Espace des modules de certains polyedres projectifs miroirs

TL;DR

This paper constructs an explicit diffeomorphism for the moduli space of certain mirror projective polyhedra with fixed dihedral angles, revealing its structure as a union of multiple Euclidean spaces based on combinatorial data.

Contribution

It introduces the class of ecimahedra and explicitly describes the moduli space of mirror projective polyhedra with fixed dihedral angles.

Findings

Explicit diffeomorphism between moduli space and union of Euclidean spaces

Introduction of ecimahedra as a new class of combinatorial polyhedra

Explicit computation of parameters n and d from combinatorics and angles

Abstract

A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in $(0,\frac{\pi}{2}]$, and the union of $n$ copies of $\R^d$, when the polyhedron has the combinatorics of an \emph{ecimahedron}, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers $n$ and $d$ can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.