T[ρ] MC = Ker(H 1 (π1 (S), g) ’ H 1 (“, g)) =

Im(H 1 (Cone• ) ’ H 1 (π1 (S), g))

g

and we have a natural pairing

B,

H 1 (Cone• ) — H 1 (Cone• ) ’’ H 2 (Cone• ) ’ R.

g g R

The fact that this pairing is non-degenerate is explained in [20]. We

denote this 2-form on the manifold MC by ω. Since Mumford™s theorem

242 CHAPTER 5. FAMILIES AND MODULI SPACES

gives a projective structure to the manifold M (r, d) MC (we continue

to assume that r and d are coprime), there is a real number c such

that the cohomology class c[ω] is integer, i.e. in the image of the map

H 2 (MC , Z) ’ H 2 (MC , R). (It actually turns out that c can be taken

equal to r(4π 2 )’1 .)

We have already mentioned that there exists a holomorphic bundle

U over M (r, d) — X such that U|V —X V for all V . Such a bundle U is

called the universal bundle. This bundle is possible to construct using

the identi¬cation between M (r, d) and MC as follows. We construct

the holomorphic bundle U on the manifold MC — X which is actually

the family of holomorphic bundles over X parametrized by the points

of MC such that for each m ∈ MC the bundle (Um )|S is the ¬‚at bundle

over S arising from a representation ρm of π1 (S) corresponding to the

point m.

˜

Let S be the universal cover of S and let us consider the following

bundle over MC — S:

˜

E = Cr — S — HomC (π1 (S), G)/(π1 (S) — G),

˜

where the group π1 (S) — G acts on Cr — S — HomC (π1 (S), G) as follows:

(a, g)(x, s, ρ) = (gρ(a)g ’1 (x), a(˜), gρg ’1 ),

˜ s

˜˜

g ∈ G, a ∈ π1 (S), s ∈ S, x ∈ Cr , ρ ∈ Hom(π1 (S), G).

Now we claim that there exists a holomorphic vector bundle U over

MC —X extending the universal bundle E over MC —S with the above

properties. (For a proof of this fact we refer to [11].) Moreover, for

each m ∈ MC the bundle U|m—X is the so-called holomorphic Deligne

extension of the ¬‚at bundle E|m—S (see [18] for more details).

Since the class c[ω] is an integral cohomology class, there exists a line

bundle L on M (r, d) whose curvature is equal to cω. Let us consider an

embedding X ’ CPN coming from the Mumford™s theorem. The dual

to the tautological line bundle L on the projective space CPN restricts

to M (r, d) and L can be taken as this restriction:

L = L|M (r,d) .

Let us give another algebraic geometric construction of the line bundle

L. Let us consider an arbitrary line bundle E on X of degree g ’ 1.

5.9. NON-ABELIAN HODGE THEORY 243

We consider the subset of M (r, d) consisting of the holomorphic vector

bundles V of rank r and degree d on X such that the bundle V — E has

at least one non-zero holomorphic section. The set of such points forms

a divisor on M (r, d) - called the generalized theta divisor. There is a

well-de¬ned line bundle L corresponding to this divisor. The sections

of L are called the generalized theta functions. It turns out that the line

bundle L does not depend upon a choice of E (up to isomorphism).

Next we shall state the Verlinde formula counting the dimension

of the space of holomorphic sections of the tensor powers of the line

bundle L. Nowadays there exist much more general formulae and each

of them has many di¬erent (at a ¬rst glance) proofs. Nevertheless, we

shall give the formula in the simplest case of rank r = 2 and degree

d = 1. The line bundle L (and its tensor powers) are of outermost

importance because of the following fact: the Picard group of M (r, d)

is isomorphic to Z and is generated by L. For the proof we refer to the

works by Beauville [6] and Drezet-Narasimhan [24].

VERLINDE FORMULA.

2k+1

k + 1 g’1

0 —k

(’1)j (

dim H (M (2, 1), L ) = ’ jπ ) .

sin2 2k+2

j=1

5.9 Non-abelian Hodge theory

Let us brie¬‚y review the abelian (standard) Hodge theory from a slightly

di¬erent angle of view. Let (X, ω) be a compact K¨hler manifold

a

m

and let H (X, R) be its m-th cohomology group with real coe¬cients.

The main theorem in abelian Hodge theory is that the vector space

H m (X, R) carries a Hodge structure of weight m. We recall

DEFINITION 5.9.1 A Hodge structure of weight m on a vector

space V is a decreasing ¬ltration

· · · ‚ F p+1 (VC ) ‚ F p (VC ) ‚ F p’1 (VC ) ‚ · · · ,

where VC = V —R C is the complexi¬cation of V . This ¬ltration must be

such that

¯

V p,q , V p,q := F p (VC ) © F q (VC ),

VC =

p+q=m

244 CHAPTER 5. FAMILIES AND MODULI SPACES

¯

where F • is the complex conjugate ¬ltration.

Indeed, H m (X, R) —R C = H m (X, C) and if we, as usual, identify

H m (X, C) with the space of complex-valued harmonic m-forms on X,

then its subspace F p (H m (X, C)) consists of harmonic m-forms which

have at least p degrees of holomorphy. The consequence of this de¬ni-

tion is that the space

¯

F p (H m (X, C)) © F q (H m (X, C))

consists exactly of harmonic forms of type (p, q) provided p + q = m.

The Hodge structure of weight m that we de¬ned is sometimes called a

pure Hodge structure of weight m as opposed to the more subtle notion

of a mixed Hodge structure. We refer the reader to [12] for review of

basic points of the Hodge-Deligne theory which deals with di¬erent

kinds of Hodge structures.

Let us consider the group S(C) := C— — C— , where C— = C \ {0}. The

action of complex conjugation on S(C) is de¬ned by (z, w) := (w, z ).

¯¯

The set of ¬xed points of this conjugation is the real algebraic group

C— and consists of elements of the form {(z, z )}. It was Deligne

S ¯

who ¬rst made the fundamental observation that to de¬ne a Hodge

structure on a real vector space V is the same as to de¬ne a morphism

of algebraic groups over R S ’ GL(V ). Indeed, if V carries a Hodge

structure of weight m then there is an induced action of S(C) = C— — C—

on VC = V — C given by

pq

(z1 , z2 )v = z1 z2 v, when v ∈ V p,q .

The invariants of the action of S on V — C are exactly the real points.

Originally, the notion of a variation of Hodge structure was de-

¬ned by Gri¬ths [30], the present de¬nition due to Deligne [19] is both

weaker and just right for our purposes. A complex variation of Hodge

structure on a K¨hler manifold X is a smooth C ∞ ¬‚at complex vector

a

bundle (V, ) satisfying the following set of conditions:

1. V comes with the decomposition V = •p+q=m V p,q into subbundles.

2. satis¬es the Gri¬ths transversality:

=d +d +¦ +¦ : (5.9.8)

5.9. NON-ABELIAN HODGE THEORY 245

V p,q ’ A1,0 (V p,q ) • A0,1 (V p,q ) • A1,0 (V p’1,q+1 ) • A0,1 (V p+1,q’1 ),

where for example A0,1 (V p,q ) is the space of di¬erential forms of type

(0, 1) with values in V p,q .

3.

¯

F r = •p≥r V p,q , F s = •q≥s V p,q (5.9.9)

are holomorphic and anti-holomorphic subbundles of V respectively

with respect to the holomorphic structure given by d and anti-holomorphic

stucture given by d .

4. A complex variation of Hodge structure is called polarized if, in

addition, there exists a hermitian form , (polarization) on V preserved

by such that

√ q’p

w, v , v ∈ V p,q

(w, v) = ’1

is a ¬‚at bilinear form on V , and such that V p,q ⊥ V s,t for (p, q) = (s, t)

and (’1)p v, v > 0 if v = 0.

We also should say that a real variation of Hodge structure VR is a

complex variation of Hodge structure V = VR —R C such that V p,q = V q,p .

Our ¬rst observation is that we saw in section 8 of part II that

the family f : X ’ Y , where f is a proper submersive holomorphic

map with K¨hler ¬bers gives rise to a polarized variation of Hodge

a

structure on Y . The polarization in this case is given by the standard

k

hermitian pairing on the primitive part of the cohomology Hprim (Xy , C):

k

if [±], [β] ∈ Hprim (Xy , C) and Xy is a K¨hler manifold with a K¨hler

a a

form ω, then we let

√ k(k’1)

¯

±§β§ω dimC Xy ’k .

[±], [β] = ’1

Xy

In this case we say that the variation of Hodge structure arises from a

geometric situation.

Now we shall move to non-abelian Hodge theory. Let X be, as

before, a compact K¨hler manifold with a K¨hler form ω. Instead of

a a

dealing with usual cohomology groups of X with coe¬cients in R or

C we de¬ne (only) the ¬rst cohomology of X with coe¬cients in some

(non-abelian) compact real Lie group G or its ”complexi¬cation” GC

(i.e. GC is a reductive algebraic group and G is its maximal compact

246 CHAPTER 5. FAMILIES AND MODULI SPACES

subgroup). To make things easier to understand and appreciate, here

and further we shall assume that GC = GL(r, C) and G = U (r). Now

we de¬ne

H 1 (X, GC ) := Hom(π1 (X), GC )/GC ,

and H 1 (X, G) := Hom(π1 (X), G)/G;

(the latter space we already saw in section 2 - it is the moduli space

of unitary representations of the fundamental group of X). Unlike

the case when the group G is abelian, for r > 1 the sets H 1 (X, GC )

and H 1 (X, G) do not have group structures. Although, by the Mum-

ford™s theorem, the latter set has the structure of projective variety and

2

using an embedding GL(r, C) ’ Cr (given e.g. by a matrix represen-

tation of GL(r, C)) one can see that the former space can be viewed

as an a¬ne variety. Let π1 (X) be presented as a group in generators

x1 , ..., xl subject to p relations ri (x1 , ..., xl ) = 1, 1 ¤ i ¤ p, where ri is

a non-commutative word in (x1 , ..., xl ). First, we can view the space

Hom(π1 (X), GC ) as a subset of

2 2

Y = GL(r, C) — · · · — GL(r, C) ‚ Cr — · · · — Cr

l

l

given by the solutions of the equations

ri (X1 , ..., Xl ) = 1, 1 ¤ i ¤ p,

where Xj is the coordinate on the j-th multiple of Y . The space

Hom(π1 (X), GC )

clearly carries the structure of an a¬ne variety as it is given by the set of

zeros of a ¬nite number of polynomials. The group GC acts algebraically

on Y by conjugating each multiple and this action in turn induces an

action of GC on Hom(π1 (X), GC ). Let O(Hom(π1 (X), GC )) be the ring

of algebraic functions on the a¬ne algebraic variety Hom(π1 (X), GC ).

The group GC has an obvious induced action on this ring. Now one can

think of the structure ring O(H 1 (X, GC )) to be equal to the subring

C

OG (Hom(π1 (X), GC ))

5.9. NON-ABELIAN HODGE THEORY 247

of GC -invariants of the ring O(Hom(π1 (X), GC )). In general, the vari-

eties H 1 (X, G) and H 1 (X, GC ) are not smooth, and the study of their

singularities is an important aspect of the Non-abelian Hodge theory.

We already know that a representation ρ of the fundamental group

π1 (X):

ρ : π1 (X) ’ GL(V ),

where V is a vector space, is the same as a vector bundle V over X

equipped with a ¬‚at connection ρ with the ¬ber over x0 (the base

point of π1 (X)) equal to V . The notion of ¬‚at bundle is closely related

to the notion of Higgs bundle introduced by Hitchin [33] for Riemann

surfaces and by Simpson [51] in general:

DEFINITION 5.9.2 A Higgs bundle (W, ¦ ) over X is a holomorphic

vector bundle W over X equipped with a holomorphic map ¦ : W ’

W — „¦1 (X) such that ¦ is a Higgs ¬eld, i.e. ¦ §¦ = 0 in End(W ) —

„¦2 (X).

Another way of saying it, a Higgs ¬eld ¦ can be considered as a

form of type (1, 0) with coe¬cients in the (complex) Lie algebra bun-

dle End(W ) satisfying some additional properties. More precisely, let

z1 , ..., zn be local holomorphic coordinates on X, if one writes ¦ =

i ¦i dzi , then ¦i are holomorphic endomorphisms of W and the condi-

tion ¦ §¦ = 0 tells us that the matrices ¦i commute with one another.

Let us think about the holomorphic bundle W as just a complex

¯

vector bundle together with an operator d : W ’ W — „¦1 (X). This

operator d de¬nes the holomorphic structure on W : the holomorphic

sections of W are exactly those annihilated by d . If we now look at

the operator

D := ¦ + d : W ’ W — A1 (X),

then it is easy to see that the conditions

2

d = 0, d (¦ ) = 0, and ¦ §¦ = 0