By exactness, i∗(δ(α)) = 0. On the other hand, c′∗ is surjective, and
therefore there is a β ∈ H1(Q∗ ⊗ S) such that c′∗(β) = δ(α). By commutativity,
c∗(β) = i∗(δ(α)) = 0. By Lemma 3.4.1, β is an extension for which c lifts to a an
element c˜. On the other hand, c′∗(β) is nonzero, so by Lemma 3.4.3 the lift c˜ is
nonvanishing at p.
As such, ker(c′∗) is strictly contained inside ker(c∗). In particular, there is a
nonempty open subset Up ⊂ ker(c∗) of extensions for which c lifts to a c˜, which does
not vanish at p.
Case 2: γ is surjective
If γ is surjective, then δ is identically 0, and i∗ is an isomorphism. In this case, ker(c′∗)
and ker(c∗) are exactly equal. Thus we expect any lift c˜ to vanish at p, regardless of
which extension we choose. However, H0(F∗ ⊗ S) is necessarily nonzero, and hence
a lift c˜ is unique only up to addition with an element in H0(F∗ ⊗ S). Since γ is
surjective, Vp = γ
−1(S|p − {0}) is a nonempty open subset of H0(F∗ ⊗ S). The sum
of c˜ plus any element in Vp is necessarily nonvanishing at p.
Finishing the proof of Proposition 3.4.4
Now, let {p1, . . . , pn} be the set of points where c vanishes. Further, assume (up
to relabelling) that {p1, . . . , pk} are the points for which a in the above diagram is
45
not surjective. Then U = ∩k1Upi is nonempty (since it is the complement of a finite
union of positive codimension closed subsets of ker(c∗)), and consists of extensions
for which c lifts to an element c˜ is nonvanishing at pi, for 1 ≤ i ≤ k. Additionally,
V = ∩nk+1Vpi is nonempty an consists of sections in H0(F∗ ⊗ S) which are non
vanishing at pi, for k + 1 ≤ i ≤ n. Picking an element γ ∈ V , γ + c˜ is a lift of c
which does not vanish at any pi, completing the proof.
Putting everything together, we have the following corollary :
Corollary 3.4.5. Let (E = E1 ⊕ Q,Φ =
 0 b
c 0
) be a parabolic U(1, 1)-Higgs
bundle of loxodromic type and let S be a rank n − 1 parabolic vector bundle. Then
there is an extension E2 of Q by S such that c lifts to a c˜ ∈ H0(E∗1 ⊗ E2 ⊗ K(D))
which is non-vanishing.
46
Chapter 4: Deligne-Simpson for PU(n, 1)
In Chapter 3, we described the structure of a parabolic Higgs bundle corresponding
to a representation into PU(p, q). In the case of PU(n, 1), we described the addi-
tional structure of the Higgs field required to guarantee the monodromies around
the punctures are loxodromic. For such a Higgs field, there are two important in-
variant subbundles, the rank n− 1 kernel and a rank 2 subbundle corresponding to
the non-zero eigenspace. In particular, the degree of the rank 2 subbundle depends
on the vanishing of the map c : E1 → E2.
Finally, we showed that a parabolic PU(n, 1)-Higgs bundle can be decomposed
into a parabolic U(1, 1)-Higgs bundle of loxodromic type and a parabolic U(n− 1)
bundle, corresponding to the kernel of Φ. The main result of chapter 3, Proposition
3.4.4, says that we can put pieces back together with c nonvanishing, thus minimizing
the degree of the rank 2 subbundle.
In this chapter, we prove our Main Theorem with the aid of the above. The
construction of the kernel of Φ comes from 2.4.1, so we are left to construct the
U(1, 1) piece. First, we show that for k ≥ 3 fixed loxodromic conjugacy classes
in U(1, 1), there is a corresponding stable parabolic U(1, 1)-Higgs bundles. As a
corollary, solutions to the Deligne-Simpson problem for loxodromic conjugacy classes
47
in U(1, 1) always exist. Second, we combine these two pieces to prove our Main
Theorem regarding the existence of stable parabolic U(n, 1)-Higgs bundles.
4.1 Constructing stable parabolic U(1, 1)-Higgs bundles
Let C1, C2, C3 be fixed loxodromic conjugacy classes in U(1, 1). The conjugacy class
Cj can be represented by matrix of the form
Aj =
 exp(2piiβj)rj 0
0 exp(2piiβj)r
−1
j
 . (4.1)
By Simpson’s Correspondence and the discussion in section 2.5.3, constructing
a solution to the Deligne-Simpson problem for C1, C2, C3 is equivalent to construct-
ing a stable parabolic U(1, 1)-Higgs bundles (not necessarily of degree 0) satisfying:
• the filtration of Epj has one rank 2 jump at βj;
• the residue of Φ acts on Grβj(Epj) with eigenvalues ±i log(rj)/4pi.
We should say something about why the eigenvalues of the residue should be
±i log(rj)/4pi. By Simpson’s Correspondence, the imaginary parts of eigenvalues
of the residue of Φ at the marked points determine the norms of the eigenvalues
of the monodromy around the marked points. In particular, if the eigenvalue of
res(Φ) = b + ci, then the corresponding eigenvalue of the monodromy has norm
exp(−4pic). Since we are interested in constructing representations with loxodromic
monodromies, then it follows from equation 4.1 that the eigenvalues of the mon-
odromy have norms r, r−1. As such, the (imaginary parts of) the eigenvalues of
res(Φ) should be ±i log(rj)/4pi.
48
Since we are interested in representations into PU(n, 1), we don’t require that
the Higgs bundle have parabolic degree 0. If (E,Φ) has nonzero degree, then after
tensoring with an appropriate line bundle (with trivial parabolic structure), we can
assume −2 < pdeg(E) ≤ 0. The trick (following [28]) is to then add an additional
marked point p, distinct from {p1, p2, p3}. The filtration at this point is given by a
single rank 2 jump of weight α = −pdeg(E)
2
at p. This gives a new parabolic bundle,
which is stable iff E is, of parabolic degree 0. This new bundle gives a representation
of pi1(P1 − {p1, p2, p3, p}) i.e. matrices Aj ∈ Cj such that A1A2A3M = 1, where M
is the monodromy around the new point p. It is not hard to see that M must be
the scalar matrix M = exp(−2
3
piipdeg(E))I, and so we end up with a representation
pi1(P1 − {p1, p2, p3})→ PU(1, 1).
In this section, we prove the following:
Proposition 4.1.1. Let C1, C2, and C3 be loxodromic conjugacy classes in U(1, 1),
represented by the matrix Aj of the form given in equation (4.1) . Then there exists
a stable parabolic U(1, 1)-Higgs bundle with three marked points D = p1 + p2 + p3
with the following properties:
• The parabolic structure at pj consists of a single rank-2 jump with weight βj
• The residue of Φ at pj has eigenvalues ±i log(rj)/4pi.
On the representation side, we have the following easy corollary:
Corollary 4.1.2. Given loxodromic conjugacy classes C1, C2, and C3 in PU(1, 1),
there is a matrix Aj ∈ Cj such that A1A2A3 = I.
49
Let (E = E1 ⊕ E2,Φ) be a stable U(1, 1)-Higgs bundle. Possibly tensoring
with a line bundle (with trivial parabolic structure), we may assume E1 = O and
E2 = O(d). With respect to this splitting of E , we have
Φ =
 0 b
c 0

where b ∈ H0(O(d)∗⊗O⊗K(D))) ' H0(O(1−d)) and c ∈ H0(O∗⊗O(d)⊗K(D))) '
H0(O(1 + d)). Since we require Φ to be of loxodromic type, each of b and c must
be nonzero at each pj, and therefore −1 ≤ d ≤ 1.
4.1.1 Constructing the Higgs field
Here we show how to construct a U(1, 1)-Higgs field with appropriate residues. From
the above, we have 3 separate cases to consider: given E = O ⊕O(d), we have the
cases d = −1, 0, 1.
We will build the components b and c of the Higgs field out of global sections
of H0(E∗2 ⊗ E1 ⊗K(D)) and H0(E∗1 ⊗ E2 ⊗K(D)). In order to verify our new Higgs
field has residues with appropriate eigenvalues, we will need to fix a choice of a local
frame and a choice of coordinates at every marked point pj. The following lemma
says the eigenvalues of res(Φ)pj do not depend on these choices:
Lemma 4.1.3. The eigenvalues of the residue of the Higgs field does not depend on
the choice of local frames and local coordinates.
Proof. Changing frames or coordinates corresponds to conjugating the Higgs field
by an invertible matrix, which preserves the eigenvalues.
50
By Simpson’s Correspondence and the discussion above, we want the residue
of Φ to have eigenvalues with imaginary parts ±i log(rj)/4pi. For a PU(n, 1)-Higgs
field, the residues have the form:  0 b
c 0

The characteristic polynomial of such a matrix is p(λ) = λ2 − bc, and therefore
the eigenvalues are ±√bc. As such we’ll construct a Φ whose residue at pj has
determinant − (log(rj)/4pi)2.
Case 1: d = ±1
We’ll focus on the case when d=1. The other case will follow a similar argument.
We have
b ∈ H0(O(1)∗ ⊗K(D)) = H0(O)
c ∈ H0(O∗ ⊗O(1)⊗K(D)) = H0(O(2)).
Let {s1, s2, s3} be a basis of H0(O(2)) such that sj is nonzero at pj and zero
at the remaining marked points. Let b ∈ H0(O) be any fixed nonzero constant and
c = xs1 +ys2 +zs3 ∈ H0(O(2). Fixing local frames and local coordinates at each pj,
denote the residue of sj by rj. Then we want to solve the following linear equations
in x, y, z:
br1x = −γ21
br2y = −γ22
br3z = −γ23
51
where γj = log(rj)/4pi. This is, of course, trivial. Using these to define Φ gives
us a Higgs field with the required residues.
Note that if we write E2 = O(−1), we can also produce a suitable Φ by
mimicking the above construction with a and c switched.
Case 2: d = 0
Now we have
b ∈ H0(O∗ ⊗K(D)) = H0(O(1))
c ∈ H0(O∗ ⊗K(D)) = H0(O(1)).
Let {s1, s2} be a basis for H0(O(1)), where s1 vanishes at p1 and s2 vanishes
at p2. Then setting b = ws1 + xs2 and c = ys1 + zs2. . Fixing local frames and
local coordinates at each pj, label the residues of s1 at p2, p3 by r12 and r13 and the
residues of s2 at p1, p3 by r21 and r23. All the residues are necessarily nonzero. Then
we want to solve the following quadratic equations in w, x, y, and z:
yzr221 = −γ21
wxr212 = γ
2
2
wxr213 + (wz + xy)r13r23 + yzr
2
23 = γ
2
3 .
Note that, since the γi’s and the rij’s are nonzero, w, x, y, and z, must also
be nonzero. For simplicity, write a =
−γ21
r221
and b =
−γ22
r212
. Then w = b
x
and y = a
z
.
Substituting this into the third equation, we want to solve:
br213 + (b
z
x
+ a
x
z
)r13r23 + ar
2
23 = −γ23
52
Finally, substituting u = z
x
and multiplying by u, we need to solve the following
quadratic equation in u:
br13r23u
2 + (γ23 + br
2
13 + ar
2
23)u+ ar13r23 = 0.
Of course, by the quadratic formula, this equation has at least one solution.
Since qr13r23 is nonzero, the solution u is nonzero. Writing z = ux and fixing a
nonzero x, we can find w and y solving the given equations.
4.1.2 Stability
To show the Higgs bundles we constructed above are stable, we must compute the
invariant subbundles and show the slope condition is satisfied. Since a and c are
both nonzero, an invariant line subbundle L must be a subsheaf of both summands
of E . For d 6= 0, this gives
pµ(L) = min(0, d) +
∑
βj
pµ(E) = d
2
+
∑
βj
As such , we have
pµ(E)− pµ(L) = d
2
−min(0, d) = 1
2
> 0
The d = 0 case is slightly trickier. The issue is that O could be invariant, in
which case the bundle is semistable. This happens when the only zero of b and c
agree, and so the Higgs bundles we’ve constructed are generically stable.
53
4.1.3 Existence of stable U(1, 1)-Higgs bundles
The above construction for d = 1 generalizes to an arbitrary number of marked
points/conjugacy classes. The general version of Proposition 4.1.1 is the following:
Theorem 4.1.4. Let C1, . . . , Ck (k ≥ 3) be fixed loxodromic conjugacy classes in
U(1, 1), represented by a matrix Aj of the form given in equation 4.1. Then there
exists a stable parabolic U(1, 1)-Higgs bundle with three marked points D = p1 +
. . .+ pk with the following properties:
• The parabolic structure at pj consists of a single rank-2 jump with weight βj
• The residue of Φ at pj has eigenvalues ±i log(rj)/4pi.
Proof. Let d = k − 2. Then (E = O ⊕ O(d),Φ) with Φ constructed similar to
the d = 1 case above is a stable parabolic U(1, 1)-Higgs bundle with the required
properties.
Again, we have the following easy corollary:
Corollary 4.1.5. Let C1, . . . , Ck (k ≥ 3) be fixed loxodromic conjugacy classes in
U(1, 1). Then there is a matrix Aj ∈ Cj such that A1 . . . Ak = I. This solution is
irreducible, in the sense that the Aj do not all preserve a nonzero subspace V ⊂ Cn
Proof. By Simpson’s Correspondence, the stable parabolic U(1, 1)-Higgs bundle in
Theorem 4.1.4 corresponds to a stable filtered local system. Since by construction
the real parts of the eigenvalues of res(Φ)j are zero, the filtration on the local
system is trivial. As such, the stable filtered local system is really an irreducible
54
representation pi1(P1 − {p1, . . . , pk}) → PU(1, 1), whose monodromy around pj is
Aj.
The construction in the case d = 0 does not, to my knowledge, generalize
easily. For the purposes of proving the main theorem, the result in theorem 4.1.4
will suffice.
4.2 Main Theorem for PU(n, 1)
Let C1, C2, . . . , Ck be k fixed loxodromic conjugacy classes in PU(n, 1). Lift each Cj
to a conjugacy class C˜j in U(n, 1), represented by a diagonal matrix of the following
form:
Aj =

rj
r−j 1
e2piiα
1
j
. . .
e2piiα
n−1
j

(4.2)
where r is a positive real number and 0 ≤ αj1 ≤ . . . ≤ αjn−1 < 1.
Theorem 4.2.1. Main Theorem: Assume the αjs defining the conjugacy classes C˜j
satisfy Σj,kα
j
k ∈ Z and the strict U(n − 1) inequalities in Theorem [4]. Then there
exists a stable parabolic U(n, 1)-Higgs bundle (E,Φ) with n ≥ 3 marked points such
that:
• the filtration of Epj has a rank 2 jump at 0 and rank 1 jumps at each of
αi1, . . . , α
i
n−1
55
• the residue of Φ at pj has eigenvalues ±i log(r)4pi, and 0 with multiplicity
n− 1.
Note that these conditions are sufficient, but not necessary. It is possible
that there are still solutions to the Deligne-Simpson problem when the U(n − 1)
inequalities are not satisfied.
4.3 Proof of Main Theorem
4.3.1 Construction
Following the procedure outlined in sections 3.3 and 3.4, we can build a parabolic
U(n, 1)-Higgs bundle out of a rank n− 1 parabolic bundle and a parabolic U(1, 1)-
Higgs bundle. The construction is as follows.
First, we construct the kernel of our Higgs field. Since the αij satisfy the
U(n − 1) inequalities, there exists by Theorem 2.4.1 a stable rank n − 1 parabolic
bundle S, with weights at pj given by the αjk.
Second, we need to ensure that the residues of our Higgs field have the appro-
priate eigenvalues. By theorem 4.1.4, there exists a (necessarily stable) parabolic
U(1, 1)-Higgs bundle (E ,Φ) of parabolic degree k − 2, such the eigenvalues of the
residue of Φ at pj are ±i log(r)/4pi. Writing E = E1 ⊕Q, Φ has the following form:
Φ =
 0 b
c 0

We will build our new Higgs bundle (E˜ = E1 ⊕ E2, Φ˜). With respect to the
56
splitting of E˜ , Φ˜ has the following form:
Φ˜ =
 0 b˜
c˜ 0

E2 is constructed by picking a special extension of Q by S. No matter which ex-
tension we pick, b : Q → E1 ⊗ K(D) extends to a morphism b˜ : E2 → E1 ⊗ K(D)
by Lemma 3.3.2. However, we still need to pick an extension for which c : E1 →
Q⊗K(D) lifts to c˜ : E1 → E2⊗K(D). For stability reasons, we must also minimize
the vanishing of the lift c˜. By Proposition 3.4.4, there is an extension such that c
lifts and is non-vanishing.
4.3.2 The parabolic structure
We haven’t mentioned the parabolic structure on E˜ . However, both the U(1, 1)
bundle E and the U(n − 1) bundle S come equipped with a parabolic structure,
and hence induce a parabolic structure on the extension E2. However, this induced
structure is not in general unique. All parabolic extensions of Q by S are classified
by elements of H1(ParHom(Q,S)).
Lemma 4.3.1. Let S and Q be two parabolic vector bundles. If Q has the trivial
parabolic structure, then ParHom(Q,S) = Hom(Q,S).
Proof. Since the parabolic structure on Q is trivial, every morphism Q → S is
parabolic, and hence ParHom(Q,S) = Hom(Q,S).
In our case, we have only fixed the extension E2. However, since by con-
struction Q has trivial parabolic structure, the parabolic structure on S uniquely
57
determines the parabolic structure on E2, and thus on E .
4.3.3 Stability
Finally we must determine if the Higgs bundle (E˜ , Φ˜) is stable. From the discussion
above, there are two invariant subbundles. The first is the kernel of b, which by
construction is S. The second invariant subbundle is the rank-2 subbundle. By
Lemma 3.2.2, this is V = E1⊕ (E1((c˜))⊗K(D)∗). Since in our construction c˜ doesn’t
vanish, we have V = E1 ⊕ (E1 ⊗K(D)∗).
When n > 2, rank(S) > 1, and since S is the kernel of Φ˜, any subbundle of
S is contained in the kernel, and hence invariant. However, S is a stable parabolic
bundle, as constructed by theorem 2.4.1. As such, if F is any subbundle of S, then
pµ(F) < pµ(S). Thus we need only verify that pµ(S) < pµ(E).
By construction, pdeg(E1) = 0, and pdeg(Q) = k − 2. By the above, 3.2.2,
pdeg(V) = 2− k. We compute the parabolic degree of E˜ below:
pdeg(E˜) = pdeg(E1) + pdeg(E2)
= pdeg(E1) + pdeg(Q) + pdeg(S)
= k − 2 + pdeg(S)
= k − 2
Finally, we compare the degrees of these subbundles to the degree of E˜ :
pµ(E˜)− pµ(S) = k − 2
n+ 1
− 0 = k − 2
n+ 1
> 0
pµ(E˜)− pµ(V) = k − 2
n+ 1
− 2− k
2
> 0
58
We conclude that the parabolic U(n, 1) bundle E˜ is stable.
Remark Note that in equation 4.2, we pick a specific lift of a loxodromic conjugacy
class in PU(n, 1) to a conjugacy class in U(n, 1). There are many other choices, for
example:
Aj =

rje
2piiβj
r−1j e
2piiβj
e2piiα
1
j
. . .
e2piiα
n−1
j

where βj is nonzero. For the purposes of stating the theorem, which lift we choose
is not important. In fact, the stability of the kernel does not depend on which lift
we choose (but the condition that the weights sum to an integer does). However,
recall that the Higgs field we construct must be parabolic with respect to the flags
chosen at the marked points. The weights on the U(1, 1) bundle are the βj’s. If
these are larger than the αkj ’s, then the morphism c constructed above, and hence
Φ, is not parabolic.
4.4 Deligne-Simpson and Representations
Reinterpreting the existence of a stable Higgs bundle in terms of the corresponding
filtered local system, we have the following the corollary:
Corollary 4.4.1. Let Cj be given loxodromic conjugacy classes in PU(n, 1), lifted
to C˜j to U(n, 1) as in equation 4.2. If the α
j
k satisfy the U(n− 1) inequalities, then
59
there is a matrix Aj ∈ Cj such that A1 . . . Ak = 1.
Proof. By Simpson’s Correspondence, the stable parabolic U(n, 1)-Higgs bundle in
Theorem 4.2.1 corresponds to a stable filtered local system L.
The filtration on L is determined by the real parts of the eigenvalues of res(Φ)j.
The nonzero eigenvalues of res(Φ)j come the U(1, 1) bundle E , as determined by
4.1.4. By construction the real parts of the eigenvalues of res(Φ)j are zero, and
hence the filtration on L is trivial. As such, the stable filtered local system L is really
an irreducible representation pi1(P1 − {p1, . . . , pk})→ PU(n, 1), whose monodromy
around pj is Aj.
4.5 Return to PU(2, 1)
The final remark is that our result generalizes the result of Falbel and Wentworth
in [12] PU(2, 1) case. We provide a separate proof, similar to that of theorem 4.2.1,
but with one key exception. In this case, we can remove the requirement that the
weights sum to an integer.
Theorem 4.5.1. Let C, . . . Ck with k ≥ 3 be fixed loxodromic conjugacy classes in
PU(2, 1), represented by a matrix of the following form:
rj
r−j 1
e2piiαj

Then there exists a stable parabolic U(n, 1)-Higgs bundle with n ≥ 3 marked points
60
and parabolic structure at pj consisting of weights 0, 0, αj, where the residues of the
Higgs field at pj have eigenvalues ±i log(r)/4pi, and 0 with multiplicity 1.
Proof. The only difference from the proof of theorem 4.2.1 is how we choose the
kernel, which in this case is a line bundle we label L = O(n). Once we pick n, the
parabolic structure on L is obvious, and the parabolic line bundle L is automatically
stable. Choose n such that
−1 < Σjαj + n ≤ 0.
Finally, we check that the stability conditions on E are satisfied.
pdeg(E)− 3pdeg(L) = (k − 2 + Σjαj + n)− 3Σjαj − 3n
= k − 2− 2Σjαj − 2n
> k − 2 > 0
By Lemma 3.2.2, the invariant rank 2 bundle is V = O⊕K(D)∗ = O⊕O(2−k).
pdeg(E)− 3
2
pdeg(V) = (k − 2 + Σjαj + n)− 3
2
(k − 2)
=
5
2
k − 3 + Σjαj + n
>
5
2
k − 4
> 0.
We conclude that E is stable.
As before, we can reinterpret the stable Higgs bundles we’ve constructed
as filtered local systems with trivial filtration, i.e. a representation pi1(P1 −
{p1, . . . , pk})→ PU(2, 1):
61
Corollary 4.5.2. Let C, . . . Ck with k ≥ 3 be fixed loxodromic conjugacy classes in
PU(2, 1). Then there exists matrices Aj ∈ Cj such that A1 . . . Ak = I.
62
sChapter 5: Solutions to the Deligne-Simpson Problem for PU(3, 1)
Our main theorem says that when some of the data defining a collection of loxo-
dromic conjugacy classes in U(n, 1) satisfies the U(n− 1) inequalities in 2.4.1, then
there is an irreducible solution to Deligne-Simpson problem. But we’ve actually
proven that solutions exist in many other cases as well. In this chapter, we study
the PU(3, 1) case with three fixed loxodromic conjugacy classes more closely.
In section 5.1, we review some material regarding the product map µ : C1 ×
C2 → PU(3, 1). First, we restrict µ to pairs (A,B) with loxodromic product. Then
we construct a new map, µ˜, which is the projection of the set of loxodromics in
PU(3, 1) onto Clox, the space of loxodromic conjugacy classes. The main result
is that the image of µ˜ consists as a collection of reducible walls and irreducible
chambers. Computing the differential dµ˜, we can show that µ˜ is a submersion at a
pair (A,B) generating an irreducible subgroup. As a consequence, any chamber is
either completely empty or completely full. This technique was used in [11] and [26]
to study the U(n) case, and used in [25] and [12] to study the elliptic and loxodromic
case for PU(2, 1), respectively.
In section 5.2 we consider the different types of reducible subgroups a pair
(A,B) can generate. With this information in mind, we compute the reducible
63
walls in Clox = T2/S2 × (1,∞).
Finally, in section 5.3, we use the Higgs bundles constructed in chapter 4 to
show that many of the chambers are full. The idea is that although there is no
universal description of an unstable parabolic bundle, we can start with a stable
parabolic bundle with known parabolic structure, and study how stability changes
as we vary the weights. In particular, we have the Harder-Narasimhan filtration of
an unstable bundle, which in rank 2 says that any destabilizing line subbundle of a
rank 2 unstable parabolic bundle is unique. In terms of the inequalities determining
stability given in Theorem 2.4.1, exactly one of them becomes invalid. By keeping
track of which inequality is invalid as we vary the weights, we can show that certain
walls are in the image of a pair (A,B) generating an irreducible. As a result, we
conclude that many chambers are full.
5.1 The product map µ
Fix two loxodromic conjugacy classes C1 and C2 in PU(3, 1). Then the group-valued
moment map µ is defined as follows:
µ : C1 × C2 → PU(3, 1)
(A,B) 7→ AB
Since in our case we are interested in (A,B) ∈ C1 × C2 with loxodromic product,
we restrict µ to (A,B) ∈ C1 × C2 whose product is loxodromic. This map is
equivariant with respect to conjugation by elements of PU(3, 1) in each factor. Thus
our restricted map descends to map onto Clox, the space of loxodromic conjugacy
64
classes:
µ˜ : (C1 × C2)
⋂
µ−1(loxodromics)→ Clox
For PU(3, 1), a loxodromic conjugacy class can be identified (up to scaling)
by two angles, α1 < α2, and a real number r > 1. Thus Clox ' T2/S2 × (1,∞). As
an example, consider a conjugacy class C in PU(3, 1). Then C is represented by a
matrix of the following form:
A =

r 0 0 0
0 r−1 0 0
0 0 e2piiα1 0
0 0 0 e2piiα2

, α1 ≤ α2, r > 1
Then the conjugacy class C is identified by the point (α1, α2, r) ∈ T2/S2 × (1,∞).
The (1,∞)-factor is not important in these circumstances, and we will usually
suppress it, and study the T2/S2 portion instead.
The next result concerns the image of pairs (A,B) generating an irreducible
subgroup. Since it involves computing the differential of the map µ˜, we must be
careuful to ensure the source and target of µ˜ are smooth manifolds. To this end, we
further restrict the map µ˜ to (A,B) with regular loxodromic product. Smoothness
is guaranteed by the following proposition. Following the proof in [15] of the well
known fact that the regular elliptic elements in PU(n, 1) form an open set, we have
the following:
Proposition 5.1.1. The regular loxodromic elements in PU(n, 1) form an open
subset.
65
Proof. Let g ∈ PU(n, 1) be a regular loxodromic element. Then by definition, we
can represent g by an element g˜ ∈ U(n, 1) with distinct eigenvalues. The elements
in GL(n,C) with distinct eigenvalues forms an open set, which when intersected
with U(n, 1) gives an open neighborhood of g˜. The projection onto PU(n, 1) gives
an open neighborhood of g.
We have the following proposition, adapted from [11] and [25] to our situation:
Proposition 5.1.2. Let C1, C2 be two fixed loxodromic conjugacy classes, and
(A,B) ∈ C1 × C2 such that AB is loxodromic and the subgroup generated by A,B
irreducible. Then the differential of µ˜ is surjective at (A,B), and hence µ˜ is locally
surjective at (A,B).
This follows immediately from the following lemma, since PU(n, 1) has trivial
center. (see for example [25], [11]):
Lemma 5.1.3. Im(d(A,B)µ) = ζ(A,B)
⊥AB
The application is that the image of µ˜ is divided into pieces by the image of
reducible points (called walls). The interiors of these walls (called chambers) are,
by the proposition, either completely full or completely empty. The main theorem
actually constructs irreducible solutions on one of the walls in our picture, and thus
implies a large number of chambers are full.
66
5.2 Description of reducible walls for PU(3, 1)
Let (A,B) ∈ C1 × C2 generate a reducible subgroup of PU(3, 1). In this case,
reducible can be interpreted as A and B both preserving a fixed subspace of C4.
Depending on the type of subspace preserved, (A,B) (after lifting to U(n, 1)) gen-
erates a subgroup of one of the groups in the following diamond:
U(3, 1)
U(1, 1)× U(1)× U(1)
U(1, 1)× U(2)U(2, 1)× U(1)
Remark While there are other possibilities, such as when A and B are simultane-
ously diagonalizable, we may assume without loss of generality that A, B generate
one of the above subgroups. The missing cases result in reducible walls which give a
relationship between the translation lengths (the r’s) of A, B, and C. In the proof
and statement of Theorem 4.2.1, the translation length r plays no role in the sta-
bility of the resulting bundle, and hence plays no role in the existence of irreducible
solutions to AB = C.
More explicitly, if AB = C is irreducible, then by Theorem 2.5.3 there is a
stable parabolic PU(3, 1)-Higgs bundle (E,Φ) with three marked points, where the
monodromy around each marked point {pa, pB, pC} are in the conjugacy class of A,
B, and C−1, respectively. Then the results of section 3.3 allow us to break (E,Φ)
into a parabolic U(1, 1)-Higgs bundle (E˜, Φ˜) and a parabolic rank 2 bundle S. In
fact, E is, up to tensoring by a line bundle, one of the 3 possibilities in section 4.1.
67
Thus we can replace the Higgs field Φ˜ with a new Higgs field Φ˜′, whose residue at
pC has different eigenvalues. Putting (E˜, Φ˜
′) and S back together via the results
in section 3.4, we a new, necessarily stable, parabolic PU(3, 1)-Higgs bundle such
that the conjugacy class of the monodromy around the point pC is the same as C
−1
except with a different translation length.
5.2.1 U(1, 1)× U(1)× U(1) walls
There are two ”vertices” (really, it’s two lines, but we’re ignoring the (1,∞) factor)
corresponding to groups< A,B > where A,B both preserve two linearly independent
1-dim elliptic subspaces.
Proposition 5.2.1. Let A,B be two loxodromic elements in PU(3, 1), which pre-
serve two 1 dimensional independent elliptic subspaces of C4. Then {γ1, γ2} are
given by one of the following:
•

γ1 = α1 + β1(mod1)
γ2 = α2 + β2(mod1)
•

γ1 = α1 + β2(mod1)
γ2 = α2 + β1(mod1)
Proof. We use the following decomposition of AB = C:
A′ 0 0
0 e2piiα1 0
0 0 e2piiα2


B′ 0 0
0 e2piiβ1 0
0 0 e2piiβ2
 =

C ′ 0 0
0 e2piiψ1 0
0 0 e2piiψ2

68
where A′, B′, C ′ ∈ U(1, 1) are loxodromic. By either Theorem 4.1.4, Theorem 4.2.1,
or [12], C ′ ranges over all possible loxodromic conjugacy classes. Diagonalizing C ′,
we have:
C ′ =
 rCe2piiψ 0
0 r−1C e
2piiψ

Finally, we compute γ1, γ2:
γ1 = ψ1 − ψ = α1 + β1
γ2 = ψ2 − ψ = α2 + β2
This gives the first case of the result. The second case follows from the above,
interchanging the roles of β1 and β2.
Note that in the above proof, rC is allowed to range over all possible values.
Thus the image of < A,B > consists of the whole line (γ1, γ2)× (1,∞).
5.2.2 U(1, 1)× U(2) walls
When A,B preserve a common 2-dimensional elliptic subspace, we have the follow-
ing:
Proposition 5.2.2. Let A, B be two loxodromic elements PU(3, 1) with angles
{α1, α2} and {β1, β2} respectively. If the group generated by A,B preserves a 2-
dimensional elliptic subspace, then the angle pair {γ1, γ2} of C = AB lie on one of
the following segments:
69
• I(ρ) = 2: γ1 + γ2 = α1 + α2 + β1 + β2
γ2 > max(α1 + β2, α2 + β1)
γ2 < min(α1 + β1 + 1, α2 + β2)
• I(ρ) = 3: γ1 + γ2 = α1 + α2 + β1 + β2 − 1
γ2 > max(α1 + β1, α2 + β2 − 1)
γ2 < min(α1 + β2, α2 + β1)
• I(ρ) = 4: γ1 + γ2 = α1 + α2 + β1 + β2 − 2
γ2 > max(α1 + β2, α1 + β2)− 1
γ2 < min(α1 + β1, α2 + β2 − 1)
In this situation, AB = C reduces to: A′ 0
0 A′′

 B′ 0
0 B′′
 =
 C ′ 0
0 C ′′
 (5.1)
where A′, B′, C ′ ∈ U(1, 1) are loxodromic, and A′′, B′′, C ′′ ∈ U(2). Me may assume
the subgroup < A′′, B′′ > of U(2) is irreducible. The case that it is reducible was
handled in the previous section. In order to prove this proposition, we will need
two results. The first is Theorem 2.4.1, regarding the existence of stable parabolic
bundles of rank 2. Fix k conjugacy classes C1, C2, . . ., Ck in U(2), each represented
by a matrix of the form:
As =
 e2piiα
j
1 0
0 e2piiα
s
2
 αs1 ≤ αs2
For convenience, we include the Theorem again here:
70
Theorem 5.2.3. (Biswas) [3]
Let S = {1, 2, . . . , k}, and assume Σs∈S(αs1 + αs2) is an odd (respectively even)
integer, say 2N (respectively 2N +1). Then there is a stable rank 2 parabolic bundle
with parabolic weights {αs1, αs2} at the marked point ps ∈ P1 if and only if for every
D ⊂ S of size 2j (resp. 2j + 1), where j is a nonnegative integer, the following
inequality holds:
−N − j + Σs∈Dαs2 + Σs∈S−Dαs1 < 0.
The second result, from [11], concerns the index of a representation. For a
representation ρ : pi1(P1 − {p1, . . . , pk}) → U(2), the index I(ρ) is the sum of the
angles defining the conjugacy class of the monodromy around each marked point
pj. From the perspective of parabolic bundles, the representation is the sum of the
parabolic weights. In either case, we have I(ρ) = Σs∈S(αs1 + α
s
2).
Proposition 5.2.4. (Falbel, Wentworth) [11]
For any representation ρ : pi1(P1 − {p1, . . . , pk})→ U(n), we have
2−N0(ρ) ≤ I(ρ) ≤ 2(k − 1) +N0(ρ)−N1(ρ)
where N0(ρ) is the number of trivial representations appearing in the decomposition
of ρ into irreducibles, and N1(ρ) is the total multiplicity of 0 among the α
s
j.
We can now return to the proof of Proposition 5.2.2.
Proof. Given the reduction in equation 5.1, we need to study the U(2) piece, i.e.
the equation A′′B′′ = C ′′. Note that this setup is slightly different than in Theorem
5.2.3 and Proposition 5.2.4. To apply these theorems to our case, simply consider
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the representation ρ corresponding to the equation A′′B′′(C ′′)−1 = I. Represent the
conjugacy class of A′′, B′′, and C ′′ by {α1, α2}, {β1, β2}, and {γ1, γ2} respectively,
with α1 ≤ α2, etc., as usual. Then using (C ′′)−1 amounts to using 1 − γ2 ≤ 1 − γ1
in the inequalities in Theorem 5.2.3 and Proposition 5.2.4.
Since we only need to worry about irreducible solutions, Proposition 5.2.4 says
the possible values of I(ρ) are 2, 3, or 4. We’ll handle the case when I(ρ) = 3. The
other two cases follow similarly.
I(ρ) = 3 gives the following:
γ1 + γ2 = α1 + α2 + β1 + β2 − 1 (5.2)
This equation is only valid, of course, when an irreducible representation ρ actu-
ally exists. Necessary and sufficient conditions for the existence of ρ are given by
Theorem 5.2.3. The inequalities take the following form:
α1 + β1 < γ2
α2 + β2 < 1 + γ2
α2 + β1 < 1 + γ1
α1 + β2 < 1 + γ1
Equation 5.2 gives a linear relationship between γ1 and γ2, depending on the
αj’s and the βj’s. In the future, we will think of these walls as giving γ1 as a function
of γ2. For this particular case, we have γ1 = α1 +α2 +β1 +β2− γ2− 1. In this case,
the above inequalities, together with 0 ≤ γ2 < 1, give the domain of this function.
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Substituting γ1 into the above inequalities, we have:
γ2 > α1 + β1
γ2 > α2 + β2 − 1
γ2 < α1 + β2
γ2 < α2 + β1
This completes the proof for the case I(ρ) = 3. The other two cases follow
similarly.
For fixed loxodromic conjugacy classes C1 and C2, at least one and at most
two of the equations in Proposition 5.2.2 have no solutions. To best understand the
geometry of these walls, we temporarily relax the restriction 0 ≤ γ2 ≤ γ1 < 1 to
0 ≤ γ1−γ2 ≤ 1. Then the wall is the shortest line segment of slope−1 line connecting
the two vertices given in 5.2.1. When two equations in 5.2.2 have solutions, the image
of this line segment in the affine chart for T2/S2 is disconnected, as seen in figure
5.1. When only one equation has solutions, the segment in T2/S2 is connected, as
in 5.2
Figure 5.1: An example of a disconnected P (U(1, 1)× U(2)) wall
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Figure 5.2: An example of a connected P (U(1, 1)× U(2)) wall
5.2.3 U(2, 1)× U(1) walls
In this section we consider the case were A,B preserve a common 1-dimensional
elliptic subspace.
Proposition 5.2.5. Let A, B be two loxodromic elements in PU(3, 1) with angles
{α1, α2} and {β1, β2} respectively. If the group generated by A, B preserves a 1-
dimensional elliptic subspace, then the angles {γ1, γ2} of AB = C are given by one
of the following lines:
• γ1 = 3γ2 + α1 + β1 − 3α2 − 3β2(mod1)
• γ1 = 3γ2 + α1 + β2 − 3α2 − 3β1(mod1)
• γ1 = 3γ2 + α2 + β1 − 3α1 − 3β2(mod1)
• γ1 = 3γ2 + α2 + β2 − 3α1 − 3β1(mod1)
where 0 ≤ γ2 ≤ γ1 < 1.
Proof. We’ll show how to derive the first equation. The argument for the other
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three is similar, and simply involves permuting the angles of A and B among the
four possibilities.
Since A, B preserve a common elliptic subspace, AB = C can be written in
the following form: A′ 0
0 e2piiα2

 B′ 0
0 e2piiβ2
 =
 C ′ 0
0 e2piiψ3

where A′, B′, C ′ ∈ U(2, 1) are loxodromic. By Theorem 4.2.1, C ′ ranges over all
possible conjugacy classes of loxodromic elements. If we write
C ′ =

re2piiψ1 0 0
0 r−1e2piiψ1 0
0 0 e2piiψ2
 ,
then diagonalizing A and B, we have the following three equalities involving γ1 and
γ2:
γ1 = ψ2 − ψ1
γ2 = ψ3 − ψ1 = α2 + β2 − ψ1
α1 + β1 = 2ψ1 + ψ2
where each equation is taken mod 1. The third equation comes from the requirement
that det(A′B′) = det(C ′). The derivation of the equation for the wall is as follows:
α1 + β1 = 2ψ1 + ψ2
α1 + β1 = 3ψ1 + γ1
γ1 = −3ψ1 + α1 + β1
γ1 = 3γ2 + α1 + β1 − 3α2 − 3β2.
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The remaining three equations can be derived similarly, by interchanging the
roles of α1 and α2, β1 and β2 in each of the four possible ways.
Though the above proof follows similarly to the corresponding result in [25]
for elliptic classes in PU(2, 1), there is one key difference. For two elliptic conjugacy
classes C1, C2 in PU(1, 1), there are restrictions on the third elliptic conjugacy class
C3, for there to be a solution AB = C. The result is that the U(1, 1)× U(1) walls
start at one of the totally reducible vertices and emanate either to the right or left.
Thus there are many different possible configurations for the U(1, 1) × U(1) walls.
For fixed loxodromic conjugacy classes, there are no such restrictions. As a result,
the walls given in 5.2.5 are in the image of µ˜ for all 0 ≤ γ1 ≤ γ2 < 1.
We should say something about the geometry of these walls. As an example,
consider the first and fourth equations. Plugging γ2 = α2 + β2 into the first wall,
we get the point (α2 + β2, α1 + β1) on the wall (compare this to our vertices in
Proposition 5.2.1. Plugging γ2 = α1 + β1 into the fourth equation, we get the point
(α2 + β2, α1 + β1) on the wall. But these are the same point with our normalization
γ1 ≤ γ2. Accounting for this, the first and fourth wall give line segments in T2/S2,
one of slope 3 and the other of slope 1/3, intersecting at exactly one point, namely
one of the two vertices.
With this understood and combining with Propositions 5.2.1 and 5.2.2, exam-
ples for possible configurations of walls are given in figures 5.3 and 5.4:
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Figure 5.3: A configuration of reducible walls with connected U(1, 1)× U(2) wall
Figure 5.4: A configuration of reducible walls with disconnected U(1, 1)×U(2) wall
5.3 Determining which chambers are full
In the proof of Theorem 4.2.1, we construct stable parabolic U(n, 1)-Higgs bundles
with stable kernel. Though the condition that the kernel be stable is not necessary,
a general rank-(n-1) parabolic bundle can be unstable in many interesting ways.
However, if we have a stable rank 2 bundle with known parabolic structure, we can
use the Harder-Narasimhan filtration to observe how stability changes as we vary
the weights of the bundle.
As a direct consequence to Theorem 4.2.1, we have:
Proposition 5.3.1. The U(1, 1)×U(2) walls in Proposition 5.2.2 are in the image
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of µ˜.
Proof. This is a direct application of our main theorem, Theorem 4.2.1.
As an application of this proposition, since the image is of µ˜ is open at irre-
ducibles, any chambers intersecting with the U(1, 1) × U(2) wall is full. See figure
5.5:
Figure 5.5: The U(1, 1)× U(2) wall is in the image of irreducibles under µ˜. Hence
the shaded chambers are full.
5.3.1 Stability conditions for the parabolic U(3, 1)-Higgs bundles
with three marked points
In this section, we explore the stability conditions for the parabolic U(3, 1)-Higgs
bundles with three marked points constructed in Theorem 4.2.1. In particular, we
lift the requirements that the weights sum to an integer and the kernel be stable, and
give new set of inequalities determining stability. However, it will still be difficult to
determine for which weights the inequalities satisfied. As such, we will instead start
with a bundle from Theorem 4.2.1, which gives a solution to AB = C whose image
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lies on the U(1, 1)×U(2) wall. Leaving the underlying parabolic bundle fixed, we’ll
then study what happens to stability as we move the weights along the wall towards
the U(1, 1) × U(1) × U(1) vertices. From here we’ll study how stability changes
along the U(2, 1)× U(1) walls.
As we move along the first wall and onto a vertex, the kernel S of Φ changes
from stable to semi-stable. Furthermore, as we move from the vertices onto the
U(2, 1) × U(1) walls, S becomes unstable. The Harder-Narasimhan filtration for
parabolic bundles tells us that a destabilizing line subbundle of a rank 2 bundle is
unique. In terms of the inequalities determining the stability of S, exactly one is
violated when S is unstable. By keeping track of which inequality is violated as we
change walls, we can easily find more walls which are in the image of µ˜.
Before going further, we should mention again that the computation and char-
acterization of the reducible walls were performed with respect to the problem
AB = C, whereas the construction of the U(n, 1)-Higgs bundles was performed
with respect to the problem ABC = 1. Thus γ1 ≤ γ2 in the AB = C problem
becomes (1 − γ2) ≤ (1 − γ1). Despite possible confusion, we also label the new
weights (1− γ2) ≤ (1− γ1) by γ1 and γ2, respectively.
Let (E = E1 ⊕ E2,Φ) be a stable parabolic U(3, 1)-Higgs bundle with stable
kernel, as constructed in Theorem 4.2.1.
Proposition 5.3.2. The U(1, 1)×U(1)×U(1) vertices in Proposition 5.2.1 are in
the image of an irreducible solutions to AB = C by µ˜.
Proof. Allow the weights of the parabolic structure on E to move along the U(1, 1)×
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U(2) wall towards the vertices. When we reach the vertices, the kernel S of Φ
becomes semi-stable. Note however, that we have changed neither the degree of S,
nor the filtration at each marked point. As such, S with the new weights is semi-
stable but remains non-split. It follows that E with the new weights is stable, and
the vertices are in the image of an irreducible under µ˜.
At first, this does not seem to give enough information to fill in chambers
incident with the vertex, since the image of µ˜ is only guaranteed to be open at
irreducibles with regular loxodromic product (where µ˜ is smooth). When the vertex
lies on the line γ1 = γ2, the regularity assumption is violated. On the other hand,
stability is an open condition, and hence we can further perturb the weights on E
slightly to find irreducibles near the vertex with regular product, thus guaranteeing
the chambers are full.
As we move away from the U(1, 1) × U(2) wall, we’ll lose the assumption
that the weights add to an integer. In order to determine the Harder-Narasimhan
filtration of S as we move away from this wall and along the U(2, 1)×U(1) wall, we
need stability conditions for this more general case. These are given in the following
proposition whose proof is almost exactly the same as in [3]. The integer d in the
proposition is determined by the index of the wall we begin on. The index can take
values 2, 3, and 4 for which S has the structure O(−1) ⊕ O(−1), O(−1) ⊕ O(−2),
and O(−2)⊕ O(−2) respectively. However the result is true for all d ≤ 0:
Proposition 5.3.3. Let S = O(d)⊕O(d−1) be a rank 2 parabolic bundle with three
marked points and weights 0 ≤ α1 ≤ α2 < 1, 0 ≤ β1 ≤ β2 < 1 and 0 ≤ γ1 ≤ γ2 < 1.
80
Then S is stable if and only if the following inequalities hold:
• −1− α1 − β1 − γ1 + α2 + β2 + γ2 > 0
• 1 + α1 + β1 − γ1 − α2 − β2 + γ2 > 0
• 1 + α1 − β1 + γ1 − α2 + β2 − γ2 > 0
• 1− α1 + β1 + γ1 + α2 − β2 − γ2 > 0
Let S = O(d)⊕O(d) be a rank-2 parabolic bundle with three marked points and
weights as above. Then S is stable if and only if the following inequalities hold:
• α1 − β1 − γ1 − α2 + β2 + γ2 > 0
• −α1 + β1 − γ1 + α2 − β2 + γ2 > 0
• −α1 − β1 + γ1 + α2 + β2 − γ2 > 0
• 2 + α1 + β1 + γ1 − α2 − β2 − γ2 > 0
Proof. See [3] for more information. For our purposes, the decomposition of S into
line bundles is fixed, and one of the above cases. Additionally the parabolic structure
is fixed, and generic. We’ll illustrate the final steps for the case S = O(d)⊕ O(d).
Since there are three marked points, the only possible destabilizing subbundles
are L = O(d) and L = O(d− 1).
An embedding O(d) ↪→ S is determined (up to scale) by two parameters. Since
the flags are generic, any embedding O(d) ↪→ S coincides with the flag at at most
one marked point. The requirement that pµ(S)− pµ(O(d)) > 0 gives the first three
inequalities.
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An embedding O(d− 1) ↪→ S on the other hand is determined (up to scale) by
four parameters. As such, there is a unique embedding O(d−1) ↪→ S which coincides
with the flag at every marked point. The requirement pµ(S) − pµ(O(d − 1)) > 0
gives the third inequality.
Let (E˜ = O⊕O(1), Φ˜) be a (necessarily stable) parabolic U(1, 1)-Higgs bundle
of loxodromic type. Let S, of either the form O(d)⊕ O(d) or O(d)⊕ O(d− 1), be a
non-split rank 2 parabolic vector bundle, with weights as in propositon 5.3.3. Then
our main construction builds a new parabolic U(3, 1)-Higgs bundle (E = O⊕ E2,Φ)
from (E˜, Φ˜) and E. The following proposition combines the analysis in the proof
of Theorem 4.2.1 with the generalized inequality in Proposition 5.3.3 to give the
stability conditions for (E,Φ).
Proposition 5.3.4. Let S be of the form O(d) ⊕ O(d) with the usual weights, and
V = O⊕ O(−1) the other invariant rank 2 subbundle. Then E is stable if and only
if the following six inequalities are satisifed:
• 1 > pdeg(S)
• pdeg(S) > −3
• 1− 2d+ α1 + β2 + γ2 − 3α2 − 3β1 − 3γ1 > 0
• 1− 2d+ α2 + β1 + γ2 − 3α1 − 3β2 − 3γ1 > 0
• 1− 2d+ α2 + β2 + γ1 − 3α1 − 3β1 − 3γ2 > 0
• 5− 2d+ α1 + β1 + γ1 − 3α2 − 3β2 − 3γ2 > 0
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Let S be of the form O(d)⊕O(d− 1) with the usual weights. Then E is stable if and
only if the following six inequalities are satisifed:
• 1 > pdeg(S)
• pdeg(S) > −3
• −2d+ α2 + β2 + γ2 − 3α1 − 3β1 − 3γ1 > 0
• 4− 2d+ α1 + β1 + γ2 − 3α2 − 3β2 − 3γ1 > 0
• 4− 2d+ α1 + β2 + γ1 − 3α2 − 3β1 − 3γ2 > 0
• 4− 2d+ α2 + β1 + γ1 − 3α1 − 3β2 − 3γ2 > 0
Proof. The proof follows similarly to that of Proposition 5.3.3, except in the context
of the PU(3, 1)-Higgs bundle (E,Φ) constructed from S and a U(1, 1)-Higgs bundle
(O⊕ O(1), Φ˜). There are two additional inequalities, coming from the requirement
that pµ(E)− pµ(S) > 0, and similarly for the other invariant rank-2 bundle V. The
other four inequalities come from the stability requirement applied to line subbundles
of the kernel S, as in 5.3.3.
Notice that unlike in the U(2) case, the parameter d plays a role in stabil-
ity. Also, the last four inequalities in each case resemble the equations giving the
U(2, 1)× U(1) walls in Proposition 5.2.5.
5.3.2 Moving along U(2, 1)× U(1) walls
Now we’re ready to study the stability of our PU(3, 1)-Higgs bundle (E,Φ) as we
move away from the vertices and onto the U(2, 1) × U(1) walls. At this point, the
83
combinatorics become a bit difficult to manage.
First, we begin on a U(1, 1)× U(2) wall, where our main theorem guarantees
the existence of a stable PU(3, 1)-Higgs bundle with stable kernel. The kernel S of
Φ is determined the index, for which there are three possibilities. Then, varying the
weights of the bundle but nothing else, we move toward a vertex, for which there
are two choices. Next, the vertices given in Proposition 5.2.1 do not account for
the normalization γ2 > γ1, resulting in two possibilities. Finally, it could be the
case that the U(1, 1) × U(2) wall is of disconnected type. We’ll show the details
completely one case. The others follow similarly.
I(ρ) = 3 case:
Let’s study the case where the index of the U(1, 1) × U(2) wall is 3. In this case,
S = O(−1)⊕O(−2). By Proposition 5.2.2, the wall is given by the equation γ1+γ2 =
α1+α2+β1+β2−1, where γ2 > max(α1+β1, α2+β2−1) and γ2 < min(α1+β2, α2+β1.
First, we’ll vary the weight γ2 in the negative direction. The kernel S becomes
semi-stable when γ2 = max(α1 +β1, α2 +β2− 1). In particular, we’ll study the case
where max(α1 + β1, α2 + β2 − 1) = α2 + β2 − 1. Combining with Proposition 5.2.1,
the vertex is given by: 
γ1 = α1 + β1
γ2 = α2 + β2 − 1
If we look at the conditions for stability in Proposition 5.3.3, the inequality
1 + α1 + β1 − γ1 − α2 − β2 + γ2 > 0 becomes equality. Continuing to move γ2 in
the negative direction, this inequality becomes invalid. Recall that this inequality
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comes from a particular embedding O(−2) ↪→ S. Therefore, this is the (unique!)
destabilizing line subbundle of S that we need to worry about. The corresponding
inequality in Proposition 5.3.4 is, given that d = −1, 6+α1+β1+γ2−3α2−3β2−3γ1 >
0. Verifying the stability of (E,Φ) is then one of verifying only this inequality and
pdeg(S) > −3, coming from pµ(E)− pµ(V) > 0. The second inequality is, of course,
trivially valid.
The first U(2, 1)×U(1) wall intersecting this vertex is γ1 = 3γ2+α1+β1−3α2−
3β2. Note that this equation is taken mod(1). Thus we can think of it as the weights
satisfying some integer relation, in particular α1+β1−γ1+−3α2−3β2+3γ2 ∈ Z. In
particular, since this wall intersects the index 3 wall at the vertex, we can determine
this integer by plugging in the values of γ1, γ2 at the vertex, we can determine this
integer:
α1 + β1 − γ1 +−3α2 − 3β2 + 3γ2
=α1 + β1 − (α1 + β1)− 3α2 − 3β2 + 3(α2 + β2 − 1)
=− 3
Finally, we convert from the AB = C problem to the ABC = I problem by
replacing γ1 with 1− γ2 and γ2 with 1− γ1. This gives:
α1 + β1 + γ2 − 3α2 − 3β2 − 3γ1 = −5
We can easily confirm then that 6 + α1 + β1 + γ2 − 3α2 − 3β2 − 3γ1 > 0.
Finally, as we move along γ1 = 3γ2 + α1 + β1 − 3α2 − 3β2, 1 + α1 + β1 − γ1 −
α2 − β2 + γ2 > 0 is only invalid on half the wall. In particular, plugging in γ1, the
85
values of γ2 for which the inequality is invalid are given by the inequality:
γ2 > α2 + β2 − 1
Now we consider the other wall passing through the vertex given in equation
5.3.2, which has formula γ1 = 3γ2 + α2 + β2 − 3α1 − 3β1. Actually, viewed as a
function γ1 of γ2, this passes through (γ2, γ1) = (α1 + β1, α2 + β2 − 1), which does
not satisfyγ1 < γ2, so we swap γ1 and γ2. Then in T2/S2, the wall has equation
γ2 = 3γ1+α2+β2−3α1−3β1. This can again be interpreted as the weights satisfying
an integer relationship. In this case, we have:
3γ1 + α2 + β2 − 3α1 − 3β1 − γ2 = −1
Applying the transformation from AB = C to ABC = I, we end up with the
following relation:
γ1 = −3 + 3α1 + 3β1 +−α2 − β2 + 3γ2
Unlike the previous case, this does not reduce the inequality 6+α1 +β1 +γ2−3α2−
3β2 − 3γ1 > 0 in a nice way. However, plugging in γ1 to the inequality:
6 + α1 + β1 + γ2 − 3α2 − 3β2 − 3γ1 = 12− 8α1 − 8β1 − 8γ2 + 6α2 + 6β2
Now, α2 > α1 and β2 > β1, and αj, βj, γj < 1, so finally
12− 8α1 − 8β1 − 8γ2 + 6α2 + 6β2 > 12− 2α1 − 2β1 − 8γ2
> 0
This is valid, complimentary to the last wall, for γ2 < α2+β2−1 (this is the direction
where O(d− 1) ↪→ S from above is still the unique destabilizing subbundle).
86
The last important thing to mention is that this analysis is valid as long as
γ1 < γ2. When we hit the outer wall γ1 = γ2, the parabolic structure of the kernel
S changes. Assuming γ1, γ2 are the weights at the marked point p, the parabolic
structure changes from having two rank 1 jumps, at γ1 and γ2, to having a single
rank 2 jump at γ1 = γ2. Unfortunately, the analysis past the γ1 = γ2 wall since we
don’t know what the filtration should look like on the other side.
The computations for other choices of index/vertex are similar. Geometrically,
the picture is as follows. Passing through each vertex are two U(2, 1)× U(1) walls.
The above computations show that the portion of the walls moving away from the
vertex are in the image of an irreducible under µ˜. Examples are shown in figures
5.6 and 5.7.
Figure 5.6: An example showing walls (highlighted in red) which are in the image
of an irreducible under µ˜
Finally, in the case that the U(1, 1)×U(2) wall is connected, we can say a little
bit more. From figure 5.6 for example a pair of U(2, 1) × U(1) walls intersecting
different vertices with reciprocal slopes could intersect each other. It would be
interesting to consider the image of µ˜ at this intersection point.
87
Figure 5.7: An example showing walls (highlighted in red) which are in the image
of an irreducible under µ˜
As we change the weights away from a vertex and along a U(2, 1)×U(1) with
angle less than pi
2
off the U(1, 1) × U(2) wall, it is difficult to determine which, if
any subbundles of S are destabilizing. In fact, none of them are! If we analyze the
inequalities in Proposition 5.3.3 in a way similar to how we determined the bounds
in the proof of Proposition 5.2.2, we can see that S remains stable if γ1, γ2 remain
in between the two lines with slope 1, passing through the vertices. See figure 5.8
for an example.
Figure 5.8: The kernel S remains stable for γ1, γ2 in between the dotted lines
Since S remains stable within this region, the inequalities determining the
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stability of E (from Proposition 5.3.4) are:
pdeg(S) > −3
pdeg(S) < 1
The first inequality is trivially valid. The second is not as obvious. We are interested
in verifying that pdeg(S) < 1 at the intersection of the two U(2, 1) × U(1) walls of
slope 1
3
from the top vertex and slope 3 from the bottom vertex. Suppose for a
contradiction that prior reaching this intersection point that pdeg(S) ≥ 1. Then
in particular there is a choice of γ1, γ2 where pdeg(S) = 1 (an integer!) and S is
stable. This implies in particular that there is another nonempty U(1, 1) × U(2)
wall, contradicting our assumption that there is one connected U(1, 1)× U(2) wall.
Therefore, when the U(1, 1)×U(2) wall is connected, the intersections (when
they exist) of U(2, 1) × U(1) walls of reciprocal slope from opposite vertices are in
the image of an irreducible under µ˜.
This argument does not work when the U(1, 1) × U(2) wall is disconnected.
Of course the contradiction above does not apply, but more importantly each piece
of the wall only has one vertex. The U(2, 1) × U(1) walls emanating from the two
vertices still intersect, but are unrelated from the perspective of our higgs bundle
construction. Therefore we would need a new strategy to show these intersections
are in the image of an irreducible.
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5.4 Conclusion
Finally we combine the descriptions of reducible walls in Propositions 5.2.1, 5.2.2,
and 5.2.5; the results regarding image of µ˜ in Propositions 5.3.1 and 5.3.2, and the
analysis along U(2, 1)×U(1) walls in section 5.3.2. Some examples of what we know
are summarized in figures 5.9, 5.10, and 5.11.
Figure 5.9: Since the red walls are in the image of an irreducible with regular
product, the chambers they bound are full.
Figure 5.10: Since the red walls are in the image of an irreducible with regular
product, the chambers they bound are full.
At this point, we have exhausted all of our tools for studying these diagrams.
For diagrams of connected type, it is difficult to determine what happens when we
90
Figure 5.11: Since the red walls are in the image of an irreducible with regular
product, the chambers they bound are full.
pass through the outer γ1 = γ2 wall, since we do not know what the parabolic
structure of our Higgs bundle looks like on the other side of the wall. For diagrams
of disconnected type, there is also the issue of what happens when U(2, 1) × U(1)
walls intersect. Our Higgs bundle techniques do not apply in this case, since each
wall corresponds to Higgs bundles of different starting index. The Higgs bundles
sitting above these intersections are therefore unrelated.
While there are still chambers which we cannot at this point guarantee are full,
we have also failed to construct any examples of fixed loxodromic conjugacy classes
in PU(3, 1) for which there is no solution. New insights are therefore required to
proceed further.
91
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