Cohomology of smooth toric varieties: naturality

TL;DR

This paper studies the naturality of cohomology ring isomorphisms for smooth toric varieties and related spaces, identifying conditions under which these isomorphisms are natural and describing deformation terms when they are not.

Contribution

It establishes when the cohomology isomorphism is natural with respect to toric morphisms, especially highlighting the role of invertibility of 2 in the coefficient ring.

Findings

Isomorphism is natural if 2 is invertible in the coefficient ring.

Deformation terms appear when 2 is not invertible, explicitly described.

Provides conditions for naturality of cohomology isomorphisms in toric geometry.

Abstract

Building on the recent computation of the cohomology rings of smooth toric varieties and partial quotients of moment-angle complexes, we investigate the naturality properties of the resulting isomorphism between the cohomology of such a space and the torsion product involving the Stanley-Reisner ring. If 2 is invertible in the chosen coefficient ring, then the isomorphism is natural with respect to toric morphisms, which for partial quotients are defined in analogy with toric varieties. In general there are deformation terms that we describe explicitly.