where ωj is a 1-form of type (0, 1) and ξj is a section of Th X. Then we

have

A·· = ωj § L(ξj )·.

j

The integrability condition then reads

0 = (d )2 = (d0 )2 + [d0 A + Ad0 ] = d0 A,

where d0 A is a 2-form of type (0, 2) with values in Th X. Therefore A

should be in the kernel of

¯ ¯

d : “(X, Th X — Th X) ’ “(X, §2 Th X — Th X).

— —

We should emphasize that CS(X) will in general be a singular sub-

space of the manifold ACS(X). What we have computed above is the

Zariski tangent space to this singular subspace, which is de¬ned as the

subspace of the tangent space to ACS(X) consisting of tangent vectors

which vanish on the de¬ning equations of CS(X). At a smooth point

of ACS(X), this yields the ordinary tangent space of a submanifold.

Actually, since we want to classify complex manifolds up to bi-

holomorphic isomorphisms, we do not wish to distinguish between two

complex structures, one of which is the image under a di¬eomorphism

of X of the other. Therefore, we shall consider the quotient space

CS(X)/Dif f (X). We will formally compute the tangent space to

this quotient. We note that even if we consider the smooth locus of

CS(X), the quotient by Dif f (X) the quotient by Dif f (X) will only

be smooth over the open set of CS(X) comprised of the J whose stabi-

lizer in Dif f (X) is trivial; this stabilizer is the group of biholomorphic

automorphisms of the complex manifold (X, J). For X compact, it is

5.3. DEFORMATION OF COMPLEX STRUCTURES 217

reasonable to consider the open set of CS(X) consisting of the com-

plex structures with trivial automorphism group. The Lie algebra of

Dif f (X) is the Lie algebra V ect(X) of smooth vector ¬elds on X. The

subspace of TJ CS(X) corresponding to the action of Dif f (X) is the

image of the mapping

¯

φ : V ect(X) ’ “(X, Hom(Th X, Th X)).

This mapping is characterized by

Re(φ(ξ)) = [L(ξ),t J].

Here we use the fact that taking the canonical isomorphism between

Th Y and T Y for a complex manifold Y is given by v ’ Re(v). Now

we claim that the operator [L(ξ),t J] on 1-forms is given by the tensor

√

A = 2 ’1d v, where ξ = v + v is the decomposition of ξ into v ∈

¯

¯¯

Th X, v ∈ Th X. This can be veri¬ed using local holomorphic coordinates

¯

(z1 , · · · , zn ). Write ξ = j fj ‚zj + j fj ‚‚ j . Then one checks that

‚

z

¯

√

t

[L(ξ), J](hdzj ) = 2 ’1hd fj

and √

[L(ξ),t J](pd¯j ) = 2 ’1pd zj .

z ¯

It follows that the subspace of TJ CS(X) spanned by V ect(X) is exactly

the image of d : “(X, Th X) ’ A(0,1) (X, Th X) . Thus we conclude that

1

TJ (CS(X)/Dif f (X)) = HDolb (X, Th X).

Let us now have a family of compact complex manifolds f : Z ’ Y ,

where f is a proper holomorphic submersive map. If Y is contractible,

then it is trivial as a smooth ¬bration and we can write Z Y — F,

where F is a ¬ber of f . Therefore, we can view f as a family of complex

structures on F and we have a well-de¬ned map

ψ : Y ’ CS(F )/Dif f (F ).

The derivative of ψ is

1

dψy : Ty Y ’ HDolb (Zy , Th Xy ).

This map is called the Kodaira-Spencer map. We have

218 CHAPTER 5. FAMILIES AND MODULI SPACES

PROPOSITION 5.3.2 The Kodaira-Spencer map ψ is complex-linear.

We also notice that for the Legendre family of elliptic curves simple

dimension considerations show that dψ» is an isomorphism.

A general question investigated by Kodaira and Spencer is when a

given class κ ∈ H 1 (X, Th X) (for X compact) arises from a family of

complex manifolds Z ’ ∆ over a disc ∆, with ¬ber at 0 equal to X.

They found an obstruction using a bracket operation

H 1 (X, Th X) — H 1 (X, Th X) ’ H 2 (X, Th X).

The existence of a family as above implies that [κ, κ] = 0. Conversely,

it was proved by Kuranishi that any such κ does arise from a family of

complex manifolds over a disc. In fact, Kuranishi proves the existence

of a family Z ’ S over a complex-analytic space S, with ¬ber at a base

point s equal to X, which is complete in the sense that the Kodaira-

Spencer map Ts S ’ H 1 (X, Th X) is injective. In case the bracket

operation is trivial, S is smooth, and the result was ¬rst proved by

Kodaira-Nirenberg-Spencer.

The case of a compact Riemann surface X is very interesting. By

Serre duality the group H 1 (X, Th X) is dual to the group “(X, §2 Th X — )

of holomorphic quadratic di¬erentials. If the genus of X is ≥ 2, then

this latter group has dimension 3g ’ 3. A universal family of compact

Riemann surfaces of genus g can be found over the so-called Teichm¨ller u

0

space Tg . The space Tg is the quotient CS(Σ)/Dif f (Σ) of the manifold

CS(Σ) of complex structures on a compact surface of genus g by the

connected component of the group Dif f (Σ). It was proved by Ahlfors

and Bers that Tg is biholomorphic to a bounded domain in C3g’3 .

We can use the study of the space CS(X) to prove statements (1)-

(3) in section 3.2 about a family f : X ’ Y of projective manifolds.

˜

First let ξ be a vector ¬eld on Y , and let ξ be any lift to a vector

¬eld on X. A section of E j of pure type (p, q) is represented by a

di¬erential form ± of type (p, q) on X which is closed on each ¬ber of

˜

f . Now we claim that L(ξ)± can only have types (p + 1, q ’ 1), (p, q),

(p ’ 1, q + 1). Indeed this can be veri¬ed locally. Pick a holomorphic

coordinate system (w1 , · · · , wr ; z1 , · · · , zd ) on X, where (w1 , · · · , wr ) are

5.4. VECTOR BUNDLES OVER AN ELLIPTIC CURVE 219

holomorphic coordinates on Y . Then write

±= fI,J dzI § d¯J ,

z

I,J

where #I = p, #J = q. We then have

˜ ˜ · fI,J )dzI § d¯J + I,J,i∈I ±fI,J d(ξzi )dzI\{i} § d¯J

˜

L(ξ)± = I,J (ξ z z

˜¯

+ I,J,j∈J ±fI,J d(ξ zj )dzI § d¯J\{j}

z

which proves the statement.

It follows from our discussion of the complex structure on ACS(X)

that a holomorphic vector ¬eld ξ on ACS(X) gives at each point x of

X an element of

Hom(Th Xx , Th Xx ) ‚ Hom(Th Xx , Th Xx ) • Hom(Th Xx , Th Xx ),

where the second vector space is the complexi¬ed tangent space to the

space of complex structures on Tx X. What this means is that ξ gives

an in¬nitesimal variation of Th X but does not move Th X ‚ T X —

C. Therefore an antiholomorphic vector ¬eld to ACS(X) will move

Th X but not Th X. In other words it is lower triangular with respect

to the decomposition T X — C = Th X • Th X. Hence the transpose

mapping is upper-triangular with respect to the decomposition T — X —

C = Th X — • Th X — . This means that a antiholomorphic vector ¬eld will

move Th X — but not Th X — . This implies that in the above formula each

˜ ˜

term d(ξzi ) = L(ξ)dzi will be of type (1, 0). Hence the whole sum will

belong to F p , which proves statement (2). Statement (3) is equivalent

to (2) by complex-conjugation.

5.4 Vector bundles over an elliptic curve

Let X be a nonsingular elliptic curve (a complex 1 - torus). The com-

plete classi¬cation of vector bundles over X is due to M. F. Atiyah

[5].

DEFINITION 5.4.1 A vector bundle V is called indecomposable if

it can not be represented as a direct sum of two nonzero vector bundles.

220 CHAPTER 5. FAMILIES AND MODULI SPACES

We denote by E(r, d) the set of indecomposable vector bundles over X

of rank r and degree d. If A is a ¬xed line bundle of degree 1 on X,

then it corresponds to a choice of a base point O on X. (So that one

has div(s) = O, for some meromorphic section s of A.) Then X can be

identi¬ed with its Jacobian variety JX as follows. For x ∈ X we take a

line bundle Lx of degree zero that corresponds to the divisor [x] ’ [O].

Thus we get a point of JX . The main assertion is that E(r, d) may be

identi¬ed with X in such a way that det : E(r, d) ’ E(1, d) corresponds

to H : X ’ X where H(x) = x + x + · · · + x and h = (r, d) is the

h

greatest common divisor (g.c.d.) of r and d. The proof of this relies on

the following

THEOREM 5.4.2 There exists a vector bundle Fr ∈ E(r, 0) (called

Atiyah bundle) unique up to isomorphism such that H 0 (X, Fr ) = 0.

Moreover there is an exact sequence

0 ’ 1X ’ Fr ’ Fr’1 ’ 0.

If V ∈ E(r, 0) then V L—Fr , where L is of degree zero and is uniquely

determined by L det(V ).

For example, when r = 1 then F1 is just the trivial line bundle, but

of course none of the extensions in the theorem is split, because Fr is

indecomposable. Next one sees that the mapping V ’ V — A—j de¬nes

a 1’1 correspondence E(r, d)’E(r, d+jr). (Hence it is enough to work

˜

in the range 0 ¤ d < r.) Besides it turns out that for each V ∈ E(r, d)

there exists an exact sequence

0 ’ Cd — X ’ V ’ V ’ 0,

such that V ∈ E(r ’ d, d) and conversely for each V ∈ E(r ’ d, d) there

is a unique V ∈ E(r, d) which is an extension of V by the trivial bundle

of rank d. Then combining those two (1 ’ 1) correspondences (in the

¬rst one we take j = 1):

E(r, d) ” E(r, d + r), and E(r, d) ” E(r ’ d, d)

with the Euclidian algorithm that allows to determine the g.c.d. of two

integers we see that there is another (1 ’ 1) correspondence

E(r, d) ←’ E(h, 0).

5.5. MODULI SPACES OF VECTOR BUNDLES 221

5.5 Moduli spaces of vector bundles on

Riemann surfaces

Let us have a Riemann surface X of genus g ≥ 2 and let V be a

holomorphic vector bundle over X. For each complex number » ∈ C

there is the corresponding endomorphism of V given by the dilation by

» in every ¬ber. We will say that such an endomorphism is scalar.

DEFINITION 5.5.1 A vector bundle V is simple if each of its endo-

morphisms is scalar.

Obviously, if a bundle is simple then it is indecomposable. In the next

statement we use the notion of holomorphic family V (m), m ∈ M of

vector bundles over X parametrized by a complex manifold M . Another

interpretation of the family V (m) can be obtained if we consider a

holomorphic vector bundle V over M —X such that locally for each m ∈

M we have the following isomorphism of holomorphic vector bundles

over X:

V|m—X V (m).

The following result is due to Narasimhan and Seshadri

THEOREM 5.5.2 There is a complex manifold Mr of complex di-

mension r2 (g ’ 1) + 1 and a holomorphic family V (m), m ∈ Mr of

simple vector bundles on X such that for each simple vector bundle V

over X there exists a unique point m ∈ Mr such that V (m) is isomor-

phic to V .

Moreover, this manifold is complete in the following sense. Let Mr be

another such manifold with the corresponding family W (m ), m ∈ Mr .

Then for each point m ∈ Mr such that W (m ) V (m) there exists a

neighbourhood U of m and unique holomorphic map β : U ’ Mr such

that the family W (m ), m ∈ U is isomorphic to the inverse image by

β of the family V (m), m ∈ Mr .

If we consider indecomposable rather then simple vector bundles

then there exists an algebraic family of vector bundles over X of rank

r and degree d parametrized by an irreducible non-singular algebraic

variety A(r, d) such that every indecomposable vector bundle of rank r

222 CHAPTER 5. FAMILIES AND MODULI SPACES

and degree d is isomorphic to a vector bundle corresponding to a point

of A(r, d).

For each vector bundle V over X we de¬ne its slope µ(V ) as the

ratio of its degree and rank: µ(V ) = deg(V )/rank(V ).

DEFINITION 5.5.3 A vector bundle V over X is called stable if for

every proper subbundle W ‚ V the condition µ(W ) < µ(V ) is satis¬ed.

A vector bundle V is called semistable if one replaces < by ¤ in the

previous inequality.

We mention a well-know fact that if V and V are two non-isomorphic

stable vector bundles of the same rank and degree then there is no non-

zero bundle homomorphism between them. Besides, if V is a stable

vector bundle then V is a simple vector bundle as well.

When we have di¬erent slopes we have the following result. For two

semistable bundles V and V such that µ(V ) > µ(V ) every homomor-

phism V ’ V is zero.

The next result is due to Harder and Narasimhan.

PROPOSITION 5.5.4 Every holomorphic bundle V over X has a

canonical ¬ltration

0 = V0 ‚ V1 ‚ · · · ‚ Vk = V

such that Di = Vi /Vi’1 is semi-stable and

µ(D1 ) > µ(D2 ) > · · · > µ(Dk ).

Clearly, if V itself is semistable then k = 1. This result allows us

to decide if two bundles are not isomorphic just by looking at this

¬ltration. For semistable bundles there is a similar result in which the

bundles Vi /Vi’1 are stable. A proper subbundle W of a semistable

bundle V is called destabilizing if µ(W ) = µ(V ).

PROPOSITION 5.5.5 If V is a semistable bundle then there exists

a Harder - Narasimhan ¬ltration

0 = V0 ‚ V1 ‚ · · · ‚ Vk = V

of destabilizing bundles of maximal rank that is unique in the sense that