Families and moduli spaces

Everything should be made as

simple as possible, but not more so.

Albert Einstein

5.1 Families of algebraic projective mani-

folds

Let Y be an arbitrary complex manifold (e.g.the punctured unit disc

∆— = {z ∈ C, 0 < |z| < 1}). Let us assume that we have a submersion

f:

’ Y — CPN

X

f p1 ,

Y

where p1 is the projection onto the ¬rst factor. It follows that ¬bers

of f are smooth projective manifolds. The basic question here is what

happens to the cohomology group of ¬ber as we vary y ∈ Y . We

introduce the notion of local system of abelian groups on Y . For each

point y ∈ Y we put into correspondence an abelian group Ay in such

a way that if γ : [0, 1] ’ Y is a path with γ(0) = y, γ(1) = y then

we have a parallel transport isomorphism Tγ : Ay Ay which depends

only on the homotopy class of the path γ with ¬xed end-points. It is

199

200 CHAPTER 5. FAMILIES AND MODULI SPACES

assumed that Tγ—γ = Tγ Tγ , where γ — γ denotes the composition of

the paths γ and γ . Let Xy stand for f ’1 (y).

LEMMA 5.1.1 The groups H m (Xy , Z) form a local system on Y .

Proof. Let γ, γ be two homotopic paths with ¬xed end-points. Then

we have a map h : ’ Y , where = [0, 1] — [0, 1] given by the

homotopy. Next we pull-back the ¬bration H m (Xy , Z) to via h— and

apply a theorem of Ehresmann which asserts that it is a trivial ¬bration

(we notice that is contractible). Since h maps the whole left side of

to y and the whole right side to y , we get the statement.

To simplify things, let us consider a smooth proper submersion f :

X ’ Y between two smooth manifolds such that the ¬bers Xy =

f ’1 (y), y ∈ Y are compact manifolds. Now and further we assume

that the base (here it is Y ) is always connected. The cohomology

groups H i (Xy ) (with Z, R, or C coe¬cients) form a local system of

abelian groups over Y . Let γ : [0, 1] ’ Y be a path with γ(0) = y1 ,

and γ(1) = y2 . The parallel transport map Tγ : H i (Xy1 ) H i (Xy2 )

has the following properties:

(1) Tγ only depends on the homotopy type of γ

(2) Tγ—γ = Tγ —¦ Tγ , where γ — γ is the natural composition of two paths

γ and γ .

Those 2 properties constitute a de¬nition of a local system of abelian

groups.

Let us pick a point y ∈ Y . One can interpret a local system in

terms of a representation ρ : π1 (Y, y) ’ Aut(H i (Xy )), which is called

the monodromy representation attached to the local system H i (Xy ).

LEMMA 5.1.2 The representation ρ completely determines the local

system.

Proof. We just need to reconstruct the group H i (Xy for any y ∈ Y .

Let us choose a path γ connecting y and y . Let us de¬ne a group Ay

by

H n (Xy ) — {γ}.

Fy =

paths γ:y’y

If we have two such paths γ and γ then γ — (γ )’1 is a loop based at y.

For ± ∈ H i (Xy ) we identify ± — {y} with ρ(γ — (γ )’1 )(±) — {γ }. Let

5.1. FAMILIES OF ALGEBRAIC PROJECTIVE MANIFOLDS 201

H i (Xy )—{γ}

Ay = Fy / ∼, where ∼ is this identi¬cation. As sets Ay

for any choice of a path γ. Thus Ay is a group, and to get a local

system, we need to construct the parallel transport map Tγ for any

path γ. Clearly, it is su¬cient to work with paths starting at y. Then

we construct

Tγ : H i (Xy ) ’ Ay H i (Xy ) — {γ}

by Tγ (±) = ± — {γ}. We conclude that the local systems of abelian

groups Ay on Y are in 1 ’ 1 correspondence with the representations

ρ : π1 (Y, y) ’ Aut(Ay ).

Example. Let us take Y = ∆— , consisting of z ∈ C such that 0 <

µ

|z| < µ, where µ is a positive real. Let f : X ’ ∆— be as above. Let

µ

—

π1 (∆µ ) Z be generated by a loop l. Then Tl ∈ Aut(H(Xy )) is called

the monodromy automorphism.

Given a local system (Ay , Tγ ) we can take the sections over any open

set U ‚ Y . The group of sections A(U ) consists of all elements

ay ∈ Ay

y∈U

satisfying the following property. For any path γ : [0, 1] ’ U and

the corresponding parallel transport map Tγ : Aγ(0) ’ Aγ(1) we have

Tγ (aγ(0) ) = aγ(1) .

Example. Let U be contractible, y ∈ U , then A(U ) Ay

Example. Let U = ∆— , then A(∆— ) = AT , the space of invariants,

µ µ y

T

i.e. Ay = {a ∈ Ay : T a = a}, where T : Ay ’ Ay is the monodromy

automorphism.

The group A(U ) is called the group of sections of the local system

over U . By viewing a local system as the law associating a group A(U )

to any open set U ‚ Y , we enter a more general setup for which we

need

DEFINITION 5.1.3 Let Y be a topological space. A sheaf F of

abelian groups on Y consists of the following data:

202 CHAPTER 5. FAMILIES AND MODULI SPACES

(i) For every open set U ‚ Y , an abelian group F(U ).

(ii) For every inclusion V ‚ U between open sets, a group homomor-

phism RU,V : F(U ) ’ F (V ), RU,V is called the restriction map (from

U to V ).

One requires the following four conditions.

1. F of the empty set is the group with one element.

2. RU,U is the identity map.

3. If W ‚ V ‚ U are open sets then RU,W = RV,W RU,V .

4. For every open covering (Ui )i∈I of U and for every family (si )i∈I ,

si ∈ F(Ui ) such that

RUi ,Ui ©Uj (si ) = RUj ,Ui ©Uj (sj ), for all i, j ∈ I

there exists a unique s ∈ F(U ) such that RU,Ui (s) = si .

An element of F(U ) is called a section of F over U .

Therefore one observes that a local system gives rise to a sheaf of

abelian groups. Let us consider the special case of local system of vector

spaces (each Ay is a vector space). It produces a sheaf of vector spaces.

(I.e. when each F(U ) is a vector space and RU,V are linear maps.)

Moreover, we shall consider a locally constant sheaf of vector spaces,

meaning that if y ∈ U and U is contractible then A(U ) Ay . An

i

example of such thing is the above situation when Ay = H (Xy ). Now

we state our main correspondence result of this section.

local systems of ←’ representations of

vector spaces π1 (Y, y)

locally constant sheaves ←’ vector bundles over Y

of vector spaces with integrable connections

In the above diagram all the correspondences are one-to-one.

Let us recall that if E is a vector bundle over Y then a connection

on E is a C-linear map “(U, E) ’ “(U, T — U — E), where U is an

open subset of Y satisfying

- (s1 + s2 ) = (s1 ) + (s2 ), si ∈ “(U, E)

5.1. FAMILIES OF ALGEBRAIC PROJECTIVE MANIFOLDS 203

s + df — s, s ∈ “(U, E), f ∈ C ∞ (U ).

- (f s) = f ·

Example Let E = Y —Rn be the trivial bundle and let A be a one-form

with values in n — n matrices over R. Let s = f = (f1 , ..., fn )t be a

section of E, then we let s = f = df + Af . One easily veri¬es that

de¬ned in such a way is a connection on E, and all connections on

E may be obtained in such a way.

The curvature K( ) of a connection is a 2-form with values in

End(E):

K( ) ∈ “(Y, Λ2 T — Y — End(E)).

The curvature K( ) is the obstruction of ¬nding locally n linearly inde-

pendent horizontal (vanishing under the application of the connection)

sections of the vector bundle. Let us write explicitly a local expression

for K( ) (so we are in the situation of the above Example). We are

trying to solve f = 0. We have 0 = f = df + Af , thus df = ’Af .

Now

0 = ddf = ’dA§f + A§df = ’(dA + A§A)§f = ’K( )§f .

Hence we de¬ned the curvature K( ) = dA + A§A.

DEFINITION 5.1.4 A connection is called integrable or ¬‚at if

and only if K( ) = 0.

Now let be an integrable connection on a vector bundle E over

Y . For an open set U ‚ Y we let A(U ) be the vector space of sections

{s ∈ “(U, E), (s) = 0}. It is clear that A de¬nes a locally constant

sheaf of vector spaces, i.e. a local system of vector spaces. Conversely,

given a local system of vector spaces we get a vector bundle with an

integrable connection (what is called ¬‚at bundle).

Next we explain the left arrow in the diagram. Let us have a rep-

resentation ρ : π1 (Y, y) ’ Aut(V ), where V is a real vector space.

˜

Let Y ’ Y be a universal covering space on which π1 (Y, y) acts

˜

by deck transformations. We take the trivial bundle Y — V with

˜

the connection f = df . The group π1 (Y, y) acts on Y — V via

˜

[γ](˜, v) = ([γ]˜, ρ([γ])v). Let us take the quotient of Y — V with re-

y y

spect to this action, which is naturally a vector bundle V over Y . Since

204 CHAPTER 5. FAMILIES AND MODULI SPACES

the connection = d is integrable and π1 (Y, y)-invariant, it descends

to an integrable connection on V .

Now let us be given a local system of real vector spaces H j (Xy , R),

and let us want to construct the corresponding ¬‚at vector bundle E over

Y . We shall do so by exhibiting the space “(U, E) as a C ∞ (U )-module

for any open U ‚ Y .

For any ¬ber Xy of a smooth proper submersion f : X ’ Y the

group H j (Xy , R) is the j-th cohomology group of the de Rham complex

d d d d

· · · ’ Aj’1 (X) ’ Aj (X) ’ Aj+1 (X) ’ · · · .

We introduce the notion of ¬berwise (or relative) di¬erential forms on

X. First we give a local de¬nition. Let (y1 , ..., yk ) be a local coordinate

system on Y and let (x1 , ..., xn’k , y1 , ..., yk ) be a local coordinate system

on X. Then we call a form ω ¬berwise if it locally looks like

ω= fi1 ...ij (x, y)dxi1 § · · · §dxij .

i1 <···<ij

We will denote the space of relative di¬erential forms by A• (X/Y ). It

is easy to see that the global de¬nition would be

A• (X/Y ) := A• (X)/I,

where I is the ideal in A• (X) generated by all dg, g ∈ f — C ∞ (Y ). Since

clearly dI ‚ I, we have the relative complex

d d d d

· · · ’ Aj’1 (X/Y ) ’ Aj (X/Y ) ’ Aj+1 (X/Y ) ’ · · · .

The cohomology of this complex H j (X/Y ) is called relative de Rham

cohomology.

Due to the fact that for g ∈ C ∞ (Y ), ± ∈ Aj (X/Y ) one has d(g±) =

gd±, the map d : Aj (X/Y ) ’ Aj+1 (X/Y ) is a map of C ∞ (Y )-modules.

This is why for any open U ‚ Y the space H j (f ’1 (U )/U ) is a C ∞ (U )-

module. Thus there exists unique vector bundle E over Y such that

the space of sections “(E, U ) = H j (f ’1 (U )/U ). Moreover, for a small

enough open set U , this space is a free C ∞ (U )-module:

H j (U — Xy /U ) = H j (Xy ) —R C ∞ (U ). (—)

5.1. FAMILIES OF ALGEBRAIC PROJECTIVE MANIFOLDS 205

We also have a ¬‚at Gauß-Manin connection on E (connections

with this name usually appear in an algebraic context). First we de¬ne

it locally using (—). We simply let (ξ —g) = ξ —dg. It glues nicely to a

globally de¬ned ¬‚at connection (since d2 = 0). The second description

of uses periods. Let ± be a section of the vector bundle E over

an open U ‚ Y . For any y ∈ Y we consider the restriction to the

¬ber ±y ∈ H j (Xy ). Let us have a locally constant family of cycles γ

parametrized by U so that γy ∈ Hj (Xy , Z). Then ± is a 1-form on U

with values in E. We have

( ±) = d( ±),

γ γ

where γ ± is a function on U : y ’ γy ±y .

We have a (complex) vector bundle E = E j over Y with ¬ber Ey =

H j (Xy , C). For an open set U ‚ Y the space of sections

“(U, E) = H j (f ’1 (U )/U ).

So any section of E j over U is represented by a di¬erential form ± on

f ’1 (U ) such that (d±)|Xy = 0 for any y ∈ U . Let us consider a vector

˜

¬eld ξ on U , and let us lift ξ to a vector ¬eld ξ on f ’1 (U ) in such a

˜

way that df (ξx ) = ξf (x) for any x ∈ f ’1 (U ). First, we do it locally

using local triviality of a ¬bration and then we apply partition of unity

to obtain a vector ¬eld over f ’1 (U ). In a situation like this the vector

˜