课程名称：Stable parabolic vector bundles on compact Riemann surfaces
授课时间：7月23日, 7月24日, 7月25日 上午 10:00-11:30
Moduli spaces of vector bundles on Riemann surfaces have been extensively studied and are still actively studied nowadays. In the construction of moduli spaces, the notion of stability plays an essential role. Narasimhan and Seshadri proved that, on a compact Riemann surface of genus > 1, a holomorphic vector bundle of degree zero is stable iff it arises from an irreducible unitary representation of the fundamental group. In this mini-course, we will introduce how to generalize this result on a finite type Riemann surface. We start from the notion of parabolic structures, which are given by Mehta and Seshadri. This notion allows us to define parabolic degree and the concept of parabolic (semi-)stable vector bundles. Then we present a correspondence between parabolic semi-stable vector bundles and unitary representations of Fuchsian groups. At last, we give a proof for the existence of the moduli space of parabolic semi-stable vector bundles. The subtitles of the three lectures are
1. The classical Narasimhan-Seshadri theorem
2. The concept of parabolic structures and a generalization of the Narasimhan-Seshadri theorem
3. Moduli space of parabolic semi-stable vector bundles.