Tilting bundles on hypertoric varieties

TL;DR

This paper provides alternative proofs that a certain tilting bundle exists on smooth hypertoric varieties and that its endomorphism ring is Koszul, simplifying previous complex constructions.

Contribution

It offers new, streamlined proofs for the existence of tilting bundles and their Koszul endomorphism rings on hypertoric varieties, using restriction and quadratic regular sequence properties.

Findings

Constructed a tilting bundle on hypertoric varieties via restriction

Proved the endomorphism ring is Koszul using quadratic regular sequences

Simplified previous proofs of tilting bundle properties

Abstract

Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open GIT substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.