Fibred toric varieties in toric hyperk\"{a}hler varieties

TL;DR

This paper introduces fibred toric varieties as special bundles over toric varieties, showing their role in the variation of hyperkähler varieties and their appearance in natural morphisms and flops.

Contribution

It defines fibred toric varieties and demonstrates their significance in hyperkähler geometry and GIT variation, linking them to exceptional sets and Mukai flops.

Findings

Fibred toric varieties are equivariant $ ext{CP}^r$ bundles over lower-dimensional toric varieties.

They naturally occur in the variation of hyperkähler varieties.

These varieties appear in the exceptional sets of hyperkähler morphisms and Mukai flops.

Abstract

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these fibred toric varieties also arise naturally in the variation of hyperk\"{a}hler varieties, namely, the fibred toric varieties are contained in the exceptional sets of the hyperk\"{a}hler natural morphisms and the Mukai flops.