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

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 geometric invariant theory (GIT) quotients 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 a 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.