˜

Next we de¬ne the Lie derivative L(ξ) : Ak (X) ’ Ak (X) by

d

˜ ˜

L(ξ)β = [exp(tξ)β]t=0 .

dt

˜

It can be also de¬ne using the interior product i(ξ)β ∈ Ak’1 (X), which

in turn is given by

˜ ˜

(i(ξ)β)(·1 , ..., ·k’1 ) = β(ξ, ·1 , ..., ·k’1 ).

Now the Cartan homotopy formula tells us that

˜ ˜ ˜

L(ξ) = d —¦ i(ξ) + i(ξ) —¦ d.

206 CHAPTER 5. FAMILIES AND MODULI SPACES

˜

We notice that if β is a closed di¬erential form on X then L(ξ)β is an

exact form.

Using Lie derivative it is possible to give another de¬nition of Gauß-

Manin connection. For ± ∈ Aj (f ’1 (U )) such that (d±)|Xy = 0 we let

˜

ξ [±] = [L(ξ)±].

LEMMA 5.1.5 The right hand side of this formula does not depend

˜

on the lift ξ.

Proof. Two possible lifts of ξ di¬er by a vector ¬eld · on X tangent to

the ¬bers of f . To prove the Lemma we need to show that L(·)±)|Xy is

exact. But this follows from the Cartan homotopy formula since L(·)

is tangent to Xy .

Example. Let Y = H = {„ ∈ C, Im(„ ) > 0} be the Lobachevsky

upper-half plane. Let us consider the following ¬bration f : X ’ Y ,

where X = (C — H)/Z2 and Z2 acts on C — H as (m, n)(z, „ ) = (z +

m + n„, „ ). A ¬ber X„ = f ’1 („ ) = C/(Z • Z.„ ) is an elliptic curve. We

claim that the relative cohomology group is

H 1 (X/Y, C) = C ∞ (Y, C)dz • C ∞ (Y, C)d¯.

z

The only thing which needs to be shown is that dz and d¯ are lin-z

1 ∞

early independent generators of H (X/Y, C) as of C (Y )-module. Let

families of cycles γ1 , γ2 locally represent the standard basis of the ¬rst

homology of the ¬ber. The linear independence follows from

√

dz d¯

z 1 1

γ1 γ1

det = det = ’2 ’1Im(„ ) = 0.

γ2 dz γ2 d¯

z „ „

¯

Let us pick a basis (e1 , e2 ) of H 1 (X/Y, C) such that ej = δij . Then

γi

one has ej = 0 and

dz)e2 = e1 + „ e2 ∈ H 1 (X/Y, C)

dz = ( dz)e1 + (

γ1 g2

√

’1

d¯ = e1 + „ e2 ,

z ¯ dz = (d„ ) — e2 = ’ [d„ — dz ’ d„ — d¯],

z

2Im(„ )

5.1. FAMILIES OF ALGEBRAIC PROJECTIVE MANIFOLDS 207

√

dz ’ d¯

z ’1(dz ’ d¯)

z

since e2 = =’ .

„ ’„¯ 2Im(„ )

We notice that dz is of type (1, 0) as a 1-form on Y .

If v is a vector ¬eld on Y = H then

v (±) = v, ±

is a section of the corresponding vector bundle. For instance,

‚f

(f („ )dz) = dz.

‚

‚„

‚„

¯

We shall further consider the case of a proper submersive holomor-

phic mapping

X

f“

Y

between two complex manifolds X and Y such that f has compact

K¨hler ¬bers. More precisely we assume the existence of a closed rel-

a

ative 2-form ω, [ω] ∈ H 2 (X/Y ), dω = 0 ∈ A3 (X/Y ) such that ωy

is K¨hler . We will refer to this situation as a holomorphic family of

a

compact K¨hler manifolds. For each y ∈ Y we have the Hodge decom-

a

position H j (Xy , C) = •p+q=j H p,q (Xy ), therefore for an open U ‚ Y

one has

H j (f ’1 (U )/U, C) = •p+q=j Hp,q (f ’1 (U )/U ),

where Hp,q (f ’1 (U )/U ) is a family of cohomology classes of type (p, q).

Let ± ∈ Hp,q (f ’1 (U )/U ) and let v be a vector ¬eld on U . We have the

following important properties

(1) Gri¬ths™ transversality theorem

∈ Hp+1,q’1 (f ’1 (U )/U ) • Hp,q (f ’1 (U )/U ) • Hp’1,q+1 (f ’1 (U )/U ).

v±

(2) If v is an anti-holomorphic vector ¬eld then

∈ Hp+1,q’1 (f ’1 (U )/U ) • Hp,q (f ’1 (U )/U ).

v±

(3) Similarly, if v is an holomorphic vector ¬eld then

∈ Hp,q (f ’1 (U )/U ) • Hp’1,q+1 (f ’1 (U )/U ).

v±

208 CHAPTER 5. FAMILIES AND MODULI SPACES

For brevity, when it is clear what ¬bration is considered we will omit

arguments in parenthesis and simply write Hp,q in place of Hp,q (f ’1 (U )/U ).

Statements (1)-(3) will be proved later after we discuss the Kodaira-

Spencer map.

We also note the following important observation. Let Y be a com-

plex manifold and let E be a complex vector bundle over Y equipped

with an integrable connection . Then E has a natural structure of

a holomorphic vector bundle. To endow E with such a structure it is

enough to produce a local basis of its holomorphic sections. For this

we simply take its local basis of horizontal sections and announce them

to be holomorphic. If we have two intersecting open sets U and V with

local basis of horizontal sections (ti ) and (sj ) respectively then we have

some transition functions (aij ) satisfying

ti = aij sj .

j

It is easy to see that each function aij is just a constant function (be-

cause ti = sj = 0). Therefore the local holomorphic sections which

we have chosen to agree on any intersection U © V in the sense that the

transition functions are holomorphic (even constant). Therefore the

vector bundle E j of j-th relative cohomology is a holomorphic vector

bundle.

Let us pick an integer p such that 0 ¤ p ¤ j. There exists a

holomorphic vector subbundle F p E j ‚ E j such that locally its space

of sections is given by

“(U, F p E j ) = Hp,q • Hp+1,q’1 • · · · • Hj,0 .

The ¬ltration {F p E j } is a holomorphic decreasing ¬ltration of the bun-

dle E j , which is called the Hodge ¬ltration of E j .

5.2 The Legendre family of elliptic curves

Let Y = C \ {0, 1}, and let f : X ’ Y be the Legendre family of

elliptic curves, where X ‚ Y — CP2 and the ¬ber X» is given by

2

Z1 Z2 = Z0 (Z0 ’ Z2 )(Z0 ’ »Z2 ), where [Z0 : Z1 : Z2 ] are homoge-

neous coordinates on CP2 and » is the parameter on C \ {0, 1}. There-

fore we have f (Z0 , Z1 , Z2 ; ») = ». If we delete the point at in¬nity

5.2. THE LEGENDRE FAMILY OF ELLIPTIC CURVES 209

0 0

O = [0 : 1 : 0] from X» we get an open curve X» . One thinks of X»

as of an a¬ne curve in C2 given by the equation y 2 = x(x ’ 1)(x ’ »).

We have x = Z0 , y = Z1 . Now we construct an algebraic model for the

Z2 Z2

cohomology of the ¬bers

H 1 (X» , C) = H 1,0 (X» ) • H 0,1 (X» ).

We notice that X» can be considered as an algebraic group with the

point O serving as the identity.

First of all, it is an easy exercise to check that ω = dx gives a global

y

1,0

holomorphic 1-form on X» . Therefore, H (X» ) = ω · C. Besides,

· = x dx is a holomorphic 1-form on the open set X» . From the exact

0

y

sequence

Res

0 ’ H 1 (X» , C) ’ H 1 (X» , C) ’ C ’ 0,

0

where Res maps the class of a form ± to the residue of ± at the point O,

we see that [·] is in the kernel of Res, because the sum of all residues

is equal to zero and · has only one pole. Therefore, · comes from

H 1 (X» , C) and moreover ([ω], [·]) is a basis of H 1 (X» , C).

It was Karl Friedrich Gauß who ¬rst computed the periods γ» ω,

where γ» is a locally constant family of cycles, |»| < µ, and γ» is a cycle

in X» which is contant as a function of » and is such that for » real γ»

is the oval going through the point (x = 1, y = 0).

LEMMA 5.2.1 There exist rational functions p(») and q(») such that

)2 [ω] + p(»)

( [ω] + q(»)[ω] = 0.

d d

d» d»

This equation relates sections of the vector bundle E 1 over C \ {0, 1}.

Besides, this equation implies the following relation on the base:

d2 d

( ω|X» ) + p(») ( ω|X» ) + q(»)( ω|X» ) = 0.

d»2 d»

γ» γ» γ»

Let ξ = d/d» be a vector ¬eld on Y = C \ {0, 1}. We can lift ξ to a

˜

meromorphic vector ¬eld ξ on X - the Legendre family. Moreover, we

can do it in such a way that all the poles will be along x = 0, 1, » or at

in¬nity. We need the following simple

210 CHAPTER 5. FAMILIES AND MODULI SPACES

LEMMA 5.2.2 Let S be a compact Riemann surface and let x1 , ..., xl

be l distinct points on S. Then the natural map

H 1 (S, C) ’ H 1 (S \ {x1 , ..., xl }, C)

is injective.

On a dense open set of X we can use (x, ») as a system of holomorphic

‚/ ‚/

coordinates so that ‚» and ‚x are linearly independent. The vector

¬eld d» on Y lifts to the vector ¬eld ‚/ on X. Since we know how y

d/

‚l

is expressed in terms of x and »: y 2 = x(x ’ 1)(x ’ ») we know that

‚

d ω is represented by the relative 1-form L( )ω where

‚»

d»

‚y ’1

‚ dx

L ω= dx = .

‚» ‚» 2y(x ’ »)

As the next step, we express this form in terms of ω and · using an

equality of the type

dx

= a· + bω + dg,

y(x ’ »)

where g is some meromorphic function. Let us take g(x) = y/(x ’ »),

then keeping in mind that » is considered as a ¬xed parameter and

y 2 = x(x ’ 1)(x ’ ») we compute

d(y 2 )

y ydx

dg(x) = d( )= ’ =

2y(x ’ ») (x ’ »)2

x’»

x2 ’ 2x» + » cdx

= dx = ’a· ’ bω + ,

2y(x ’ ») y(x ’ »)

where a = ’1/2, b = »/2, c = »(1 ’ »)/2. Therefore in H 1 (X» ) or in

the vecto bundles E we have

dx [·] [ω]

[ω] = [ ]=’ + . (5.2.1)

d

2y(x ’ ») 2»(1 ’ ») 2(1 ’ »)

d»