Hamiltonian actions of unipotent groups on compact K\"ahler manifolds

TL;DR

This paper investigates how unipotent complex Lie groups act meromorphically on compact Kähler manifolds, establishing stability conditions and constructing geometric quotients with compatible Kähler structures via symplectic reduction.

Contribution

It introduces natural stability criteria for unipotent group actions on Kähler manifolds and demonstrates the existence of geometric quotients with Kähler structures, linking complex-analytic and GIT approaches.

Findings

Sets of semistable points are Zariski-open.

Existence of geometric quotients with compactifiable Kähler structures.

Connection between complex-analytic theory and GIT for unipotent actions.

Abstract

We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable K\"ahler structures obtained by symplectic reduction. The relation of our complex-analytic theory to the work of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group actions on projective varieties is discussed in detail.