Now we look for functions p(») and q(») such that

)2 [ω] + p(»)

( [ω] + q(»)[ω] = 0.

d d

d» d»

5.2. THE LEGENDRE FAMILY OF ELLIPTIC CURVES 211

For this we use that · = xω and thus

‚ ‚ xdx dx »dx

( ·) = L( )· = xL( )ω = = + .

d

‚» ‚» 2y(x ’ ») 2y 2y(x ’ »)

d»

On the level of cohomology

[ω] [·] »[ω] [ω] [·]

[·] = ’ + = ’ .

d

2 2(1 ’ ») 2(1 ’ ») 2(1 ’ ») 2(1 ’ »)

d»

We then have

[ω] [·] 1 ’ 2»[·] [ω]

)2 [ω] = ’

( + +2 + ’

d

4»(1 ’ »)2 4»(1 ’ »)2 2» (1 ’ »)2 2(1 ’ »)2

d»

[·] [ω] ’1 + 3» 1 ’ 2»

’ + = [ω] + 2 [·].

4»(1 ’ »)2 4(1 ’ »)2 4»(1 ’ »)2 2» (1 ’ »)2

We use now Equation 5.2.1 and simple transformations to get the fol-

lowing desired equation:

1

)2 [ω] + (1 ’ 2»)

»(1 ’ »)( [ω] ’ [ω] = 0.

d d

4

d» d»

Now for small |»| let us look at a function F (») de¬ned as a period

dx dx

∞

F (») = ω= =2 ,

y x(x ’ 1)(x ’ »)

γ» γ» 1

where γ» varies continuously with ». The consequence of the above

Lemma is that F (») satis¬es

d2 F dF F

»(1 ’ ») 2 + (1 ’ 2») ’ = 0.

d» d» 4

We invoke the Gauß hypergeometric equation

d2 y dy

»(1 ’ ») 2 + (c ’ (a + b + 1)») ’ aby = 0, (5.2.2)

dx dx

where a, b, c are parameters which in our case are

1

a=b= c = 1.

2

212 CHAPTER 5. FAMILIES AND MODULI SPACES

It was Gauß again who wrote a holomorphic solution of (3.2) (which is

unique up to a scalar multiple) in the disk |»| < 1, which is given by

the following power series

∞

(a)i (b)i i

F (a, b, c; ») = 1 + »,

i!(c)i

i=1

where the symbol (d)i stands for d(d + 1) · · · (d + i ’ 1). Since the

function F (») we are looking for is indeed holomorphic, we have

11

F (») = κF ( , , 1; »),

22

where κ is a number. To ¬nd κ we compute

dx

∞

√

κ = F (0) = 2 = 2π.

x x’1

1

In fact, F (») can be continued to a multi-valued meromorphic function.

The hypergeometric equation 5.2.2 is an example of a so-called al-

gebraic di¬erential equation, since all its coe¬cients are polynomials in

». This equation gives rise to a local system S of complex vector spaces

over C \ {0, 1}. Equivalently, we can describe a locally constant sheaf S

of vector spaces over C \ {0, 1}. Let us take an open set U ‚ C \ {0, 1},

then S(U ) consists of holomorphic functions y(») on U which are solu-

tions of 5.2.2. If U is path-connected then dim S(U ) ¤ 2. For a small

contractible disc ∆µ of radius µ whose center is di¬erent from 0 and 1,

S(∆µ ) has dimension 2 since 5.2.2 is of order 2.

Let us understand what is the ¬ber S» of the local system at some

point ». If we take µ small enough then S» = S(∆µ ), or, the same, S»

consists of germs of holomorphic functions at » which are solutions of

5.2.2, so dim S» = 2. If we have two points » and µ connected by a

path γ then we have the parallel transport isomorphism Tγ : S» Sµ

de¬ned as follows. Let us cover γ by a ¬nite number of small discs and

using the Cauchy analytic continuation propagate the solution of 5.2.2

from the ¬rst disc (containing ») to the last disc (containing µ). The

same thing to say, one can take the universal covering space C \ {0, 1}

˜

of C\{0, 1}, which is in fact contractible. Then we choose » ∈ C \ {0, 1}

˜

which is a lift of » and we propagate the solution of 5.2.2 from » to the

whole C \ {0, 1} unambiguously.

5.2. THE LEGENDRE FAMILY OF ELLIPTIC CURVES 213

The fundamental group of C\{0, 1} is a free group in two generators

γ1 and γ2 . Here we take γ1 to be a loop around 0 and γ2 to be a loop

around ∞. For our local system S on C \ {0, 1} we have a monodromy

representation

ρ : π1 (C \ {0, 1}, ») ’ Aut(S» ) GL(2, C).

If we return to the geometric situation, then we have the Legendre

family of elliptic curves

X» ’ X

“ “ ,

» ∈ C \ {0, 1}

then we have a local system G over C \ {0, 1} with the ¬ber G» =

H1 (X» , C). Now we would like to identify the local systems S and G.

We note that the monodromy of the local system G is explicitly

computable. We have the following commutative diagram

ρ

π1 (C \ {0, 1}, ») ’ GL(2, C)

∪

GL(2, Z)

10 12

with ρ(γ1 ) = , ρ(γ2 ) = .

21 01

We note here the following relation with the theory of modular

forms. We may identify C \ {0, ∞} with the quotient of the upper half

plane H by the group “(2)/ ± 1, where “(2) ‚ SL(2, Z) consists of the

matrices congruent to Id modulo 2. This identi¬cation is accomplished

by a modular function » with respect to “(2). The group “(2) is gen-

10 12

erated by the matrices and , which each generate the

21 01

stabilizer of a cusp (the fundamental domain for “(2) has three cusps).

We refer to the book of Chandrasekharan [15] for details.

Now we construct an isomorphism of local systems φ : G ’S. Such

˜

an isomorphism amounts to isomorphisms φ» : G» ’S» for all » ∈

˜

C \ {0, 1}. It is required that for any », µ ∈ C \ {0, 1} and any path γ

214 CHAPTER 5. FAMILIES AND MODULI SPACES

connecting them we have

φ»

G» S»

Tγ “ “ Tγ .

φµ

Gµ Sµ

We shall use the relative holomorphic 1-form ω = dx/y. We have that

φ» (γ» ) is the function y(µ) = γµ ω holomorphic in a disc |µ ’ »| < µ.

Also y(µ) is a solution of 5.2.2 and we use

d

ω= [ω].

d

dµ dµ

γµ γµ

It is obviously compatible with Tγ . Now to see the isomorphism, it

is enough to show that φ» is injective, because both G» and S» are

complex vector spaces of dimension 2. Assume that γ» is in the kernel

of φ» . Then

ω = 0 for any µ s.t. |µ ’ »| < µ.

γµ

It follows that γµ d [ω] = 0 as well. We recall that d [ω] is a linear

dµ dµ

combination of [ω] and [·] with non-zero coe¬cients. Therefore [ω] and

1

d [ω] generate H (Xµ , C) and we conclude that

γµ [±] = 0 for any

dµ

[±] ∈ H 1 (Xµ , C). Thus γµ = 0 and the two local systems S and G have

the same monodromy representations.

5.3 Deformation of complex structures on

a complex manifold.

To start with we shall study the complex-linear structures on a vector

R2n . If we identify V Cn then we have a complex structure

space V

J, J 2 = ’Id. Let us denote by CS(V ) the set of complex structures

on V . We have

CS(V ) AutR (V )/AutC (V ) GL(2n, R)/GL(n, C),

5.3. DEFORMATION OF COMPLEX STRUCTURES 215

so that dim CS(V ) = 4n2 ’ 2n2 = 2n2 . In fact, CS(V ) is a complex

manifold. We will show this by embedding ψ : CS(V ) ’ G(n, 2n) into

n-dimensional complex subspaces of C2n by letting

the grassmanian of √

ψ(J ) = Ker(J + ’1Id). We have the direct sum decomposition

¯

C2n = H • H into the sum of ”holomorphic” and ”anti-holomorphic”

√ √

¯

parts, where H = Ker(J + ’1Id) and H = Ker(J ’ ’1Id). Then

the tangent space to CS(V ) at a point J is given by

¯

TJ CS(V ) = Hom(H, H).

Now let (X, J) be a complex manifold, and let us consider the set

ACS(X) of almost complex structures on X (i.e. bundle maps J :

T X ’ T X with J 2 = ’Id). We may view ACS(X) as an in¬nite-

dimensional Fr´chet manifold. Its tangent space is described similarly

e

to the above as

TJ ACS(X) = Hom(Th X, Th X),

where Th X by de¬nition is the holomorphic tangent bundle to X.

Equivalently, TJ ACS(X) consists of 1-forms of type (0, 1) with values

in Th X. A complex structure J : T X ’ T X de¬nes by transposi-

tion a complex structure t J on the cotangent bundle T — X. Now the

corresponding d operator is de¬ned on functions as

√

(Id + ’1 t J)

d= d,

2

√

(Id+ ’1 t J)

since is the projection operator of the space of 1-forms onto

2

the 1-forms of type (0, 1). The operator d determines the complex

structure J. We have the elementary

LEMMA 5.3.1 The map J ’ dJ from the space ACS(X) to the

vector space of operators C ∞ (X) ’ A1 (X) is holomorphic.

Now if we consider the set of integrable complex structures CS(X) ‚

ACS(X), it is easy to see that J ∈ CS(X) if and only the operator

d has square 0. From this we can see that TJ CS(X) is the set of

1-forms ± of type (0, 1) with values in Th X such that d ± = 0. Indeed,

216 CHAPTER 5. FAMILIES AND MODULI SPACES

let us consider an in¬nitesimal family J = J + ± of almost complex

structures. The corresponding operator is

d = d + A,

where A operates on di¬erential forms as follows: write

A= ωj — ξj ,