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j

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

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