The ´¬ürst examples of hyper-K┬Ęhler manifolds can be already met
in real dimension 4. Apart from the obvious example of the Euclidean
space R4 , there is the 4-torus T 4 with the ´¬‚at metric induced by the
covering R4 Ôć’ T 4 . Another example of a compact 4-dimensional hyper-
K┬Ęhler manifold will be a K3-surface, where the existence of hyper-
K┬Ęhler structure was proved by S.T. Yau.
254 CHAPTER 5. FAMILIES AND MODULI SPACES
Now let ╬ú be a complete Riemann surface and let
MC (╬ú) := Homirr (¤Ç1 (╬ú), GC )/GC
be the moduli space of irreducible GC -representations of the fundamen-
tal group of ╬ú, which is a smooth manifold. It was established by
Hitchin  that the space MC (╬ú) has natural hyper-K┬Ęhler structure.
In fact, one complex structure (let us call it I) can be seen right away
and is induced from the complex structure on GC . Another complex
structure J appears when we make the identi´¬ücation between MC (╬ú)
and the smooth part of the moduli space of (semi-stable) Higgs bundles
with the vanishing ´¬ürst Chern class on ╬ú, which is naturally a complex
Now let X Ôć’ CPM be a projective manifold and let MC (X) be
the moduli space of irreducible GC -representations of ¤Ç1 (X). We will
think that MC (X) is a smooth manifold. As we have seen, for this it
is enough to require the vanishing of the cup-product
H 1 (X, ╦ť) ├— H 1 (X, ╦ť) Ôć’ H 2 (X, ╦ť).
g g g
If X is of (complex) dimension n then let us consider the intersection
╬ú of X with (n Ôł’ 1) hyperplanes in CPM . The projective manifold ╬ú
is a smooth Riemann surface and each loop in S can be considered as
a loop in X too. Moreover, by a theorem of Lefschetz the natural map
╬ż : ¤Ç1 (S) Ôć’ ¤Ç1 (X) is surjective. This means that each loop in X is
homotopic to one in S. The map ╬ż induces an embedding of manifolds
MC (X) Ôć’ MC (╬ú). This embedding is compatible with complex and
K┬Ęhler structures, and therefore MC (X) has a hyper-K┬Ęhler structure
too. It was proven by Fujiki  that the moduli space MC (X) has a
hyper-K┬Ęhler structure for a general K┬Ęhler manifold X (not necessarily
At the end of this section, we provide another important example
of hyper-K┬Ęhler manifolds due to Kronheimer . Let gC be the Lie
algebra of GC and gÔł— be the dual space to gC . The group GC acts on the
complex Lie algebra gC by the adjoint representation. In terms of the
usual representation of GC as the multiplicative group of complex r ├— r
invertible matrices and gC as the algebra of all complex r ├— r matrices,
C Ôłł GC acts on x Ôłł gC as CxC Ôł’1 . If A Ôłł gÔł— , then A is just a linear
5.11. MONODROMY GROUPS 255
functional on the space gC . Then we de´¬üne the coadjoint action of GC
(C Ô—¦ A)(x) = A(C Ôł’1 xC), x Ôłł gC , C Ôłł GC .
The space gÔł— is then strati´¬üed by the orbits of gC -action, and each orbit
is a hyper-K┬Ęhler manifold.
5.11 Monodromy groups
In this section we shall discuss works of Simpson and others which
throw a bridge between complex variations of Hodge structures and
moduli spaces of representations of fundamental groups of compact
K┬Ęhler manifolds in the context of non-abelian Hodge theory. There is
a natural action of the group CÔł— on the space of Higgs bundles. The
action is easy to describe: ╬» Ôłł CÔł— maps (W, ╬¦ ) to (W, ╬»╬¦ ). This
important CÔł— -action plays similar r╦ćle to the action of CÔł— on the usual
LEMMA 5.11.1 The Higgs bundle (W, ╬»╬¦ ) is isomorphic to (W, ╬¦ )
for all ╬» Ôłł CÔł— if and only if there exists a decomposition
W i s.t. ╬¦ W i ÔŐ‚ W iÔł’1 ÔŐ— Ô„¦1 (X).
Let us recall that if (V, ) is a complex variation of Hodge structure
on our K┬Ęhler manifold X then as a C Ôł× vector bundle V is isomorphic
to V = ÔŐ•V p , where V r = F r /F rÔł’1 and F r is as in 5.9.9 then ╬¦ from
5.9.8 and the Gri´¬âths transversality give us exactly
╬¦ V r ÔŐ‚ V rÔł’1 ÔŐ— Ô„¦1 (X).
Now if we take the correspondence between representations of ¤Ç1 (X)
(or, equivalently, ´¬‚at bundles on X) and Higgs bundles previously noted
by us then the above Lemma amounts to
PROPOSITION 5.11.2  A ´¬‚at bundle (V, ) (corresponding to
a semi-simple representation of ¤Ç1 (X)) is a complex variation of Hodge
structure if and only if the corresponding Higgs bundle (W, ╬¦ ) is iso-
morphic to (W, ╬»╬¦ ) for all ╬» Ôłł CÔł— .
256 CHAPTER 5. FAMILIES AND MODULI SPACES
We see that this property of Higgs bundles simply tells us that the CÔł—
action on the moduli space MHiggs of (semi-stable) Higgs bundles with
vanishing Chern classes has a ´¬üxed point - the isomorphism class of
(W, ╬¦ ).
It happens next that the correspondence between Higgs bundles and
´¬‚at bundles as well as the CÔł— -action on MHiggs are both functorial with
respect to holomorphic mappings between two K┬Ęhler manifolds.
THEOREM 5.11.3  Let f : Y Ôć’ X be a holomorphic mapping
between two compact K┬Ęhler manifolds such that the induced map of
fÔł— : ¤Ç1 (Y ) Ôć’ ¤Ç1 (X)
is surjective. Then a ´¬‚at bundle (V, ) on X is a complex variation of
Hodge structure if and only if (f Ôł— V, f Ôł— ) is.
In spite of the obvious non-compactness of the space MHiggs it turns
out that the action of CÔł— extends to an action of C . We can conclude
from this that the limit
(W0 , ╬¦0 ) = lim [(W, ╬»╬¦ )]
is a point of MHiggs and the Higgs bundle (W0 , ╬¦0 ) is clearly CÔł— -
invariant. From this and the above Proposition Simpson concludes:
THEOREM 5.11.4 If X is a projective manifold, then every con-
nected component of H 1 (X, GC ) contains a class [¤ü] of a representation
¤ü such that the corresponding ´¬‚at bundle (V, ) is a complex variation
of Hodge structure.
If we assume that a representation ¤ü of ¤Ç1 (X) is rigid, meaning that
every smooth one-parameter family of representations ¤üt , 0 Ôë¤ t Ôë¤ 1
starting at ¤ü0 = ¤ü has the property that ¤ü1 is conjugate to ¤ü, then
it follows that [¤ü] is a connected component of H 1 (X, GC ) and thus
the corresponding ´¬‚at bundle (V, ) is a complex variation of Hodge
structure. These circumstances allowed Simpson to draw some conclu-
sions regarding which groups can not occur as fundamental groups of
compact K┬Ęhler manifolds. We shall call such groups NFGK (Not the
Fundamental Group of a compact K┬Ęhler manifold). The ´¬ürst steps
5.11. MONODROMY GROUPS 257
in this direction were made by Siu  and Sampson  who used
harmonic analysis on compact K┬Ęhler manifolds to get such topological
restrictions. We should also mention an empirical rule (which also has
strong supportive arguments) that a ´¬ünitely generated group is generi-
cally NFGK if in a minimal set of generators and relations the number
of generators exceeds by far the number of relations.
Returning to exact science, a result of Carlson and Toledo  tells
us that a group is NFGK if it is a discrete cocompact lattice in SO(n, 1).
We need the following de´¬ünition due to Simpson
DEFINITION 5.11.5 A real Lie group G is said to be of Hodge type
if it occurs as the real Zariski closure of the monodromy group of a
complex variation of Hodge structure.
This can be formulated in terms of Lie theory: a group G is of Hodge
type if there is an action of CÔł— on GC such that U (1) preserves G and
the element Ôł’1 is a Cartan involution. We recall (chapter II) that
the correspondence between ´¬‚at bundles on X and representations of
¤Ç1 (X) occurs exactly via the monodromy representations ¤Ç1 (X) Ôć’ G.
Therefore a rigid lattice in a group which is not of Hodge type is NFGK.
The combinatorics of the Dynkin diagrams allows one to decide whether
a group is of Hodge type. Let us list some of the simple groups which
are not of Hodge type :
Sl(n, R), n > 2.
SU Ôł— (2n), n > 2.
SO(p, q), p, q odd.
Any complex group.
Thus, for example any cocompact lattice in these groups (like SL(n, Z)
for n > 2) is NFGK.
We also have to mention that the technique of taking the Malcev
completion of a discrete group and applying real homotopy theory ap-
proach (originated by Sullivan, see also , and Morgan ) gives
many more examples of the groups which are NFGK (see  chapter
3 for a comprehensive report on this topic). Kapovich and Millson in
 exhibit in´¬ünitely many Artin groups which are NFGK.
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