Space of Unitary Vector Bundles on a Compact Riemann Surface

Abstract

Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this paper that on the space of isomorphic classes of unitary vector bundles on X of a given rank, there is a natural structure of a normal projective variety (Theorem 8.1). We recall that Mumford has proved that, on the space of (isomorphic classes) stable vector bundles on X of a given rank and degree, there is a natural structure of a non-singular quasi-projective variety (cf. [7]); further, it was proved in [9] that a vector bundle on X of degree zero is stable if and only if it is associated to an irreducible unitary representation of the fundamental group of X. Thus our result shows the existence of a canonical compactification (as an algebraic variety) of the space of stable bundles on X of a given rank and degree zero. We shall now give a brief outline of the proof. It consists in a refinement of the proof of Mumford for the existence of a natural structure of a quasiprojective variety on the space of stable bundles of a given rank and degree (loc. cit.). Let us fix a very ample invertible sheaf OX(1) on X; then if m is a positive integer which is sufficiently large, we have H'(V(m)) 0 0 and H0( V(m)) generates V(m) for any Ve Or,, where Or, stands for the category of unitary vector bundles on X of rank r. Then the rank of H0(V(m)) is the same whatever be V e OR9; let this be p. The Hilbert polynomial of V(m), is also the same whatever be V e OR,; let this be P. Let Q = Quot(E/P) be the scheme in the sense of Grothendieck; E being the free coherent sheaf of rank p on X (cf. [4]). Let R be the open subscheme of Q consisting of the points which represent quotients of E which are locally free, and whose sections can be canonically identified with H0(E). Thus one has a family of vector bundles {Fq}qeR on X such that every Fq can be canonically considered as a quotient vector bundle of the trivial bundle E on X of rank p. The linear group G = Aut E acts on Q, and R is a G-invariant subscheme; further given V e Or there is a q e R such that Fq V, and the set of such points q con-