Symplectic implosion and non-reductive quotients

TL;DR

This paper generalizes the symplectic implosion technique to construct GIT-like quotients for actions of unipotent radicals of parabolic subgroups on projective varieties, extending previous work related to compact group actions.

Contribution

It introduces a symplectic construction for GIT-like quotients by unipotent radicals of parabolic subgroups, broadening the scope of symplectic implosion methods.

Findings

Provides a symplectic framework for non-reductive GIT quotients

Extends symplectic implosion to unipotent radical actions

Connects symplectic geometry with non-reductive GIT theory

Abstract

The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates to a Hamiltonian action of a compact group K on a symplectic manifold X its 'imploded cross section'. When X is a complex projective variety and K acts linearly on X, this construction is closely related to geometric invariant theory (GIT) for the action on X of a maximal unipotent subgroup U of the complexification G of K. The aim of this paper is to generalise symplectic implosion to give a symplectic construction for GIT-like quotients by unipotent radicals U of arbitrary parabolic subgroups P of the complex reductive group G acting linearly on the projective variety X.