where R is considered as the trivial π1 (X, x0 )-module. Therefore we

have a non-degenerate pairing

H i (π1 (X, x0 ), g) — H 2’i (π1 (X, x0 ), g ) ’ H 2 (π1 (X, x0 ), R) R

given by cup product in group cohomology together with the natural

coe¬cient pairing g — g ’ R. There is a non-degenerate symmetric

bilinear form B : g — g ’ R given as B(x, y) = T r(xy). This form is

clearly invariant under the adjoint representation and hence it gives as

an isomorphism g g . This allows us to conclude that

dim H 2 (π1 (X, x0 ), g) = dim H 0 (π1 (X, x0 ), g).

The latter group is the submodule consisting of the π1 (X, x0 )-invariants

in g, i.e. the centralizer of ρ(π1 (X, x0 )) in g. Since G = SU (r) and the

center of g is trivial, the irreducibility of ρ implies that this submodule

is zero submodule and Lemma follows.

In our attempts to solve 5.6.3 inductively on each successive step

we would have an equation like

‚fk+1 (g1 , g2 ) = Hk (f1 , ..., fk )(g1 , g2 ). (5.6.5)

230 CHAPTER 5. FAMILIES AND MODULI SPACES

In order to solve this equation for fk+1 each time we need to show that

the right hand side is a 2-cocycle. This can be done in several ways. It is

possible to use the correspondence between representations of π1 (X, x0 )

and ¬‚at bundles on X. (Since we know that the vector bundle Vρ comes

with a natural ¬‚at unitary connection which descends from d on

˜ r

X — C .) This approach was pursued in [28] where the following much

more general result was proved

THEOREM 5.6.3 (Goldman, Millson) Let X be a compact K¨hler

a

manifold and let G be a compact Lie group. A representation ρ is a

smooth point of Hom(π, G) if the pairing

Z 1 (π1 (X, x0 ), g) — Z 1 (π1 (X, x0 ), g) ’ H 2 (π1 (X, x0 ), g)

given by cup product using the Lie bracket on g as a coe¬cient pairing

is identically zero.

Here we shall use di¬erent method to show that for a Riemann

surface X the expression Hk (g1 , g2 ) is a 2-cocycle. Let us consider

a free group F generated by A1 , ..., Ag , B1 , ..., Bg . This group comes

with the natural surjective map F ’ π1 (X, x0 ) given by Ai ’ Ai and

Bj ’ Bj . The space

Hom(F, G) G2g

is smooth and therefore there is no obstruction to ¬nding ρt : F ’ G up

to order k + 1 inducing a k-th order formal homomorphism. Therefore

for a free group F it is possible to ¬nd a solution of the lift of 5.6.5

to the free group F . This means that not only the right hand side of

the lift of 5.6.5 but also Hk (f1 , ..., fk )(g1 , g2 ) is a 2-cocycle. Therefore

there exists fk+1 satisfying 5.6.5 and we therefore have completed the

inductive step.

To summarize, we showed that the space Y = Homirr (π1 (X, x0 ), G)

has a smooth manifold structure. Now we are ready to de¬ne the moduli

space M of irreducible representations π1 (X, x0 ) ’ G. We notice that

the group G acts on the space Y by conjugation. There is really no

distinction in terms of character theory between two representations

(two points of Y ) obtained by conjugation from one another. Therefore

we shall identify two such representations and consider the quotient

5.6. UNITARY BUNDLES AND REPRESENTATIONS OF π1 231

M := Y /G. A priori there is no reason that M is non-singular, because

we take a quotient, but since we have proved that Y is a manifold and

we consider only irreducible representations, all the G-orbits in Y have

the same dimension. Besides, the action of the group G is locally free

on Y .

Let us consider a deformation ρt obtained from ρ which represents

a tangent vector to the orbit of ρ in Y . For this we take a path gt in

’1

G such that ρt = gt ρgt . If we expand gt = exp(at + bt2 + · · ·) with

a, b ∈ g then the cocycle φ : π1 (X, x0 ) ’ g corresponding to ρt is

φ(g) = Adρ(g)a ’ a, g ∈ G.

This shows that the tangent space to the orbit of ρ consists exactly

of the subspace B 1 (π1 (X, x0 ), g) ‚ Z 1 (π1 (X, x0 ), g) of coboundaries. It

follows that when we pass to the quotient M = Y /G the tangent space

to the equivalence class [ρ] of ρ is exactly the ¬rst group cohomology

group H 1 (π1 (X, x0 ), g).

LEMMA 5.6.4 The moduli space M does not depend upon the choice

of a base point x0 .

Proof. Let us have two base points x0 and x. Let γ be a path in

X coming from x0 to x. The two fundamental groups π1 (X, x0 ) and

π1 (X, x) are isomorphic via the conjugation by γ. I. e. if l is a loop

on X based at x0 representing the class [l] ∈ π1 (X, x0 ) then the loop

γlγ ’1 is based at x and represents the class [l] ∈ π1 (X, x). Let γ

be another path from x0 to x and let κ = γ ’1 γ be the corresponding

loop. On the ¬rst glance the isomorphism between Hom(π1 (X, x0 ), G)

and Hom(π1 (X, x), G) given by virtue of the path γ is non-canonical,

because we would get a di¬erent isomorphism by applying the path

γ . If ρ(l) = A, where ρ ∈ Hom(π1 (X, x0 ), G) then if under the ¬rst

isomorphism mentioned we have ρ ’ ρ ∈ Hom(π1 (X, x), G) and thus

ρ (γlγ ’1 ) = A. Now we compute

ρ (γ l(γ )’1 ) = ρ (γκlκ’1 γ ’1 ) = ρ(κlκ’1 ) = ρ(κ)Aρ’1 (κ).

This means that the two isomorphisms we consider and which corre-

spond to paths γ and γ are conjugate and give the same point at the

moduli space M.

232 CHAPTER 5. FAMILIES AND MODULI SPACES

This lemma allows us to skip any reference thereafter to a base point

x0 ∈ X when we are dealing with the moduli space M.

5.7 Symplectic structure on moduli spaces

We will exhibit the symplectic nature of the moduli space

M := Homirr (π1 (X), G)/G

by explicit construction of a non-degenerate closed 2-form ω. Of course,

we already know that such a structure should exist. By the Mum-

ford theorem stated in the preceding section, the non-singular locus

of M (r, 0) is a quasi-projective manifold and therefore it inherits a

K¨hler structure from a projective space. The Narasimhan-Seshadri

a

theorem then gives us the identi¬cation between the non-singular locus

of M (r, 0) and the moduli space M we consider.

Let ρ be an irreducible representation π1 (X) ’ G and let B , be

an invariant non-degenerate bilinear pairing on g. The tangent space

to

Hom(π1 , G)/G

at ρ can be identi¬ed with the space H 1 (π1 (X), g), where g is a π1 (X)

-module via the adjoint action of G followed by ρ. We have a natural

bilinear non-degenerate skew-symmetric form ω:

H 1 (π1 (X), g) — H 1 (π1 (X), g) ’ R.

This form uses cup product in group cohomology, the pairing B , and

the fact that H 2 (π1 (X), R) H 2 (X, R) R because X is a compact

Riemann surface. More precisely, let c1 , c2 be two group 1-cocycles:

c1 , c2 ∈ M ap(π1 (X), g),

ci (g1 g2 ) = ci (g1 ) + Adρ(g1 )ci (g2 ), g1 , g2 ∈ π1 (X), i = 1, 2.

They represent the cohomology classes [c1 ], [c2 ] ∈ H 1 (π1 (X), g) respec-

tively. Then the cup product combined with B is by de¬nition the

cohomology class of the following two cocycle with coe¬cients in R:

ω([c1 ], [c2 ])(g1 , g2 ) = B c1 (g1 ), Adρ(g1 )c2 (g2 ) .

5.7. SYMPLECTIC STRUCTURE ON MODULI SPACES 233

Again it will be a useful exercise for the reader to check that

B c1 (g1 ), Adρ(g1 )c2 (g2 )

is a 2-cocycle with coe¬cients in R. The basic fact needed for this is

that the pairing B , is Ad-invariant. We also point out that we used

basically the same computations to check that the right hand side of

5.6.4 is a 2-cocycle.

THEOREM 5.7.1 (Goldman, [27]) The form ω gives a symplectic

structure on the moduli space M.

In order to show the statement we need to establish two conditions:

the form ω is non-degenerate and closed. Non-degeneracy easily fol-

lows from the fact that the pairing B , is non-degenerate and thus is

equivalent to the non-degeneracy of the pairing

∪ : H 1 (X, R) — H 1 (X, R) ’ H 2 (X, R) R

which in turn is a consequence of the Poincar´ duality.

e

To work out the question of the closedness of the form ω we shall use

the correspondence between representations of π1 (X) and local systems

on X (or, equivalently, ¬‚at bundles on X.) Let ˜ be the (unitary)

g

local system on X corresponding to our irreducible representation ρ.

We shall denote by Vρ the corresponding ¬‚at bundle (with the ¬‚at

connection ρ ) on X. The ¬‚at bundle End(Vρ ) corresponding to the

local system ˜ admits the following explicit construction. Let X — G be

g

the trivial principal G-bundle on X. (Since X is a Riemann surface and

G = SU (r) is simply connected all principal G-buundles on X are in

fact trivial.) Now, we treat g as G-module via the adjoint representation

and we make the ¬ber change: End(Vρ ) = (X — G) —G g. To see where

the ¬‚at connection on End(Vρ ) is coming from we consider a universal

˜ ˜

cover X of X and the π1 (X)-action on X — G given by

γ(˜, g) = (γ x, ρ(γ)g).

x ˜

˜

The trivial connection d on X — G determines a ¬‚at connection on

X — G and descends to a ¬‚at connection on End(Vρ ).

234 CHAPTER 5. FAMILIES AND MODULI SPACES

Next, we would like to de¬ne di¬erential forms on X with so-called

twisted coe¬cients. A di¬erential k-form ± with coe¬cients in ˜ can g

be described using local trivialization of ˜ in the basis of ¬‚at sections

g

of ˜. We de¬ne ± as a tensor product β — v, where β is a usual k-form

g

on X and v is a section of ˜. The di¬erential of ± is de¬ned by

g

d± = d(β — v) = dβ — v + β — v,

where is the connection de¬ning the ¬‚at structure. The collection of

˜-valued di¬erential k-forms on X is denoted by Ak (X, ˜). If we take

g g

all such forms, we get the following complex of di¬erential forms:

d d d d

· · · ’ Ak’1 (X, ˜) ’ Ak (X, ˜) ’ Ak+1 (X, ˜) ’ · · · . (5.7.6)

g g g

The cohomology groups of this complex will be denoted by H k (X, ˜).

g

Again, the fact that X is K(π1 (X), 1) space amounts to the isomor-

phisms in cohomology:

H i (π1 (X), g) H i (X, ˜).

g

Let us exhibit the 2-form ω in this new context. Let us take two ˜-

g

valued 1-forms ± — v and β — w (de¬ned locally). The Ad-invariant

symmetric bilinear pairing B , de¬nes in turn a pairing

˜ — ˜ ’ 1X ’

gg

the trivial real line bundle on X. Therefore the cup product of di¬er-

ential forms together with B , produce a usual di¬erential two-form

written locally as

B v, w ±§β.

After we integrate this two-form over X we get a number and thus we

constructed a pairing

A1 (X, ˜) — A1 (X, ˜) ’ R,

g g

which induces our two-form ω on M.

In fact, the easiest way now to see that the form ω is closed is to

produce following Atiyah and Bott [3] a two-form ω on an in¬nite-

dimensional a¬ne space A of all connections which is naturally closed

5.7. SYMPLECTIC STRUCTURE ON MODULI SPACES 235

and then obtain ω by means of a reduction process. Let us explain

in detail what we mean. Let A be the a¬ne space of all unitary con-

nections on the bundle Vρ . Of course, ρ ∈ A and if a ∈ A then the

di¬erence A = a ’ ρ is a ˜-valued 1-form. (By abuse of notation

g

we will denote by ρ both the point in A corresponding to ρ and the

corresponding covariant derivative.) Therefore, the tangent space to A

at a point a shall be Ta A = A1 (X, ˜). We also can identify the space

g

A itself with the in¬nite-dimensional vector space A1 (X, ˜) by choosing

g

1

ρ as its origin and any A ∈ A (X, ˜) de¬nes unambiguously the point

g

ρ + A in A. Inside A there lies the space F of all ¬‚at connections.

Identifying A A1 (X, ˜) as above the equation for F is

g

1

ρA + [A, A] = 0 (5.7.7)

2

and easily can be deduced from the ¬‚atness condition combined with

the fact that ρ itself is ¬‚at. Clearly, F is not a linear subspace of A,

but if we take the linear part of the equation 5.7.7, namely, ρ A = 0

it gives us the (linear) subspace of the tangent space to A at ρ which

is the tangent space to F at ρ . We do not claim, however, that F is

a submanifold of A. Thus T ρ F exactly coincides with Z 1 (X, ˜) - the

g

space of 1-cocycles of the complex 5.7.6. The form ω on the space A

de¬ned as above:

ω (± — v, β — w) = B v, w ±§β

X