Limiting Distributions of Scaled Eigensections in a GIT-Setting

Abstract

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

67 pages - This work is the author's doctoral thesis submitted in partial fulfillment of the requirements for the degree of Doktor rer. nat. at the Ruhr-Universität Bochum