Homeomorphism type of the non-negative part of a complete toric variety

Abstract

In this note we show that the nonnegative part of a proper complex toric variety has the homeomorphism type of a sphere, and consequently that the nonnegative part has a natural structure of a cell complex. This extends previous results of Ehlers and Jurkiewicz. The proof also provides a simplicial decomposition of the nonnegative part, and a parameterization of each maximal simplex. This result is needed in arXiv:2504.12903 as part of an argument constructing a torus-stable reduced Čech complex for any semi-proper toric variety.