Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety.In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup.Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively.Our main result is that there are natural correspondences among quotients of these three actions.In the end, we mention a possible application to the moduli spaces of parabolic bundles over algebraic curves for further research.