11 - Dualities on T^*SU_X (2, O_X)

Summary

To P.E. Newstead with best wishes, in gratitude, celebration of the past and anticipation of his future leadership.

Abstract. The notion of Algebraic Complete Integrability (ACI) of certain mechanical systems, introduced in the early 1980s, has given great impetus to the study of moduli spaces of holomorphic vector bundles over an algebraic curve (or a higher-dimensional variety, still at a much less developed stage). Several notions of ‘duality’ have been the object of much interest in both theories. There is one example, however, that appears to be a beautiful isolated feature of genus-2 curves. In this note such example of duality, which belongs to a ‘universal’ class of ACIs, namely (generalized) Hitchin systems, is interpreted in the setting of the classical geometry of Klein's quadratic complex, following the Newstead and Narasimhan-Ramanan programme of studying moduli spaces through explicit projective models.

Introduction

In this volume's conference, dedicated to Peter Newstead and his work, one of the prominent objects was SU_X (2, ξ), the moduli space of (semi)stable, rank-2 vector bundles over a Riemann surface X of genus g ≥ 2, with fixed determinant ξ. The cases of degree(ξ) even, odd respectively, give rise to isomorphic varieties (by tensoring with a line bundle, since Jac(X) is a divisible group), usually denoted by SU_X (2, 0), SU_X (2, 1) respectively, when ξ is not important. When the rank is coprime with the degree, a semistable bundle must be stable and the variety is nonsingular.