Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

TL;DR

This paper establishes a link between analytic Hilbert quotients and GIT stability for algebraic varieties with group actions, providing new insights into the structure and classification of such quotients.

Contribution

It proves that open subsets with analytic Hilbert quotients correspond to semistable points for some G-linearised Weil divisor, connecting analytic and algebraic stability notions.

Findings

Semistability with respect to a momentum map equals GIT-semistability.

Finiteness of compact momentum map quotients for a given algebraic Hamiltonian G-variety.

A projectivity criterion for varieties with compact Kähler quotients.

Abstract

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest.