Moduli Space Techniques in Algebraic Geometry … · Abstract Moduli Space Techniques in Algebraic Geometry and Symplectic Geometry Kevin Ka Hang Luk Master of Science Graduate Department

Moduli Space Techniques in Algebraic Geometry and SymplecticGeometry

by

Kevin Ka Hang Luk

A thesis submitted in conformity with the requirementsfor the degree of Master of Science

Graduate Department of MathematicsUniversity of Toronto

c© Copyright 2012 by Kevin Ka Hang Luk

Abstract

Moduli Space Techniques in Algebraic Geometry and Symplectic Geometry

Kevin Ka Hang Luk

Master of Science

Graduate Department of Mathematics

University of Toronto

2012

The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is

divided into the following two parts: the first part is devoted to presenting moduli problems in algebraic

geometry using a modern perspective, via the language of stacks and the second part is devoted to

studying moduli problems from the perspective of symplectic geometry. The key motivation to the first

part is to present the theorem of Keel and Mori [20] which answers the classical question of under what

circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two

of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

ii

Dedication

I would like to dedicate this to my late mother who was the first to instill in me the importance of

knowledge.

iii

Acknowledgements

First and foremost, I would like to thank my supervisor, Lisa Jeffrey, for her permission for me to

pursue this topic for my Master’s thesis. Her constant patience and instructive suggestions were critical

throughout both the research and the writing phase of this thesis. Almost all of Part II of this thesis

would never have been possible without her careful direction.

I would also like to thank two other professors here at UofT, Marco Gualtieri and Ragnar Olaf-

Buchweitz, both of whom I have learned a lot of mathematics from over the past year. I would like to

thank the former for sacrificing lots of personal time to guide me towards interesting new research direc-

tions and thank the latter for warmly accommodating me in his various research seminars throughout

the past year.

Lastly, I would like to thank my family and friends for their love, support, and encouragement. All

of the small accomplishments in my life, including the writing of this M.Sc. thesis, would not have been

possible without them.

iv

Contents

1 Algebraic Geometry Approach: Via Stacks 1

1.1 Motivation for Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Groupoids and Categories fibered over Groupoids . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Categories fibered over Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Properties of CFGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Deligne-Mumford Stacks and Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 Deligne-Mumford Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.2 Some remarks on Deligne-Mumford Stacks . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Theorem of Keel-Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.2 Proof of the Keel-Mori Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.3 Consequences of the Keel-Mori Theorem . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Symplectic Geometry Approach 22

2.1 Connection between Symplectic Quotients and G.I.T. Quotients . . . . . . . . . . . . . . . 23

2.1.1 The Symplectic Quotient and the G.I.T. Quotient . . . . . . . . . . . . . . . . . . 23

2.1.2 Theorem of Kirwan-Kempf-Ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Hyperkahler Manifolds and Hyperkahler Quotients . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Hyperkahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Hyperkahler Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Cohomology of Symplectic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Quick Overview of Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Minimally Degenerate Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3 Inductive Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.4 Atiyah-Bott: Cohomology of Moduli Space of Vector Bundles . . . . . . . . . . . . 35

2.4 Moduli Space of Higgs Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2 Relationship to Higgs Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 Moduli Space of Higgs Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

v

3 Appendix: Grothendieck Topologies and Algebraic Spaces 41

3.1 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Algebraic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography 43

vi

Chapter 1

Algebraic Geometry Approach: Via

Stacks

The first part of this thesis is devoted mainly to presenting moduli problems in algebraic geometry using

the modern language of stacks. We first introduce what a “moduli problem” in algebraic geometry

should be formally, that is, through Grothendieck’s approach using representable functors. Using the

simple examples of the moduli problem of classifying triangles and elliptic curves, we will see that the

moduli functor for both of these moduli problems is not representable in the category of schemes. There

is essentially two methods to circumvent this problem: the first is to look for an “approximation” to the

moduli functor; this is the idea of the coarse moduli space and the second is to look at a larger category

where the moduli functor is representable. We will look at both of these scenarios but focus on the latter

as this will be the main motivation to introduce algebraic stacks.

In Section 1.2, we will define the notion of groupoids and what it means to have a category fibered in

groupoids; both of which are necessary ingredients to define stacks. It is also essential in understanding

the Keel-Mori theorem in Section 1.5 as most of the proof of the theorem is written in the language

of groupoids. We give a concise survey of groupoids as well as a number of examples in order to

give the reader familiarity with these new notions. It turns out that a stack is simply a category

fibered in groupoids satisfying certain conditions: one of which is descent datum. This condition can be

naively interpreted as a sophisticated method of standard “gluing” constructions rephrased in categorical

language. We will examine this idea more clearly in Section 1.3.

In Section 1.4, we look at certain classes of stacks which are the most important ones in algebraic ge-

ometry: Deligne-Mumford and algebraic stacks. Many famous moduli problems, such as the classification

of genus g curves, principally polarized abelian varieties, etc., can be interpreted as Deligne-Mumford

stacks. We will look extensively at how Mg is a Deligne-Mumford stack.

We finally come to the main theorem of Part I: the theorem of Keel-Mori. An important consequence

of the theorem is that it gives (under certain hypothesis) the existence of quotients of a flat group

scheme G acting on a finite type scheme X. However, the outcome is that the quotient exists not in the

category of schemes, but rather in the category of algebraic spaces which illustrates that in some sense,

the category of schemes is not “geometric” enough. To present the theorem, we will first introduce what

“quotients” should mean in the world of groupoids and following [20], define what geometric quotients,

categorical quotients, etc. are in this abstract setting. We will then give a simple overview of the

1

Chapter 1. Algebraic Geometry Approach: Via Stacks 2

techniques used to prove the theorem and also we will discuss about the important corollaries of the

Keel Mori theorem. A particularly important corollary is the existence of coarse moduli spaces for a

stack satisfying certain hypothesis. This allows us to determine existence of the coarse moduli space

without explicitly constructing the coarse moduli space itself (which is often done via G.I.T. techniques)

1.1 Motivation for Stacks

A “moduli problem” in algebraic geometry formally is a contravariant functor F from the category of

schemes to the category of sets defined as:

F(B) = S(B)/ ∼

where S(B) is the set of all families over B and ∼ an equivalence relation on the set S(B) of all such

families over B. For example, the moduli problem of classifying genus g curves (which we will explore

more extensively later), has a formal description where F(B) is defined to be a proper, smooth family

C → B whose geometric fibers are smooth, connected 1-dimensional schemes of arithmetic genus g.

Another example would be if we want to classify vector bundles of rank n over B; the formal description

would to simply define F(B) to be the set of vector bundles of rank n where two vector bundles of rank

n are in the same equivalence class if they are isomorphic in the usual sense.

To study moduli problems, what we would like to do is to construct some kind of space (variety,

scheme, etc) whose geometric points correspond to isomorphism classes of the objects that we are

interested in parametrizing. In the case of the moduli of genus g curves, what we would like is to

construct a moduli space Mg whose geometric points correspond to isomorphism classes of smooth

curves of genus g. We come to the definition of fine moduli spaces and coarse moduli spaces; the latter

of which is more congruent with the naive interpretation of what a moduli space should be whereas the

former, as we shall see, is a far stronger condition.

Definition 1.1.1. Let F be a moduli functor. If F is representable by a scheme M (i.e., there exists an

isomorphism between F and the functor of points of M), then we say M is a fine moduli space for F .

Perhaps the most powerful consequence of having a fine moduli space for F is that there is a notion

of a universal family : there exists a family C →M (which will be called the universal family) such that

any family D → B in S(B) can be obtained as a pullback of C via a unique map B → M . To see the

technical details behind this, please see [14]. What is important is that having a fine moduli space for

our moduli problem gives us a way of translating information between the geometry of families of our

moduli problem along with the geometry of the moduli space M itself: one of the most fundamental

reasons why moduli theory is studied.

However, it will turn out that fine moduli spaces rarely exist, as the following examples indicate:

Example 1.1.2. We start with a very simple example: the moduli problem of classifying triangles. Fix

S a topological space, a family of triangles over S will be a continuous and proper map X → S where X

is a fiber bundle over S with a continuously varying metric on fibers such that each fiber is a triangle.

What the desired notion of the moduli space of triangles is natural: the candidate should be T/S3

where:

T =

(a, b, c) ∈ R3 : a+ b > c, b+ c > a, c+ a > b


and S3 is the symmetric group which acts on T via coordinate permutation. Let us denote T/S3 by

T . Any family of triangles X → S will give a map from S → T , but the family might be not isomorphic

to the pullback of Y → T . To see this, consider the example where S = S1 and X → S is a family of

equilateral triangles that rotates the triangle by 120 degrees in one revolution around the circle. The

result is that this is not a constant family even though the map S → T is constant; of course, the reason

behind this is we can “twist” the triangles. Hence, we do not get a fine moduli space for the problem of

classifying triangles.

Remark 1.1.3. If we consider ordered triangles, T is a fine moduli space for the moduli problem.

Example 1.1.4. Consider the moduli problem of classifying elliptic curves. A fine moduli space is

impossible for this problem. There are a number of ways of show this but we shall discuss two of them.

The first method comes from [11]. Consider the family of elliptic curves:

X =y2 = x(x− 1)(x− λ)

⊂ P2

x,y × A1λ

over A1\0, 1 where (x, y, λ) 7→ λ and we have that Xλ∼= Xλ′ if and only if:

λ′ ∈λ, 1− λ, 1

λ,

1

1− λ,λ− 1

λ,

λ

λ− 1

The natural candidate for the moduli space of elliptic curves is A1 where we have the j-map (given

by the j-invariant) going from A1λ → A1

j . Suppose this is a fine moduli space, then we have the following:

X J //

ϕ

C

1

A1λ

j // A1j

where 1 : C → A1j is the universal curve. If we try to lift the involution λ 7→ 1 − λ, using a simple

computation, we show that this is not an involution on X .

A more sophisticated way to see that there can not be a fine moduli space for classifying elliptic

curves is due to [27]. In [27], Mumford proves the existence of nontrivial line bundles on schemes S given

any family of elliptic curves over S. The natural candidate for the moduli space of elliptic curves is A1

of course, but any line bundle (in fact vector bundle) over A1 is trivial so we can not obtain any of these

nontrivial bundles over S via a morphism S → A1. The paper of [27] is one of the earliest papers in

algebraic geometry that present moduli problems as a description in the language of stacks (despite the

fact that the term stack never appears in the paper!). Formally, the main theorem of [27] can be stated

as:

Pic(M1,1) ∼= Z/12Z

where M1,1 is the moduli stack of elliptic curves, a notion that we will define later.

Remark 1.1.5. One formally can study the moduli of elliptic curves without going to the language

of stacks. Instead of considering representable functors, one can work with the idea of relatively repre-

sentable functors; a weaker notion of representability. By adding “extra structure” to the moduli problem

of classifying elliptic curves, it can be shown that the moduli functors used for classifying elliptic curves


are relatively representable. We will not pursue any of these methods here though but refer to [19] for

an extensive treatment of this.

We will make one more remark about the moduli problem of elliptic curves, via the complex-analytic

viewpoint:

Remark 1.1.6. We can work with moduli of elliptic curves in the complex-analytic viewpoint. An elliptic

curve is simply a one dimensional torus C/Γ where Γ = z1Z + z2Z is a lattice in C. Via coordinate

change, we can write Γ = Z+ τZ where τ ∈ H. We say that two elliptic curves will be isomorphic if and

only if:

C/(Z + τZ) ∼= C/(Z +A(τ)Z)

for A ∈ SL(2,Z). Then, the moduli space of elliptic curves is simply the quotient:

H/SL(2,Z)

We know from differential topology that this quotient admits a smooth manifold structure if the SL(2,Z)

action on H is free. However, the action is not free at two points (excluding the identity): i and

e2πi/3. This means that the quotient does not have a smooth manifold structure, but rather an orbifold.

Note that once again this is related to the existence of objects in our moduli problem with non-trivial

automorphisms: the points i and e2πi/3 have nontrivial automorphism groups. We shall not pursue

orbifold methods to study the moduli problem of elliptic curves but refer to [10] for an extensive treatment

in this direction.

In both examples given above, we notice that the obstruction to having a fine moduli space was the

presence of nontrivial automorphisms that certain triangles and elliptic curves possess. This problem was

first mentioned by Grothendieck to Serre in 1959 in his study of moduli problems in algebraic geometry.

The ultimate solution to this, as we will see later, is to study moduli problems using stacks. However,

before going to stacks, we shall mention the idea of coarse moduli spaces which is a weaker version of

the idea of fine moduli spaces. The key is that we do not demand that there is an isomorphism between

the moduli functor and the functor of points of the moduli space. The formal definition is as follows:

Definition 1.1.7. A scheme M and a natural transformation ΨM from the moduli functor F to the

functor of points of M are a coarse moduli space for the functor F if:

1. The map ΨSpec(C) : F(Spec(C))→ Mor(Spec(C),M) is a set theoretic bijection

2. Given another scheme M ′ and a natural transformation ΨM ′ from F → MorM ′ , there is an unique

morphism π : M → M ′ such that the associated natural transformation Π : MorM → MorM ′

satisfies ΨM ′ = Π ΨM

Of course, a fine moduli space is a coarse moduli space. In fact, in that particular example, condition

(2) of the above definition is exactly the statement of the Yoneda lemma.

Example 1.1.8. We return to the examples of triangles and elliptic curves. For the moduli problem

of triangles, T is a coarse moduli space and for the moduli problem of elliptic curves, A1 is the coarse

moduli space. Given a morphism X → S of elliptic curves, the map S → A1 is given by the j-invariant.

We shall return to the idea of coarse moduli spaces later but for now, we shall describe the main

goal of the subsequent sections; that is, study moduli problems in algebraic geometry via stacks. Let us


see what a stack should (very) naively be. Consider the example of the moduli of vector bundles; the

stack associated to this problem (or simply moduli stack of vector bundles) can naively be thought of

as a category where the objects are families E → S, E′ → S′ of such bundles over a schemes S, S′ and

a morphism between these objects would be a morphism of schemes from S′ → S with the isomorphism

of E′ with the pullback bundle with the fiber product:

E′ ∼= f∗E //

E

S′

f // S

In essence, we are ensuring by definition that we have the property that any family E′ → S′ is obtained

via a pullback of E → S via a map S′ → S. As a result, “twisting” by nontrivial automorphisms is

impossible in the world of stacks. We shall now proceed to formally define stacks and understand to

what extent this naive definition of stacks is the correct notion for stacks.

1.2 Groupoids and Categories fibered over Groupoids

1.2.1 Groupoids

We start by defining what groupoids are abstractly. Fix a base category S, a groupoid in S (or a

S-groupoid) is defined as a pair of objects R,X in S together with the following morphisms:

• Source and Target morphisms:

s, t : R→ X

• Multiplication morphism:

m : R×(s,t) R→ R

• Identity morphism:

e : X → R

• Inverse morphism:

i : R→ R

which satisfies the following axioms:

• The composites s e and t e are identity maps on X

• If pr1 and pr2 are projections from R×(s,t) R to R, then s m = s pr1 and t m = t pr2

• (Associativity) The maps m (1R ×m) and m (m× 1R) are equal

• (Identity) The maps m (e s, 1R) and m (1R, e t) from R→ R are equal to the identity on R

• (Inverse) i i = 1R, s i = t, t i = s, m (1R, i) = e s, and m (i, 1R) = e t

For shorthanded purposes, we write (R,X) for a groupoid with spaces R,X involved. We should also

note that in literature, the space R is sometimes called the space of arrows and X is sometimes called


the space of objects. Also, when S is the category of topological spaces, so R,X are topological spaces

and all the structure maps are continuous maps between topological spaces. We call such groupoids

topological groupoids. Similarly, if S is the category of smooth manifolds, we have that R,X are smooth

manifolds and all structure maps are smooth maps. These are referred to as Lie groupoids. An important

class of groupoids in algebraic geometry are algebraic groupoids (but they are usually called groupoid

schemes instead) where the underlying category S is the category of schemes.

Definition 1.2.1. A morphism of groupoids (R,X) and (R′, X ′) is given by a pair of morphisms (φ,Φ)

where φ : X ′ → X, Φ : R′ → R which commutes with all the structure maps of the associated groupoids.

We also define what it means to have a subgroupoid:

Definition 1.2.2. A subgroupoid of (R,X) is a subobject P ⊂ R containing the identity section such

that the induced structure maps on P define a groupoid itself.

We now present a series of examples of groupoids:

Example 1.2.3. The first example is the fundamental groupoid of a topological space X. We define it

as (R,X) where X is just the topological space given to us and elements of R are triples (x, y, ϕ) with

x, y ∈ X and ϕ a homotopy class of paths in X starting at x and ending at y. The structure morphisms

are as follows: s takes the triple (x, y, ϕ) to x and t it to y. The multiplication morphism m is defined

by: m((x, y, ϕ), (y, z, ϕ′)) = (x, z, ϕ ∗ ϕ′) where ∗ is just path concatenation.

The next example illustrates how standard group actions can be rephrased in this high-tech language.

It will be especially important later on to think of group actions as groupoids instead when we discuss

the Keel-Mori theorem later.

Example 1.2.4. Suppose that we are working in the algebraic category so we have an algebraic group

action G on a scheme X. This, of course, gives an equivalence relation on X: two points x, y ∈ X are

equivalent if there exists some g ∈ G such that y = g ·x. If we set R = G×X and define structure maps

s(g, x) = x, t(g, x) = g · x, m((g, x), (g′, g · x)) = (g′ · g, x), e(x) = (1G, x), and i(g, x) = (g−1, g · x), the

groupoid (R,X) encodes exactly the group action G on X. We call (R,X) a transformation groupoid.

The last example shows that atlases can be constructed using groupoids:

Example 1.2.5. In differential topology and algebraic geometry, a smooth manifold (or a scheme) X

can be constructed from an open covering Uα of X. How to properly glue these Uα’s together, i.e.,

construct an atlas where the Uα’s satisfy compatibility conditions, has a groupoid description. Consider

the groupoid (R,U) where U = qUα and R = qUα ∩ Uβ , the disjoint union of all intersections over all

ordered pairs (α, β). Then, we define the structure maps: let s take a point in Uα ∩Uβ to a Uα and the

map t taking it to Uβ ; e takes a point in Uα to the same point in Uα ∩ Uβ ;i takes a point in Uα ∩ Uβ to

the same point in Uβ ∩ Uα. For the multiplication map m, if u is in Uα ∩ Uβ , v is in Uδ ∩ Uγ , requiring

t(u) = s(v) means that β = δ and u = v so we set m(u, v) = u = v in Uα∩γ . As we can see, the function

of this groupoid is to give a gluing atlas for X coming from the data of the open covering Uα.

1.2.2 Categories fibered over Groupoids

Let us fix a base category S. For simplicity purposes, though it will not matter right now, we let S be

the category of schemes.


We first define what it means to have a category over S and what it means to have a morphism

between such objects:

Definition 1.2.6. A category over S is a category X together with a covariant functor p : X → S. Let

X and N be two categories over S; a morphism from X to N is a functor f : X → N which commutes

with the given functors of X ,N to S. Furthermore, a morphism from X to N is an isomorphism if it

induces an equivalence of categories.

We see some examples of categories over S:

Example 1.2.7. Let X be a scheme; we see how it determines a category over S (even though X is an

element of S!). Consider the category X where the objects are pairs (S, f), where S is an object in Sand f : S → X is a morphism of schemes. A morphism between objects (S, f) and (S′, f ′) is given by a

morphism g : S′ → S such that f g = f ′. The covariant functor p : X → S simply takes (S, f) to the

scheme S and the morphisms between (S, f) and (S′, f ′) to a morphism between S′ and S. As we can

see, X is a category over S.

Remark 1.2.8. Note this example stems from the idea of the functor of points for schemes. The basic idea

of the functor of points is that given a schemeX, we can replace it with its functor of points MorSch(S,X);

Yoneda’s lemma is what is used to show that the functor of points is well-defined. Functor of points

are a powerful method to study schemes; many proofs in scheme theory (for example, proving that the

fiber product of two schemes remains a scheme) is simplified using the functor of points approach. For

a more detailed exposition, we recommend the reader to the last chapter of [8].

Example 1.2.9. Consider the category Mg where the objects are Mg are smooth proper morphisms

π : C → S whose geomeric fibers are connected curves of genus g. The morphisms between objects

π : C → S and π′ : C ′ → S′ is a morphism from C → C ′ and S → S′ such that:

C //

π

C ′

π′

S // S′

is a Cartesian square, i.e., C ′ ∼= C ×S S′. To see thatMg is a category over groupoids; just consider the

functorMg to S defined as taking a family C → S to S and a morphism to its constituent map S′ → S.

We look at the example of principal G-bundles (or also more common referred in algebraic geometry

as G-torsors):

Example 1.2.10. Fix G an algebraic group and consider the category BG where objects of BG are

principal G-bundles. A morphism from a G-torsor E′ → S′ to a G-torsor E → S is given by a morphism

S′ → S and a G-equivariant morphism E′ → E such that the diagram:

E′ //

E

S′ // S

is Cartesian. The morphism from BG to S is exactly the same as the previous example of Mg.


The next example is a slight generalization of the previous example:

Example 1.2.11. Let G be an algebraic group acting on a scheme X. There is a category [X/G]

whose objects are G-torsors E → S, together with an equivariant map E to X. A morphism from

E′ → S′, E′ → X ′ to E → S,E → X is given by a map of torsors as before and in addition, the

composite E′ → E → X is required to be equal to the given map from E′ → X.

Now, if we take the functor p : [X/G]→ S where it takes (E → S,E → X) to S. This makes [X/G]

into a category over S.

The last example is the case of vector bundles:

Example 1.2.12. Let Vn be the category of vector bundles of rank n. Precisely, the objects of Vn are

vector bundles E → S and the morphism between two objects is given by a Cartesian diagram as in the

previous two examples. The functor Vn to S is just taking the object E → S to S.

Before defining categories over groupoids, we give a very simple example of morphisms between

categories over S:

Example 1.2.13. We saw in a previous example that given schemes X,Y ; they determine categories

over S. Let us call these X and Y respectively. Then, a morphism of schemes X → Y gives rise to a

morphism between X and Y. It simply takes an object in X : a scheme S equipped with a morphism of

schemes S → X to the composite S → X → Y . We can do this procedure backwards: given a morphism

φ between categories over S, X and Y. Apply φ to the identity map X → X gives us a map of schemes

X → Y .

We now define the notion of a category fibered in groupoids. As we will see in the next section, this

will bring us very close to the definition of stacks. In fact, a stack will be nothing more than a category

fibered in groupoids satisfying a few more axioms.

Definition 1.2.14. Let p : X → S be a category over S. X will be called a category fibered in groupoids

(or abbreviated CFG) if the functor p satisfies the following two axioms:

• For every morphism f : T → S in S, and object s in X with p(s) = S, there is an object t in X ,

with p(t) = T , and a morphism ϕ : t→ s in X such that p(ϕ) = f

• Given the following commutative diagram in S:

U

h

g

S

T

f

??

with ϕ : t→ s in X mapping to f : T → S, and η : u→ s in X mapping to h : U → S, there is a


unique morphism γ : u→ t in X mapping to g : U → T such that η = ϕ γ:

uη

!γ

s

t

ϕ

??

Let us see what these technical axioms are telling us. Axiom (1) is saying basically that pullbacks

of objects exist and axiom (2) asserts the pullbacks obtained are unique up to canonical isomorphism.

Also, if we set X (S) to be the subcategory consisting of all objects s such that p(s) = S and morphisms

f such that p(f) = idS , the axioms give us that X (S) is a category where all morphisms are essentially

isomorphisms, i.e., a groupoid. This justifies the terminology given for these objects. All of the examples

of categories over S that were given previously (Mg, Vn, etc.) are CFGs over S .

We now present a theorem which might be thought of as a “functor of points” for category over

groupoids:

Theorem 1.2.15. Let X be a category fibered in groupoids over S and let X be an object of S. Let X

be the category fibered in groupoids determined by X as usual, then the functor HOM(X,X ) to X (X)

induces an equivalence of categories.

Remark 1.2.16. There is a weaker notion of a fibered category in which the second axiom of the definition

for CFGs is replaced with a weaker condition (the precise definition can be found in [32]). We will not

delve into any theory about fibered categories but note that a reason why they are important is that

descent theory (which will come into play when we define stacks) can be studied in fibered categories.

For explicit details, we refer to [32].

1.2.3 Properties of CFGs

We would like to understand the “category” of CFGs over S in this subsection. It turns out that the

“category” of CFGs over S is a 2-category; which possesses a richer structure than a category. Let us

define what 2-categories in general are following the definition found in [9]:

Definition 1.2.17. A 2-category C consists of the following data:

• A collection Ob C, called the objects of C

• For every pair (X,Y ) of objects of C, a category HomC(X,Y ), whose objects are 1-morphisms of

C. Arrows (morphisms) in HomC(X,Y ) are called 2-morphisms of C.

• For every object X of C, a distinguished object idX ∈ Ob Hom(X,X)

• For every triple (X,Y, Z) of objects of C, a functor:

µX,Y,Z : Hom(X,Y )×Hom(Y,Z)→ Hom(X,Z)

which satisfies the following conditions:


1. For every triple (X,Y, Z) of objects of C, we have:

µX,X,Y (idX ,−) = µX,Y,Y (−, idY ) = idHom(X,Y )

2. For every quadruple (X,Y, Z,W ) of objects of C, we have:

µX,Z,W (µX,Y,Z × idHom(Z,W )) = µX,Y,W (idHom(X,Y ) × µY,Z,W )

Despite the very technical definition of 2-categories, there is a very simple example that comes to

mind immediately. Consider the collection of all categories; this forms a 2-category where the objects

are categories, Hom(X,Y ) for X,Y categories (objects in 2-category) itself is a category whose objects

are functors from X to Y and morphisms are natural transformations of these functors.

Indeed, the collection of CFGs forms a 2-category as well, we will refer to Chapter 2 of [2] for explicit

details. The key reason why we have to introduce 2-categories when studying CFGs is that many

intuitive categorical notions are different when we work with a 2-category. For example, the notion of

fiber products in a CFG differs from the standard notion of a fiber product in a category. We would like

to have a well-defined notion of fiber products in CFGs. This is very important as similar to the case

of ordinary schemes, we would like to have a well-defined idea of what the fiber product of stacks would

be.

Definition 1.2.18. Given X ,Y, and Z CFGs over S, and morphisms f : X → Z and g : Y → Z,

the fiber product X ×Z Y is the category (x, y, α), where x is an object in X , y is an object in Y (over

the same scheme S ∈ S) and α is an isomorphism from f(x) to g(y) in Z (over the identity on S). A

morphism from (x′, y′, α′) to (x, y, α) is given by morphisms x′ → x in X and y′ → y in Y (over the

same scheme morphism S′ → S in S) such that the diagram:

f(x′) //

α′

f(x)

α

g(y′) // g(y)

commutes.

Note that the fiber product X ×Z Y is a CFG over S itself. Also, there are canonical projections

p : X ×Z Y → X and q : X ×Z Y → Y such that the following diagram:

X ×Z Yq

##

p

X

f##

Y

g

Z

is not commutative but rather 2-commutative. This description of the fiber product brings us closer to

our usual intuition of fiber products. We now give a very easy example of a fiber product of CFGs:

Example 1.2.19. Suppose X,Y, Z are objects in S (i.e., schemes) and X → Z, Y → Z are morphisms


in S (i.e., morphisms of schemes). Let X ,Y,Z be CFGs determined by X,Y, Z. Then, we have that the

fiber product of the CFGs:

X ×Z Y

is the isomorphic to the CFG of the usual fiber product X×ZY (which certainly exists in the category

of schemes S).

1.3 Stacks

We begin by insisting that the base category S is now equipped with a Grothendieck topology (see the

Appendix), we call this a site S. This is the common terminology for a category to be endowed with a

Grothendieck topology. Once again, we suppose that S is just the category of schemes. Let X → S be a

CFG over S, T a scheme (object in S) and a map f : T → S, if x, y ∈ X , the definition of a CFG over Smeans that pullbacks f∗(x) and f∗(y) exists up to canonical isomorphism. Define a functor IsomX (x, y)

as follows:

IsomX (x, y) : S → (Sets)

(f : T → S) 7→ isomorphisms from f∗(x)to f∗(y)in X (T )

We now state the definition of a prestack:

Definition 1.3.1. Let X be a category fibered in groupoids over a site S. Then X is a prestack if the

functor IsomX (x, y) is a sheaf on the site S. Commonly, we will be working with the etale topology, so

in other words, we can rephrase to say that IsomX (x, y) is a sheaf in the etale topology.

For a prestack to be a stack, we would like to say that there is a certain “gluing” condition that

happens in X , i.e., objects in X can be obtained via local gluing.

Definition 1.3.2. Let X be a CFG over the site S. A descent datum for X over S ∈ S (i.e., a scheme) is

the following: an open covering Si → S such that for every i, a lifting Xi of Si to X with isomorphisms:

φij : Xi |Sij→ Xj |Sij

for every i, j. Furthermore, the φij ’s satisfy the cocycle condition: φik = φjk φij over Sijk.

We call a descent datum effective if there exists a lifting X of S to X with isomorphism φi : X |Si

∼=→Xi inducing the isomorphisms φij above.

Now, we give the definition of a stack:

Definition 1.3.3. A prestack X is a stack if the descent datum is effective.

We will note that all of the examples of CFGs given before are stacks. However, we will not delve

into technical details on checking the stack axioms as it requires far too much descent theory. In the next

section, we will take a look at some of these examples and show briefly that they are Deligne-Mumford

stacks.

Before concluding this section, we like to make a few comments. The first comment is to revisit the

“naive” definition of the moduli stack of vector bundles given at the end of Section 1.1. We mentioned


that the moduli stack of vector bundles should naively be a category whose objects are families of bundles

E → S, E′ → S′ and morphism defined between the objects as:

E′ ∼= f∗E //

E

S′

f // S

This can be re-interpreted as the CFG Vn as the definition of CFGs means that pullbacks exist and

are unique up to canonical isomorphism. The essential point of the stack axioms is that that we can

“glue” together vector bundles. Let us look at Axiom (1) (the prestack axiom), what this is basically

telling us is that we can “glue” together isomorphisms between vector bundles, i.e., once we define the

isomorphisms on the local covering that agree on overlaps, it is possible to “glue” them together in an

unique way. Formally, suppose Si → S is an etale cover of the scheme S, and we have vector bundles

E,E′ over S with isomorphisms:

E |Si∼=E′ |Si

such that for all i, j: ϕi |Sij= ϕj |Sij , the fact that Isom is a sheaf means that by the sheaf axioms, there

exists a unique ϕ : E → E′ such that ϕ |Si= ϕi.

Axiom (2) tells us that not only can we “glue” together isomorphisms between vector bundles, we

can also “glue” together the vector bundles themselves. This is very useful as given an open covering

Si → S, Ei is the pullback of E to Si, we would like to know if we can reconstruct E from the Ei’s

only. In general, this is of course not possible as there are different vector bundles that can be trivialize

on the same open cover. However, the descent datum condition tells us that we can do so. Formally,

suppose once again that Si → S is an etale cover of the scheme S and

Ei |Sij∼=Ej |Sij

such that for all i, j, k, we have the cocycle condition ϕjk ϕij = ϕik over Sijk. Then, the fact that the

descent datum is effective tells us that there is a vector bundle E over S where ϕi : E |Si→ Ei inducing

the isomorphisms ϕij .

The last comment we wish to make before concluding this section is to relate stacks to moduli

problems. Given a stack X → S, we can think of this as a moduli problem. Formally, what we would

like to do is see if we can find a scheme X, such that X ∼= X (where X is the stack defined by the scheme

X). However, as before, this is of course not possible if the objects in X has nontrivial automorphisms.

Similar to the classical case, there is a notion of coarse moduli spaces for stacks:

Definition 1.3.4. Let X be a stack. A scheme X is a coarse moduli space for X if given π : X → X such

that for all X → Y there exists a unique map φ : X → Y such that the following diagram commutes:

X π //

X

φ

Y

and also for every algebraically closed field k, there is a bijection between the set of isomorphism classes

of objects in the groupoid X (k) and X(k).


Coarse moduli spaces are often constructed using G.I.T. techniques. However, in Section 1.5, when

we discuss the theorem of Keel-Mori, we will see the existence of a coarse moduli space (without using

G.I.T.) for a stack which satisfies certain properties.

1.4 Deligne-Mumford Stacks and Algebraic Stacks

In this section, we will turn our attention to two classes of stacks that are most studied in algebraic

geometry; the Deligne-Mumford and algebraic (or sometimes referred to as Artin) stacks. We will focus

primarily on the former and only discuss briefly about the latter. Many of the moduli problems that

have been introduced previously have their realizations as Deligne-Mumford or algebraic stacks. We will

show that the moduli stack of genus g curves are Deligne-Mumford stacks. Before introducing any formal

definition of Deligne-Mumford or algebraic stacks, we explain the idea of representable morphisms:

Definition 1.4.1. A morphism X → Y of stacks is said to be representable if for any scheme T and

morphism T → Y, the fiber product X ×Y T is isomorphic to a stack associated to a scheme.

Given a morphism of schemes X → Y , the corresponding morphism of their associated stacks X → Y

is clearly a representable morphism. In the theory of schemes, we have that many properties of scheme

morphisms (such as proper, flat, finite type, separated, etc.) are both local properties and are preserved

under base change. However, by “local” here, we will mean that it is etale locally instead of Zariski

locally. We have a similiar analogue for stacks:

Definition 1.4.2. A representable morphism of stacks X → Y has property P if for any scheme T and

any morphism T → Y, the corresponding morphism:

X ×Y T → T

has property P. Of course, X → Y is representable, the fiber product X ×Y T is isomorphic to a scheme

so the morphism X ×Y T → T can be identified with a morphism of schemes.

1.4.1 Deligne-Mumford Stacks

We come now to the definition of Deligne-Mumford stacks which was first introduced in [5]. These are

referred to as “algebraic stacks” in [5] but the term “algebraic stack” is now reserved for a more general

object as we will see later.

Definition 1.4.3. Let X be a stack. X is called a Deligne-Mumford stack if it satisfies the following

two properties:

1. The diagonal 4X : X → X × X is representable, quasi-compact, and separated. As usual if the

diagonal 4X is proper, then we say that the stack is separated.

2. There exists a scheme U and a morphism U → X which is etale and surjective.

The next theorem will tell us the automorphisms of a Deligne-Mumford stack:

Theorem 1.4.4. If X is a Deligne-Mumford stack, B quasi-compact, and X ∈ X (B), then X has only

finitely many automorphisms.


Proof. We follow [7]. By definition, there is a scheme U with the map U → X corresponding to X.

Compose this map with the diagonal to get the map: U → X ×S X where S is some fixed base scheme,

then the pullback U ×X×SX X can be identified with the scheme IsomX (X,X). X is a Deligne-Mumford

stack means that the map IsomX (X,X) → X is unramified over X (Prop 7.15 of [33]). Also, X quasi-

compact means that IsomX (X,X) has only finitely many sections so we get that X has only finitely

many automorphisms.

Before proceeding to some examples of Deligne-Mumford stacks, we would like to state a criterion

that is commonly used to check whether a given stack X is actually Deligne-Mumford:

Fact 1.4.5. Let X be a stack over over a Noetherian scheme S. Assume that the diagonal is repre-

sentable, quasi-compact, separated and unramified. Also, assume that there exists a scheme U of finite

type over S and a smooth surjective S-morphism U →X . Then, X is a Deligne-Mumford stack.

A proof of this can be found in [5]. To illustrate how this fact can be important, consider X/S a

Noetherian scheme of finite type and G/S a smooth affine group scheme of finite type over S acting on X

where the stabilizers of geometric points are finite and reduced, we get that [X/G] is a Deligne-Mumford

stack. To see this, the fact that G acts on X in this way means that IsomX (E,E) is unramified over E

for any map U → [X/G] corresponding to the principal G-bundle E → U . So, the diagonal is unramified

and hence the first condition of the above fact is satisfied. Now, the projection morphism X → [X/G]

is a smooth morphism and also representable (p. 12 of [7]) so it the second condition is satisfied as well.

Hence, we get that [X/G] is a Deligne-Mumford stack.

We will now use this to show that Mg and Mg are of Deligne-Mumford stacks.

We start by defining what a stable curve following [5]:

Definition 1.4.6. Let S be a scheme. A stable curve of genus g over a scheme S is a proper flat

morphism C → S whose geometric fibers are reduced, connected, 1-dimensional schemes Cs such that:

1. Cs has only ordinary double points as singularities

2. If E is a non-singular rational component of Cs, then E meets the other components of Cs in more

than 2 points

3. dimH1 (OCs) = g i.e., Cs has arithmetic genus g

We define the functor FMgto be the groupoid over SpecZ whose sections over a scheme B are families

of stable curves X → B. Similar to the groupoid of genus g curves, a morphism from X ′ → B′ to X → B

will be a fiber product diagram:

X ′ //

X

B′ // B

The method to prove that Mg and Mg are Deligne-Mumford stacks is to show that the respective

groupoids FMgand FMg

are isomorphic to some quotient stack and hence we can use the previous

result. In fact, many of the ideas that will be involved in carrying out this method is similar to what is

done in Chapter 5 of [28] showing the construction of the coarse moduli space of genus g curves Mg.


First, let π : C → S be a stable curve, there is a canonical invertible dualizing sheaf ωC/S on C.

Moreover, if C/S is smooth, this is the relative cotangent bundle on C. By a theorem in [5], we have

that ω⊗nC/S is relatively very ample for n ≥ 3 and π∗(ω⊗nC/S) is a locally free sheaf of rank (2n− 1)(g− 1).

Using these results, we can embed our stable curve C → S into the projective space. The canonical

embedding will be C → PN where PN ∼= P(H0(C;ω⊗nC/S)) and N = (2n − 1)(g − 1) − 1. However, the

isomorphism PN ∼= P(H0(C;ω⊗nC/S)) is not a canonical one, it depends upon a choice of basis for the

space H0(C;ω⊗nC/S). Hence, what we need to parametrize is pairs which consist of both C and the data

of the embedding C → PN . Define a functor Hg which sends a scheme S to a family of stable curves

π : C → S of genus g and an isomorphism P(π∗(ω⊗nC/S)) ∼= PN × S. The theory of Hilbert schemes gives

us that there is some closed subscheme Hg of the Hilbert scheme of PN that represents the functor Hg.Also, there is a subscheme Hg ⊂ Hg which corresponds to the canonically embedded smooth genus g

curves.

In Chapter 5 of [28], a coarse moduli space of genus g curves Mg is constructed as a G.I.T. quotient

Hg/PGL(N + 1). We will do the same here to prove that Mg and Mg are Deligne-Mumford stacks;

that is, show that they are isomorphic to the quotient stacks [Hg/PGL(N + 1)] and[Hg/PGL(N + 1)

]respectively.

Construct a functor from the groupoid of stable curves FMgto the quotient stack

[Hg/PGL(N + 1)

]as follows: suppose we have a family of stable curves π : C → S, consider the principal PGL(N + 1)-

bundle E → S associated to the bundle P(π∗(ω⊗nC/S)). Let π′ : C ×S E → E be the pullback family;

pulling back the bundle P(π∗(ω⊗nC/S)) to E is trivial and isomorphic to P(π

′

∗(ωC×SE/E)) so there is a

map from E → Hg which is PGL(N + 1)-invariant. Thus, at the level of objects of the two categories,

we have defined how to send objects of FMgto objects of

[Hg/PGL(N + 1)

].

At the level of morphisms, take a morphism in FMg:

C ′ //

π′

C

π

S′

φ // S

We have that π′

∗(ωC′/S′) ∼= φ∗(π∗(ωC/S)) so there is a morphism of the PGL(N + 1)-bundles:

E′ //

E

S′ // S

Thus, we have defined a functor from FMgto[Hg/PGL(N + 1)

]. We will leave the details on how

this functor is faithful and full, as well as the existence of a functor from[Hg/PGL(N + 1)

]to FMg

(such that it induces an equivalence of categories with the functor that we have already defined) to pages

21-22 in [7].

Remark 1.4.7. In [5], what is also proven is that the coarse moduli space Mg is irreducible over any

algebraically closed field k. This is deduced from analyzing properties of the moduli stack Mg; the

important property being that Mg has irreducible geometric fibers over SpecZ, a full account of this is

in Chapter 5 of [5]. There is another proof of the irreducibility of Mg using the stable reduction theorem

given in [5] but the proof using stacks is much more powerful.


1.4.2 Some remarks on Deligne-Mumford Stacks

We will briefly discuss some properties of Deligne-Mumford stacks. Previously, we have seen what it

means for a morphism of Deligne-Mumford stacks to have a certain property P (i.e., finite type, proper,

flat, etc.). Many theorems concerning the morphisms of schemes such as Chow’s Lemma, Valuative

criterion for separation, Valuative criterion for properness (see 2.4 of [12]) can be recasted in the world

of Deligne-Mumford stacks. A full account of these statements (without proof however) can be found in

Chapter 4 of [5].

A final remark on Deligne-Mumford stacks we would like to make is that Deligne-Mumford stacks

are essentially equivalent to a groupoid scheme subjected to “stackification”. We will not stress any

technicalities here; the idea is that given a groupoid scheme R ⇒ U with some additional structure,

we can “stackify” and get that the resulting stack [R⇒ U ] is a Deligne-Mumford stack. Conversely,

given a Deligne-Mumford stack X , we know that there is a scheme U such that the morphism U → Xis etale and surjective. We can construct a groupoid R ⇒ U such that X ∼= [R⇒ U ]. Hence, it makes

sense to call the scheme U as the etale atlas of X . In this way, we see that Deligne-Mumford stacks is

the true algebro-geometric counterpart to orbifolds in differential geometry as orbifolds similarily have

a presentation by smooth etale groupoids (see [13]).

1.4.3 Algebraic Stacks

There is a more general class of stacks that are also widely used in algebraic geometry called algebraic

stacks (or Artin stacks):

Definition 1.4.8. Let X be a stack (but here we will need to insist that it is not over an etale site but

rather a fppf site). We call X an algebraic stack if:

1. The diagonal 4X : X → X × X is quasi-compact, representable, and separated (but we do not

insist it is unramified)

2. There exists a scheme U and a morphism U → X which is surjective and smooth (we do not insist

it is etale)

Similar to the etale topology, the fppf topology is a Grothendieck topology where the covering

Ui → U are a collection of flat maps locally of finite presentation (refer to [32]).

There are some fundamental differences between algebraic stacks and Deligne-Mumford stacks. The

first key difference is that objects in X can have infinite automorphism groups whereas in Deligne-

Mumford stacks these must be finite. Similar to Deligne-Mumford stacks, algebraic stacks have a pre-

sentation via groupoids; but it is not a groupoid scheme rather a groupoid in the category of algebraic

spaces.

We will conclude by listing some examples of stacks that are algebraic but not Deligne-Mumford:

• The moduli stack Vn of vector bundles of rank n

• The classifying stack BG where G is an algebraic group

• The moduli stack of conics; it can be shown in Chapter 1 of [2] that the stack has an interpretation

as[P5/PGL(3)

]where we know PGL(3) is an algebraic group


1.5 Theorem of Keel-Mori

1.5.1 Preliminaries

We work with a groupoid (R,X) in the category of algebraic spaces. Following Definition 1.8 in [20], we

introduce what different notions of quotients should be in the language of groupoids:

• (G): For any geometric point ξ, the natural map:

X(ξ)/R(ξ)→ Y (ξ)

is a bijection.

• (C): ϕ is universal among maps from X/R to algebraic spaces.

• (UC): For every flat morphism Y ′ → Y , the diagram:

R×Y Y ′ ⇒ X ×Y Y ′ → Y ′

satisfies (C).

• (US): ϕ is a universal submersion (i.e., U ⊂ Y is open if and only if ϕ−1(U) ⊂ X is open and this

remains true after any base change on Y ).

• (F): The sequence of sheaves in the etale topology:

OY → ϕ∗OX ⇒ ϕ∗OR

is exact. So, the sheaf of functions on Y consists exactly of the R-invariant functions on X.

Definition 1.5.1. A map ϕ satisfying (C) is a categorical quotient, and if it satisfies (UC) is a uniform

categorical quotient. If ϕ satisfies (G) and (C), it is called a coarse moduli space. If ϕ satisfies (G), (US),

and (F), it is called a geometric quotient. If ϕ satisfies (G), (UC), and (US), then it is called a GC

quotient.

We should note that these definitions were most likely inspired by the conditions required in the defini-

tions of categorical and geometric quotients of group actions schemes given in Chapter 0 of [28]. Now

condition (UC) implies (F), so the strongest version of a quotient given here will be the GC quotient.

We should also warn that since we are working in the category of algebraic spaces, having a geometric

quotient does not imply the existence of a categorical quotient as in the case of schemes (Proposition

0.1 of [28]).1

As we want to relate groupoids to group actions, it is natural to understand what a stabilizer in the

groupoid scenario will be. The formal definition is:

1A counterexample can be found in Example 2.18 in [23]. In fact, their example is exactly the same as Example 0.4 in[28]. The idea is that the proof in [28] does not work in the category of algebraic spaces.


Definition 1.5.2. The stabilizer of the groupoid (R,X) is the fiber product:

I //

X

4X

R // X ×X

If we impose that I → X be a finite morphism, then (R,X) is defined to have finite stabilizer.

Following the notation in [20], we set j = (t, s) and we will denote the stabilizer as j−1(4X)→ X.

There is also an abstract definition of orbits:

Definition 1.5.3. Given a geometric point x → X, the orbit of x, denoted o(x), is the set of points

t(s−1(x)) ⊂ X.

1.5.2 Proof of the Keel-Mori Theorem

We start off by formally state the Keel-Mori theorem; Theorem 1.1 in [20]:

Theorem 1.5.4. Let (R,X) be a flat groupoid such that its stabilizer j−1(4X) → X is finite. Then

there is an algebraic space which is a GC quotient.

Note that of course when we say (R,X) is a flat groupoid, we mean that the structure maps s, t are

flat morphisms.

We are now ready to give an step-by-step outline of the proof of the Keel-Mori theorem. The bulk

of the proof is very technical and we will focus only on the main ideas of the proof rather than the

technicalities.

Localization of Quotients

First, we introduce what it means to slice a groupoid:

Definition 1.5.5. Given a map ϕ : W → X, define R |W→ W ×W by R |W= R ×X×X W ×W . The

groupoid (R |W ,W ) will be called the slice along ϕ (or slice along W when there is no ambiguity for ϕ)

of the groupoid (R,X)

The ability to slice a groupoid is one of the main reasons why groupoids are our objects of considera-

tion instead of just group actions. Lemma 3.2 from [20] tells us that to construct a GC quotient X → Y ,

it suffices to do so on an etale cover of X. The idea is that we can construct the quotient after slicing.

Formally:

Lemma 1.5.6. Assume that t, s : R → X are universally open. Suppose Ui is a finite etale cover of

X. Suppose GC quotients φi : Ui/(R |Ui)→ Yi exist for all i. Then, a GC quotient φ : X → Y exists.

So, we know that it basically suffices to construct the GC quotient at an etale neighbourhood. Fur-

thermore, if we insist that j = (t, s) is quasi-finite, we will be allowed to make the following assumptions

courtesy of Lemma 3.3 of [20]:

• R,X are separated schemes

• s, t : R→ X are quasi-finite


Splitting the groupoid

Let us define what it means to split a groupoid first due to Definition 4.1 in [20]:

Definition 1.5.7. A flat groupoid (R,X) is split over a point x ∈ X if R is a disjoint union of open and

closed subschemes R = P qR2 with (P,X) a finite flat subgroupoid and j−1(x, x) ⊂ P (i.e., P contains

the stabilizer)

Then, we have the following key result from Proposition 4.2 of [20]:

Lemma 1.5.8. Let s, t : R⇒ X be a quasi-finite flat groupoid of separated schemes. Then, every point

x ∈ X has an affine etale neighbourhood (W,w) such that the slice R |W is split over w.

In the previous part, we have seen that it is permissible to assume that we are working in the

hypothesis of the above theorem. The key now will be to analyze the finite flat part P that results from

the splitting.

GC Quotients for Finite Flat Groupoids with Affine Base

We now work with a very specific scenario: (R,X) is a finite flat groupoid with X,R affine, as well as

X of finite type. If we write X = Spec(A), R = Spec(B), and let AB be the subring of A equalizing the

two maps A→ B. Then, Proposition 5.1 of [20] gives us that:

Lemma 1.5.9. The ring AB is of finite type and the map:

Spec(A)→ Spec(AB)

is a GC quotient of the groupoid (R,X).

The proof of this uses extensive commutative algebra techniques which we will refer to Chapter 5 of

[20]. What is important here is that when we have reduced to the level of finite flat groupoids with affine

base, we have essentially solved our problem, i.e., determined a GC quotient for the groupoid (R,X).

Boot-Strap

Recall earlier what it means to split a flat groupoid (R,X); we have that (R,X) splits if we can write

R = P qR2 with (P,X) being a finite flat subgroupoid and j−1(x, x) ⊂ P . We also saw that there exists

a GC quotient for the finite flat part (P,X), what we would like to understand more is the other piece

of the splitting R2. It turns out that from 7.2, 7.3 of [20] that there exists an induced action of P on R2

by composition on both sides yielding a finite flat groupoid:

P ′ := P ×(s,t) R2 ×(s,t) P → R2 ×R2

Now, we continue to work with X affine and R2 → X quasi-finite, then Zariski’s Main Theorem gives

us that R2 is quasi-affine. Hence, if we work with the groupoid (P ′, R2) the result of the previous case

gives us that there exists a GC quotient R′′ for (P ′, R2). Of course, we know that a GC quotient for

(P,X) certainly exists, let us denote this by X ′. Furthermore, Part 1 of Theorem 7.8 in [20] gives us

that the map R′′ → X ′ ×X ′ is an etale equivalence relation, which means that X ′/R′′ is an algebraic

space by definition and in fact, it is the GC quotient for X/R . Moreover, Part 2 of Theorem 7.8 in [20]


gives us that X/R and X ′/R′′ are isomorphic, they define the same GC quotient problem in an etale

neighbourhood of x ∈ X.

Putting it all together

Let us start off by working with (R,X) a flat quasi-finite groupoid with finite stabilizer. By the localiza-

tion of quotients, we can reduce the problem to constructing a GC quotient on a etale neighbourhood of

x ∈ X. But we can split the groupoid (R,X) and assume that we are working with X,R affine schemes,

then we know that GC quotients exist for finite flat groupoids over affine base. Using the boot-strap

theorem, we get the desired result.

For the general case of (R,X) being a flat groupoid with finite stabilizer j−1(4X) → X, we have

that j is quasi-finite. So, we simply run the argument as before and the fact that above that (R,X) a

flat quasi-finite groupoid with finite stabilizer admits GC quotients thus we get the result in the general

case.

1.5.3 Consequences of the Keel-Mori Theorem

One immediate corollary is an answer to the classical question of the existence of quotients of a group

scheme G acting on a scheme X:

Corollary 1.5.10. Let G be a linear algebraic group acting properly on a finite type scheme X with

finite stabilizers, then a GC quotient for X/G exists in the category of algebraic spaces.

As discussed previously, the GC quotient is the “finest” type of quotient that we can possibly have.

It illustrates that the for quotient problems involving the category of schemes, the category of algebraic

spaces is the ideal setting.

Another important corollary is an answer to the question of existence of coarse moduli spaces for

stacks. In general, to prove that the existence of a coarse moduli space for a moduli stack, we would

need to use G.I.T. to explicitly construct the coarse moduli space. The theorem of Keel-Mori gives us a

method to prove existence without constructing the coarse moduli space explicitly.

To see this, let X be a Deligne-Mumford stack and we insist that X is smooth and of finite type over

a base field k. There is a notion of stabilizer of X , called the inertia stack I(X ) which is defined as the

fiber product diagram:

I(X ) //

X

4X

X

4X

// X × X

We will say that X has finite stabilizer if the projection I(X )→ X is a finite morphism. Note that

this is quite similar to the case of stabilizer defined previously for groupoids but we should note that

the fiber product diagram for Deligne-Mumford stacks are 2-commutative diagrams since stacks form a

2-category rather than a usual category. If we insist that X is a separated Deligne-Mumford stack, then

certainly we get that I(X )→ X is a finite morphism. The corollary of the Keel-Mori theorem is:

Corollary 1.5.11. A separated Deligne-Mumford stack of finite type has a coarse moduli space.

Prior to the Keel-Mori theorem, the above result had a “folk-lore” status in algebraic geometry. It

appears in [9] without proof but was probably first known by Deligne and Artin.


As an example, one can get the existence of Mg and Mg,n, the coarse moduli space parametrizing

stable curves of genus g and stable curves of genus g with n marked points respectively. By the Keel-Mori

theorem, one only needs to determine the existence of a flat groupoid with finite stabilizer parametrizing

such curves (i.e., a flat groupoid whose associated stack is the moduli stack Mg and Mg,n). However,

finding such groupoids is not an easy task, we shall refer to [29] for details of such a construction.

There is, however, a relationship between the GC quotients constructed from the Keel-Mori theorem

and G.I.T. quotients. We know that a GC quotient exists for a flat group action G on properly on a

finite type scheme X with finite stabilizer. From Theorem 9.1 of [20], we get that:

Theorem 1.5.12. Let G be a reductive group scheme acting on a finite type scheme X and we linearize

this action on an invertible sheaf L. Denote Xs to be the stable points for the linearized action. Then,

the G.I.T. quotient Xs//G is isomorphic to the GC quotient Xs/G.

Xs//G is not the conventional notation for the G.I.T. quotient of the stable locus but we do so to

avoid confusion with the GC quotient.

Remark 1.5.13. We conclude with several remarks. A proof of the Keel-Mori theorem (in an even more

general situation) using the language of stacks rather than groupoids can be found in [4]. There is also a

weaker version of the first corollary stated above due to [23]; what Kollar proved in [23] is that given an

affine reductive algebraic group G acting properly on an algebraic space X, a geometric quotient X/G

exists in the category of algebraic spaces. The techniques in [23] do not involve any groupoid methods.

Another remark we would like to make is that if we drop the assumption that the Deligne-Mumford

stack X is separated, a coarse moduli space might not necessarily exist; for details, we refer to Example

6.14 in [31]. The final comment that we would like to make here is that while there is a correspondence

between the GC quotient and the G.I.T. quotient of the properly stable locus, such a correspondence

is unknown if we replace with the semistable locus. Hence, the G.I.T. program for groupoids remains

incomplete.

Chapter 2

Symplectic Geometry Approach

This part of the thesis is a more detailed examination of the topics found in Chapter 8 of [28]. The

main idea is that there is an important connection between the G.I.T. quotient for a reductive group G

acting on a complex projective variety X and the symplectic quotient µ−1(0)/K where K is the maximal

torus of G, µ is the moment map for the unitary action of G on X. This is precisely the theorem of

Kirwan-Kempf-Ness which states that there exists a homeomorphism between the symplectic quotient

µ−1(0)/K and quotient of the semistable locus X//G. We will examine this in greater detail in Section

2.1.

We will then explore hyperkahler manifolds, an extension of Kahler manifolds. It turns out that

hyperkahler manifolds admit a quotient operation, called a hyperkahler quotient, which is somewhat

similar to the Marsden-Weinstein reduction for symplectic manifolds. This was pioneered in [17] and is

now an indispensable tool in the study of moduli theory in geometry. We will present this idea in Section

2.2 as well as give some examples of hyperkahler quotients, the most important of which is the moduli

space of Higgs bundles which was first introduced in [16]. We will discuss more about this moduli space

in the last section of the thesis.

The next part is devoted to the theory of computing the cohomology of symplectic quotients involving

finite dimensional group actions following Chapters 2-5 of [22]. This approach was originally inspired

by [1] which attempted to apply equivariant Morse theory to the Yang-Mills functional to calculate the

cohomology of the moduli space of stable vector bundles over a compact Riemann surface. The Morse

function that is considered in [22] is the norm-square of the moment map. It turns out that this is not

even a Morse-Bott function in general but rather what Kirwan calls a minimally degenerate function.

The interesting phenomena here is that this weaker notion of minimally degenerate functions is enough

to induce the standard Morse inequalities in equivariant cohomology. By proving that the norm-square

of the moment map is a perfect minimally degenerate function, we can successfully obtain the Betti

numbers of symplectic quotients involving finite dimensional group actions. This will be the primary

focus of Section 2.3. We shall also make a slight excursion at the end of section and look briefly at the

techniques used in [1] to study the cohomology of vector bundles.

The last section will be a more detailed description of the moduli space of Higgs bundles which was

first mentioned in Section 2.2. We shall look at how these are intimately tied to the solutions of the

self-duality equations presented in [16] as well as mention briefly the geometry of the moduli space of

Higgs bundles.

22

Chapter 2. Symplectic Geometry Approach 23

2.1 Connection between Symplectic Quotients and G.I.T. Quo-

tients

2.1.1 The Symplectic Quotient and the G.I.T. Quotient

We first mention the two objects that the Kempf-Ness theorem will associate together: the symplectic

quotient (sometimes called the Marsden-Weinstein reduction) and the G.I.T. quotient from algebraic

geometry. The setup here will be G a compact complex reductive Lie group, X a nonsingular projective

variety and G will act on X via a representation:

ρ : G→ GL(n+ 1)

Conjugating ρ with a suitable element of GL(n+ 1), we may assume that ρ(G) is contained in U(n+ 1).

In the symplectic geometry perspective, there is a moment map µ : X → g∗ and suppose that 0 is a

regular value of µ and that G acts freely on µ−1(0), we have that the quotient:

µ−1(0)/G

is itself a symplectic manifold. On the other hand, from the G.I.T. perspective, considering the semistable

points of X with respect to the G-action, we have the existence of a good quotient (see [28] Chapter 0

Remark 6)1:

Xss//G

which is a projective variety. Furthermore, the set of stable points Xs of X with respect to the G-action

is contained in Xss as an open Zariski subset and the resulting quotient:

Xs/G

is a geometric quotient. The theorem of Kirwan-Kempf-Ness, as we will see below, will relate the

symplectic quotients and G.I.T. quotients together

2.1.2 Theorem of Kirwan-Kempf-Ness

We state the theorem of Kirwan-Kempf-Ness:

Theorem 2.1.1. Any x ∈ X is semistable if and only if the closure of its orbit meets µ−1(0); i.e., if

and only if:

OG(x) ∩ µ−1(0) 6= 0

Furthermore, the inclusion of µ−1(0) into Xss induces a homeomorphism:

µ−1(0)/K → Xss//G

where K is the maximal compact subgroup of G.

1Mumford does not explicitly use the term “good quotient” to describe the conditions listed in Remark 6 of Chapter 0in [28]; the terminology is rather due to Seshadri and is now conventional


We first collect some results from [21] which be vital to proving the above theorem. Let v ∈ Cn+1

and denote Gv as the stabilizer of v in G; consider the function pv on G defined by:

pv(g) = ‖g · v‖2

This function is the main object of study in [21]. The results that we will need to use in the proof of the

Kirwan-Kempf-Ness theorem will be Theorem 0.1 a),b) and Theorem 0.2 in [21] which are respectively

stated as follows:

1. Any critical point of pv is a point where pv attains its minimum value

2. If pv attains a minimum value, it does so on exactly one double coset KgGv ∈ K\G/Gv

3. pv attains a minimum value if and only if the orbit OG(v) is closed in Cn+1

Let us prove the first part of the Kirwan-Kempf-Ness theorem. Suppose x ∈ Xss and x ∈ Cn+1\0lies over x. The definition of semistability in the affine scenario (Proposition 2.2 in [28]) means that

0 /∈ OG(x), this implies that there exists a nonzero closed orbit whose image in X is contained in OG(x).

We note that for a vector v ∈ Cn+1\0 that lies over v ∈ X, dpv(g) = 0 if and only if µ(gv) = 0. Now,

we can prove one implication of the first part of the Kirwan-Kempf-Ness theorem. (3) above states that

p evaluated on this nonzero closed orbit must attain a minimum value but by what we have just noted,

this means that µ evaluated on this orbit is 0. Hence, we get:

OG(x) ∩ µ−1(0) 6= ∅

Conversely, suppose that OG(x) ∩ µ−1(0) 6= ∅. The fact that dpv(g) = 0 if and only if µ(gv) = 0

for any v lying over v implies that µ−1(0) must be a critical value for the function pv. But by (1), we

get that these must be where pv attains its minimum value. By (3), we get that this means the orbit is

closed in Cn+1. Appealing once again to the definition of semistability in the affine case, we have that:

µ−1(0) ⊆ Xss

Thus, x ∈ Xss whenever OG(x) ∩ µ−1(0) 6= ∅.To prove the second part of the Kirwan-Kempf-Ness theorem, we define the natural quotient map:

φ : Xss → Xss//G

φ is continuous and G-invariant; we have an inclusion µ−1(0) into Xss as shown above, so this induces

a continuous map:

ψ : µ−1(0)/K → Xss//G

ψ is a bijection (see 8.1 of [28]) which by standard point set topology results, will yield a homeomorphism

between µ−1(0)/K and Xss//G.

Remark 2.1.2. In [21], fact (3) stated above (Theorem 0.2 in [21]) is stated as saying that v is a stable

point. This has exactly the same meaning but we should note that this definition of stability implied in

[21] is not the common definition of stability used. There is usually an extra condition that the stabilizer

of v is finite in the usual definition of stability. Also, [21] works with a slightly more general setup than


what we have done; instead of Cn+1 they work with V a finite dimensional complex representation of

G. We do not need this generality here.

The last part of this section is to see a version of the Kirwan-Kempf-Ness theorem regarding the

stable points of X, Xs. We use a simple symplectic geometry fact first: dµ(x) at x is surjective if and

only if the stabilizer Kx in K is finite. Furthermore, this means that the stabilizer of x in G is finite

(Theorem 7.2 in [22]),

Lemma 2.1.3. If x ∈ µ−1(0), then x ∈ Xs if and only if dµ(x) is surjective.

Proof. x ∈ µ−1(0) means as before that the orbit OG(x) is closed in Cn+1 where x is the element of

Cn+1\0 lying over x. For x ∈ Xs, we just need that dimOG(x) = dimG, i.e., the stabilizer is finite.

But this is exactly equivalent to dµ being surjective as noted above.

What this lemma gives us is that the homeomorphism:

ψ : µ−1(0)/K → Xss//G

obtained previously restricts to give the homeomorphism:

µ−1reg(0)/K → Xs/G

where µ−1reg (0) is the set of points in µ−1(0) where dµ is surjective.

2.2 Hyperkahler Manifolds and Hyperkahler Quotients

2.2.1 Hyperkahler Manifolds

Definition 2.2.1. A hyperkahler manifold is a smooth manifold X equipped with a Riemannian metric

g and three complex structures Ji, i = 1, 2, 3 where the Ji’s satisfy the quaternion relations and such

that if we define ωi(·, ·) = g(Ji·, ·), (g, Ji, ωi) is a Kahler structure on X.

Before proceeding with examples of hyperkahler manifolds, we should understand why studying

Kahler manifolds and hyperkahler manifolds are very different. In Kahler geometry, there is the well-

known ∂∂-lemma which asserts that on a complex manifold, by adding ∂∂f where f is a C∞ function, we

can get from one Kahler metric to another. What this means is that on a complex manifold, the number

of Kahler metrics is infinite. However, this is not the case with hyperkahler metrics; on a compact

complex manifold, there are only finitely many hyperkahler metrics up to isometry ([15]). This means

that examples of hyperkahler manifolds are much harder to find than Kahler manifolds in general.

Example 2.2.2. The basic example of a hyperkahler manifold is Hn where H is the space of quaternions

with the standard flat metric.

Before introducing other examples, we would like to understand the holonomy group of a hyperkahler

manifold better. Since the three complex structures Ji, i = 1, 2, 3 are covariantly constant (thus perserved

by parallel translation) and the fact that parallel translation commutes with quaternion multiplication,

the holonomy group of a hyperkahler manifold is contained in the intersection O(4n) ∩GL(n,H). This

intersection is isomorphic to Sp(n). Immediately, we get a non-trivial example:


Example 2.2.3. Every Calabi-Yau surface is hyperkahler. Indeed, since Sp(1) ∼= SU(2) and that SU(n)-

holonomy implies the existence of a Ricci-flat Kahler metric, i.e., a Calabi-Yau manifold. Conversely,

since Sp(n) ⊂ SU(2n), we get that every hyperkahler manifold is Calabi-Yau.

It turns out that hyperkahler manifolds are closely related to holomorphic symplectic manifolds;

complex manifolds equipped with a holomorphic symplectic form (i.e. a non-degenerate holomorphic

2-form):

Theorem 2.2.4. Every hyperkahler manifold is a holomorphic symplectic manifold. Conversely, every

compact Kahler manifold with a holomorphic symplectic form is a hyperkahler manifold.

Proof. From the definition of hyperkahler manifolds, we have three Kahler forms ω1, ω2, ω3 where each

of which are defined as ωi(·, ·) = g(Ji·, ·). Define ω = ω2 + iω3, ω is certainly closed, non-degenerate and

covariantly constant. It can also be check that it is holomorphic with respect to J1. This gives that any

hyperkahler manifold is holomorphic symplectic.

Conversely, suppose that M is compact Kahler with a holomorphic symplectic form ω. Then, ωn is

nowhere vanishing and by definition, a nowhere vanishing section of the canonical bundle of M . This

means that the canonical bundle of M is holomorphically trivial. Appealing to the Calabi-Yau theorem

([18] Proposition 4.B.21), there exists an unique Kahler metric with vanishing Ricci tensor. A result of

Bochner states that a holomorphic form on a compact Kahler manifold with vanishing Ricci tensor is

covariantly constant. This yields that the holonomy group of M is contained in Sp(2n)∩U(2n) ∼= Sp(n),

i.e., M is hyperkahler.

There are two main approaches to constructing hyperkahler metrics: the first is using the twistor

space approach and the second is via the hyperkahler quotient approach. We shall not delve into the

first but refer to [15] and discuss more about the second approach in the next sub-section. The study of

hyperkahler manifolds is vast and we have only touch on a very small portion of the theory here; more

details as well as more complicated examples of hyperkahler manifolds can be found in [15].

2.2.2 Hyperkahler Quotients

Similar to the Marsden-Weinstein reduction for symplectic manifolds, there is a method for taking the

quotient of a hyperkahler manifold such that the resulting quotient remains as a hyperkahler manifold.

This was first proved in [17] which arose out of questions in supersymmetry. The setup is as follows:

let G be a compact Lie group acting on a hyperkahler manifold X which preserves the three Kahler

structures (gi, Ji, ωi) for i = 1, 2, 3. There exists moment maps: µi : X → g∗ for i = 1, 2, 3. We can

combine all three of these together as:

µ : X → g∗ ⊗ R3

defined by µ = (µ1, µ2, µ3). Let λi ∈ g∗ i = 1, 2, 3 be regular values and consider µ−1(λ) where

λ = (λ1, λ2, λ3). The G-action on µ−1(λ) is free and discontinuous as well as µ−1(λ) is G-invariant. The

central result of [17] is that:

µ−1(λ)/G

is a hyperkahler manifold.

Our example of a hyperkahler quotient will be the moduli space of Higgs bundles; a concept which

will be discussed in more detail in Section 2.4:


Example 2.2.5. LetX be a compact Riemann surface. A Higgs bundle overX is a pair (E, φ) where E is

a holomorphic vector bundle over X together with the Higgs field : a sheaf homomorphism φ : E → E⊗Kwhere K is the canonical line bundle of X. Similar to the case of vector bundles over a Riemann surface,

there is an analogous notion of stability for Higgs bundles over a Riemann surface: we call (E, φ) a stable

Higgs bundle if µ(F ) < µ(E) for every proper subbundle F of E such that φ(F ) ⊂ F ⊗K.

It turns out that the existence of stable Higgs bundles is related to the existence of the solutions of

a certain equation: the Hitchin equation. These can be expressed as follows: let h be a hermitian metric

on a C∞ vector bundle E, A be the set of all connections on E that are compatible with h, then the

solutions of the Hitchin equations are the pairs (A, φ) ∈ A×Ω1,0(X,End E) that satisfies the following

equation (the Hitchin equation):

FA + [φ, φ∗] = 0

∂Eφ = 0

where FA is the curvature of the connection A and ∂E = A0,1. The second condition above means that

φ is holomorphic with respect to the holomorphic structure defined by A. This tells us that the pair

(E, φ) is a Higgs bundle.

In [16], it is proven that the moduli space of irreducible solutions to the Hitchin equations: X ∗0 /Gwhere X ∗0 consists of the irreducible pairs that satisfy the Hitchin equations and G is the gauge group

of the hermitian vector bundle E is homeomorphic to the moduli space of stable Higgs bundles.

We now see how to use this to construct a hyperkahler quotient version of the moduli space of stable

Higgs bundles. From above, we see that it is intimately tied to the solutions of the Hitchin’s equations.

We first define a hyperkahler structure on the space X = A×Ω1,0(X,End E); define the following three

complex structures on X :

J1(α,ψ) = (iα, iψ)

J2(α,ψ) = (iψ∗,−iα∗)

J3(α,ψ) = (−ψ∗, α∗)

where α∗ and ψ∗ is defined using the hermitian metric h on E. The three complex structures defined on

X satisfy the quarternion relations which gives X a hyperkahler manifold structure. Consider the gauge

group G acting on X with moment maps:

µ1(A, φ) = FA + [φ, φ∗]

µ2(A, φ) = Re(∂Eφ)

µ3(A, φ) = Im(∂Eφ)

Then, if we take λ = (0, 0, 0), we get that the hyperkahler quotient µ−1(0)/G (where µ = (µ1, µ2, µ3))

is the moduli space of solutions to the Hitchin’s equations. If we restrict to the irreducible solutions, as

mentioned before, the hyperkahler quotient is the moduli space of stable Higgs bundles.

There are a vast number of spaces which have interpretations as hyperkahler quotients; some of these

are ALE spaces, hyperpolygon spaces, quiver varieties, moduli space of instantons etc. These are all

beyond the scope of this thesis and we shall not pursue in these directions further.


2.3 Cohomology of Symplectic Quotients

2.3.1 Quick Overview of Morse Theory

We will be using techniques from equivariant Morse theory to compute the cohomology of symplectic

quotients. The purpose of this subsection is to simply review very briefly the basic fundamental facts on

Morse theory as well as any Morse theory machinery needed later in this section. We will merely state

most of the relevant theorems and refer to [26], [22] for explicit details of their proofs.

Let M be a smooth manifold and f : M → R be a smooth real-valued function. We call a point

p ∈M to be a critical point if df(p) = 0. Consider the Hessian of f , Hpf , a well defined quadratic form

on TpM . If we use local coordinates xi around p, the matrix of Hpf with respect to the basis ∂∂xi at p

is given by:

Hpf =

∥∥∥∥ ∂2f

∂xi∂xj

∥∥∥∥We call p ∈ M a non-degenerate critical point of f if df(p) = 0 and detHpf 6= 0. A smooth real

valued function f : M → R is called Morse if all the critical points of f are non-degnerate.

The index of a critical point p of f , λp(f), is defined to be the number of negative eigenvalues in a

diagonalization of Hpf and we can define the Morse polynomial of a Morse function f as:

Mt(f) =∑p

tλp(f)

where p is such that df(p) = 0. We also have the Morse inequalities that relate the Morse polynomial of

f and the Poincare series for M . The Poincare series for M with coefficient field K is defined as:

Pt(M ;K) =∑

ti dimHi(M ;K)

We often just write Pt(M) if the field K is understood. The Morse inequalities states that for a

Morse function f on M , there exists a polynomial R(t) with nonnegative coefficients such that:

Mt(f)− Pt(M ;K) = (1 + t)R(t)

We call a function f a perfect Morse function if equality holds, that is:

Mt(f) = Pt(M ;K)

This will eventually be the method of attack when we compute the cohomology of symplectic quo-

tients. We will find a Morse function on the symplectic quotient that is perfect which will give us

complete information about the Betti numbers of the symplectic quotient as seen above.

Returning to the basics, the “Morse lemma” of Morse theory basically states that a Morse function

f has a very specific local expression near a non-degenerate critical point p of f . We give the precise

statement:

Proposition 2.3.1. If p is a non-degenerate critical point of f of index k, then there is a coordinate

system x1, . . . , xn near p such that:

f = f(p)− x21 − . . .− x2k + x2k+1 + . . .+ x2n


Proof. See [26] page 6

One of the key uses of the Morse lemma is to prove the famous Theorem B of Morse theory, which

along with Theorem A, is the two central theorems of Morse theory.

Theorem 2.3.2. Let f : M → R be a smooth real valued function. Suppose a < b, f−1 ([a, b]) is compact

and contains no critical points of f . Define Ma = f−1 ((−∞, a]) and Mb similarily define. Then, Ma

has the same homotopy type as Mb.

This is the Theorem A of Morse theory. Theorem B is stated as follows:

Theorem 2.3.3. Let f be as in the above theorem and suppose that f−1 ([a, b]) contains precisely one

critical point of f in its interior, which is non-degenerate of index k, then Mb has the same homotopy

type as Ma ∪ ek.

We will refer to Chapter 17 of [3] for a nice exposition of the proofs of the above two theorems.

We now discuss about Morse stratification on manifolds. A stratification of a smooth manifold M is

a finite collection Sβ : β ∈ B of subsets whose disjoint union is M and there is a strict partial order >

on the indexing set B such that:

Sβ ⊆ ∪γ≥βSγ

for every β ∈ B. The stratification Sβ : β ∈ B is called smooth if every Sβ is a locally closed subman-

ifold of M .

Consider a smooth real valued function f : M → R, we suppose that the function f is not necessarily

Morse but rather Morse-Bott. Morse-Bott functions are a weaker notion of Morse functions; we define a

function f to be Morse-Bott if its set of critical points is a finite disjoint union of connected submanifolds

C ∈ C of M and the kernel of the Hessian at a critical point p is TpS.

Now, let ω(x) be defined as the set of limit points of the path of steepest descent of f starting from

x ∈ M . ω(x) is connected ([22] p. 14-15) so there exists a unique C such that ω(x) is contained in C.

We define the Morse stratum SC corresponding to any C ∈ C to consist of x ∈ M with ω(x) contained

in C. The stratum SC retracts to the corresponding submanifolds C and form a smooth stratification

of M . Note that the partial order here is given by C > C ′ if f(C) > f(C ′). Hence, we have seen how a

Morse-Bott function determines a Morse stratification SC of M .

We see why stratifications are useful when studying cohomology; it turns out that the cohomology of

M can be build up from the cohomology of the constituent pieces in the strata. The method to do this is

to use the Thom-Gysin sequences; in fact, the Thom-Gysin sequences will recover the Morse inequalities

as we shall see.

Let β ∈ B, assume that each component of the stratum Sβ has the same codimension d(β) in M . We

use the Thom-Gysin sequences to relate the cohomology groups of Sβ and the two open subsets of M :

∪γ<βSγ ,∪γ≤βSβ

What the Thom-Gysin sequences gives us is the following Morse inequalities for a stratification:∑β

td(β)Pt(Sβ)− Pt(M) = (1 + t)R(t)


where R(t) has nonnegative coefficients and d(β) is the codimension of Sβ in M . As before, we call a

smooth stratification perfect if the Morse inequalities are equalities, i.e.:

∑β

td(β)Pt(Sβ) = Pt(M)

We would like to mention here that we do not necessarily need to have a Morse function to induce

Morse inequalities. As we have seen above, a Morse-Bott function gives rise to Morse inequalities.

What we will do subsequently in this section is actually find a function on the symplectic quotient

which has weaker conditions that even a Morse-Bott function but yet the Morse inequalities still hold.

Furthermore, it will be shown that such a function is perfect. This will allow us to understand the

cohomology of the symplectic quotient completely. However, there is one slight caveat, we will have to

work in equivariant cohomology instead of ordinary cohomology. We discuss a little bit of this before

concluding this subsection.

Suppose G is a compact Lie group acting on the smooth manifold M (in fact, we can use weaker

notions involving topological groups and topological spaces), the equivariant cohomology H∗G(M ;Q) is

defined by:

H∗G(M ;Q) = H∗(EG×GM ;Q)

where EG→ BG is the universal classifying bundle for G and EG×GM is the quotient of EG×M by

the diagonal action of G acting on EG on the right and on M on the left.

Similar to the previous scenario, we have equivariant Morse inequalities. Suppose that we have a

smooth stratification Sβ of M whose strata are all invariant under an action of the group K on M ,

we have the equivariant Morse inequalities:∑β

td(β)PKt (Sβ)− PKt (M) = (1 + t)R(t)

where R(t) has nonnegative coefficients and PKt is the equivariant Poincare series. We call the stratifi-

cation equivariantly perfect if these are equalities.

2.3.2 Minimally Degenerate Morse Functions

The goal of this subsection is to introduce Kirwan’s notion of minimally degenerate Morse functions.

These are functions that are weaker than the notion of Morse-Bott functions yet the Morse inequalities

still hold. Let us define this notion precisely:

Definition 2.3.4. Let M be a compact Riemannian manifold. We call a smooth function f : M → Rto be a minimally degenerate Morse function if the set of critical points of f is a finite disjoint union of

closed subsets Cβ : β ∈ B of M , along each of which there exists a minimizing submanifold for f in

the following sense:

A locally closed submanifold Σβ of X with a orientable normal bundle, which contains Cβ and is

closed in a neighbourhood of Cβ , is a minimising submanifold for f along Cβ if the following holds:

1. The restriction of f to Σβ achieves its minimum value exactly on Cβ

2. The tangent space to Σβ at any x ∈ Cβ is maximal among the subspaces of TxX on which the

Hessian Hx(f) of f at x is nonnegative semi-definite.


A Morse-Bott function is a minimally degenerate Morse function. Indeed, the critical set of a Morse-Bott

function is by definition a disjoint union of connected submanifolds. We will take this as the critical

subsets Cβ . The minimising submanifolds for a Morse-Bott function is locally given by the image of N+C

under the exponential map. Note that N+C is where f is positive definite on NC and NC ∼= N+

C ⊕ N−C

which is induced by the Hessian of f once we choose a metric on M .

The motivation behind introducing minimally degenerate Morse functions is to give a Morse theoretic

framework for smooth real-valued functions that fail to be even Morse-Bott; for example, if the critical

set contains singularities. We introduce now the main theorem of this subsection which is Theorem 10.2

from [22]:

Theorem 2.3.5. Let f : M → R be a minimally degenerate Morse function with critical subsets Cβ :

β ∈ B. Then, we have the Morse inequalities:∑β∈B

tλ(β)Pt(Cβ)− Pt(M) = (1 + t)R(t)

where R(t) has nonnegative coefficients and λ(β) is the index of f along Cβ defined as codim∑β .

The technical details of the proof of this theorem can be found in Chapter 10 of [22]. We will also

note that the Poincare series defined in the above theorem uses Cech cohomology.

Our eventual objective is to use Morse theoretic methods to understand the Betti numbers of sym-

plectic quotients. In the next subsection, we will be working with a compact symplectic manifold M

acted on by a compact Lie group K. We will study the function f = ‖µ‖2 where µ is the moment map

for the K action on M . It turns out that f is not a Morse or even a Morse-Bott function, but rather a

minimally degenerate Morse function so by the results mentioned in this subsection, we have that the

existence of the Morse inequalities for f . Furthermore, if we use equivariant Morse theory, we will also

show in the next subsection that f is a equivariantly perfect Morse function, which will give us complete

knowledge of Betti numbers of M .

2.3.3 Inductive Formulas

Throughout this subsection, we will let K be a compact Lie group acting on a compact symplectic

manfiold M . Denote the moment map of this action by:

µ : M → k∗

We consider the function f : M → R defined by f = ‖µ‖2 where ‖·‖ is a norm associated to the

inner product of k∗ which is invariant under the adjoint action of K. Our first goal is to prove that f is

a minimally degenerate Morse function as defined in the previous subsection. To do this, we begin by

understanding better what the critical sets of f are explicitly.

Let T be the maximal torus of K, then via reduction by stages, we have a moment map for T -action

on M :

µT : M → k∗ → t∗

We can think of µT as an orthogonal projection of µ onto t (using inner products) so for µ(x) ∈ t

the function f is critical at x if and only if it is the function fT = ‖µT ‖2 is critical at x.


However, for torus actions on a symplectic manifold, we have the following result of Atiyah-Guillemin-

Sternberg which states that the image of the fixed set of points in X by T under the moment map µT

is the convex hull of a finite set of points in t∗ (these points will be called the weights of the T -action

on X).

For β ∈ t∗, we denote Tβ = expRβ so Tβ is a subtorus of T . We also denote µβ : X → R by

µβ(x) = µ(x).β and the critical points of µβ is precisely the fixed point set of the subtorus Tβ .

Now µβ itself can be thought of as a Morse function (in fact, it is Morse-Bott), set Zβ be the union

of those connected components of the critical set of µβ on which µβ takes the value ‖β‖2. Then, Lemma

3.12 in [22] gives that x ∈ M is critical for fT = ‖µT ‖2 if and only if x ∈ Zβ . However, there is

a geometric interpretation of this which is that β is the closest point to the origin of the convex hull

of weights of the T -action. β having this property will be referred to as the minimal combination of

weights.

Definition 2.3.6. Let t+ be a fixed positive Weyl chamber in t and B the set of all minimal combinations

of weights which lie in t+. For β ∈ B, define Cβ as:

Cβ = K(Zβ ∩ µ−1(β))

Now, Lemma 3.15 in [22] gives us that the critical set of f = ‖µ‖2 is the disjoint union of the closed

subsets Cβ : β ∈ B defined above. This gives us a complete understanding of what the critical sets of

f = ‖µ‖2 is.

To prove that f is indeed a minimally degenerate Morse function, we need to determine minimizing

submanifolds Σβ for f along the critical sets Cβ . Recall that we had the Morse-Bott functions µβ , so we

have a Morse stratum Yβ associated to Zβ consisting of all points of X where −gradµβ has limit points

in Zβ . Let us consider KYβ , by Corollary 4.11, Lemma 4.12, Lemma 4.13 in [22], it is proven that if we

take the intersection Σβ of KYβ with a sufficiently small neighbourhood of Cβ , we get that Σβ is the

minimizing submanifold for f along Cβ . Therefore, f = ‖µ‖2 is a minimally degenerate Morse functions

and by previous discussions, we have the existence of Morse and equivariant Morse inequalities for f .

Summarizing, we have the following theorem (Theorem 4.16 of [22]):

Theorem 2.3.7. Let M be a compact symplectic manifold acted on by a compact Lie group K and

suppose µ : M → k∗ a moment map for this action. Fix an invariant inner product on k and consider

f = ‖µ‖2. Then, the set of critical points for f is a disjoint union of closed subsets Cβ : β ∈ B on each

of which f takes a constant value. There is a smooth stratification Sβ : β ∈ B of X such that a point

of x ∈ X lies in Sβ if and only if the limit set of the path of steepest descent for f from x is contained

in Cβ. For each β, the inclusion Cβ ⊂ Sβ induces isomorphisms of Cech cohomology and K-equivariant

cohomology.

The smooth stratification Sβ is of course the Morse stratification of the minimally degenerate

Morse function f . Before proceeding further to prove that the equivariant Morse inequalities for f are

equivariantly perfect, we need to describe the critical set a little differently. The reason is that the index

of the Hessian of f at points of Cβ may not necessarily be constant. To remedy this, we make the

following definition:

Definition 2.3.8. For any integer m ≥ 0, let Zβ,m be the union of those connected components of Zβ


along which the index of µβ is m. Let:

Cβ,m = K(Zβ,m ∩ µ−1(β))

so the set of critical points for f is the disjoint union of the closed subsets Cβ,m : 0 ≤ m ≤ dimX.

Furthermore, by Lemma 4.20 of [22], we have that the index of the Hessian Hx(f) of f at any point

x ∈ Cβ,m is:

d(β,m) = m− dimK + dim Stabβ

where Stabβ is is the stabilizer of β under the adjoint K-action.

Also, we have that Cβ,m is homeomorphic to:

K ×Stabβ (Zβ,m ∩ µ−1(β))

and by [1], we have the following isomorphism in cohomology:

H∗K(Cβ,m;Q) ∼= H∗Stabβ(Zβ,m ∩ µ−1(β);Q)

With the aid of these results, let us return to the original problem. We have that f is a minimally

degenerate Morse function so we have the existence of the equivariant Morse inequalities:∑β∈B

td(β,m)PKt (Cβ,m)− PKt (M) = (1 + t)QK(t)

where QK(t) has nonnegative coefficients.

But the fact that H∗K(Cβ,m;Q) ∼= H∗Stabβ(Zβ,m ∩ µ−1(β);Q) means that we can change the above

statement to: ∑β,m

td(β,m)P Stabβt (Zβ,m ∩ µ−1(β))− PKt (M) = (1 + t)QK(t)

What we would like to do is prove that the equivariant Morse inequalities are in fact equalities.

The technique to do this to use the Morse stratification Sβ : β ∈ B corresponding to Cβ : β ∈ B and

as noted in the first subsection, the Thom-Gysin sequences allows us to recover the Morse inequalities

defined with respect to Sβ ’s.

To show that the stratification is equivariantly perfect over Q, there is a very useful criterion that

can be used which is due to [1]:

Proposition 2.3.9. Suppose Sβ : β ∈ B is a smooth K-invariant stratification of M such that for

each β the equivariant Euler class of the normal bundle to Sβ in M is not a zero divisor in H∗K(Sβ ;Q).

Then, the stratification is equivariantly perfect over Q.

Now, we have all the tools necessary to prove that the function f is equivariantly perfect. We first

state the theorem (Theorem 5.4 in [22]) formally and then prove it:

Theorem 2.3.10. Let M be a symplectic manifold acted on by a compact Lie group K with moment

map µ : M → k∗ and give k a fixed invariant inner product. Then, the function f = ‖µ‖2 on M is

equivariantly perfect over Q. Moreover, the equivariant Poincare series of M is given by:


PKt (M) =∑β,m

td(β,m)PKt (Cβ,m) =∑β,m

td(β,m)PStabβt (Zβ,m ∩ µ−1(β))

Proof. Let Sβ,m be the components of Sβ which have codimension d(β,m). Now,

Sβ,m ⊆ Sβ,m ∪⋃γ>β

Sγ

Sβ =⋃m≥0

Sβ,m

so Sβ,m : β ∈ B, 0 ≤ m ≤ dimX itself is a smooth stratification such that:

H∗K(Sβ,m,Q) ∼= H∗K(Cβ,m,Q)

as the inclusion Cβ,m into Sβ,m induces isomorphisms of K-equivariant cohomology.

The only thing that remains to be shown is that the equivariant Euler class of the normal bundle to

each Sβ,m is not a zero divisor in H∗K(Sβ,m). We have:

H∗K(Cβ,m,Q) ∼= HStabβ(Zβ,m ∩ µ−1(β))

so the equivariant Euler class of the normal bundle to Sβ,m can be identified with Stabβ-equivariant

Euler class of its restriction N to Zβ,m∩µ−1(β). From (4.17) of[22], N is a quotient of the restriction to

Zβ,m∩µ−1(β) of the normal bundle to Zβ,m. Zβ,m by definition is just the union of components of fixed

point set of Tβ of Stabβ, by (3.8) of [22], the action of Tβ on the normal bundle to Zβ,m has no nonzero

fixed vectors and this is also true for the action of Tβ on N . (5.3) of[22] gives us that the equivariant

Euler class of N is not a zero divisor in HStabβ(Zβ,m ∩ µ−1(β)) which is precisely what we want.

Thus, we get that Sβ,m is an equivariantly perfect stratification over Q; since Sβ,m is the Morse

stratum associated to the critical set Cβ,m, we conclude that f = ‖µ‖2 is equivariantly perfect over

Q.

Before concluding this subsection, we would like to find a similar inductive formula for computing

Betti numbers of the case of the symplectic quotient µ−1(0)/K.

Consider µ−1(0) and suppose it is nonempty. Then, µ−1(0) is a submanifold of X where the function

f = ‖µ‖2 attains its minimum so the above theorem gives a formula for PKt (µ−1(0)) in terms of PKt (M)

and P StabβK (Zβ,m ∩ µ−1(β)).

Similarily, if we look at each Zβ,m, it is a compact symplectic submanifold on which Stabβ acts on.

The moment map for this action is:

x 7→ µ(x)− β

Now, the inverse image of this moment map is Zβ,m ∩ µ−1(β) so appealing to the above theorem

again, the conclusion is that we have the formula:

PKt (µ−1(0)) = PKt (M)−∑β∈B

td(β,m)P Stabβt (Zβ,m ∩ µ−1(β))


However, we can make reductions to this formula: we can replace PKt (X) by Pt(X)Pt(BK); in

other words, the rational equivariant cohomology of X is the tensor product of the ordinary rational

cohomology of X and the ordinary rational cohomology of the classifying space BK. The technical

details of this fact can be found in Proposition 5.8 of [22] but the rough idea is that if we consider the

fibration X×KEK → BK with X as the fibre, one can show that the Serre spectral sequence associated

to this fibration degenerates to give the desired result.

So, our formula now reads:

PKt (µ−1(0)) = Pt(M)Pt(BK)−∑β∈B

td(β,m)P Stabβt (Zβ,m ∩ µ−1(β))

which gives an inductive formula for the equivariant Betti numbers of µ−1(0).

For the symplectic quotient µ−1(0)/K, by definition, K will act freely (to ensure a manifold structure

for the symplectic quotient) and that the stabilizers of every point x ∈ X is finite. Then, in this case,

the natural map:

µ−1(0)×K EK → µ−1(0)/K

induces an isomorphism:

H∗(µ−1(0)/K;Q)→ H∗K(µ−1(0);Q)

Thus, the final conclusion is that all of the equivariant Betti numbers of µ−1(0) obtained from the

inductive formula above are precisely the Betti numbers of the symplectic quotient µ−1(0)/K. This

gives us complete understanding of the rational cohomology of the symplectic quotient.

2.3.4 Atiyah-Bott: Cohomology of Moduli Space of Vector Bundles

In this subsection, we will briefly explore some of the arguments used in [1] to compute the moduli

space of stable holomorphic vector bundles over a Riemann surface. We setup the notation: let M

be a compact Riemann surface and consider the moduli space of stable holomorphic vector bundles on

M which we will denote by Ms(n, d). Now, the theorem of Narasimhan-Seshadri states that a stable

holomorphic bundle is equivalent to an irreducible unitary connection with constant central curvature

so we get that:

Ms(n, d) = A∗0/G

where G is the gauge group of automorphisms. The description above gives us a gauge theoretic inter-

pretation of the moduli space of stable holomorphic vector bundles.

What we would like to do is to use the results of the previous subsection to compute the cohomology

of Ms(n, d). Since we have this gauge theoretic description of Ms(n, d), we would like to apply the

techniques of the previous section; that is, prove that the Yang-Mills functional is a perfect Morse

function on the space of connections A which allows us to deduce the Betti numbers of Ms(n, d). This

was the original goal of [1]. Unfortunately, analytical issues arise when using this method due to the

fact that we are dealing with an infinite dimensional problem (A is infinite dimensional) rather than a

finite dimensional problem as before.

Instead of using Morse theory directly, in Chapter 7 of [1], Atiyah and Bott introduces the Shatz

stratification on the space C of holomorphic structures on a fixed smooth vector bundle E of rank n,

degree d over M . We shall present the Shatz stratification here. Take E a holomorphic vector bundle


over M : there exists a Harder-Narasimhan filtration on E :

0 = E0 ⊂ E1 ⊂ . . . ⊂ Er = E

with each quotient Di = Ei/Ei−1 semistable and

µ(D1) > µ(D2) > . . . > µ(Dr)

for 1 < i ≤ r.Following [1], if Di has rank ni and Chern class ki so that n =

∑ni and k =

∑ki, we call the

sequence of pairs (ni, ki) i = 1, . . . r the type of E. We can re-write this as a single n-vector µ whose

components are the ratios ki/ni each represented ni times and arranged in decreasing order, i.e.,

µ = (µ1, . . . , µn)

If we define Cµ the subspace of C consisting of all holomorphic bundles of type µ, [1] tells us that

Cµ forms a stratification of C. Futhermore, it turns out that the stratification Cµ is sufficiently well-

behaved, meaning that we can use it as if we had a Morse stratification coming from a Morse function

f .

We now present the two main theorems in Section 7 of (Theorem 7.12 and 7.14 in [1] respectively):

Theorem 2.3.11. The equivariant cohomology of the stratum Cµ(E) is isomorphic to the tensor product

of the equivariant cohomology of the semistable strata for the quotients Di.

The next theorem says that this Morse stratification created out of the Shatz stratification is equiv-

ariantly perfect, i.e., we can understood the cohomology of C via Cµ :

Theorem 2.3.12. The stratification of C by Cµ is equivariantly perfect so that for the equivariant

Poincare series, we have:

Pt(C) =∑µ

t2d(µ)Pt(Cµ)

where d(µ) is the codimension of Cµ computed explicitly as:

d(µ) =∑µi>µj

(µi − µj + (g − 1))

If we combine the results of the two theorems together, we obtain that:

Pt(C) =∑µ

t2d(µ)∏

1≤i≤r

Pt(C(ni, di)ss)

which gives us an inductive formula.

Remark 2.3.13. We note that the problem of determining Betti numbers for the moduli space of vector

bundles was first studied by Harder-Narasimhan. The proof of Harder-Narasimhan is number-theoretic

and based on the Weil conjectures whose proof was completed by Deligne one year earlier. We will refer to

Section 11 in [1] for a comparison between the methods used in [1] and the Harder-Narasimhan approach.


2.4 Moduli Space of Higgs Bundles

2.4.1 Motivation

We follow Chapter 1 of [16] where Higgs bundles were first introduced for the purpose of studying self-

duality equations over a compact Riemann surface. Let G be a compact connected Lie group and P be

a principal G-bundle over R4. Consider the vector bundle:

ad(P ) = P ×G g

This bundle may be thought of as the associated bundle of P via the adjoint action of G on its

tangent space TeG at the identity element e. There is a graded Lie algebra of differential forms on R4

valued in ad(P ):

Ω∗(R4, ad(P ))

which is obtained via extending the space of sections Γad(P ) (which has a natural Lie algebra structure).

Let A be a connection on P and F (A) its curvature; F (A) ∈ Ω2(R4, ad(P ))2. Then, the connection

A satisfies the self-duality equations if: F (A) = ∗F (A) where ∗ is the Hodge star operator:

∗ : Ω2(R4, ad(P ))→ Ω2(R4, ad(P ))

If we trivialize P over R4, we can write the curvature F (A), a Lie algebra valued 2-form, in coordi-

nates:

F (A) =∑i<j

Fijdxi ∧ dxj

and the self-duality equations become the system:

F12 = F34

F13 = F24

F14 = F23

The connection A, a Lie algebra valued 1-form, in coordinates:

A = A1dx1 +A2dx2 +A3dx3 +A4dx4

By introducing a covariant derivative:

Oi =∂

∂xi+Ai

Fij = [Oi,Oj ]

we assume that the Lie algebra valued functions Ai are defined only by (x1, x2) ∈ R2 and independent

of x3, x4. We have an induced connection A on R2 defined by:

A = A1dx1 +A2dx2

2For a more detail description of connections and curvature on a principal bundle P , we refer to [1] p.547-548


and re-label A3, A4 by φ1, φ2. Also, if we write φ = φ1 − iφ2; the self-duality equations above can be

re-written as:

F =1

2i [φ, φ∗]

[O1 + iO2, φ] = 0

From now on, we work with this induced connection on R2 and its curvature F . If we write φ1 =

O3 − L∂/∂x3, φ2 = O4 − L∂/∂x4

(we are considering the invariant solutions to the self-duality equations

over R4 where invariance is respect to the translation action of the additive group R2). Furthermore,

writing z = x1 + ix2, and consider:

Φ =1

2φdz ∈ Ω1,0(R2, ad(P )⊗ C)

Φ∗ =1

2φ∗dz ∈ Ω0,1(R2, ad(P )⊗ C)

The self-duality equations therefore become:

F = − [Φ,Φ∗]

d′′

AΦ = 0

where d′′

A is the (0, 1)-part of dA. These are also referred to as Hitchin’s equations. In fact, instead of R2,

consider a compact Riemann surface M and a principal G-bundle P over M . The pair (A,Φ) consisting

of a connection A on P and Φ ∈ Ω1,0(M ; ad(P ) ⊗ C) (called the Higgs field) satisfies the self-duality

equations if:

F = − [Φ,Φ∗]

d′′

AΦ = 0

Example 2.4.1. Suppose Φ = 0; this implies that the curvature F (A) = 0, i.e., unitary flat connection.

The theorem of Narasimhan and Seshadri states that this is equivalent to stable holomorphic vector

bundles.

2.4.2 Relationship to Higgs Bundles

Recall that we defined what it means to have a Higgs bundle over a compact Riemann surface in Section

2.2; a Higgs bundle is essentially a pair (E, φ) where E is a holomorphic vector bundle over the compact

Riemann surface M and φ ∈ H0(End E⊗K). We would like to know under what circumstances having

a Higgs bundle is equivalent to giving a solution to Hitchin’s self-duality equations. A complete answer,

at least in the case of rank 2 bundles is provided in [16].

We adopt the notations and definitions as in [16]:

Definition 2.4.2. Let V be a rank 2 holomorphic vector bundle over a compact Riemann surface M

and Φ ∈ H0(End E ⊗K) where K is the canonical bundle of M . The pair (V,Φ) is called stable if, for


every Φ-invariant rank 1 subbundle L of V (i.e. Φ(L) ⊂ L⊗K),

degL <1

2deg

(∧2V

)Of course, the above definition is just a special case of the more general definition of a stable Higgs

bundle given in Section 9 previously. In [16], these are called stable pairs rather than the conventional

term Higgs bundle. Note also that we call the pair (V,Φ) semistable if the inequality < is replaced by

≤.

It is possible for stable Higgs bundles to not exist; in the case of P1, there are no stable rank 2 Higgs

bundle on P1 (Remarks 3.2 in [16]). Before proceeding to the relationship between Higgs bundles and

solutions to self-duality equations. We shall give a criterion which states under what circumstance does

a rank 2 vector bundle over a compact Riemann surface occurs as a stable Higgs bundle (Theorem 3.4

in [16]):

Theorem 2.4.3. Let M be a compact Riemann surface of genus g > 1. A rank 2 vector bundle V occurs

in a stable pair (V,Φ) if and only if there is a Zariski-open dense subset U ⊆ H0(M ; End E ⊗K) such

that if Φ ∈ U , then Φ leaves invariant no proper subbundle.

We now state the theorems that relate stable Higgs bundles and solutions to self-duality equations.

This is important as it gives an algebra-geometric interpretation of an essentially gauge-theoretic object.

Nitsure in [30] explores this idea further as he constructs the moduli space of Higgs bundles via a purely

algebra-geometric approach whereas Hitchin’s construction of the moduli space is purely analytical. We

will discuss this briefly in the next subsection.

The first theorem we state is (Theorem 2.1 in [16]):

Theorem 2.4.4. Let (A,Φ) be an irreducible solution to the SO(3) self duality equations on a compact

Riemann surface M and let V be the associated rank 2 vector bundle. If L ⊂ V is a Φ-invariant

subbundle, then:

degL <1

2deg(∧2V )

i.e., the associated rank 2 bundle V with Φ is a stable Higgs bundle.

The converse is given by the following (Theorem 4.3 in [16]):

Theorem 2.4.5. Let A be a SO(3) connection on a principal SO(3)-bundle on a compact Riemann

surface M of genus g > 1, and let Φ ∈ Ω1,0(M ; ad P ⊗ C) satisfy d′′

AΦ = 0. Let V be the associated

rank 2 bundle with holomorphic structure determined by the connection A. If (V,Φ) is a stable pair,

then there exists an automorphism of V of determinant 1, unique modulo SO(3) transformations, which

takes (A,Φ) to a solution of the equation F (A) + [Φ,Φ∗] = 0.

Of course, this means that a rank 2 stable Higgs bundle is essentially equivalent to a pair (A,Φ) satis-

fying the self-duality equations. The proof of the converse is somewhat of the same spirit as Donaldson’s

proof of the Narasimhan-Seshadri theorem [6] as the central tool in both proofs is the Uhlenbeck’s com-

pactness theorem. The full details of the above theorem can be found in Chapter 4 of [16]; a bulk of

which is heavily analytical in flavour as it involves work with infinite dimensional Banach manifolds.

Note also that if we set Φ = 0, we obtain the classical theorem of Narasimhan and Seshadri.


2.4.3 Moduli Space of Higgs Bundles

In this subsection, we will discuss very briefly about the moduli space of Higgs bundles. There are two

approaches: the analytical approach taken in [16] and the algebro-geometric approach taken in [30]. We

first look at Hitchin’s approach.

Using infinite dimensional analytic techniques involving Banach manifolds, Hitchin obtains that the

moduli space of Higgs bundles over a Riemann surface of rank 2 with fixed determinant and odd degree

is a smooth manifold of dimension 12(g − 1) where g is the genus of M (Theorem 5.8 in [16]).

In the previous section, we have saw that this moduli space has a hyperkahler metric, i.e., it carries

the structure of a hyperkahler manifold. So, the moduli space M has the three complex structures

J1, J2, J3 as defined before. It is shown in Section 7 of [16] that we can use Morse theoretic methods

to understand the topology of M. Note that if (A,Φ) is a solution to the self-duality equations, it is

clear that (A, eiθΦ) is a solution as well and this circle action preserves the complex structure J1 on

A×Ω1,0(M ; adP ). Moreover, this circle action descends to the moduli spaceM with the moment map:

µ ((A,Φ)) = −1

2‖Φ‖2L2

where:

‖Φ‖2L2 = 2i

ˆTr(ΦΦ∗)

In Section 7 of [16], there are a number of properties that can be deduced in ‖Φ‖2L2 which gives us

various topological information about M such as the Betti numbers of M.

We now discuss the algebro-geometric method of the construction of the moduli space of Higgs

bundles due to Nitsure. In [30], Nitsure works in a more general setting: instead of just considering

Higgs bundles, Nitsure constructs a moduli space for pairs (E, φ) where E is a vector bundle over a

smooth projective curve X (over algebraically closed field k) together with a morphism φ : E → L⊗ Ewhere L is any fixed line bundle on X rather than imposing it to be ΩX as in the usual case. The notions

of stability and semistability defined for pairs is the same as in the case of Higgs bundles.

The techniques used in [30] is very similar to the G.I.T. techniques used in the construction of the

moduli space of vector bundles over a smooth projective curve in which M will arise as a quotient of a

Grassmannian variety. To conclude, we will simply list some of the results obtained by Nitsure:

• There exists a coarse moduli schemeM(r, d) for S-equivalent classes of semistable pairs (E, φ) on

X where the rank of E is r and the degree of E is d. Recall of course that two semistable bundles

are S-equivalent if they have the same Jordan-Holder filtration.

• In the case of rank 2 and degree d odd, M(2, d) is a non-singular variety.

Chapter 3

Appendix: Grothendieck Topologies

and Algebraic Spaces

3.1 Grothendieck Topologies

Definition 3.1.1. Let C be a category. A Grothendieck topology is a category C admitting all finite

fiber products along with a collection of sets of maps Ui → X called coverings such that:

1. Any isomorphism is a covering

2. Given Y → X and a covering Ui → X, the pullback Ui ×X Y → Y is a covering

3. Given a covering Ui → X and coverings Vij → Ui, the set Vij → X is a covering

There is a basic example that immediately comes to mind: consider C the category of topological

spaces and let U be a topological space, then a covering of U will be just be a collection of open

embeddings Ui → U where the set-theoretic union of their images is exactly U . Another example is if we

consider S to be the category of schemes over some fixed scheme S, and U be an element of this category,

there is a covering Ui → U a collection of open embeddings covering U . By “open embedding”, we

mean a morphism V → U that gives an isomorphism of V with an open subscheme of U , not simply the

embedding of an open subscheme. In principle, the usual idea of an open set is replaced by a covering

in the world of Grothendieck topologies.

As we can see, the Grothendieck topology is the more general object of which the classical topologies

(Euclidean, Zariski, etc.) are simply special cases. In Part I of the thesis, we have worked extensively

with the etale topology, a special case of a Grothendieck topology on S where we take Si → S a

collection of etale morphisms to be a covering when the union of the maps covers S.

We explain briefly about the idea of etale topology and why they are the correct notion to use.

The idea of etale topology, in a naive sense, is to transport the idea of “local isomorphism” in the

differentiable and analytic category to the world of algebraic geometry. In the Zariski topology, there is

no analogue of the usual implicit function theorem. To see this, consider the very simple example of a

projective map:

V (x2 − y) → A1

41

Chapter 3. Appendix: Grothendieck Topologies and Algebraic Spaces 42

(x, y) 7→ y

At the point (1, 1), we have that:∂

∂x(x2 − y) = 2 6= 0

but the projection is not one-to-one in any Zariski open set U of V (x2 − y) since for infinitely many

values y = a, U will contain both points (√a, a) and (−

√a, a). Another example would be if we took

the map:

Ank → Ankx 7→ xn

The n-th square root map is of course not an algebraic map so we have no possibility of any notion of

“local isomorphism”.

To “remedy” this problem in algebraic geometry, let us first introduce the idea of an etale morphism,

which generalizes the notion of a “local diffeomorphism” from differential topology. For intuition pur-

poses, we restrict our attention to smooth varieties over an algebraically closed field k instead of working

with arbitrary schemes. We shall also impose that the characteristic of the field k is 0.

Definition 3.1.2. Let X,Y be smooth varieties over an algebraically closed field k. A map ϕ : X → Y

is etale at x ∈ X if the differential:

dϕ : TxX → Tϕ(x)Y

is an isomorphism. We say that ϕ is etale if it holds for every x ∈ X.

In the differentiable category, of course, the inverse function theorem states that if M,N were smooth

manifolds such that a smooth map ψ : M → N has the property that dψ : TmM → Tf(m)N is an

isomorphism, then ψ is a local diffeomorphism. However, in the realm of algebraic geometry, an etale

map is not necessarily a local isomorphism. For example, the map x 7→ xn is etale but not a local

isomorphism.

However, in the etale topology, the notion of etale morphisms are indeed “local isomorphisms”. For

this and more about etale morphisms/topology, we refer to [25].

3.2 Algebraic Spaces

The last topic we discuss will be algebraic spaces, which we can think of as gluing affine schemes using

the etale topology rather than the Zariski topology in the usual case of schemes. Formally:

Definition 3.2.1. Let (R,X) be a groupoid scheme (or algebraic groupoid) and fix S a base scheme.

We call a relation to mean any morphism j : R → X ×X. An equivalence relation is a relation j such

that the image of j(T ) : R(T )→ X(T )×X(T ) for all schemes T is a monomorphism. A sheaf Q is said

to be an algebraic space over S if Q = U/V for some schemes U, V over S and an equivalence relation

j : V → U × U is a morphism of finite type such that each of the projections pi : V → U is etale.

Another way to think of algebraic spaces is the following equivalent definition which is perhaps more

intuitive:

Chapter 3. Appendix: Grothendieck Topologies and Algebraic Spaces 43

Definition 3.2.2. Let S be a scheme. An algebraic space X over S is a contravariant functor X :

(Sch/S)→ Sets such that:

1. X is a sheaf in the etale topology

2. The diagonal 4 : X → X ×S X is representable

3. There exists a scheme over S, U → S together with an etale and surjective morphism U → X

The details of the proof of the equivalence of the two definitions can be found in [24].

One of the key interests in algebraic spaces is that they are much suited for quotient problems than

the category of schemes. As shown in Section 1.5, the Keel-Mori theorem asserts that given a flat group

scheme G acting properly on X with finite stabilizer, there is a geometric and categorical quotient for

X/G that exists not as a scheme, but rather a algebraic space.

Another motivation for algebraic spaces is that similar to Serre’s GAGA principle which relates

schemes to complex analytic spaces, there is correspondence between algebraic spaces and Moishezon

manifolds, a special class of compact complex manifolds. More discussion of Moishezon manifolds can

be found in Appendix B of [12].

Similar to schemes, there are well-defined notions of quasi-compact, finite type, Noetherian, sepa-

rated, smooth, regular, etc. for algebraic spaces. For more details of this as well as general theory of

algebraic spaces, the comprehensive reference is [25].

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Moduli Space Techniques in Algebraic Geometry … · Abstract Moduli Space Techniques in Algebraic Geometry and Symplectic Geometry Kevin Ka Hang Luk Master of Science Graduate Department

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