A Kleiman criterion for GIT stack quotients

TL;DR

This paper extends Kleiman's ampleness criterion to GIT quotients, relating the GIT chamber structure to a positivity condition involving quasimaps, thereby providing a new geometric criterion for stability.

Contribution

It introduces an analog of Kleiman's criterion for GIT quotients, connecting GIT chamber decomposition with positivity conditions on quasimaps.

Findings

GIT chambers correspond to cells in the variation of GIT decomposition.

A positivity condition on quasimaps characterizes GIT stability.

The criterion generalizes classical ampleness to the GIT setting.

Abstract

Kleiman's criterion states that, for $X$ a projective scheme, a divisor $D$ is ample if and only if it pairs positively with every non-zero element of the closure of the cone of curves. In other words, the cone of ample divisors in $N^1(X)$ is the interior of the nef cone. In this paper we present an analogous statement for a variety $X$ acted on by a reductive group $G$ with a choice of $G$-linearization $L \to X$. In this new context, the ample cone of $X$ is replaced by a cell in the variation of GIT decomposition of the G-ample cone, and curves in $X$ are replaced by quasimaps to $[X/G]$.