A Minimal Triangulation of Complex Projective Plane Admitting a Chess Colouring of Four-Dimensional Simplices

We construct and study a new 15-vertex triangulation $X$ of the complex projective plane $\mathbb C\mathrm P^2$. The automorphism group of $X$ is isomorphic to $S_4\times S_3$. We prove that the triangulation $X$ is the minimal (with respect to the number of vertices) triangulation of $\mathbb C\mathrm P^2$ admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of $X$ and show that the automorphism group of $X$ can be realized as a group of isometries of the Fubini–Study metric. We find a 33-vertex subdivision $\overline X$ of the triangulation $X$ such that the classical moment mapping $\mu\colon\mathbb C\mathrm P^2\to\Delta^2$ is a simplicial mapping of the triangulation $\overline X$ onto the barycentric subdivision of the triangle $\Delta^2$. We study the relationship of the triangulation $X$ with complex crystallographic groups.