Limiting Distributions of Scaled Eigensections in a GIT-Setting

TL;DR

This paper studies the asymptotic distribution of scaled eigensections of a line bundle over a compact variety with a torus action, linking their behavior to the geometry of the associated GIT quotient.

Contribution

It characterizes the limiting distributions of eigensections in a GIT setting, extending previous work by connecting asymptotics to geometric invariant theory.

Findings

Describes the limiting distribution of eigensections as the tensor power grows large.

Establishes a connection between eigensection asymptotics and the geometry of the Hilbert quotient.

Provides a framework for understanding the asymptotic behavior in GIT contexts.

Abstract

Let $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ be a base point free $\mathrm{\mathbb{T}=T^{\mathbb{C}}}$-linearized hermitian line bundle over a compact variety $\mathrm{\mathbf{X}}$ where $\mathrm{T=\left(S^{1}\right)^{m}}$ is a real torus. The main focus of this paper is to describe the asymptotic behavior of a certain class of sequences $\mathrm{\left(s_{n}\right)_{n}}$ of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ as $\mathrm{n\rightarrow \infty}$, introduced by Shiffman, Tate and Zelditch, and its connection to the geometry of the Hilbert quotient $\mathrm{\pi\!:\!\bf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ where $\mathrm{\xi\in \mathfrak{t}^{*}}$.