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

Abstract

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.

Dedicated to Robert MacPherson on the occasion of his 60th birthday. MacPherson's special issue, Pure and Applied Mathematics Quarterly (to appear)