Acta Applicandae Mathematicae 75: 183194, 2003.

2003 Kluwer Academic Publishers. Printed in the Netherlands.

183

Monodromy of Variations of Hodge Structure

C. A. M. PETERS1 and J. H. M. STEENBRINK2,
1 Department of Mathematics, University of Grenoble I, UMR 5582 CNRS-UJF,

38402-Saint-Martin dHres, France. e-mail: chris.peters@ujf-grenoble.fr

2 Department of Mathematics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen,
The Netherlands. e-mail: steenbri@sci.kun.nl
(Received: 4 April 2002)
Abstract. We present a survey of the properties of the monodromy of local systems on quasiprojective varieties which underlie a variation of Hodge structure. In the last section, a less widely
known version of a NoetherLefschetz-type theorem is discussed.
Mathematics Subject Classifications (2000): 14D07, 32G20.
Key words: variation of Hodge structure, monodromy group.

1. Introduction
k
(X) of a smooth projective variety X
The kth primitive cohomology group Hprim
is known to carry a weight k polarized rational Hodge structure. For a family of
those over a smooth connected base manifold S, the cohomology groups glue
together to a local system on S. Such a local system is completely determined
by its monodromy representation, i.e. the induced representation of 1 (S, s0 ) on
the kth cohomology group V of the fibre at s0 . For a projective family, i.e. one
which is locally embeddable in S PN (projection onto the first factor giving the
family), the polarizations glue together so that the monodromy representation lands
in Aut(V , Q), where Q is the polarization. The polarized Hodge structures on the
primitive kth rational cohomology groups of the fibre glue to a polarized variation
of weight k rational Hodge structures. All of these notions will be reviewed in
Sections 2, 3 and 4. For the moment, think of the latter as a certain filtration on
the complexification of V that varies holomorphically with s (and satisfies some
additional properties to be made explicit later).
The main question that we are considering is what are the restrictions on a local
system if it underlies a polarized variation of weight k Hodge structures?
In examples coming from geometry, the base manifold S is often a smooth
quasi-projective variety. In that case, it is customary to choose a smooth projective
compactification S such that the complement D = S \ S is a divisor with normal
crossings, i.e. locally D has an equation z1 z2 za = 0, where z1 , . . . , zs are local
 The second author thanks the University of Grenoble I for its hospitality and financial support.

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coordinates in S. Then S looks like U = ( )a sa , where  is a unit disk

in C and  =  \ {0}. The local monodromy operators are then given by the
images in Aut(V , Q) under the monodromy representation of the a loops around
the origin (in counter clockwise direction) that generate the fundamental group
of this open set U . In fact, for most of what follows S could be Zariski-open in
a compact analytic manifold and we can then choose a smooth compact analytic
compactification S by adding a suitable normal crossing divisor. Any such S will
be called a compactifiable complex manifold. Note that a given complex manifold
may have several compactifications, which are not even bimeromorphic! For an
example due to Serre, see [11, Ex. VI.3.2].
In the setting of variations of polarized rational Hodge structure over a compactifiable manifold S, there are serious restrictions on the monodromy which we
recall in Section 5 such as the Monodromy Theorem, the Theorem of the Fixed
Part and a very restrictive finiteness result saying that there are at most finitely
many conjugacy-classes of N-dimensional representations that yield local systems
underlying a polarized variation of Hodge structure (of any weight) over a given
S. In Section 6 we briefly discuss the proof from [2] that the locus where a given
integral cycle of type (p, p) stays of type (p, p) is algebraic when S is algebraic.
We also relate the monodromy group and the MumfordTate group of the Hodge
structure at a very general stalk.
The classical NoetherLefschetz theorem tells us that the only algebraic cycles
on a generic surface in P3 of degree
4 are the hypersurface sections. In Section 7, we generalize this to polarizable variations of Hodge structures with big
monodromy. See Theorem 17. This result is folklore among experts, but we think
that it is useful to dispose of an elementary proof.

2. Local Systems and Monodromy

Given a ring A, a local system of A-modules can be defined for any topological
space S: it is a sheaf V of AS -modules with the property that every point x S has
a neighbourhood U such that the natural map V(U ) Vx is an isomorphism. This
is a functorial construction: any continuous map f : T S induces a morphism
of ringed spaces (T , AT ) (S, AS ) and for any local system V of A-modules on
S, the pull back f (V) is a local system of A-modules on T .
If S is locally and globally arcwise connected the monodromy representation of
V can be defined as follows. For any arc : [0, 1] S the pull-back V is a local
system of A-modules on [0, 1]. In particular, we obtain a canonical identification
of ( V)0  V (0) with ( V)1  V (1) . Choosing a base point s0 S this yields
the monodromy representation
V : 1 (S, s0 ) AutA (Vs0 ).
The image of is called the monodromy group of the local system V.

MONODROMY OF VARIATIONS OF HODGE STRUCTURE

185

The construction of the monodromy representation from a local system has an

inverse: let us start with a representation
: 1 (S, s0 ) AutA (V )
s0 ) (S, s0 ) be a universal covering space for
for some A-module V . Let p: (S,
(S, s0 ). Then we define a sheaf of AS -modules V on S as the sheaf associated to the
presheaf which to U S open associates the set of all locally constant functions f :
p 1 (U ) V such that f ( (y)) = ( )f (y) for all y p 1 (U ), 1 (S, s0 ).
These two constructions define an equivalence between the category of local
systems of A-modules on S and the category of modules over the group ring
A[1 (S, s0 )]. We refer to [3] for more details.
EXAMPLE 1. A family p: X S of compact differentiable manifolds is nothing
but a proper submersion between differentiable manifolds. The kth cohomology
group V = H k (Xs ) of the fibre Xs at s defines a local system of Abelian groups
of finite type on S. If S is connected, fixing s0 we obtain the monodromy group of
the family as the associated representation of 1 (S, s0 ) on H k (Xs ).
Monodromy representations need not be fully reducible, but in the sequel we
mainly consider those. Irreducible representations give indecomposable local systems. If we have a representation on a Q-vector space which stays irreducible under
field-extensions we say that the representation is absolutely irreducible.
There is one particular type of such representations, namely representations
with big monodromy group in the following sense.
DEFINITION 2. Let V be a local system of Q-vector spaces on S with monodromy representation
: 1 (S, s0 ) Gl(V ),

V := Vs0 .

(1) The smallest algebraic subgroup of Gl(V ) containing the monodromy group
(1 (S, s0 )), i.e. the Zariski-closure of the monodromy group, is denoted
(1 (S, s0 ))cl .
(2) Suppose that in addition V carries a nondegenerate bilinear form Q which
is either symmetric or anti-symmetric and which is preserved by the monodromy
group. The latter is said to be big if the connected component Hs0 of (1 (S, s0 ))cl
acts irreducibly on VC .
The proof of the following lemma is left to the reader:
LEMMA 3. Suppose V is a local system on a connected complex manifold S with
big monodromy group. Suppose that : S  S is a finite covering map. Then V
has also big monodromy.

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C. A. M. PETERS AND J. H. M. STEENBRINK

Indeed, the reader can check that the inclusion 1 (S  , s0 ) 1 (S, s0 ) (where
(s0 ) = s) induces an isomorphism Hs0  Hs0 . It follows that a local system with
big monodromy group stays indecomposable after pull-back by any finite covering
map.
To determine which monodromy groups are big we need a way to determine the
Zariski-closure of subgroups of Aut(V , Q); we cite from [7]:
LEMMA 4 Let V be a finite-dimensional complex vector space of dimension n
equipped with a nondegenerate symmetric bilinear form Q which is either symmetric or anti-symmetric. Let M Aut(V , Q) be an algebraic subgroup.
(1) If Q is anti-symmetric we suppose that M contains the transvections T : v 
v + Q(v, ), where runs over an M-orbit R which spans V . Then M =
Aut(V , Q) (= Sp(V )).
(2) If Q is symmetric, suppose that M contains the reflections R : v  v
Q(v, ) in roots , i.e. with Q(, ) = 2 which form an R-orbit spanning
V . Then either M is finite or M = Aut(V , Q) (= O(V )).
SPECIAL CASES. (1) Let V = VZ C, where VZ is a free finite rank Z-module
equipped with a nondegenerate anti-symmetric bilinear form. The Zariski-closure
inside Sp(V ) of the group Sp(VZ ) of symplectic automorphisms of the lattice VZ is
the full group Sp(V ). This follows from the fact that Sp(VZ ) contains all symplectic
transvections Tv , v VZ and for given nonzero VZ , the elements Tv , v VZ
span already V . It follows that the Zariski-closure of any subgroup of finite index
in Sp(VZ ) is also the full symplectic group.
(2) Let V = VZ C, where VZ is a free finite Z-module equipped with a nondegenerate symmetric bilinear form Q. If Q is definite, the orthogonal group preserving the lattice VZ is of course finite and equals its Zariski-closure. Hence it is
never big. In general it will contain reflections R in all roots VZ . Assuming
that these roots contain at least one orbit which spans the lattice, we conclude in the
indefinite case that the Zariski closure of Aut(VZ , Q) is the full orthogonal group.
These special cases can be used in the following geometric context.
EXAMPLE 5. Consider a smooth subvariety X of PN (C) of dimension n + 1
spanning PN (C). Its dual variety X consists of the hyperplanes H PN (C) such
that X H is singular. It is an irreducible hypersurface in the dual projective space
PN (C) . Let S denote its complement. Over S we have the tautological family of
hyperplane sections {Ys } of X. We have an orthogonal splitting (with respect to cup
product)
n
n
(Ys ; Q) Hfixed
(Ys ; Q),
H n (Ys ; Q) = Hvar
n
(Ys ; Q) is the part of H n (Ys ; Q) which is invariant under the monwhere Hfixed
odromy group of the family. By the invariant cycle theorem ([4, Th. 4.1.1])
n
(Ys ; Q) = Im[H n (X, Q) H n (Ys ; Q)].
Hfixed

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187

A Lefschetz pencil of hyperplane sections of X is a family {Yt }t $ where $ is a line

in PN (C) which intersects X transversely. By a theorem of Van Kampen [12],
the map 1 (S $, s) 1 (S, s) is surjective for a Lefschetz pencil. Hence the
monodromy groups for the families over S and over $ S coincide.
We take as a special set of generators for 1 (S $, s) the classes of loops based
at s surrounding each of the intersection points of $ and X in counterclockwise direction. The PicardLefschetz formula computes the action of such a generator and
n
(Ys ; Q)
the absolute irreducibility of the action of the monodromy group on Hvar
follows from the fact that all the associated vanishing cycles are conjugate. See [13,
Sect. 3] for details. In fact, Lemma 4 implies that the variations on the variable part
of the cohomology groups of the families under consideration have big monodromy
groups, unless the monodromy group is finite.
For Lefschetz pencils of hypersurfaces of Pn+1 or complete intersections of
n
(Ys ; C), Q the intercodimension p in Pn+p we can apply the above to V = Hvar
n/2
section pairing (if n is odd) or (1) times the intersection pairing (if n is even).
By the criterion, (1 (S, s))cl is the full symplectic group if n is odd and either
finite or the full orthogonal group if n is even. If n is even and (1 (S, s)) is finite,
Q must be definite. Observing that the variable cohomology in this case is just the
primitive cohomology, the Hodge Riemann bilinear relations tell us however that
p,q
the signature is (a, b) with a, resp. b the sum of the Hodge numbers dim Hprim with
n,n
= 0,
p even, resp. odd. Since even-dimensional hypersurfaces always have Hprim
p,q
the form Q can only be definite if all Hodge numbers h with p = q are zero. It
is not too hard to see [5] that this only happens for a quadric hypersurface, a cubic
surface or an even-dimensional intersection of two quadrics. So apart from these
three classes, the primitive cohomology of a complete intersection of projective
space gives a local system with big monodromy group.
3. Hodge Structures
Let us recall some definitions concerning Hodge structures. A Hodge structure
of weight k is a finitely generated Abelian group V together with a direct sum
decomposition


V p,q , with V p,q = V q,p the Hodge decomposition
VC =
p+q=k

on the complexification VC = V C. The decreasing Hodge filtration on VC is

defined by


V r,kr .
F p VC =
r
p

From this we see that

p

V p,kp  GrF VC = F p /F p+1

p

under the projection F p GrF .

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C. A. M. PETERS AND J. H. M. STEENBRINK

For a Hodge structure V we define its Weil operator as the automorphism C of

VC defined by Cv = i pq v for v V p,q . A polarization of a Hodge structure V of
weight k is a bilinear form Q: V V Z which is (1)k -symmetric and such
that for its C-bilinear extension to VC
(1) The orthogonal complement of F p is F kp+1 ;
(2) The Hermitian form (u, v)  Q(Cu, v) is positive definite.
The first condition is equivalent with Q(V p,q , V r,s ) = 0 for (p, q) = (s, r) and
that Q gives a perfect pairing between V p,q and V q,p .
In a similar way one can define Q-Hodge structures and R-Hodge structures.
The notion of morphism of Hodge structures is the obvious one: the complexification of the given group homomorphism should respect the Hodge decompositions.
It is sufficient to require that the Hodge filtrations be respected.
The category of Hodge structures has tensor products and internal Homs. The
category of polarized Q-Hodge structures is abelian and semisimple.
Let Z(k) denote the subgroup (2 i)k Z C = Z(k)k,k which is a Hodge
structure of weight 2k. Then a polarization Q of a Hodge structure V of weight
k induces a morphism of Hodge structures V V Z(k).
The cohomology groups of compact Khler manifolds are Hodge structures.
Their primitive cohomology groups are polarized R-Hodge structures. For projective manifolds the primitive cohomology groups are even polarized Q-Hodge
structures. Algebraic cycles of codimension p define (p, p)-classes in cohomology
group and the Hodge conjecture predicts that the rational (p, p)-classes are carried
by algebraic cycles. This motivates the following definition.
DEFINITION 6. Let V be a rational Hodge structure of even weight 2p. Any
v V of pure type (p, p) is called a Hodge element.
Given any rational Hodge structure V , define for u C the operator C(u)
Aut(VC ) as multiplication by up u q on V p,q . Note that C = C(i). This defines
a rational representation of the algebraic group C on VC with image S(V ). The
special MumfordTate group MT (V ) of any rational Hodge structure V = VQ is
the smallest algebraic subgroup of Gl(V ) whose set of real points contains S(V ).
The MumfordTate group MT(V ) is defined as the smallest algebraic subgroup of
Gl(V ) Gm defined over Q whose set of real points contains the image of the map
C N: C Gl(VR ) Gm (R) where N(z) = zz.
The MumfordTate group is useful to describe Hodge elements inside twisted
tensor spaces related to V defined by
T m,n V (p) = V m (V )n Q(p),

(m n)k 2p = 0.

Indeed, this is a Hodge structure of weight (m n)k 2p = 0 by assumption.

On the other hand this is a representation for MT(V ) GL(V ) Gm as follows:
GL(V ) acts in the obvious way on V m (V )n and trivially on Q(p), whereas
Gm acts trivially on V m (V )n and by the character z  zp on Q(p).

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189

THEOREM 7. The MumfordTate group is exactly the (largest) algebraic subgroup of Gl(VC ) C which fixes all Hodge elements inside T m,n V (p) for all
(m, n, p) such that (m n)k 2p = 0. Conversely, an element of T m,n V (p) with
(m, n, p) such that (m n)k 2p = 0 is a Hodge element if and only if it is
fixed by MT(V ). Every subspace of T m,n V (p) invariant under MT(V ) is a Hodge
substructure.
See [9, Prop. 3.4].
4. Variations of Hodge Structure
Let S be a complex manifold. A variation of Hodge structure of weight k on S
consists of the following data:
(1) a local system VZ of finitely generated Abelian groups on S;
(2) a finite decreasing filtration {F p } of the holomorphic vector bundle V :=
VZ Z OS by holomorphic subbundles (the Hodge filtration).
These data must satisfy the following conditions:
(1) for each s S the filtration {F p (s)} of V(s)  VZ,s Z C defines on the
finitely generated Abelian group VZ,s a Hodge structure of weight k;
(2) the connection : V V OS /1S whose sheaf of horizontal sections is VC
satisfies the Griffiths transversality condition (F p ) F p1 /1S .
EXAMPLES. (1) Let V be a Hodge structure of weight k and s0 S a base
point. Suppose that one has a representation : 1 (S, s0 ) Aut(V ). Then the
local system V() associated to underlies a locally constant variation of Hodge
structure. In this case the Hodge bundles F p are even locally constant, so that
(F p ) F p /1S . This property characterizes the local systems of Hodge structures among the variations of Hodge structure. In case is the trivial representation,
we denote the corresponding variation by VS .
(2) Let f : X S be a proper and smooth morphism of complex algebraic
manifolds. We have seen that the cohomology groups H k (Xs ) of the fibres Xs
fit together into a local system. This local system, by the fundamental results of
Griffiths underlies a variation of Hodge structure on S such that the Hodge structure
at s is just the Hodge structure we have on H k (Xs ). This case will be referred to as
the geometric case.
The notion of a morphism of variations of Hodge structure is defined in the
obvious way.
A polarization of a variation of Hodge structure V of weight k on X is a morphism of variations Q: V V Z(k)S which induces on each fibre a polarization of the corresponding Hodge structure of weight k.

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EXAMPLE 8. Let V be a polarized variation of Hodge structure on a connected

complex manifold which is purely of type (p, p). Then V has a finite monodromy
group.
Indeed, the Hodge structure plays no role here; one just needs the fact that the
isometry group of a lattice is finite.

5. Properties of Variations of Hodge Structure

There is another way of describing variations of Hodge structure using period domains as introduced by Griffiths which we briefly recall. Fixing a rational vector
space V , a weight k, a nondegenerate (1)k -symmetric rational bilinear
 i form Q
i
ki
such
that
h
=
h
and
h = dim V ,
on V , and a vector (h0 , . . . , hk ) Zk+1

0
the possible Hodge structures of weight k polarized by Q having Hodge numbers
hp,q = hp are parametrized by the homogenous space D = Aut(VR , Q)/H , where
H is the subgroup of Aut(VR , Q) stabilizing any fixed Hodge structure of the given
type. The space D turns out to be a complex manifold. Given any variation of
Hodge structure of the same type over a base S, the map which to s S assigns the
point in D corresponding to the Hodge structure on V given by x yields a multiply
valued holomorphic map S D, which becomes uni-valued after dividing out by
the action of the monodromy group 0. In this way one arrives at the period map
p: S D/ 0. Period maps have the property that they are locally liftable, i.e.
locally in a contractible open subset U S, the map lifts to U D. Moreover
such lifts are horizontal. This is a translation of the Griffiths transversality relation.
Conversely, locally liftable holomorphic maps S D/ 0 with horizontal local
lifts are period maps for variations of Hodge structures on S (of the given type).
As shown by Griffiths [10, Theorem (10.1)] period domains have invariant
metrics with special curvature properties which force period maps with source a
polydisk (equipped with the Poincar metric) to be distance decreasing.
First of all this imposes a strong restriction on local monodromy transformations
as shown by Borel, see [14, Lemma 4.5, Thm 6.1].
THEOREM 9 (Monodromy theorem). Let V be a polarized variation of Hodge
structure on the punctured disc  . Then the monodromy operator T is quasip,q
unipotent. More precisely: if $ = max({p q | Vt = 0}) and T = Ts Tu is the
Jordan decomposition if T with Tu unipotent, then (Tu I )$+1 = 0.
Secondly [10, Theorem (9.5)], given a compactifiable base, say S S compactified to a smooth S by a normal crossing divisor, this implies that we may extend
the period map across points s  S all of whose local monodromy-operators have
finite order. This yields a compactifiable S  S whose image under the period
map is a closed analytic subset of D/ 0. See [10, Theorem (9.6)].
As a third implication we have Delignes finiteness result from [8]:

MONODROMY OF VARIATIONS OF HODGE STRUCTURE

191

THEOREM 10. Fix a connected compactifiable S and an integer N. There are at

most finitely many conjugacy classes of rational representations of 1 (S) of dimension N giving local systems that occur as a direct factor of a polarized variation
of Hodge structure on S (of any weight).
A result whose proof requires much more Hodge theory is the following.
THEOREM 11. Let V be a variation of Q-Hodge structure of weight k on a
compactifiable complex manifold S. Then H 0 (S, V) has a Q-Hodge structure of
weight k in such a way that for each s S the restriction map H 0 (S, V) 2 Vs is
a morphism of Q-Hodge structures.
As H 0 (S, V) is the fixed part of Vs under the action of 1 (S, s), this theorem
is mostly called the Theorem of the fixed part. It was first proved by Griffiths [10]
in the case where S is compact, then by Deligne [4] in the algebraic case where
f : X S is a proper and smooth morphism and V = R k f QX , and finally by
Schmid [14] in general.
This theorem has several interesting consequences. A first obvious consequence
is that any variation of Hodge structure on a simply-connected compactifiable
complex manifold is in fact constant.
Secondly we deduce that a global horizontal section of VC which at one point
p,q
is of type (p, q), i.e. lies in Vs for some s S, is everywhere of type (p, q).
Applying this to Hom(V, V ) where V and V are variations of Hodge structures of
the same weight k, and (p, q) = (0, 0) we get the
COROLLARY 12 (Rigidity theorem). Every morphism of Hodge structures Vs
Vs which intertwines the actions of 1 (S, s) extends to a morphism of variations of
Hodge structure V V .
Finally we conclude:
COROLLARY 13. The category of polarizable variations of Q-Hodge structures
on a fixed compactifiable base is semisimple.
Proof. Suppose that V is a subvariation of V and suppose that V is polarized.
Take s S. The Hodge structure Vs is polarized, and this polarization induces an
orthogonal projector in End(Vs ) with image equal to Vs , which commutes with
the action of the monodromy group. Hence, it extends to a projector in EndVHS (V)

with image V .

6. The Locus of a Hodge Element

Let us consider a variation of Hodge structure over any smooth connected complex
base S of even weight k = 2p. We consider a section v of VZ on the universal

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C. A. M. PETERS AND J. H. M. STEENBRINK

cover of S and we let Yv be the locus of all s S where some determination

v(s  ), s   s is of type (p, p). This locus is an analytic subvariety of S since the
condition to belong to the Hodge bundle F p is analytic and a local section v of
VZ is a Hodge element in Vs precisely when v(s) F p . In case Yv = S we call
v special. The union of all Yv , with v special forms a thin subset of S. We call
s S very general with respect to V if it lies in the complement of this set. The
very general points of S with respect to V form a dense subset. Now, if s S is
very general, by definition any Hodge element in Vs extends to give a multivalued
horizontal section of V everywhere of type (p, p).
We can now show how the monodromy group is related to the MumfordTate
group of the Hodge structure at a very general s S using the characterisation
(Proposition 7) of MT(Vs ) as the largest rationally defined algebraic subgroup of
Gl(Vs ) C fixing the Hodge elements in Vm,n
s (p), for all triples (m, n, p) with
(m n)k 2p = 0. So we look at s S which is very general for all local systems
V(m, n)(p) with (m n)k 2p = 0. Then there is a local system H(m, n, p) on
S whose stalk at s is Hodge(Vm,n
s (p)). Using this we deduce:
PROPOSITION 14. Let S be a smooth complex variety. For very general s S
a finite index subgroup of the monodromy group is contained in the MumfordTate
group of the Hodge structure on Vs .
Proof. The Hodge structure on H(m, n, p)s is polarizable and so there is a positive definite quadratic form on this space invariant under monodromy. Hence, the
monodromy acts on H(m, n, p) through a finite group. The MumfordTate group
being algebraic, the Noetherian property then implies that finitely many triples
(m, n, p) determine the MumfordTate group and so a finite index subgroup  of
the fundamental group has its image in the MumfordTate group.

In case S is quasi-projective, the validity of the Hodge conjecture would imply

that analytic sets Yv in fact are algebraic. Surprisingly there is an independent
proof; it is a consequence of the following result due to Cattani, Deligne and
Kaplan [2]:
THEOREM 15. Fix a natural number m. Suppose that we have a polarized variation V of even weight k = 2p on a compactifiable S S whose compactifying
divisor has normal crossings. Define
S (m) = {s S | (VZ )s contains a Hodge element v such that Q(v, v)  m}.
Then the closure of S (m) in S is a finite union of closed analytic subspaces.
As we noted before, the fact that S (m) a finite disjoint union of analytic subspaces of S is not hard; the difficult point is the assertion about the behaviour near
the boundary.
The following result is due to Andr [1], who stated it for variations of mixed
Hodge structure:

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193

THEOREM 16. Let S be a compactifiable complex manifold and let V be a polarizable variation of Hodge structure on S. Let 3 S denote the subset of points
which are not very general with respect to V. Then
(1) For all s S \ 3 the connected component Hs of the algebraic monodromy
group is a normal subgroup of the derived group M der of the generic Mumford
Tate group M = MT(Vs );
(2) If MT(Vs ) is Abelian for some s S then Hs = M der for every s S \ 3.

7. NoetherLefschetz Properties
In this section S is any connected complex manifold and V is a polarizable variation
of Hodge structure on S and we assume that the monodromy representation is
big. We recall that if q: S  S is a finite unramified cover, q V still has a big
monodromy group.
The fact that V underlies a variation of Hodge structure imposes severe restrictions of NoetherLefschetz type:
THEOREM 17. Let there be given a (rational) polarized weight k variation of
Hodge structure V over S with big monodromy. If s S is very general with
respect to Hom(V, V), then Vs has no nontrivial rational Hodge substructures.
Proof. Any projector p: Vs Vs onto a Hodge substructure extends to a multivalued flat section of Hom(V, V) everywhere of type (0, 0). This flat section
generates a sub Hodge structure of type (0, 0) which by Example 8. has finite
monodromy. Replacing S by a finite unramified cover, we may assume that the
flat section is uni-valued, i.e. invariant under the monodromy. This means that the
projector p intertwines every element from the monodromy group and thus defines
a subsystem of V. Since the latter is irreducible, this subsystem is either zero or all
of V.

Remark. The proof from [6] asserting the truth of the theorem for certain variations related to K3-surfaces can be applied to our setting. The crucial result to use
here is Proposition 14. Clearly this gives a much less elementary proof.
Using the above proposition and Example 5, we deduce:
COROLLARY 18. Except for quadric hypersurfaces, cubic surfaces and evendimensional intersections of two quadrics, the primitive cohomology of a generic
smooth complete intersection in complex projective space does not contain nontrivial Hodge substructures.

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Acknowledgement
The authors thank Ben Moonen for valuable comments.
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