Maximality of the Moduli Stacks for Vector Bundles over Riemann Surfaces

We explore the concept of maximality for a moduli stack, giving a new definition for this property. This definition is given by imposing conditions on the Borel fibration for the Galois action, mimicking the classical construction for real varieties. It is applied in the context of the moduli stack of vector bundles over a Riemann surface equipped with an antiholomorphic involution induced by the Galois action. We prove that the moduli stack of vector bundles of fixed rank and degree is maximal if and only if the Riemann surface is maximal in the classical sense. This research was motivated by the classical result for the coarse moduli space of vector bundles, and we link back to it when the rank and degree are coprime. This is a joint work with Florent Schaffhauser.