re¬‚ecting respectively the facts that the holomorphic structure d is

integrable, ¦ is holomorphic, and ¦ is a Higgs ¬eld, are equivalent to

the one simple condition D2 = 0.

248 CHAPTER 5. FAMILIES AND MODULI SPACES

We recall that in the abelian Hodge theory a very important rˆle

o

was played by the harmonic di¬erential forms on X. More precisely,

we have the decomposition of the m-th cohomology group

H m (X, C) = H p,q (X)

p+q=m

into the direct sum of the spaces of harmonic forms on X of type (p, q).

By de¬nition, a harmonic form ± on X is one which is annihilated by

the Laplace operator: ∆± = 0. The K¨hler metric on X (this de¬nition

a

as we saw works for any Riemannian manifold) allows us to say that

harmonic forms minimize the functional

F (±) = d±, d± + δ±, δ± ,

X X

where the integrals are taken with respect to the volume form on X.

Now let us take the quotient H = GC /G, which is the symmetric

space of GC - a manifold parametrizing the coset space for the maximal

compact subgroup G ‚ GC . The manifold H is a homogeneous space

for the group GC , i.e. the group GC acts on H and the stabilizer of

each point is isomorphic to G. As we mentioned in section 2, one also

considers a principal GC -bundle P over X corresponding to a complex

vector bundle V over X of rank r. The bundle V can be obtained from

P by the standard procedure of the change of ¬ber:

C

V = P —G C r .

To de¬ne this operation, let us take Cr regarded as the standard rep-

resentation space of GC and let us consider the action of GC on the

product P — Cr given by

g(p, v) = (pg ’1 , gv), g ∈ GC , p ∈ P, v ∈ Cr .

(The group GC by the standard convention acts on P on the right.)

The quotient space with respect to this action is naturally a complex

vector bundle V over X with the structure group GC . A reduction of the

structure group from GC to G for the principal GC -bundle P is a smooth

principal G-subbundle P ‚ P such that P = P —G GC . It is clear from

the above discussion what is a connection on a principal bundle P : it

5.9. NON-ABELIAN HODGE THEORY 249

is a GC -equivariant one-form q on P with values in gC (the Lie algebra

vert

of GC , which can be identi¬ed with the vertical tangent space Tp P

vert

at a point p ∈ P ) such that ι(Z)q = Z, where Z ∈ Tp P gC . Inside

T P we have the horizontal tangent bundle T hor P de¬ned as the kernel

of q. We have a direct sum decomposition

T P = T hor P • T vert P.

We have an isomorphism

dp ψ

hor

Tp P Tψ(p) X,

where ψ : P ’ X is the projection.

Next we will say what does it mean for the 1-form q to be GC -

equivariant. Formally, it means that for g ∈ GC we have

q(dg(v)) = Ad(g)’1 q(v), v ∈ Tp P,

where dg : Tp P ’ Tpg P is the di¬erential for the right action of g ∈ GC

on P . As we know, representation ρ : π1 (X) ’ GC gives rise to a ¬‚at

principal GC -bundle (P, q) over X as well as to a ¬‚at bundle ˜:g

˜

˜ := X —π1 (X) g,

g

˜

where X is the universal cover of X and g is viewed as a π1 (X)-module

via Ad —¦ ρ. We also notice that the representation ρ induces a π1 (X)-

action on the homogeneous space H. Such a connection q then can be

extended to an operator acting from GC -invariant k-forms with coe¬-

cients in ˜C to GC -invariant (k + 1)-forms with coe¬cients in gC and it

g

is called ¬‚at if q 2 = 0.

˜

Let β : X ’ H be a smooth π1 (X)-equivariant map. Any such map

C

de¬nes a smooth section s of the ¬ber bundle P —G H over X as follows.

˜

We recall that the principal bundle P is obtained from X —GC by taking

the quotient with respect to the action of π1 (X). The fundamental

group of X acts on the second factor GC via the representation ρ.

˜

C

Therefore, one can identify the ¬ber bundle P —G H with X —H/π1 (X),

where π1 (X) acts on the ¬rst factor by deck transformations and on the

second factor also via ρ. Now it is clear that only π1 (X)-equivariant

˜

map β : X ’ H will descend to the quotient.

250 CHAPTER 5. FAMILIES AND MODULI SPACES

In fact, such a section s is completely de¬ned by a choice of a G-

reduction of P . A principal GC -bundle (P, q) induces a bundle H ’ X

with ¬ber H which can be written as

C

H = P —G H.

In H = GC /G there is a distinguished point eG - the coset correspond-

ing to G itself. Let P ‚ P be a G-reduction for P then the inclusion

C

P — (eG) ‚ P —G GC de¬nes the smooth section s of P —G H.

Let us choose a metric on H by identifying TeG H with the space of

hermitian matrices Herm. If H1 , H2 ∈ Herm, we let

H1 , H2 = T r(H1 H2 ).

This metric is clearly G-invariant.

DEFINITION 5.9.3 The energy functional corresponding to such a

˜

π1 (X)-equivariant map β : X ’ H (or such a reduction P ‚ P ) is

||dβ||2 V olX ,

E(β) =

X

where V olX is the volume form on X de¬ned by the K¨hler form ω and

a

|| · || is de¬ned by the above invariant metric on H.

Such an equivariant map β or such a reduction P ‚ P is called

harmonic if it is a critical point of the functional E.

Among the major tasks of the non-abelian Hodge theory is to show

˜

how the concept of equivariant harmonic map X ’ H helps to under-

stand the interrelations between Higgs bundles and the space H 1 (X, GC ).

THEOREM 5.9.4 (Corlette, [16]) There exists such a harmonic

equivariant map β if and only if the Zariski closure of ρ(π1 (X)) is

reductive.

This theorem was proved independently by Donaldson [23] in the rank

2 case. In fact, this theorem works not only for GL(r, C) but also for

any other reductive algebraic groups. Another important aspect in this

5.9. NON-ABELIAN HODGE THEORY 251

theory is the identi¬cation between H 1 (X, GC ) and the moduli spaces

of Higgs bundles with vanishing Chern classes:

ch1 (W )[ω]d’1 = 0, ch2 (W )[ω]d’2 = 0, d = dimC (X)

(see [53] for the construction of this space when X is projective). In-

deed, let P be a ¬‚at principal GC -bundle corresponding to an irreducible

representation ρ de¬ning a point [ρ] ∈ H 1 (X, GC ). We can decompose

the ¬‚at connection in P according to types as = + . Let

P ‚ P be a G-reduction and let = δ + ¦ be such a decomposition

of that δ preserves P and ¦ restricts to 0 on P . Let also

δ = δ + δ , and ¦ = ¦ + ¦

be the decompositions of δ and ¦ according to types so that = δ +¦

and =δ +¦ .

The number

deg(ch1 (W )[ω]d’1 /rk(W )

is called the slope of W . A Higgs bundle (W, ¦ ) is called stable if every

its proper Higgs subbundle has strictly smaller slope. A Higgs bundle

(W, ¦ ) is called polystable if it is a direct sum of stable Higgs bundles

of the same slope.

Now we assume that P is actually a harmonic G-reduction. Then

one can see that the harmonicity condition amounts to the fact that

(P, δ ) has a structure of holomorphic principal bundle on X, i.e. (δ )2 =

0. Moreover, if W is a holomorphic vector bundle corresponding to P

(obtained from P by the change of ¬ber using the standard represen-

tation GC ’ Aut(Cr )) then (W, ¦ ) is a Higgs bundle.

Conversely, given a Higgs bundle (W, ¦ ) (with vanishing Chern

classes) we shall look for a ¬‚at connection on the C ∞ bundle underlying

W . Let h be a hermitian metric on W ; as we know there always exists

a unique connection on W respecting the holomorphic structure and

the hermitian metric. We decompose = + according to types;

we also denote by ¦ the operator of type (0, 1) which is h-adjoint to ¦

in the operator sense. Next, we de¬ne a new complex structure on W

using the operator ¦ + , which is of type (0, 1). Now let us consider

the connection

(¦ + ) + (¦ + ).

252 CHAPTER 5. FAMILIES AND MODULI SPACES

˜

If h is a harmonic metric (i.e. induced by a harmonic map X ’ H)

then this connection is ¬‚at. We refer to a work by Simpson [50] for

details on this procedure which amounts to

THEOREM 5.9.5 The following categories are equivalent:

- the category of semisimple ¬‚at bundles on X

- the category of (polystable) Higgs bundles on X with vanishing Chern

classes

- the category of harmonic bundles on X

5.10 Hyper-K¨hler manifolds

a

We recall that the non-commutative division ring of quaternions H is

a vector space over R spanned by 1, I, J, K which satisfy the standard

quaternion identities:

I 2 = J 2 = K 2 = IJK = ’Id.

We begin the study of hyper-K¨hler manifolds with the following

a

simple example. Let V be a vector space over R endowed with linear

endomorphisms I, J, and K satisfying the standard quaternion identi-

ties:

I 2 = J 2 = K 2 = IJK = ’Id,

or, equivalently, these four equalities can be replaced by a single one:

(aI + bJ + cK)2 = ’(a2 + b2 + c2 ), a, b, c ∈ R.

If this happens, then the dimension of the space V must be divisible

by 4 (say, equal to 4d), because V is a vector space over H, and it is

convenient to think of I, J, and K in terms of the following matrix

representation:

« «

0d 0d 1d 0d 0d 0d 0d 1d

¬0 · ¬0 0d ·

0d 0d 1d · 0d ’1d

I=¬ d ¬d ·

·, J = ¬ ·,

¬

’1d 0d 0d 0d 0d 1d 0d 0d

0d ’1d 0d 0d ’1d 0d 0d 0d

¨

5.10. HYPER-KAHLER MANIFOLDS 253

«

0d 1d 0d 0d

¬ ’1 0d 0d 0d ·

¬ ·

d

K=¬ ·,

0d 0d 0d ’1d

0d 0d 1d 0d

where 1d is the identity d — d matrix and ’1d is the minus identity

d — d matrix and 0d is zero d — d matrix. The group which preserves

the relations between I, J, and K as well as the Euclidean metric on

R4d is the intersection of the orthogonal group O(4d) with the

V

quaternionic general linear group GL(d, H). This intersection is the

group of quaternionic unitary matrices Sp(d). Now we are ready for

the following

DEFINITION 5.10.1 (Calabi, [13]) A Riemannian manifold (X, g)

is called hyper-K¨hler if there are three integrable endomorphisms I, J,

a

and K of the tangent bundle T X satisfying the above identities and

such that

ω1 (X, Y ) = ’g(IX, Y ), ω2 (X, Y ) = ’g(JX, Y ), ω1 (X, Y ) = ’g(KX, Y )

are three K¨hler forms on X.

a

The dimension of such a manifold X again must be divisible by 4, and

I, J, and K de¬ne three integrable complex structures on X. It is not

hard to check that for r = (a, b, c) ∈ R3 such that a2 + b2 + c2 = 1,

the expression Ir = aI + bJ + cK is an integrable complex structure

on X as well. Therefore one has a family of complex structures on a

hyper-K¨hler manifold X of dimension 4d parametrized by CP1 (viewed

a

as the unit sphere in R3 ).

The twistor space of X is the product X — CP1 , where X — {r} is

regarded as a complex manifold with the complex structure de¬ned by

Ir . It turns out that the twistor space has an integrable complex struc-

ture induced by these complex structures and the complex structure