Projective Moduli for Hitchin Pairs

In the paper [2], Hitchin studied pairs (E,φ), where E is a vector bundle of rank two with a fixed determinant on a curve C and φ : E −→ E ⊗ KC is a trace free homomorphism, and constructed a moduli space for them. This moduli space carries the structure of a non-complete, quasi-projective algebraic variety. Later, Nitsure [5] gave an algebraic construction of moduli spaces of pairs (E,φ) over a curve C consisting of a vector bundle E of fixed degree and rank and a homomorphism φ : E −→ E ⊗ L where L is some previously chosen line bundle. He also obtained non-complete moduli spaces. The most general results were obtained by Yokogawa [7]. In his paper, C is replaced by a relative scheme f : X −→ S where f is a smooth, projective, geometrically integral morphism and S is a scheme of finite type over a universally Japanese ring, and L by a locally free sheaf F on X. It is the aim of our paper to compactify some of the spaces obtained by Yokogawa, namely those where S = SpecC and F is again a line bundle. In order to avoid confusion with the objects studied e.g. by Simpson, we will call our objects (oriented) Hitchin pairs. We shall also mention that, only recently, T. Hausel compactified the space of oriented Hitchin pairs of rank two with fixed determinant over a curve C, using methods from symplectic geometry. This result and a detailed investigation of the resulting spaces will appear in a forthcoming preprint of his. The structure of this note is as follows: In the first section we treat the case where X is a point. This case shows how to define Hitchin pairs correctly and suggests the definition of (semi)stability. Then we prove a boundedness result following [5], construct a projective parameter space for semistable Hitchin pairs and a universal family on this parameter space, and finally define a linearized SL(V )-action on this parameter space such that the moduli space is given as parameter space// SL(V ). After these constructions, we prove the (semi)stability criterion.