The universal Poisson deformation of hypertoric varieties and some classification results

TL;DR

This paper explores the universal Poisson deformation space of hypertoric varieties, classifies affine hypertoric varieties via matroids, and analyzes their resolutions and relations to quiver varieties.

Contribution

It provides a concrete description of the universal Poisson deformation space and classifies affine hypertoric varieties through matroid theory, extending prior results.

Findings

Affine hypertoric varieties are classified by regular matroids.

Criteria established for isomorphisms between certain quiver varieties.

Explicit enumeration of projective crepant resolutions for specific hypertoric varieties.

Abstract

In this paper, we study and describe the universal Poisson deformation space of hypertoric varieties concretely. In the first application, we show that affine hypertoric varieties as conical symplectic varieties are classified by the associated regular matroids (this is a partial generalization of the result by Arbo and Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose dimension vector have all coordinates equal to one are isomorphic to each other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as quiver varieties and give some examples of 8-dimensional hypertoric varieties which cannot be raised as such quiver varieties. In the second application, we compute explicitly the number of all projective crepant resolutions of some 4-dimensional hypertoric varieties by using the combinatorics of hyperplane arrangements.