GIT stability and biquotients of $SU(3)$

TL;DR

This work identifies when a double-sided $(\mathbb{C}^*)^2$-quotient of $SL(3,\mathbb{C})/U$ agrees with a GIT quotient and when the $SL(3,\mathbb{C})/U$-quotient matches the $\chi$-stable locus in its affine closure. By passing to the quasi-affine model $M$ and its closure, the authors derive a precise, cone-based criterion—the Japanese fan—for when $M = \overline{M}^{\chi\text{-s}}$, ensuring the quotients coincide and are projective. This leads to a concrete application: constructing $T\times T$-invariant complex structures on $SU(3)$ that are not left-invariant and realizing associated biquotients as projective GIT quotients, thereby linking GIT, Cox rings, and Lie-group geometry. The results provide a robust geometric-analytic bridge between stability conditions and the projective realizability of biquotients of $SU(3)$, with potential implications for the study of non-left-invariant complex structures on compact Lie groups.

Abstract

We study double-sided actions of $(\mathbb{C}^*)^2$ on $SL(3,\mathbb{C})/U$ and the associated quotients, where $U$ is a maximal unipotent subgroup of $SL(3,\mathbb{C})$. The main results of this paper are a sufficient condition for the double-sided quotient to agree with the quotient in terms of the geometric invariant theory (GIT), and an explicit necessary and sufficient condition for $SL(3,\mathbb{C})/U$ to agree with the $χ$-stable locus in its affine closure. We apply this result to characterize certain complex structures on $SU(3)$ which are not left invariant by means of the GIT quotient.

Figures (3)

• Figure 1: Illustration of the condition ($\star$).
• Figure 2: Japanese traditional fan 'sensu'
• Figure 3: Lemma \ref{['lemm:forchiss']} fails for a cone without an apex.

Theorems & Definitions (25)

• Corollary : Corollary \ref{['coro']}
• Lemma 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• proof