A symplectic fourfold

TL;DR

This paper introduces a method to construct four-dimensional irreducible symplectic varieties with specific properties, expanding the known examples and highlighting cases where the global Torelli theorem does not apply.

Contribution

It provides explicit constructions of irreducible symplectic varieties with non-quotient singularities and small second Betti number, not previously documented.

Findings

Constructed irreducible symplectic varieties of dimension 4 with b2=4

Produced examples with non-quotient singularities

Showed Torelli theorem's limitations for these varieties

Abstract

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-K\"ahler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2 = 4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.