5.5. MODULI SPACES OF VECTOR BUNDLES 223

We say that two semistable bundles V and V are S-equivalent if G(V ) =

G(V ). We notice in this case that if V is stable then k = 1. Two stable

bundles are S-equivalent if and only if they are isomorphic. The notion

of S-equivalence helps to overcome the phenomenon of degeneration of

bundles (Section 3.2) and construct a good moduli space.

THEOREM 5.5.6 (Mumford). There exists a connected moduli

space

M (r, d)

of S-equivalent vector bundles of rank r and degree d over X which is

a complex projective variety of dimension r2 (g ’ 1) + 1.

The space M (r, d) is ¬bered over the Jacobian variety J(X) and the

map M (r, d) ’ J(X) is de¬ned by V ’ det(V ). It turns out that the

¬bration M (r, d) ’ J(X) is a trivial ¬bration and therefore the ¬ber

M (r, d) is a well-de¬ned projective variety of dimension (r2 ’ 1)(g ’ 1).

Equivalently, M (r, d) is the moduli space of holomorphic vector bundles

on X of rank r, degree d and ¬xed determinant. One notices that when

r and d are coprime then every semistable bundle is actually stable. In

this case as we shall see later the moduli space in question is smooth and

compact. Another observation is that if V is stable (resp. semistable)

then so is V — L for any line bundle L. Thus similarly to the case

g = 1 the moduli space is the same for the degrees d and d + rj:

M (r, d) M (r, d + rj), j ∈ Z.

When r and d are coprime there exists a universal bundle U over

M (r, d) — X such that U|V —X V for all V . However when r and d are

not coprime there is no universal bundle even on a Zariski open subset

as it was shown by Ramanan.

One would like to have a description of the tangent space Tm M (r, d)

to the moduli space M (r, d) at some point m. We know that all the

topological invariants of a smooth complex vector bundle over X are

given by its degree and rank. So let us ¬x the topological type of a

bundle V over X and see how V can be endowed with a holomorphic

structure. Apparently we have to decide which local sections of V are

holomorphic so we introduce the operator that acts (locally) on sections

of V :

D : “(V ) ’ „¦0,1 (V ),

224 CHAPTER 5. FAMILIES AND MODULI SPACES

where „¦0,1 (V ) is a space of local C ∞ V -valued (0, 1)- forms. We choose

a local C ∞ basis of V and let z be a local coordinate on X. Locally

each section is a Cr - valued function of z: (f1 (z), ..., fr (z)). As in

Section 4.4 we can write D = d + B, where B is a matrix of (0, 1)

forms on X and d fi = ‚f¯i d¯. The matrix B can be chosen arbitrarily

z

‚z

because dim X = 1 and there is no integrability condition. So the

space C of all holomorphic structures on V is a complex a¬ne space

with the underlying space of translation „¦0,1 (End(V )), where End(V )

is the C ∞ -bundle of complex endomorphisms of V .

One has to factor out isomorphic bundles, i.e. identify two holo-

morphic structures that are di¬erent by a choice of trivialization. Let

Aut(V ) stand for the group of all automorphisms of V . Locally an

element of this group is a smooth map from X to GL(r, C) (this corre-

sponds to a change of trivialization). The group Aut(V ) acts on C and

the space of orbits is our moduli space. Any orbit is locally a manifold

of ¬nite codimension in C and the tangent space to it is the image of

d : „¦0 (End(V )) ’ „¦0,1 (End(V )),

because any smooth endomorphism e of V changes d by the addition of

d e. Thus the tangent space to the moduli space itself is the cokernel of

the above map, namely H 1 (X, End(V )). (In the case when V is simple

we can use Riemann-Roch theorem to see that dim H 1 (X, End(V )) =

r2 (g ’ 1) + dim H 0 (X, End(V )) = r2 (g ’ 1) + 1.)

5.6 Relation with unitary bundles and rep-

resentations of the fundamental group.

˜

Let π1 (X, x0 ) be the fundamental group of X, let X be the universal

cover of X and let ρ : π1 (X, x0 ) ’ U (r) be a unitary representation.

˜

We have a natural action of π1 (X, x0 ) on the space X — Cr given by

˜

a ∈ π1 (X), x ∈ X, y ∈ Cr .

a(˜, y) = (a˜, ρ(a)y),

x x ˜

˜

The quotient Vρ = (X — Cr )/π1 (X, x0 ) is a ¬‚at vector bundle of rank r

over X and it has a natural holomorphic structure. The bundle Vρ is

with necessity of degree zero.

5.6. UNITARY BUNDLES AND REPRESENTATIONS OF π1 225

THEOREM 5.6.1 (Narasimhan, Seshadri) (I) A unitary repre-

sentation ρ is irreducible if and only if the corresponding vector bundle

Vρ is stable.

(II) For any ρ there exists a semistable bundle V such that Vρ G(V )

and conversely, for any semistable bundle V of degree 0 there exists a

unitary representation ρ de¬ned uniquely up to conjugation such that

Vρ G(V ).

Therefore, instead of considering the moduli space of stable holo-

morphic vector bundles on X of rank r and degree zero, one can consider

the moduli space M of irreducible representations π1 (X, x0 ) ’ U (r).

However, if we do not restrict ourselves to irreducible representations,

then two semistable bundles V1 , V2 such that G(V1 ) = G(V2 ) will corre-

spond to the same point [ρ] at the moduli space of all representations.

From now on we will study unitary representations of π1 (X, x0 )

with the image lying in the special unitary group and we let G :=

SU (r). One can represent π1 (X, x0 ) as the group given by 2g generators

A1 , ..., Ag , B1 , ..., Bg and one single relation

R = R(A1 , ..., Ag , B1 , ..., Bg ) = [A1 , A2 , ..., Ag , B1 , B2 , ..., Bg ] = Id,

where [A, B] = ABA’1 B ’1 is the commutator of A and B and

[A1 , A2 , ..., Ag , B1 , ..., Bg ] = A1 A2 · · · Ag B1 B2 · · · Bg A’1 · · · A’1 B1 · · · Bg

’1 ’1

1 g

is the generalized commutator of 2g elements of G. Let us ¬x an ir-

reducible representation ρ : π1 (X, x0 ) ’ G; this allows us to think of

Ai , Bj as of elements of the group SU (r) obeying the relation R. We

recall the adjoint representation of a Lie group: Ad : G ’ Aut(g),

where g is the Lie algebra of a Lie group G. If we think of g as of the

tangent space to the identity Id of G, this representation is given by

Ad(A) = (dκA )Id , where κ : G ’ G is the automorphism of G given

by the conjugation by A. This shows that we can think of g as of a

π1 (X, x0 )-module and the action of π1 (X, x0 ) is given by the adjoint

representation followed by ρ.

Let us give an explicit construction of the moduli space M. First of

all, we consider the space Hom(π1 (X, x0 ), G) of all unitary representa-

tions of π1 (X, x0 ) of rank r. This space lies inside G2g = G — · · · — G

2g

226 CHAPTER 5. FAMILIES AND MODULI SPACES

and is given by the equation

[X1 , X2 , . . . X2g’1 , X2g ] = Id,

where Xj belongs to the j-th multiple in G2g . Now we would like

to restrict ourselves and consider only irreducible representations; we

denote

Y = Homirr (π1 (X, x0 ), G) ‚ Hom(π1 (X, x0 ), G)

the subspace consisting of irreducible representations. In fact, Y has

a manifold structure. Let us have a smooth path ρt in Y such that

ρ0 = ρ. Since for each t ∈ R ρt is a representation, we have

ρt (g1 g2 ) = ρt (g1 )ρt (g2 ), g1 , g2 ∈ π1 (X, x0 ).

Let us ¬nd out the tangent space Tρ Y by employing the fact that each

· ∈ Tρ Y is tangent to an analytic path ρt starting at ρ. We can write

∞

fi (g)ti )ρ0 (g),

ρt (g) = exp(

i=1

where fi (g) ∈ g.

We need to use the language of group cohomology for proper for-

mulation of the results. Let us have a group π and a π-module M . The

1-cochains of π with coe¬cients in M are just the maps π ’ M . We

say that φ : π ’ M is a 1-cocycle if

‚φ(g1 , g2 ) = φ(g1 ) ’ φ(g1 g2 ) + g1 —¦ φ(g2 ), g1 , g2 ∈ π

where —¦ is the action of π on M . Let S = Z 1 (π, M ) be the space of

1-cocycles. We say that φ is a 1-coboundary if φ(g) = ‚x(g) = g —¦ x ’ x

for some x ∈ M . Let B 1 (π, M ) be the space of 1-coboundaries. It is

easy to check that each 1-coboundary is a 1-cocyle as well, therefore

the cohomology group

H 1 (π, M ) := Z 1 (π, M )/B 1 (π, M )

is well-de¬ned. This allows us to think of fi as of 1-cochains.

5.6. UNITARY BUNDLES AND REPRESENTATIONS OF π1 227

By de¬nition, we also have

H 0 (π, M ) := M π ,

i.e. the submodule of π-invariants consisting of all m ∈ M such that

‚m(γ) = γ —¦ m ’ m = 0

for all γ ∈ π. Now, the group of 2-cocycles Z 2 (π, M ) consists of maps

ψ : π — π ’ M such that

‚ψ(g1 , g2 , g3 ) = ψ(g1 , g2 ) ’ ψ(g1 , g2 g3 ) + ψ(g1 g2 , g3 ) ’ g1 —¦ ψ(g2 , g3 ),

where g1 , g2 , g3 ∈ π. A 2-cochain ψ is a coboundary if

ψ(g1 , g2 ) = ‚φ(g1 , g2 ) = φ(g1 ) ’ φ(g1 g2 ) + g1 —¦ φ(g2 ), g1 , g2 ∈ π

for some 1-cochain φ. Again one easily checks the inclusion B 2 (π, M ) ‚

Z 2 (π, M ), where B 2 (π, M ) is the group of 2-coboundaries and one de-

¬nes the second group cohomology group as

H 2 (π, M ) = Z 2 (π, M )/B 2 (π, M ).

It is apparent how to extend our formulae to de¬ne higher group coho-

mology groups, but we shall not use them in the present discussion.

Returning to our situation, the condition ρt (g1 g2 ) = ρt (g1 )ρt (g2 )

implies that

∞ ∞ ∞

i i

fi (g2 )ti )ρ0 (g2 ).

exp( fi (g1 g2 )t )ρ0 (g1 g2 ) = exp( fi (g1 )t )ρ0 (g1 ) exp(

i=1 i=1 i=1

Since ρ0 (g1 g2 ) = ρ0 (g1 )ρ0 (g2 ) we have

∞ ∞ ∞

i i

fi (g2 )ti ).

exp( fi (g1 g2 )t ) = exp( fi (g1 )t ) exp(Ad(ρ0 (g1 ))

i=1 i=1 i=1

(5.6.3)

The Campbell-Hausdor¬ formula

1 1

eX eY = exp(X + Y + [X, Y ] + ([X, [X, Y ]] + [Y, [Y, X]]) + · · ·)

2 12

228 CHAPTER 5. FAMILIES AND MODULI SPACES

implies that up to ¬rst order of t we have

f1 (g1 g2 ) = f1 (g1 ) + Ad(ρ0 (g1 ))f1 (g2 ), g1 , g2 ∈ π1 (X, x0 ).

This exactly means that f1 is a group 1-cocycle with coe¬cients in g. If

· ∈ Tρ Y is tangent to the path ρt then we should have · = (dρt )|t=0 . It is

easy to see that · = f1 and thus we showed that Tρ Y ‚ Z 1 (π1 (X, x0 ), g).

In fact, those two spaces coincide. To see this, we have to prove that

each 1-cocycle f1 ∈ Z 1 (π1 (X, x0 ), g) is tangent to an analytic path in

Y starting at ρ. We shall tacitly use a theorem by M. Artin [2] which

tells us that a vector · ∈ Z 1 (π1 (X, x0 ), g) is tangent to an analytic path

in X if and only if there exists a formal power series deformation of ρ

with ¬rst term equal to ·. This means that we have to solve 5.6.3. In

fact, we have to take f1 = · and we actually can do so since we saw

that f1 only has to be a group 1-cocycle.

If we write then 5.6.3 to order 2 in t, we will have

1

‚f2 (g1 , g2 ) = ’ [Adρ(g1 )h1 (g2 ), h1 (g1 )]. (5.6.4)

2

Let us check that the right hand side of this expression is a 2-cocycle.

For this we shall verify the identity

[Adρ(g1 )f1 (g2 ), f1 (g1 ] ’ [Adρ(g1 )f1 (g2 g3 ), f1 (g1 )]+

+[Adρ(g1 g2 )f1 (g3 ), f1 (g1 g2 )] ’ Adρ(g1 )[Adρ(g2 )f1 (g3 ), f1 (g2 )] = 0.

This identity is equivalent, using the fact that f1 is a 1-cocycle and thus

f1 (xy) = f1 (x) + x —¦ f1 (y), to the following:

[Adρ(g1 )f1 (g2 ), f1 (g1 )] ’ [Adρ(g1 )f1 (g2 ), f1 (g1 )]’

’[Adρ(g1 )Adρ(g2 )f1 (g3 ), f1 (g1 )] + [Adρ(g1 g2 )f1 (g3 ), f1 (g1 )]+

+[Adρ(g1 g2 )f1 (g3 ), Adρ(g1 )f1 (g2 )]’Adρ(g1 )[Adρ(g2 )f1 (g3 ), f1 (g2 )] = 0.

Now we employ the simple facts that Adρ(gi )Adρ(gj ) = Adρ(gi gj ) and

that Adρ(gi )[x, y] = [Adρ(gi )x, Adρ(gi )y] to see that all the terms can

be pairwise cancelled out. Thus the right hand side of 5.6.4 is really a

2-cocycle.

Therefore we are able to ¬nd such an f2 if the right hand side will

be the coboundary as well. One of the possibilities to do so is to prove

5.6. UNITARY BUNDLES AND REPRESENTATIONS OF π1 229

LEMMA 5.6.2 If ρ is an irreducible representation then

H 2 (π1 (X, x0 ), g) = 0.

Proof. We will use an important observation that X is a so-called

K(π1 (X, x0 ), 1)

space meaning that X as a Riemann surface of genus greater than 1 has

only one non-zero homotopy group π1 (X, x0 ). The vanishing of higher

homotopy groups π2 , ... follows from the fact that the Lobachevsky

upper-half plane is a covering space for X and is contractible. Therefore

their higher homotopy groups vanish simultaneously.

The basic property about K(π1 (X, x0 ), 1) spaces is the equality be-

tween usual (de Rham) cohomology and group cohomology: