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with such ‘‘as bad as possible’’ normal crossing

singularities. Smoothing a local model (in x

= 1)

i = 1

= 0, we can see the tori in f

i = 1

= g:



j¼

; jx

j¼

;

j¼

; x





½14

These are even Lagrangian in the standard symplec-

tic form on the local model, and fiber the smoothing

over the base {(

, 

)}. It turns out that,

metrically, these tori (which vanish into the normal

crossings singularity at the LCLP) actually form a

large part of the smooth Calabi–Yau. This

enlightens the apparent paradox between the SYZ

conjecture and the Batyrev construction, that is, why

a vertex of the original moment polytope (corre-

sponding to the deepest type of singularity

(0, 0, 0, 0) 2 {

i = 1

= 0}) can be replaced by the

dual three-dimensional face in the dual polytope.

This was first suggested by Leung and Vafa.

Gross and Siebert (2003) exploit this to extend SYZ

and Batyrev’s construction to nontoric LCLP Calabi-

Yau manifolds; it is only the local toric nature of the

normal crossing singularities of the LCLP that they

use. It seems possible that their construction will give

the mirrors of all Calabi–Yau manifolds with LCLPs.

Much of mirror symmetry should soon be reduced to

graphs (the discriminant locus of a Lagrangian torus

fibration) in spheres, and further graphs over which D-

branes (such as holomorphic curves) fiber, as in recent

conjectures of Kontsevich and Soibelman and Fukaya

(2005). It may soon be possible to write down a

triangulated category in terms of such data. The full

geometric story (involving Joyce’s description of sLag

fibrations, for instance) is still some way off, however;

we cannot even write down an explicit Ricci-flat

metric on a compact Calabi–Yau.

Monodromy around the LCLP

As well as the SYZ torus fiber [14] we can also see a

Lagrangian 0-section on the quintic and its mirror as a

component of the real locus of [12] for >5.

Remarkably, like the torus [14],thiscyclewasalready

described and used by Candelas et al. (1991),long

before the relevance of torus fibrations was suspected.

Gross and Ruan have been able to describe the

quintic and its mirror (at least topologically or

symplectically) very explicitly as a simple torus

fibration over this S

with a natural integral affine

structure and codimension-two graph discriminant

locus (see, e.g., Gross et al. (2003)).

Under monodromy about  = 1, the 0-section is

moved to another section T(O), and T is given by

translation by T(O) using the group structure on the

fibers. This is the analog of the Dehn twist [10], and

one can choose a basis of H

(

Q) (with first element

the invariant cycle, the T

-fiber, second element

a cycle fibered over a curve in S

, third fibered over

a surface, and last the 0-section itself) such that



11

01

001

0001

½15

Like the Dehn twist [10], it turns out that T



maximally unipotent; that is, we have in n-dimensions,

ðT



 1Þ

nþ1

¼ 0 but ðT



 1Þ

6¼ 0

Again, this is a general feature of LCLPs as formulated

by Morrison (1993) as part of the definition.

This should be compared with the Lefschetz

operator L = [ ! on the cohomology of the mirror,

which also satisfies L

6¼ 0, L

nþ1

= 0 (or, more

relevantly, exp (L), which satisfies (e

 1)

6¼ 0,

 1)

nþ1

= 0). Their similarity was noticed by the

Griffiths school working on VHS in the late 1960s!

Now we know that for Calabi–Yau manifolds at an

LCLP dual to an LKLP along a ray ! = c

(L) on the

mirror, they should be considered mirror operators

(up to some factors of the Todd class of the

underlying Calabi–Yau, to do with the relationship

between the Chern character e

of the line bundle L

(see [11]) and the Riemann–Roch formula).

Both, by linear algebra of the nilpotent operator

N = log T



k = 1



 1)

, induce a natural

filtration W



:0 W

W

= H on the coho-

mology on which they operate (which is H = H

for

N = log T



and H = H

for N = L = [ !):

0  imðN

ÞimðN

n1

Þ\kerðNÞ

 kerðN

n1

ÞþimðNÞkerðN

ÞH

½16

For a discussion of the construction of this mono-

dromy weight filtration, the reader is referred to the

furtherreadingsection.Itplaysakeyroleinstudying

degenerations of varieties and Hodge structures, in this

case as we approach the LCLP. It is a beautiful result of

Gross that this filtration coincides with the Leray

filtration on H

induced by the fibration. That is,

under Poincare´ duality, the weight filtration on cycles

is by the minimal dimension (over all homologous

cycles) of the image in the base over which the cycle is

fibered. So, the first graded piece is spanned by the

invariant cycle, the T

fiber, supported over a point,

and the last by the 0-section; cf. [15]. (Similarly on the

mirror, the filtration for the Lefschetz operator [e

has first piece spanned by the cohomology class of a

Mirror Symmetry: A Geometric Survey 447

point, which is invariant under the monodromy action

L of [11],etc.)

Letting 

be the class of a fiber and 

span

(which is one-dimensional) over the inte-

gers, then T





= 

þ 

. It follows that

q ¼ exp 2i









is invariant under monodromy. This is the higher-

dimensional analog of the coordinate exp (2i)on

the moduli space of elliptic curves, where  is the

period point. It is this coordinate q that is mirror to

the coordinate

line

on the Ka¨ hler moduli space on the mirror quintic,

which allows one to compute the correspondence

between VHS and Gromov–Witten invariants men-

tioned in the introduction.

More generally, following Morrison (1993), one

can make a rigorous definition of an LCLP using

features noted above extended to the case of h

2, 1

> 0

(see, e.g., Cox and Katz (1999). Roughly, the

upshot is that M

(of dimension s = h

2, 1

(

X)) should

be compactified with s divisors (D

)

i = 1

(parametriz-

ing singular varieties) forming a normal crossings

divisor meeting at the LCLP, with monodromies T

about them. There should be a unique (up to

multiples) integral cycle 

(our torus fiber) invariant

under all T

, and cycles (

)

i = 1

such that











is logarithmic at D

; that is 

= (1=(2 i)) log (z

where z

is a local parameter for D

= {z

= 0}.

So, z

= exp (2i

) form local coordinates for

moduli space, mirror to the polydisk coordinates [2]

on K

. The direction of approach to the LKLP in that

section corresponds to the holomorphic curve z

= z

[3] we take through the LCLP (z

= 0 8i), and the

monodromy

varies accordingly, but the

corresponding weight filtration W



remains constant

if k

6¼ 0 8i, by a theorem of Cattani and Kaplan.

Morrison then requires that the (

)

i = 0

should

form an integral basis for W

= W

(with 

a basis

of W

= W

). Finally, part definition and part

conjecture, we should be able to make a choice

such that they satisfy the condition log T

(

) = 



Of course, as has been emphasized, Morrison’s

definition of an LCLP is really where the mathematics

and geometry of mirror symmetry begin, and should

have been the starting point of this article. But that

would have required appreciable knowledge of

abstract VHS that are best understood, in this context,

through the new geometry of Lagrangian torus

fibrations that mirror symmetry has inspired.

See also: AdS/CFT Correspondence; Calibrated

Geometry and Special Lagrangian Submanifolds;

Derived Categories; Fourier–Mukai Transform in String

Theory; Geometric Analysis and General Relativity;

Geometric Flows and the Penrose Inequality; Geometric

Measure Theory; Geometric Phases; Number Theory in

Physics; Riemann Surfaces; Several Complex Variables:

Compact Manifolds; Topological Gravity, Two-

Dimensional; Topological Sigma Models; WDVV

Equations and Frobenius Manifolds.

Further Reading

Candelas P, de la Ossa X, Green P, and Parkes L (1991) A pair of

Calabi–Yau manifolds as an exactly soluble superconformal

theory. Nuclear Physics B 359: 21–74.

Cox D and Katz S (1999) Mirror Symmetry and Algebraic

Geometry. Mathematical Surveys and Monographs, vol. 68.

Providence, RI: American Mathematical Society.

Fukaya K (2005) Multivalued Morse theory, asymptotic analysis

and mirror symmetry. In: Mikhail Lyubich and Leon

Takhtajan (eds.) Proceedings of the conference dedicated to

Dennis Sullivan on his 60th birthday held at Stony Brook

University, Stony Brook, NY, June 14–21, 2001. Proceedings

of Symposia in Pure Mathematics, in Graphs and patterns in

mathematics and theoretical physics, vol. 73, pp. 205–278.

Providence, RI: American Mathematical Society.

Gross M (1998) Special Lagrangian fibrations I: topology. In:

Ueno K, Saito M-H, and Shimizu Y (eds.) Integrable Systems

and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 156–193.

River Edge, NJ: World Scientific Publishing.

Gross M, Huybrechts D, and Joyce D (2003) Universitext. Calabi–

Yau Manifolds and Related Geometries. Berlin: Springer.

Gross M and Siebert B (2003) Mirror symmetry via logarithmic

degeneration data I. Preprint math.AG/0309070.

Hori K, Katz S, Klemm A, Pandharipande R, Thomas R et al.

(2003) Mirror Symmetry. Clay Mathematics Monographs,

vol. 1. Providence, RI: American Mathematical Society;

Cambridge, MA: Clay Mathematics Institute.

Kontsevich M (1995) Homological Algebra of Mirror Symmetry.

International Congress of Mathematicians. Zu¨ rich: Birkha¨user.

Morrison D (1993) Compactifications of moduli spaces inspired

by mirror symmetry. Aste´risque 218: 243–271.

Strominger A, Yau S-T, and Zaslow E (1996) Mirror symmetry is

T-duality. Nuclear Physics B 479: 243–259.

Voisin C (1999) (Translated from the 1996 French original by Roger

Cooke.) SMF/AMS Texts and Monographs, vol. 1. Mirror

Symmetry. Providence, RI: American Mathematical Society.

Modular Tensor Categories see Braided and Modular Tensor Categories

448 Mirror Symmetry: A Geometric Survey

Moduli Spaces: An Introduction

F Kirwan, University of Oxford, Oxford, UK

An earlier version of this article was originally published in

Proceedings of the Workshop on Moduli Spaces, Oxford, 2–3

July 1998, Eds. Kirwan F, Paycha S, Tsou S T (1998). Cairo,

Egypt: Hindawi Publishing Corporation.

The con cept of a moduli space has been used by

mathematicians for nearly 150 years, although it was

not until the 1960s that Mumford (1965) gave precise

definitions of moduli spaces and methods for con-

structing them. The use of the word ‘‘moduli’ ’ in this

context goes back to Riemann in a paper of 1857, in

which he observed that an isomorphism class of

compact Riemann surfaces of genus g ‘‘ha¨ngt ...

von 3g  3 stetig vera¨ nderlichen Gro¨ ssen ab, welche

die Moduln dieser Klasse genannt werden sollen.’’

The idea of moduli as parameters in some sense

measuring or describing the variation of geometric

objects has been of fundamental importance in

geometry ever since.

Moduli spaces arise naturally in classification

problems in geometry, particularly in algebraic

geometry (Mumford 1965, Newstead 1978, Popp

1977, Seshadri 1975, Sundaramanan 1980, Viehweg

1995). Algebraic geometry is, roughly speaking, the

study of solutions of systems of polynomial equa-

tions in many variables; the solutions to such a

system form an algebrai c variety. A simple example

of an algebraic variety is a hypersurface, consisting

of the solutions to a single polynomial equation in

some number of variables. We can try to classify

hypersurfaces by their degree and their dimension;

these are ‘‘discrete invariants’’ for the classification

problem, but of course they do not determine

hypersurfaces completely, even if we regard two

hypersurfaces as equivalent when one is obtained

from the other after making a change of coordinates.

It is typical of classification pro blems in algebraic

geometry (and other areas of geometry) that there

are not enough discrete invariants to classify objects

sufficiently finely, and this is where the concept of a

moduli space arises.

In complex algebraic geometry, discrete invariants

often come from topology. For exampl e, a non-

singular complex curve (i.e., a complex algebraic

variety which is a connected complex manifold of

dimension 1, in other words a Riemann surface)

which is projective (i.e., points have been added at

infinity to make it compact) is topologically just a

sphere with a number of handles attached to it; the

number of handles is called the genus of the curve

and is a discrete invariant. Nonsingular complex

projective curves (or equivalently compact Riemann

surfaces) are not classified completely by their genus

g; they are determined by g when regarded simply as

topological surfaces, but the genus does not deter-

mine their complex structure when g > 0.

A classification problem such as this one (the

classification of nonsingular complex projective

curves up to isomorphism, or, equivalently, compact

Riemann surfaces up to biholomorphism), can be

resolved into two basic steps.

Step 1 is to find as many discrete invariants as possible

(in the case of nonsingular complex projective

curves the only discrete invariant is the genus).

Step 2 is to fix the values of all the discrete invariants

and try to construct a ‘‘moduli space’’; that is, a

complex manifold (or an algebraic variety) whose

points correspond in a natural way to the

equivalence classes of the objects to be classified.

What is meant by ‘‘natural’’ here can be made

precise (as we shall see shortly) given suitable notions

of families of objects parametrized by base spaces and

of equivalence of families. A ‘‘fine moduli space’’ is

then a base space for a universal family of the objects

to be classified (any family is equivalent to the

pullback of the universal family along a unique map

into the moduli space). If no universal family exists

there may still be a ‘‘coarse moduli space’’ satisfying

slightly weaker conditions, which are nonetheless

strong enough to ensure that if a moduli space exists it

will be unique up to canonical isomorphism.

It is often the case that not even a coarse moduli

space will exist. Typically, particularly ‘‘bad’’ objects

must be left out of the classification in order for a

moduli space to exist. For example, a coarse moduli

space of nonsingular complex projective curves exists

(although to have a fine moduli space we must give the

curves some extra structure, such as a level structure),

but if we want to include singular curves (which is

often important so that we can understand how

nonsingular curves can degenerate to singular ones)

we must leave out the so-called ‘‘unstable curves’’ to

get a moduli space. However all nonsingular curves

are stable, so the moduli space of stable curves of genus

g is then a compactification of the moduli space of

nonsingular projective curves of genus g.

Moduli spaces are often constructed and studied as

orbit spaces for group actions (using Mumford’s

geometric invariant theory or more recently ideas due

to Kolla´r (1997) and Keel and Mori (1997);geometric

invariant theoretic quotients can also often be described

Moduli Spaces: An Introduction 449

naturally as symplectic reductions, and it is in this guise

that many moduli spaces in physics appear. Another

technique involves period maps, Torelli theorems and

variations of Hodge structures, initiated by Griffiths

(1984) and others. In the special case of moduli spaces

of compact Riemann surfaces, Teichmu¨ ller theory can

also be used (see e.g., Lehto (1987)).

Remark 1 Recall that a compact Riemann surface

(i.e., a compact complex manifold of complex dimen-

sion 1) can be thought of as a nonsingular complex

projective curve, in the sense that every compact

Riemann surface can be embedded in some

complex projective space

¼ C

nþ1

f0g=ðmultiplication by nonzero

complex scalars)

as the solution space of a set of homogeneous

polynomial equations. Moreover, two nonsingular

complex projective curves are biholomorphic if and

only if they are algebraically isomorphic. So, there is

a natural identification between the moduli space of

compact Riemann surfaces of genus g up to

biholomorphism and the moduli space of nonsingu-

lar complex projective curves up to isomorphism.

There are other situations where an ‘‘algebraic’’

moduli space can be naturally identified with the

corresponding ‘‘complex analytic’’ moduli space, but

this is not always the case. For example, if we

consider K3 surfaces (compact complex manifolds

of complex dimension 2 with first Betti number and

first Chern class both zero), we find that the moduli

space of all K3 surfaces has complex dimension 20,

whereas the moduli spaces of algebraic K3 surfaces

(which have one more discrete invariant, the degree,

to be fixed) are 19-dimensional.

This problem of algebraic moduli spaces versus

nonalgebraic ones is one reason why the question of

classifying n-folds (i.e., compact complex manifolds –

or, in the algebraic category, nonsingular projective

varieties – of dimension n) becomes much harder

when n > 1 than in the case n = 1 (which is the case of

compact Riemann surfaces or nonsingular projective

curves). Another difficulty is that families of n-folds

can be ‘‘blown up’’ along families of subvarieties to

produce ever more complicated families.

Remark 2 Recall that we blow up a complex

manifold X along a closed complex submanifold Y

by removing the submanifold Y from X and glueing

in the projective normal bundle of Y in its place. We

get a complex manifold

X with a holomorphic

surjection  :

X ! X such that  is an isomorphism

over X  Y and if y 2 Y then 

1

(y) is the complex

projective space associated to the normal space

X=T

Y to Y in X at y.IfX = C

nþ1

and Y = {0}

and we identify P

with the set of one-dimensional

linear subspaces of C

nþ1

, then

X ¼fðv; wÞ2C

nþ1

 P

: v 2 wg

with (v, w) = v.

Again this problem does not arise when n = 1,

because blowing up a 1-fold makes no difference unless

the 1-fold has singularities (in which case blowing up

may help to ‘ ‘resolve’ ’ the singularities; for example,

when we blow up the origin {0} in C

, then the singular

curve C in C

defined by y

= x

þ x

is tranformed

into a nonsingular curve

C with the origin in C replaced

by two points, corresponding to the two complex

‘ ‘tangent directions’’ in C at 0).

Thus, the classification of n-folds when n > 1

requires a preliminary step before there is any hope

of carrying out the two steps described above.

Step 0 (the ‘‘minimal model programme’’ of Mori

(1987) and others): Instead of all the objects to be

classified, consider only specially ‘‘good’’ objects,

such that every object is obtained from one of these

specially good objects by a sequence of blow-ups

(or similar carefully prescribed operations).

How to carry out Mori’s minimal model program

is well understood for algebraic surfaces and 3-folds,

but in higher dimensions is incom plete as yet (Kolla´r

and Mori 1998). We shall ignore both step 0 and

step 1 from now on, and concentrate on step 2, the

construction of moduli spaces.

Ingredients of a Moduli Problem

Formally before posing a moduli problem, we need

to fix the category in which we are working; that is,

we need to specify what we mean by ‘‘space’’ and

‘‘map’’ in the description below. If, for example, we

are working in complex analytic geometry then we

might take ‘‘space’’ to mean a complex manifold (or

more generally we might allow singularities) and

take ‘‘map’’ to mean a co mplex analytic map,

whereas in algebraic geometry ‘‘space’’ might mean

an algebraic variety, or a scheme, or even a stack,

with ‘‘map’’ interpreted as a morphism of algebraic

varieties (or schemes, or stacks).

Once this is fixed, the ingredients of a moduli

problem are:

1. a set A of objects to be classified,

2. an equivalence relation  on A,

3. the concept of a family of objects in A with base

space S (or parametrized by S), and sometimes

4. the concept of equivalence of families.

450 Moduli Spaces: An Introduction

These ingredients must satisfy:

1. a family parametrized by a single point {p} is just

an object in A (and equivalence of objec ts is

equivalence of families over {p}) an d

2. given a family X parametrized by a space S and a

map  :

S ! S, there is a family 



X parametrized

S (the ‘‘pullback of X along ’’), with

pullback being functorial and preserving

equivalence.

In particular, for any family X parametrized by S

and any s 2 S, there is an object X

given by pulling

back X along the inclusion of {s}inS. We think of

as the object in the family X whose parameter is

the point s in the base space S.

Example 1 A family of compact Riemann surfaces

parametrized by a complex manifold S is a surjective

holomorphic map

 : T ! S

from a complex manifold T of (complex) dimen-

sion dim (T) = dim (S) þ 1toS, such that  is

proper (i.e., the inverse image 

1

(C)ofany

compact subset C of S under  is compact) and

has maximal rank (i.e., its derivative is everywhere

surjective). Then 

1

(s) is a compact R iemann

surface for each s 2 S, and is the object in the

family with parameter s.

The family defined by  is an algebraic family if

 is a morphism of nonsingular complex projective

varieties.

Example 2 A family of nonsingular compl ex

projective varieties parametrize d by a nonsingular

complex variety S is a proper surjective morphism

 : T ! S

with T nonsingular and  having maximal rank. We

can also allow T and S to be singular, but then we

require an extra technical condition (that  must be

flat with reduced fibers).

In the above example, equivalence of families



: T

! S

and 

: T

! S

is given by isomorph-

isms f : T

! T

and g : S

! S

such that g  



 f . Equivalence of families in the first example is

similar.

Definition 1 A ‘‘deformation’’ of a nonsingular

projective variety or compact complex manifold M

is given by a family  : T ! S together with an

isomorphism



1

ðs

ÞﬃM

for some s

2 S.

Strictly speaking, the deformation is the germ at

of such a ; that is, the restriction of  over any

open neighborhood of s

in S determines the same

deformation of M as  does.

A study of deform ations leads to information

about the local structure of moduli spaces. Let

 : X ! S be a deformation of a compact complex

manifold M = 

1

) where s

2 S. We can cover M

(thought of as a subset of X) with open subsets W

of X such that there exist isomorphisms

: W

! U

 V

where V

= (W

) is open in S and U

= M \ W

open in M = 

1

) and the projection of h

onto V

is just  : W

! V

. For each i 6¼ j, we then get a

holomorphic vector field 

on U

\ U

by differ-

entiating h

 h

1

in the direction of any tangent

vector v 2 T

S. These holomorphic vector fields

define a 1-cocycle in the tangent sheaf  of M. This

gives us the ‘‘Kodaira–Spencer map’’





: T

S ! H

ðM; Þ

Theorem 1 (Kuranishi). If M is a compact com-

plex manifold, then it has a deformation  : X ! S

with 

1

) = M such that

(i) the Kodaira–Spencer map 



: T

S ! H

(M, )

is an isomorphism,

(ii)  has the local universal property for deforma-

tions (i.e., any deformation of M is locally the

pullback of  along a map f into S),

(iii) if H

(M, ) = 0, then the map f in (ii) is unique,

and

(iv) if H

(M, ) = 0, then S is nonsingular at s

and

so dim S = dim H

(M, ).

This deformation  is called the ‘‘Kuranishi

deformation’’ of M (its germ at s

is unique up to

isomorphism), and S is called the ‘‘Kuranishi space’’

of M.

Example 3 A family of holomorphic (or algebraic)

vector bundles over a compact Riemann surface (or

nonsingular complex projective curve)  is a vector

bundle over   S where S is the base space (see e.g.,

Verdier and Le Potier (1985)). A deformation of a

vector bundle E

over  is then given by a vector

bundle E over a product   S together with an

isomorphism

fs

ﬃ E

for some s

2 S (strictly speaking it is the germ at s

of such a family of vector bundles).

Moduli Spaces: An Introduction 451

Fine and Coarse Moduli Spaces

For definiteness, ex cept when it is specified other-

wise, let us consider moduli problems in algebraic

geometry with ‘‘space’’ meaning algebraic variety

(over some fixed field k which is usually C) and

‘‘map’’ meaning morphism of algebraic varieties.

Definition 2 A ‘‘fine moduli space’’ for a given

(algebro-geometric) moduli problem is an algebraic

variety M with a family U parametrize d by M

having the following (universal) property: for every

family X parametrized by a base space S, there exists

a unique map  : S ! M such that

X  



U is then called a ‘‘universal family’’ for the given

moduli problem.

Many moduli problems have no fine moduli

space, but nonetheless there may be a moduli space

satisfying slightly weaker conditions, called a coarse

moduli space. If a fine moduli space does exist, it

will automatically satisfy the conditions to be a

coarse moduli space. Both fine and coarse moduli

spaces, when they exist, are unique up to canonical

isomorphism.

Definition 3 A ‘‘coarse moduli space’’ for a given

moduli problem is an algebraic vari ety M with a

bijection

 : A=!M

(where as before A is the set of objects to be

classified up to the equivalence relation ) from the

set A= of equivalence classes in A to M such that:

(i) For every family X with base space S, the

composition of the given bijection  : A=!M

with the function



: S ! A=

which sends s 2 S to the equivalence class [X

]

of the object X

with parameter s in the family

X, is a morphism.

(ii) When N is any other variety with  : A=!N

such that for each family X parametrized by a

base space S the composition   

: S ! N is a

morphism, then

  

1

: M ! N

is a morphism.

Remark 3 For some moduli problems, a family X

with base space S which is connected and of

dimension strictly greater than zero may exist such

that for some s

2 S we have

(i) X

 X

for all s, t 2 S  {s

} and

(ii) X

6 X

for all s 2 S  {s

This is the ‘‘jump phenomenon,’’ and when it

occurs we cannot construct a moduli space including

the equivalence class of the object X

. Typically, to

construct a moduli space, some objects (often called

‘‘unstable’’) must be left out because of the jump

phenomenon and we only get a moduli space of

‘‘stable’’ objects. This happens, for example, in the

construction of moduli spaces of complex projective

curves, if we want to include singular curves, or

moduli spaces of vector bundles.

Example 4 The Jacobian J() of a compact Rie-

mann surface  is a fine moduli space for holo-

morphic line bundles (i.e., vector bundles of rank 1)

of fixed degree over  up to isomorphism. As a

complex manifold

JðÞﬃC

=

where g is the genus of  and  is a lattice of maximal

rank in C

(in other words J() is a complex torus).

Since J () is also a complex projective variety, it is an

‘ ‘abelian variety.’ ’

More precisely, J() is the quotient of the

complex vector space H

(, K



) of dimension g by

the lattice H

(, Z) ﬃ Z

. Here K



is the complex

cotangent bundle of  and H

(, K



) is the space of

its holomorphic sections, that is, the space of

holomorphic differentials on . If we choose a

basis !

, ..., !

of holomorphic differentials and a

standard basis 

, ..., 

for H

(, Z) such that



:

iþg

¼ 1 ¼

iþg

:

when 1  i  g and all other intersection pairings



.

are zero, then we can associate to  the g  2g

‘‘period matrix’’ P() given by integrating the

holomorphic differentials !

around the 1-cycles 

The Jacobian J() can then be ident ified with the

quotient of C

by the lattice spanned by the columns

of this period matrix.

We can in fact always choose the basis !

, ..., !

of holomorphic differentials so that the period

matrix P() is of the form

ðI

ZÞ

where I

is the g  g identity matrix. This period

matrix is called a ‘‘normalized period matrix.’’ The

Riemann bilinear relations tell us that Z is sym-

metric and its imaginary part is positive definite.

452 Moduli Spaces: An Introduction

Example 5 The moduli space A

of all abelian

varieties of dimension g was one of the first moduli

spaces to be constructed. We have

ﬃH

=Spð2g; ZÞ

where H

is Siegel’s upper half space, which consists

of the symmetric g  g complex matrices with

positive-definite imagina ry part.

Example 6 One way to construct and study the

moduli space M

of compac t Riemann surfaces of

genus g is via the ‘‘Torelli map’’

 : M

given by

 7!JðÞ

Torelli’s theorem tells us that  is injective (cf.

Griffiths (198 4)). Describing the image of M

in A

is known as the Schottky problem.

We can calculate the dimension of the moduli

space M

using Kuranishi theory as in the previous

section: we get

dim M

¼ dim H

ð; Þ¼3g  3

for any compact Riemann surface  of genus g  2.

In fact, if M is any compact complex manifold and

there exists a fine moduli space of complex mani-

folds diffeomorphic to M, then the moduli space is

locally isomorphic near [M] to the Kuranishi space

near s

. More often, there is only a coarse moduli

space (as in the case of M

), and then the moduli

space is locally isomorphic near [M] to the quotient

of the Kuranishi space by the action of the group of

automorphisms of M.

For the Teichmu¨ ller approach to M

(cf. Lehto

(1987)), we consider the space of all pairs consisting

of a compact Riemann surface of genus g and a basis



, ..., 

for H

(, Z) as abov e such that



:

iþg

¼ 1 ¼

iþg

:

if 1  i  g and all other intersectio n pairings 

.

are zero. If g  2, this space (called Teichmu¨ ller

space) is naturally homeom orphic to an open ball in

3g3

(by a theorem of Bers). The mapping class

group 

(which consists of the diffeomorphisms of

the surface modulo isotopy) acts discretely on

Teichmu¨ ller space, and the quotient can be identi-

fied with the moduli space M

. This gives us a

description of M

as a complex analytic space, but

not as an algebraic variety.

To construct the moduli space M

as an algebraic

variety, we can use the fact that every compact

Riemann surface of genus g can be embedded

canonically as a curve of degree 6(g  1) in a

projective space of dimension 5g  6. The use of

the word ‘‘canonical’’ here is a rather poor pun; it

refers both to the canonical line bundle (or

cotangent bundle) of the Riemann surface, although

here ‘‘tricanonical’’ would be more accurate, and

also to the fact that no choices are involved, except

that a choice of basis is needed to identify the

projective space with the standard one P

5g6

. This

enables us to identify M

with the quotient of an

algebraic variety by the group PGL(n þ 1; C). How-

ever, here we do not have a discrete group action,

and to construct the quotient we must use Mum-

ford’s geometric invariant theory (see below), which

was developed in the 1960s in order to provide

algebraic constructions of this moduli space and

others.

In fact, geometric invariant theory also provides a

beautiful compactification of M

known as the

Deligne–Mumford (1969) compactification



This compactification is itself a moduli space: it is

the moduli space of (Deligne–Mumford) stable

curves, which are complex projective curves with

only nodal singularities and at most finitely many

automorphisms.



is singular but in a relatively

mild way; it is the quotient of a nonsingular variety

by a finite group action.

The moduli space M

g, n

of nonsingul ar complex

projective curves of genus g with n marked points

has a similar compactification



g, n

which is the

moduli space of complex projective curves with n

marked nonsingul ar points and with only nodal

singularities and finitely many automorphisms.

Finiteness of the automorphism group of such a

curve  is equivalent to the requirement that any

irreducible component of genus 0 (respectively 1)

has at least 3 (respectively 1) special points, where

‘‘special’’ means either marked or singular in .

The construction of M

using the period matrices

of curves and the Torelli theorem leads to a different

compactification

of M

known as the Satake

(or Satake–Baily–Borel) compactification. Like the

Deligne–Mumford compactification,

is a com-

plex projective variety, but the boundary of M

has (complex) codimension 2 for g  3 whereas

the bounda ry  of M



has codimension 1.

Each of the irreducible components 

, ..., 

[g=2]

 is the closure of a locus of curves with exact ly one

node (irreducible curves with one node in the case of



, and in the case of any other 

the union of two

nonsingular curves of genus i and g  i meeting at a

single point). The divisors 

meet transversely in



, and their intersections define a natural decom-

position of  into connected strata which parame-

trize stable curves of a fixed topological type.

Moduli Spaces: An Introduction 453

For a recent guide to many different aspects of the

moduli spaces M

, see Harris and Morrison (1998).

Example 7 Given any nonsingular complex pro-

jective variety X, we can study the moduli spaces of

maps from curves to X considered by Kont sevich.

Intersection theory on these moduli spaces leads to

Gromov–Witten theory and the quantum cohomol-

ogy of X , with many applications, for example, to

enumerative geometry (cf. Cox and Katz (1999),

Fulton and Pandharipande (1997), Dijkgraaf et al.

(1995)).

More precisely, if 2g  2 þ n > 0 then for any

 2 H

(X; Z) there is a moduli space M

g, n

(X, )of

n-pointed nonsingular complex projective curves 

of genus g equipped with maps f :  ! X sat isfying



[] = . This moduli space has a compactification



g, n

(X, ) which classifies ‘‘stable maps’’ of type 

from n-pointed curves of genus g into X (Fulton

and Pandharipande 1997). Here, a map f :  ! X

from an n-pointed complex projective curve 

satisfying f



[] =  is called stable if  has only

nodal singularities and f :  ! X has only finitely

many automorphisms, or equivalently every irre-

ducible component of  of genus 0 (respectively

genus 1) which is mapped to a single point in X by

f contains at least three (respectively 1) special

points. The forgetful map from M

g, n

(X, )toM

g, n

which sends [, p

,...,p

,f : !X]to[, p

,...,p

]

extends to a forgetful map :



g,n

(X,) !



g,n

which collapses components of  with genus 0 and

at most two special points.

Of course, when X is itself a single point,

g, n

(X, )and



g, n

(X, ) are simply the moduli

spaces M

g, n

and



g, n

. In general



g, n

(X, )has

more serious singularities than



g, n

and may indeed

have many different irreducible components with

different dimensions. In spite of this,



g, n

(X, )has

a ‘‘virtual fundamental class’’ [



g, n

(X, )]

vir

lying in

the expected dimension

3g  3 þ n þð1  gÞdim X þ



ðTXÞ



g, n

(X, ). Gromov–Witten invariants (origin-

ally developed mainly in the case g = 0 when



g, n

(X, ) is more tractable, but now also studied

when g > 0) are obtained by evaluating cohomology

classes on



g, n

(X, ) against this virtual funda-

mental class.

Moduli Spaces as Orbit Spaces

Example 8 As a simple example, let us consider the

moduli space of ‘‘hyperelliptic’’ curves of genus g.

By a hyperelliptic curve of genus g, we mean a

nonsingular complex projective curve C with a

double cover f : C ! P

branched ov er 2g þ 2 points

in the complex projective line P

Let S be the set of unordered sequences of 2g þ 2

distinct points in P

, which we can identify with an

open subset of the complex projective space P

2gþ2

associating to an unordered sequence a

, ..., a

2gþ2

points in P

the coefficients of the polynomial whose

roots are a

, ..., a

2gþ2

. Then, it is not hard to

construct a family X of hyperelliptic curves of genus

g with base space S such that the curve parametrized

by a

, ..., a

2gþ2

is a double cover of P

branched

over a

, ..., a

2gþ2

. This family is not quite a universal

family, but it does have the following two properties.

(i) The hyperelliptic curves X

and X

parametrized

by elements s and t of the base space S are

isomorphic if and only if s and t lie in the same

orbit of the natural action of G = SL(2; C)onS.

(ii) (Local universal property) Any family of hyper-

elliptic curves of genus g is locally equivalent to

the pullback of X along a morphism to S.

These properties (i) and (ii) imply that a (coarse)

moduli space M exists if and only if there is an

‘‘orbit space’’ for the action of G on S (Newstead

1978). Here, by an orbit space we mean a

G-invariant morphism  : S ! M such that every

other G-invariant morphism : S ! M factors

uniquely through , and moreover 

1

(m) is a single

G-orbit for each m 2 M. (We can think of an orbit

space as the set of G-orbits endowed in a natural

way with the structure of an algebraic variety.)

This sort of situation arises quite often in moduli

problems, and the construction of a moduli space is

then reduced to the construction of an orbit space.

Unfortunately, such orbit spaces do not in general

exist. The main problem (which is closely related to

the jump phenomenon discussed above) is that there

may be orbits contained in the closures of other

orbits, which means that the natural topology on the

set of all orbits is not Hausdorff, so this set cannot

be endowed naturally with the structure of a variety.

This is the situation the geometric invariant theory

of Mumford (1965) attempts to deal with, telling us

how to throw out certain ‘‘unstable’’ orbits in order

to be able to construct an orbit space. For more

general constructions of orbit spaces which can be

used for moduli problems where geometric invariant

theory may not be of use, see Keel and Mori (1997)

and Kolla´r (1997).

Example 9 Let G = SL(2; C) act on (P

)

via

Mo¨ bius transformations on the Riemann sphere

¼ C [ f1g

454 Moduli Spaces: An Introduction

Then,

fðx

; x

Þ2ðP

: x

¼ x

is a single orbit which is contained in the closure of

every other orbit. On the other hand, the open subset

fðx

; x

Þ2ðP

: x

; x

distinctg

of (P

)

has an orbit space which can be identified

with

f0; 1; 1g

via the cross ratio.

In order to describe Mumford’s geometric invar-

iant theory, let X be a complex projective variety

(i.e., a subset of a complex projective space defined

by the vanishing of homogeneous polynomial

equations), and let G be a complex reductive group

acting on X. We a lso require a ‘‘linearization’’ of the

action; that is, an ample line bundle L on X and a

lift of the action of G to L. We lose very little

generality in assuming that for some projective

embedding X  P

the action of G on X extends

to an action on P

given by a representation

 : G ! GLðn þ 1Þ

and taking for L the hyperplane line bundle on P

Algebraic geometry associates to X  P

its homo-

geneous coordinate ring

AðXÞ¼

k0

ðX; L

k

Þ¼C½x

; ...; x

=I

which is the quotient of the polynomial ring

C[x

, ..., x

]inn þ 1 variables by the ideal I

generated by the homogeneous polynomials vanish-

ing on X. Since the action of G on X is given by a

representation  : G ! GL(n þ 1), we get an induced

action of G on C[x

, ..., x

] and on A(X), and we

can therefore consider the subring A(X)

of A(X)

consisting of the elements of A(X) left invariant by

G. This subring A( X)

is a graded complex algebra,

and because G is reductive it is finitely generated

(Mumford 1965). To any finitely generated graded

complex algebra we can associate a complex

projective variety, and so we can define X== G to

be the variety associated to the ring of invariants

A(X)

. The inclusion of A(X)

in A( X) defines a

‘‘rational’’ map  from X to X== G, but because

there may be points of X  P

where every

G-invariant polynomial vanishes, this map will not

in general be well defined everywhere on X (i.e., it

will not be a morphism).

We define the set X

of ‘‘semistable’’ points in X

to be the set of those x 2 X for which there exists

some f 2 A(X)

not vanishing at x. Then, the

rational map  restricts to a surjective G-invariant

morphism from the open subset X

of X to the

quotient variety X== G. However,  : X

! X== G is

still not in general an orbit space: when x and y are

semistable points of X, we have (x) = (y) if and

only if the closures

(x) and O

(y) of the G-orbits

of x and y meet in X

. Topologically, X== G is the

quotient of X

by the equivalence relation for which

x and y in X

are equivalent if and only if O

(x)

and

(y) meet in X

We define a ‘‘stable’’ point of X to be a point x of

with a neighbourhood in X

such that every

G-orbit meeting this neighborhood is closed in X

and is of maximal dimension equal to the dimension

of G.IfU is any G-invariant open subset of the set

of stable points of X,then(U) is an open subset

of X== G and the restriction j

: U ! (U)of to U

is an orbit space for the action of G on U in the sense

described above, so that it makes sense to write U=G

for (U). In particular, there is an orbit space X

for the action of G on X

,andX== G can be thought

of as a compactification of this orbit space.

 X

open open



= ¼X== G

open

Example 10 Let us return to hyperelliptic curves

of genus g. We have s een that the construction of a

moduli space reduces to the construction of an

orbit space for the action of G = SL(2; C)onan

open subset S of P

2gþ2

. If we identify P

2gþ2

with the

space of unordered sequences of 2g þ 2 points in

,thenS is the subset consisting of unordered

sequences of distinct points. When the action of G

on P

2gþ2

is linearized in the obvious way, then an

unordered sequence of 2g þ 2 points in P

semistable if and only if at most g þ 1 of the points

coincide anywhere on P

, and is stable if and only

if at most g of the points coincide anywhere on P

(cf. Kirwan (1985), chapter 16). Thus, S is an open

subset of P

2gþ2

,soanorbitspaceS=G exists with

compactification the projective variety P

2gþ2

== G.

This orbit space is then the moduli space of

hyperelliptic curves of genus g.

Other moduli spaces (such as moduli spaces of

curves and of vector bundl es; see e.g., Donaldson

(1984), Gieseker (1983), Mumford (1965, 1977),

and Newstead (1978)) can be constructed as orbit

spaces via geometric invariant theory in a similar

way.

Moduli Spaces: An Introduction 455

Symplectic Reduction and Moduli Spaces

of Vector Bundles

Geometric invariant theoretic quotients are clos ely

related to the process of reduction in symplectic

geometry, and thus many moduli spaces can be

described as symplectic reductions.

Suppose that a compact, connected Lie group K

with Lie algebra k acts smoothly on a symplectic

manifold X and preserves the symplectic form !. Let

us denote the vector field on X defined by the

infinitesimal action of a 2 k by

x 7!a

By a moment map for the action of K on X we mean

a smooth map

 : X ! k



which satisfies

dðxÞðÞ:a ¼ !

ð; a

for all x 2 X ,  2 T

X and a 2 k. In other words, if



: X ! R denotes the component of  along a 2 k

defined for all x 2 X by the pairing



ðxÞ¼ðxÞ:a

between ( x) 2 k



and a 2 k,then

is a Hamiltonian

function for the vector field on X induced by a.We

shall assume that all our moment maps are equivariant

moment maps; that is,  : X ! k



is K-equivariant

with respect to the given action of K on X and the

co-adjoint action of K on k



It follows directly from the definition of a

moment map  : X ! k



that if the stabilizer K



any  2 k



acts freely on 

1

(), then 

1

()isa

submanifold of X and the symplectic form ! induces

a symplectic structure on the quotient 

1

()=K



With this symplectic structure, the quotient



1

()=K



is called the Marsden–Weinstein reduc-

tion, or symplectic quotient, at  of the action of K

on X. We can also consider the quotient 

1

()=K



when the action of K



on 

1

() is not free, but in

this case it is likely to have singularities.

Example 11 Consider the cotangent bundle T



Y of

any n-dimensional manifold Y with its canonical

symplectic form ! which is given by the standard

symplectic form

! ¼

j¼1

^ dq

½1

with respect to any local coordinates (q

, ..., q

)on

Y and the induce d coordinates (p

, ..., p

) on its

cotangent spaces. If Y is the configuration space of a

classical mechanical system, then T



Y is the phase

space of the system and the coordinates p =

, ..., p

) 2 T 

Y are traditionally called the

momenta of the system.

If Y is acted on by a Lie group K , the induced

action on T



Y preserves ! and there is a moment

map  : T



Y ! k



whose components 

along a 2

k are given by pairing the moment coordinates p

with the vector fields on X induced by the

infinitesimal action of K; that is,



ðp; qÞ¼p:a

for all q 2 Y and p 2 T

Y. When K = SO(3) acts by

rotations on Y = R

, then  is the angular momen-

tum, or moment of momentum, abou t the origin.

The connection with geometric invariant theory

arises as follows. Let X be a nonsingular complex

projective variety embedded in complex projective

space P

, and let G be a complex Lie group acting

on X via a complex linear representation  : G !

GL(n þ 1; C). A necessary and sufficient condition

for G to be reductive is that it is the complex-

ification of a maximal compact subgroup K (e.g.,

G = GL(m; C) is the complexification of the unitary

group U(m)). By an appropriate choice of coordi-

nates on P

, we may assume that  maps K into the

unitary group U(n þ 1). Then, the action of K

preserves the Fubini–Study form ! on P

, which

restricts to a symplectic form on X. There is a

moment map  : X ! k



defined (up to multiplica-

tion by a constant scalar factor depending on

differences in convention on the normalization of

the Fubini–Study form) by

ðxÞ:a ¼





ðaÞ

2ijj

xjj

½2

for all a 2 k, where

x 2 C

nþ1

 {0} is a representa-

tive vector for x 2 P

and the representation  : K !

U(n þ 1) induces 



: k ! u (n þ 1) and dually





: u(n þ 1)



! k



In this situation, we have two possible quotient

constructions, givin g us the geometric invariant

theory quotient X== G if we want to work in

algebraic geometry and the symplectic reduction



1

(0)=K if we want to work in symplectic geome-

try. In fact, these give us the same quotient space, at

least up to homeomorphism (and diffeomorphism

away from the singularities). More precisely, any

x 2 X is semistable if and only if the closure of its

G-orbit meets 

1

(0), and the inclusion of 

1

(0)

into X

induces a homeomorphism



1

ð0Þ=K ! X== G

There are other quotient constructions closely related

to symplectic reduction and geometric invariant

456 Moduli Spaces: An Introduction