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Chapter 5

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-
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
’ Y — CPN
f p1 ,
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


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
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

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
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

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

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

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

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
DEFINITION 5.1.3 Let Y be a topological space. A sheaf F of
abelian groups on Y consists of the following data:

(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
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)

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
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 .
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

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
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 ). (—)

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


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