Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g ⩾ 3 and let M 0 be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M 0 have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan–Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwanʹs desingularization to Narasimhan–Ramananʹs, and prove that the Narasimhan–Ramananʹs desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwanʹs and Seshadriʹs as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.