Sharp bounds on fake weighted projective spaces with canonical singularities

TL;DR

This paper establishes precise upper bounds on the multiplicity of fake weighted projective spaces with canonical singularities, characterizing cases of equality and exploring related conjectures.

Contribution

It provides the first sharp bounds on multiplicity and lattice index for these spaces, with a complete characterization of equality cases.

Findings

Sharp upper bounds on multiplicity established

Complete characterization of equality cases provided

Discussion of related conjectures included

Abstract

We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a lattice simplex containing only the origin as an interior lattice point. We also completely characterise when equality occurs and discuss related questions and conjectures.