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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.

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

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
as we saw works for any Riemannian manifold) allows us to say that
harmonic forms minimize the functional

F (±) = d±, d± + δ±, δ± ,

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:
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

is a GC -equivariant one-form q on P with values in gC (the Lie algebra
of GC , which can be identi¬ed with the vertical tangent space Tp P
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
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,

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
is called ¬‚at if q 2 = 0.
Let β : X ’ H be a smooth π1 (X)-equivariant map. Any such map
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.
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.

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
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
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) =

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

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

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
(¦ + ) + (¦ + ).

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
- the category of harmonic bundles on X

5.10 Hyper-K¨hler manifolds
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
simple example. Let V be a vector space over R endowed with linear
endomorphisms I, J, and K satisfying the standard quaternion identi-
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
«  « 
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
« 
0d 1d 0d 0d
¬ ’1 0d 0d 0d ·
¬ ·
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
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,
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.
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
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


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