A K\"ahler structure for the ${\rm PU}(2, 1)$ configuration space of four points in $S^3$

TL;DR

This paper establishes a Kähler structure on a subset of the configuration space of four points in the 3-sphere, linking it to Sasakian and Kähler geometry through a bijection with a product space.

Contribution

It introduces a novel Kähler structure on a configuration space of four points in $S^3$, derived from the geometry of affine-rotational groups and Sasakian manifolds.

Findings

Identifies a bijection between an open subset of the configuration space and a Kähler manifold.

Shows the inherited Kähler structure from the cone over a Sasakian manifold.

Connects configuration space geometry with complex and Sasakian geometry.

Abstract

We show that an open subset ${\mathfrak F}_4''$ of the ${\rm PU}(2,1)$ configuration space of four points in $S^3$ is in bijection with an open subset of %with a K\"ahler structure which is inherited from the one 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 K\"ahler and thus ${\mathfrak F}_4''$ inherits this K\"ahler structure.