A classification of $\mathbb{C}$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group ${\mathrm{PSU}}_{1,2}({\mathcal{O}}_K)$ preserving a complex geodesic in the complex hyperbolic plane ${\mathbb{H}}_{\mathbb{C}}^2$. Complementing work of Holzapfel, Chinburg-Stover and M"oller-Toledo, we show that these maximal $\mathbb{C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\left(\text{\hspace{0.17em}⁣}\begin{array}{c}D,D_K\\ \mathbb{Q}Q\end{array}\text{\hspace{0.17em}⁣}\right)$ for some explicit $D\in \mathbb{N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of ${\mathbb{P}}_2(\mathbb{C})$.