2π√

−1 Xr j=1

λ_{ij}u_{j}

if we recall thatu_{j}∈t^{∗} are the coordinates oft. Then we have
e_{T}(ν_{F}) =Y

i

Xr j=1

λ_{ij}u_{j} ∈R.
Denote the previous polynomial by f_{F} and let f := Q

F⊆M^{T}f_{F} ∈ R. Then, by
definition and by (3.3), the map induced byι^{∗}ι_{∗}in the localization byfis invertible,
and for all classφ∈H_{T}(M), we have

φ= X

F⊆M^{T}

ι^{F}_{∗}ι^{∗}_{F}φ

e_{T}(ν_{F}) (3.4)

with self-evident notation for ι_{F} and ι^{F}. Applying the push-forward to a point
π_{∗} : H_{T}(M) → R (which is by definition the integral R

M) to (3.4) we obtain the result.

**3.4** **Examples**

We summarize here a few remarkable examples which will be useful in the fol-lowing sections.

**3.4.1** **Compact Riemann surfaces**

LetM=Σ_{g} be a compact Riemann surface of genusg>0. Then it is well known
thatH^{1}(Σ_{g})is generated as vector space by cyclesa_{i}andb_{i}fori=1, . . . ,g, which
satisfy the intersection properties

a_{i}a_{j} =b_{i}b_{j}=0, a_{i}b_{j}=δ_{i,j}ω

where ω ∈ H^{2}(Σ_{g}) is the Poincaré dual to[Σ_{g}] ∈ H^{0}(Σ_{g}), so that R

Σgω = 1 by definition.

Notice that since there is only one generator of even cohomology, namely ω,
the Chern class of any complex vector bundle overΣ_{g}must be an integer multiple
ofω. LettingLbe a line bundle over Σ_{g}, we have

c_{1}(L) = (degL)ω,

so that the Chern character ofL is 1+ (degL)ω. It is well-known that the degree
of the tangent bundleT Σ_{g} isχ(M) =2−2g, so that

c_{1}(T Σ_{g}) = (2−2g)ω.

In particular the Todd class ofΣ_{g} is 1+ (1−g)ω. The Hirzebruch-Riemann-Roch
formula then reads

H^{0}(Σ_{g};L) −H^{1}(Σ_{g};L) =
Z

Σg

ch(L)td(M) =degL−g+1,

which is the classical Riemann-Roch formula for line bundles on a Riemann sur-face.

**3.4.2** **Grassmannians**

Consider theGrassmannianmanifold Gr(n,k), which is the space ofk-dimensional subspaces of a fixed complex vector space V of dimension n. It is a compact complex manifold of dimension k(n−k), which comes with two distinguished vector bundles: thetautological bundle S, whose fiber at a point[H] ∈ Gr(n,k) is the space H itself, and the trivial bundle V of rank n, of which S is naturally a subbundle. Thequotient bundleQis the one fitting into the exact sequence

0→S→V →Q→0. (3.5)

We write the total Chern classes ofSandQas
c(S) =1+s_{1}+. . .+s_{k},
c(Q) =1+q_{1}+. . .+q_{n−k},

which are related by c(S)c(Q) = 1, because of (3.5). This allows, recursively, to
compute theq_{i}’s as polynomial expressions in thes_{i}’s: here are the first ones

q_{1} = −s_{1}, q_{2}=s^{2}_{1}−s_{2}, q_{3} = −s^{3}_{1}+2s_{1}s_{2}−s_{3}.

Notice that such expressions forq_{i}are completely formal, therefore are valid even
if i > n−k, for which q_{i} = 0 trivially. This gives relations in the cohomology
ring of Gr(n,k), and the main theorem in the study of such ring is that these are a
complete set of relations, i.e.

H^{∗}(Gr(n,k))=∼ **C**[s_{1}, . . . ,s_{k}]/(q_{n−k+1}, . . . ,q_{n}).

In particular, the cohomology ring of the Grassmannian is generated by the Chern
classess_{i}fori=1, . . . ,n−k.

In order to compute geometrically the classess_{i}, we need to pick generic
sec-tionsγ_{1}, . . . ,γ_{k−i+1}of S, and check the locus the subspace they generate is not of
maximal dimension. To do this, letλ_{i} for 16i6kbe integers such that

n−k>λ_{1}>. . .>λ_{k} >0 (3.6)

and define, forλ:= (λ_{1}, . . . ,λ_{k}) and forF= (V_{i})^{n}_{i=0}an arbitrary complete flag of
V, the locus

Σ_{λ}(V) ={[H]∈Gr(n,k) : dim(V_{n−k+i−λ}_{i}∩H)>i, for alli>0}.

These loci are calledSchubert cells, and their definition can be considered an
assign-ment of incidence conditions of a subspace Hagainst the spaces of the complete
flag F. The study of the products of the cohomology classes σ_{λ} := [Σ_{λ}(F)] is at
the core ofSchubert calculus, one of the first and most important landmark in
in-tersection theory. As for the Chern classes, it can be shown (see for instance [EH])
that

s_{i}= (−1)^{i}σ_{(1,...,1)}

where on the subscript there areiones (and k−izeros, which are omitted in the
writing). Similarly, we haveq_{i} = σ_{i}. Notice that eachσ_{λ} ∈ H^{2|}^{λ}^{|}(Gr(n,k)), thus
the top class isω:=σ(n−k,...,n−k)(withkentries) so that

Z

Gr(n,k)

σ(n−k,...,n−k)=1.

Notice that since the cohomology ring of the Grassmannian is generated by the
Chern classess_{i}, all intersection theory on Gr(n,k)is encoded into the generating
function

F_{n,k}(t_{1}, . . . ,t_{k}) :=

Z

Gr(n,k)

(1−t_{1}s_{1}−. . .−t_{k}s_{k})^{−1}.
Now we choosendistinct integers

a_{1} < a_{2}<. . .< a_{n}
and letT =**C**^{∗} act on**C**^{n}by

z·(x_{1}, . . . ,x_{n}) = (z^{a}^{1}x_{1}, . . . ,z^{a}^{n}x_{n}),

an action which naturally descends to the Grassmannian Gr(n,k). There are ^{n}_{k}
fixed points corresponding to the subspaces spanned byv_{i}_{1}, . . . ,v_{i}_{k} where thev_{i}’s
are the vectors of the canonical basis of **C**^{n}. We want to use the localization
for-mula of Theorem 3.3.2 to deal with the intersection theory on Gr(n,k). Notice that
S and Q are naturallyT-equivariant bundles: we will denote the corresponding
equivariant chern classes still bys_{i}andq_{i}.

Then it can be shown as in [Zi] that the localization formula gives
F_{n,k}(t_{1}, . . . ,t_{k}) = 1

k! Res

z_{1},...,zk=∞

Q

i6=j(z_{i}−z_{j})
(1−t_{1}e_{1}−. . .−t_{k}e_{k})Q_{n}

i=1

Q_{k}

j=1(a_{i}−z_{j})
where e_{i} = e_{i}(z_{1}, . . . ,z_{k}) are the elementary symmetric polynomials. It is rather
surprising that such expression does not depend on the particular choice of the
weights a_{i}. Since we are taking residues at only one point, the order of the
residues will not matter. As mentioned, this formula contains all intersection
numbers for Gr(n,k), and can be used to extract information about its
cohomol-ogy ring. However, the level of difficulty of Schubert calculus suggests that using
the expression ofF_{n,k} in practice is not so easy.

**3.4.3** **Symmetric products**

In our study of the moduli space of Higgs bundles, the symmetric products of
the Riemann surface Σ_{g} will play a key role, they being identified with some
connected components of the fixed locus with respect to the natural **C**^{∗}-action.

What follows are the basic results on their cohomology ring, due to Macdonald
[Mac] and also found in [Th1]. For the sake of ease of notation, we will denote the
Riemann surfaceΣ_{g} byX, and itsi-th symmetric product byX^{(i)}. The generators
of the first cohomology group H^{1}(X;**Z**) are a_{i}, b_{i} fori = 1, . . . ,g. We define the
universal divisor∆⊆X_{i}×Xas

∆= [{(S,p)∈X_{i}×X: p∈S}]∈H^{2}(X_{i}×X;**Z**)
which we decompose in Künneth components as

∆=η+X

j

(ζ_{i}a_{i}−ξ_{i}b_{i}) +i[X]

thus yielding classes η ∈ H^{2}(X_{i};**Z**) and ξ_{i}, ζ_{i} ∈ H^{1}(X_{i};**Z**) which generate the
cohomology ring of X_{i}; we write σ_{j} := ξ_{j}ζ_{j} andσ = P

jσ_{j}. Notice that σ^{2}_{j} = 0,
thus ifk>0,σ^{k}/k! is thek-th elementary symmetric polynomial in theσ_{j}’s, which
is a sum of ^{g}_{k}

monomials. Now, for any multi-indexIwithout repeats, we have Z

Xi

η^{i−}^{|}^{I}^{|}σ_{I}=1
so that for any two formal power seriesA(x)andB(x),

Z

X_{i}

A(η)exp(B(η)σ) = X∞ k=0

Z

X_{i}

A(η)B(η)^{k}σ^{k}
k! =

= Xg k=0

g k

Resη=0

A(η)B(η)^{k}
η^{i−k+1} dη

=Res

η=0

dη

η^{i+1}A(η)(1+ηB(η))^{g}. (3.7)
Since the even cohomology of X_{i} is generated by η andσ, formula (3.7) encodes
all intersection numbers onX_{i}. It will be used in a following section to prove the
residue formulas for equivariant integrals on the moduli space of Higgs bundles.

**Chapter 4**

**Vector Bundles**

In this section, we will introduce the main concepts related to vector bundles which will be used in the present thesis. In particular, the focus will be on the definition and basic properties of the moduli space of stable bundles over Riemann surfaces, in the case the rank and the degree are coprime: in this case, the moduli space will be a smooth compact complex manifold. We will then briefly investigate the intersection theory on such space through the celebrated Witten’s intersection formulas, introduced in [Wi] and proved in several different ways in [Wi], [Th1]

and [JK].

**4.1** **Chern classes**

LetVbe a rankrvector bundle over a compact smooth manifoldM. We have seen
in Definition 2.1.4 and in the following Theorem 2.1.5 that we can associate classes
ci(V) ∈ A^{i}(M) in the Chow ring of M. Through the homomorphism of Remark
2.2.1 we can see that there are corresponding cohomology classes, still calledChern
classesand still denoted by c_{i}(V)∈H^{2i}(M;**Z**).

IfM=Σ_{g}is a compact Riemann surface of genusg>0, we make the following
definition.

**Definition 4.1.1.** LetV be a vector bundle over a Riemann surfaceΣ_{g}. The degree
ofV is defined as

deg(V) :=c_{1}(V)∈**Z**

under the isomorphismH^{2}(Σ_{g},**Z**)=∼ **Z**given by the complex structure ofΣ_{g}.
**Remark 4.1.2.** One can also define the degree more in general, for a vector bundle
over a projective variety. However, the definition above is sufficient for the scope
of the present thesis.

One of the most important feature of a vector bundle is itsstability, which tells us which vector bundles should be considered in the moduli problem.

**Definition 4.1.3.** LetV be a vector bundle over a Riemann surfaceΣ_{g}. Itsslopeis
defined as the ratio between the degree and the rank ofV:

µ(V) := deg(V) rk(V) .

We say thatV issemi-stableif for all subbundles W⊆V, we haveµ(W) 6µ(V). If strict inequality holds for all proper subbundles, we say thatV isstable.

Stability is essentially a condition about automorphisms of a vector bundle.

**Proposition 4.1.4.** LetV be a stable vector bundle. Then the only automorphisms of V
are multiplication by a non-zero scalar.

Proof. The essential step is proving that if Uand V are semi-stable bundles and µ(U) > µ(V), then there are no nonzero homomorphisms φ : U → V. Indeed, lettingW be the image ofφ, by (semi-)stability ofVand by definition of slope we have

µ(U)6µ(W)6µ(V) (4.1)

thus bringing to a contradiction. Assume now thatµ(U) =µ(V); ifUis stable then (4.1) forces φ to be an injective, and if V is stable (4.1) forces φ to be surjective.

Thus we can conclude that if V is stable, any endomorphism of V is either 0 or an
isomorphism. This means that End(V) is a finite-dimensional division algebra over
**C, therefore End**(V)=∼ **C**because**C**is algebraically closed.

**Lemma 2.** LetV be a vector bundle such thatdeg(V)andrk(V)are coprime. ThenVis
semi-stable if and only if it is stable.

Proof. Assume V is not stable. Then there is a proper U ⊆ V such that µ(U) = µ(V), which means rk(U)<rk(V)and

deg(U)·rk(V) =deg(V)·rk(U) which implies that deg(V)and rk(V)have a common factor.

The set of line bundles of degree 0 over Σ_{g} forms a group under the tensor
product; this is a dimensiongvariety called the Jacobianand denoted by Jac(Σ_{g}).
Its importance in our setting lies in the fact that tensorizing a vector bundleEover
Σ_{g} by an element of the Jacobian does not change the rank and degree of E, and
preserves its stability.