On a smooth compactification of PSL(n, C)/T

TL;DR

This paper constructs a smooth compactification of the quotient of PSL(n,C) by a maximal torus using geometric invariant theory, and analyzes its automorphism group for n ≥ 4.

Contribution

It provides a new smooth compactification of PSL(n,C)/T as a GIT quotient of the wonderful compactification, for n ≥ 4.

Findings

The compactification is smooth and projective.

The automorphism group contains PSL(n,C) as the connected component.

The construction applies for all n ≥ 4.

Abstract

Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of ${\rm PSL}(n, \mathbb C)/T$ is ${\rm PSL}(n, \mathbb C)$ itself.