A Kähler structure for the $$\text {PU}(2, 1)$$ configuration space of four points in $$S^3$$

We show that an open subset $${\mathfrak {F}}_4''$$ of the $$\mathrm{PU}(2,1)$$ configuration space of four points in $$S^3$$ is in bijection with an open subset of $${\mathfrak {H}}^{\star }\times {\mathbb {R}}_{>0}$$ , where $${\mathfrak {H}}^\star $$ is the affine-rotational group. Since the latter is a Sasakian manifold, the cone $${\mathfrak {H}}^\star \times {\mathbb {R}}_{>0}$$ is Kahler and thus $${\mathfrak {F}}_4''$$ inherits this Kahler structure.