Thirty-three deformation classes of compact hyperk\"ahler orbifolds

TL;DR

This paper classifies 33 deformation classes of compact hyperk"ahler orbifolds, analyzing their singularities and providing explicit examples in dimensions 4 and 6.

Contribution

It introduces a classification of hyperk"ahler orbifolds via deformation classes and computes singularities for numerous examples, expanding understanding of their structure.

Findings

29 orbifold examples in dimension 4 with computed singularities

4 orbifold examples in dimension 6

these examples are deformation independent

Abstract

As their smooth analogue the irreducible symplectic varieties appear as elementary bricks in the generalizations of the Bogomolov decomposition theorem (arXiv:math/0402243, arXiv:2012.00441). Let $S$ be a K3 surface; generalizing the Fujiki construction, we investigate the irreducible symplectic varieties with simply connected smooth locus that can be obtained as terminalizations of quotients of the product $S^{n}$. In dimension 4, we compute the singularities for 29 orbifolds examples which appear to be independent under deformation. We also provide 4 additional orbifolds examples in dimension 6.