A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

In this paper we construct and study a new 15-vertex triangulation $X$ of the complex projective plane $\CP^2$. The automorphism group of $X$ is isomorphic to $S_4\times S_3$. We prove that the triangulation $X$ is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for 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 provide a 33-vertex subdivision $\bX$ of the triangulation $X$ such that the classical moment mapping $\mu:\CP^2\to\Delta^2$ is a simplicial mapping of the triangulation $\bX$ onto the barycentric subdivision of the triangle $\Delta^2$. We study the relationship of the triangulation $X$ with complex crystallographic groups.