Stable Configurations of Linear Subspaces and Quotient Coherent Sheaves

TL;DR

This paper establishes stability criteria for systems of linear subspaces and quotient coherent sheaves using geometric invariant theory and moment maps, generalizing known correspondences and exploring the structure of G-ample cones.

Contribution

It generalizes the Gelfand-MacPherson correspondence to subspaces and provides new stability criteria for sheaves and subspace systems.

Findings

Derived stability criteria using Hilbert-Mumford and moment map techniques.

Generalized Gelfand-MacPherson correspondence to linear subspaces.

Identified examples of G-ample cones lacking top chambers.

Abstract

In this paper we provide some stability criteria for systems of linear subspaces of $V \otimes W$ and for systems of quotient coherent sheaves, using, respectively, the Hilbert-Mumford numerical criterion and moment map. Along the way, we generalize the Gelfand-MacPherson correspondence [11] from point sets to sets of linear subspaces (of various dimensions). And, as an application, we provide some examples of $G$-ample cones without any top chambers. The results of this paper are based upon and/or generalize some earlier works of Klyachko [18], Totaro [28], Gelfand-MacPherson [11], Kapranov [17], Foth-Lozano [8], Simpson [24], Wang [30], Phong-Sturm [22], Zhang [32] and Luo [20], among others.