After reviewing the current status of the Kojima-Mizushima-Tan conjecture concerning circle packings on compact complex projective surfaces, we pivot to the non-compact setting and examine circle packings on complex projective surfaces with punctures. For the thrice punctured sphere we describe a three-dimensional slice of the infinite dimensional moduli space that parameterizes the elementary triangular surfaces and ones obtained from these by grafting. We then construct circle packings on all those of hyperbolic and parabolic type. This falsifies the Kojima-Mizushima-Tan conjecture in the non-compact setting.
This is joint work with Sam Ballas, Alex Casella, and Lorenzo Ruffoni.