The ¬rst examples of hyper-K¨hler manifolds can be already met

a

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 ’ T 4 . Another example of a compact 4-dimensional hyper-

K¨hler manifold will be a K3-surface, where the existence of hyper-

a

K¨hler structure was proved by S.T. Yau.

a

254 CHAPTER 5. FAMILIES AND MODULI SPACES

Now let Σ be a complete Riemann surface and let

MC (Σ) := Homirr (π1 (Σ), GC )/GC

be the moduli space of irreducible GC -representations of the fundamen-

tal group of Σ, which is a smooth manifold. It was established by

Hitchin [34] that the space MC (Σ) has natural hyper-K¨hler structure.

a

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¬cation between MC (Σ)

and the smooth part of the moduli space of (semi-stable) Higgs bundles

with the vanishing ¬rst Chern class on Σ, which is naturally a complex

manifold.

Now let X ’ CPM be a projective manifold and let MC (X) be

the moduli space of irreducible GC -representations of π1 (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, ˜) — H 1 (X, ˜) ’ H 2 (X, ˜).

g g g

If X is of (complex) dimension n then let us consider the intersection

Σ of X with (n ’ 1) hyperplanes in CPM . The projective manifold Σ

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

ξ : π1 (S) ’ π1 (X) is surjective. This means that each loop in X is

homotopic to one in S. The map ξ induces an embedding of manifolds

MC (X) ’ MC (Σ). This embedding is compatible with complex and

K¨hler structures, and therefore MC (X) has a hyper-K¨hler structure

a a

too. It was proven by Fujiki [26] that the moduli space MC (X) has a

hyper-K¨hler structure for a general K¨hler manifold X (not necessarily

a a

projective).

At the end of this section, we provide another important example

of hyper-K¨hler manifolds due to Kronheimer [40]. Let gC be the Lie

a

algebra of GC and g— be the dual space to gC . The group GC acts on the

C

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 ∈ GC acts on x ∈ gC as CxC ’1 . If A ∈ g— , then A is just a linear

C

5.11. MONODROMY GROUPS 255

functional on the space gC . Then we de¬ne the coadjoint action of GC

as

(C —¦ A)(x) = A(C ’1 xC), x ∈ gC , C ∈ GC .

The space g— is then strati¬ed by the orbits of gC -action, and each orbit

C

is a hyper-K¨hler manifold.

a

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¨hler manifolds in the context of non-abelian Hodge theory. There is

a

a natural action of the group C— on the space of Higgs bundles. The

action is easy to describe: » ∈ C— maps (W, ¦ ) to (W, »¦ ). This

important C— -action plays similar rˆle to the action of C— on the usual

o

cohomology.

LEMMA 5.11.1 The Higgs bundle (W, »¦ ) is isomorphic to (W, ¦ )

for all » ∈ C— if and only if there exists a decomposition

q

W i s.t. ¦ W i ‚ W i’1 — „¦1 (X).

W=

i=1

Let us recall that if (V, ) is a complex variation of Hodge structure

on our K¨hler manifold X then as a C ∞ vector bundle V is isomorphic

a

to V = •V p , where V r = F r /F r’1 and F r is as in 5.9.9 then ¦ from

5.9.8 and the Gri¬ths transversality give us exactly

¦ V r ‚ V r’1 — „¦1 (X).

Now if we take the correspondence between representations of π1 (X)

(or, equivalently, ¬‚at bundles on X) and Higgs bundles previously noted

by us then the above Lemma amounts to

PROPOSITION 5.11.2 [51] A ¬‚at bundle (V, ) (corresponding to

a semi-simple representation of π1 (X)) is a complex variation of Hodge

structure if and only if the corresponding Higgs bundle (W, ¦ ) is iso-

morphic to (W, »¦ ) for all » ∈ C— .

256 CHAPTER 5. FAMILIES AND MODULI SPACES

We see that this property of Higgs bundles simply tells us that the C—

action on the moduli space MHiggs of (semi-stable) Higgs bundles with

vanishing Chern classes has a ¬xed point - the isomorphism class of

(W, ¦ ).

It happens next that the correspondence between Higgs bundles and

¬‚at bundles as well as the C— -action on MHiggs are both functorial with

respect to holomorphic mappings between two K¨hler manifolds.

a

THEOREM 5.11.3 [51] Let f : Y ’ X be a holomorphic mapping

between two compact K¨hler manifolds such that the induced map of

a

fundamental groups

f— : π1 (Y ) ’ π1 (X)

is surjective. Then a ¬‚at bundle (V, ) on X is a complex variation of

Hodge structure if and only if (f — V, f — ) is.

In spite of the obvious non-compactness of the space MHiggs it turns

out that the action of C— extends to an action of C [51]. We can conclude

from this that the limit

(W0 , ¦0 ) = lim [(W, »¦ )]

»’0

is a point of MHiggs and the Higgs bundle (W0 , ¦0 ) is clearly C— -

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 [ρ] of a representation

ρ such that the corresponding ¬‚at bundle (V, ) is a complex variation

of Hodge structure.

If we assume that a representation ρ of π1 (X) is rigid, meaning that

every smooth one-parameter family of representations ρt , 0 ¤ t ¤ 1

starting at ρ0 = ρ has the property that ρ1 is conjugate to ρ, then

it follows that [ρ] 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¨hler manifolds. We shall call such groups NFGK (Not the

a

Fundamental Group of a compact K¨hler manifold). The ¬rst steps

a

5.11. MONODROMY GROUPS 257

in this direction were made by Siu [54] and Sampson [49] who used

harmonic analysis on compact K¨hler manifolds to get such topological

a

restrictions. We should also mention an empirical rule (which also has

strong supportive arguments) that a ¬nitely 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 [14] tells

us that a group is NFGK if it is a discrete cocompact lattice in SO(n, 1).

We need the following de¬nition 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 C— on GC such that U (1) preserves G and

the element ’1 is a Cartan involution. We recall (chapter II) that

the correspondence between ¬‚at bundles on X and representations of

π1 (X) occurs exactly via the monodromy representations π1 (X) ’ 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 [51]:

Sl(n, R), n > 2.

SU — (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 [17], and Morgan [43]) gives

many more examples of the groups which are NFGK (see [1] chapter

3 for a comprehensive report on this topic). Kapovich and Millson in

[36] exhibit in¬nitely many Artin groups which are NFGK.

¦

258 CHAPTER 5. FAMILIES AND MODULI SPACES

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