3.2 Moduli spaces of holomorphic bundles

In the abelian case we have already used the classical result that H'(Z, (7(1)) can be identified with the Jacobian of ZT, once a complex structure r has been fixed on 2. Similar results hold in the non-abelian case. First, however, we have to describe the analogues of the Jacobian. These are the moduli spaces of holomorphic Gc bundles over £T, where Gc is the complexification of G. For simplicity we shall restrict our­selves here to the case G = SU(n) and refer to [5] for the more general case.

The important notion here is that of stability of holomorphic bundles. A holomorphic vector bundle e over a Riemann

surface 2T is said to be stable if, for all holomorphic sub- bundles F, we have

(3.2.1)

Here degree means the value of the first Chern class. For an SL(n, C)-bundle C!=0 and so (3.2.1) simply amounts to deg F < 0. Semi-stable bundles are defined similarly by requir­ing deg F< 0.

A theorem of Narasimhan and Seshadri [22] asserts that the isomorphism classes of stable holomorphic bundles of rank n form a non-singular Zariski open set Ms(n) in a projective algebraic variety M(n). Moreover M(n) is obtained from the semi-stable bundles by an equivalence relation (stronger than just isomorphism).

Since every flat bundle is automatically holomorphic it is not surprising that there is a natural map

 

 

The main theorem of Narasimhan and Seshadri [22] is that this map is a homeomorphism. The significance of this result will become clearer in Chapter 4. At the level of tangent spaces, at an irreducible point a, it corresponds to the natural isomorphism

 

 

(3.2.2)

where End0 denotes trace-free endomorphisms and the two sheaves are:

End0 (e ) = locally constant skew-Hermitian endomorphisms

End0 (e) = holomorphic endomorphisms.

 

This is the obvious generalization of the result used in Chapter 2 for the Jacobian. However, in that case, since the manifold is a torus, the whole map is linear. Here the manifolds are

non-linear and only the linearized tangent map can be easily identified by sheaf cohomology.

From (3.2.2) it is clear that the complex structure induced on H\Z, SU(n)) by a complex structure on I depends only on the isomorphism class of h (modulo the identity com­ponent of Diff+ (2)), i.e. on the point r in Teichmiiller space. This is the generalization of the fact that the complex structure on H\l, (7(1)) depends only on the period matrix of the Riemann surface. This gives a certain rigidity to our moduli spaces, a property not shared by the family of Riemann surfaces.

The moduli space M(n) has a natural holomorphic line- bundle L and the space of sections of Lk will give the quantiz­ation at level k. We shall postpone a discussion of these questions until Chapter 5 where they will appear in proper context. However, it may be worth remarking at this stage that L generates the group of holomorphic line-bundles on M{n) as shown by Drezet and Narasimhan [12]. Unlike the Jacobian case there are no flat line-bundles and for this reason spin structures on 1 are not needed.

There is a generalization of stability and of the moduli space M(n) to take account of marked points. This is due to Seshadri [30] and involves assigning weights a1,...,ar at each marked point. Seshadri proves that his moduli space (for given weights) is naturally homeomorphic to the space of unitary representations of ttAZ -(P|U"-uPr) where the loop around a marked point P is represented by a matrix with eigenvalues

exp (2iriaj), j=l,...,n.

This is the moduli space of representations we met in the preceding section, except that we restricted the eigenvalues to be fcth roots of unity, i.e.

 

In addition we described the case of SU(n), rather than U(n).

All these moduli spaces have a natural line-bundle Lk whose holomorphic sections give the quantization at level k. Note that here the moduli space itself depends on k, whereas in the absence of marked points the moduli space is independent of k and Lk = Lk is the fcth power of a fixed line-bundle L.

 

We first define the degree of E, relative to these weights a, by

 

 

E (or better (E, L)) is then defined to be stable, relative to a, if for all F

 

As an example we give Seshadri's definition of stability for a rank 2 vector bundle with just one marked point P. There are just two weights (assumed distinct) and we choose them so that

Next we fix a line L (one-dimensional subspace) of the fibre EP which we regard as part of the structure of E. Given any holomorphic sub-line-bundle F of £ we then define