On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32

TL;DR

This paper explicitly constructs 81 symplectic resolutions of a specific 4-dimensional quotient singularity arising from a group of order 32, revealing detailed geometric structures and symplectic properties.

Contribution

It provides an explicit GIT-based construction of all 81 symplectic resolutions for the quotient singularity, and introduces a new Kummer-type symplectic 4-fold.

Findings

Explicit construction of 81 symplectic resolutions

Description of geometric structures and flops

Representation of the group as automorphisms of an abelian 4-fold

Abstract

We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via GIT quotient of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As the result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.