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

TL;DR

This paper introduces a minimal 15-vertex triangulation of the complex projective plane with a chess coloring, detailing its automorphisms, explicit simplices parametrization, and relation to complex crystallographic groups.

Contribution

It constructs the first minimal vertex triangulation of P^2 with a chess coloring and analyzes its automorphism group and geometric properties.

Findings

Triangulation has 15 vertices, minimal for this type.

Automorphism group is isomorphic to S_4 d S_3.

Provides explicit parametrizations and geometric realizations.

Abstract

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.