6 - Parabolic G-Bundles and Equivariant G-Bundles

Summary

We define the stability, semistability and polystability of vector bundles over any smooth curve? and extend these notions to G-bundles over ?. More generally, we define the parabolic stability and parabolic semistability for parabolic G-bundles over an s-pointed curve. We further extend the notions of stability, semistability and polystability to A-stability, A-semistability and A-polystability in the case a finite group A acts faithfully on a smooth projective curve ?’. Then, we prove an equivalence between the groupoid fibration of A-equivariant G-bundles on ?’ and quasi-parabolic G-bundles on an s-pointed curve ? = ?’/A consisting of the A-ramification points. We prove the existence and uniqueness of the Harder--Narasimhan reduction of any G-bundle. The main highlight of this chapter is to prove the celebrated Narasimhan--Seshadri theorem asserting that any polystable vector bundle over any smooth curve ? is obtained through a topological construction via unitary representation of the fundamental group of the curve. We also prove its G-bundle generalization and, in fact, A-equivariant G-bundle generalization.