Double sided torus actions and complex geometry on $SU(3)$

TL;DR

This paper constructs explicit complex structures and holomorphic foliations on SU(3), generalizing known structures and producing orbifold variants of the flag variety through double-sided torus actions.

Contribution

It introduces new explicit complex structures and foliations on SU(3), extending previous invariant structures and creating orbifold quotients of the flag variety.

Findings

Explicit complex structures on SU(3) constructed

Holomorphic foliations with transverse Kähler structures developed

Orbifold quotients of the flag variety obtained

Abstract

We construct explicit complex structures and transversely K\"ahler holomorphic foliations on $SU(3)$ corresponding to variations of real quadratic equations on a complex quadric in $\mathbb{C}^{6}$ as generalizations of left-invariant complex structures on $SU(3) $ and an invariant K\"ahler structure on the flag variety $SU(3)/T$.Consequently, we obtain orbifold variants of the flag variety $SU(3)/T$ as quotients of double sided torus actions.