Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic $2$-space are completely determined, up to conjugation in ${\rm PU}(2,1)$, by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.

MSC Classifications:

32G05, 32M05 show english descriptions Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]
Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10] 32G05 - Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]
32M05 - Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]

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