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

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.