The universal covers of hypertoric varieties and Bogomolov's decomposition

TL;DR

This paper investigates the universal covers of affine hypertoric varieties, establishing their structure, fundamental groups, and a Bogomolov-type decomposition, linking symplectic structures and combinatorial classifications.

Contribution

It provides a detailed description of the universal cover of affine hypertoric varieties and establishes a Bogomolov decomposition analogue for these varieties.

Findings

Universal cover corresponds to hyperplane arrangement simplification.

Fundamental group of the regular locus is described.

Unique symplectic structures imply indecomposability.

Abstract

In this paper, we study the (singular) universal cover of an affine hypertoric variety. We show that it is given by another affine hypertoric variety, and taking the universal cover corresponds to taking the simplification of the associated hyperplane arrangement. Also, we describe the fundamental group of the regular locus of an affine hypertoric variety in general. In the latter part, we show that the hamiltonian torus action is block indecomposable if and only if $\mathbb{C}^*$-equivariant symplectic structures on the associated hypertoric variety are unique up to scalar. In particular, we establish the analogue of Bogomolov's decomposition for hypertoric varieties, which is proposed by Namikawa for general conical symplectic varieties. As a byproduct, we show that if two affine (or smooth) hypertoric varieties are $\mathbb{C}^*$-equivariant isomorphic as varieties, then they are also the hamiltonian torus action equivariant isomorphic as symplectic varieties. This implies that the combinatorial classification actually gives the classification of these varieties up to $\mathbb{C}^*$-equivariant isomorphisms.