On the Kähler cone of irreducible symplectic orbifolds
Grégoire Menet, Ulrike Rieß
TL;DR
This work extends the Hodge version of the global Torelli theorem to irreducible symplectic orbifolds and develops a robust Kähler-cone theory in the orbifold context, including a generalized Boucksom criterion and a wall-divisor framework that is invariant under parallel transport. By connecting period domains, twistor constructions, and birational geometry, the authors derive consequences for orbifold singularities, non-separated moduli points, and the Morrison–Kawamata cone conjecture in the $b_2=4$ setting. A key innovation is defining a lattice-based mirror symmetry as an involution on a sub-moduli space via an isomorphism of period domains, enabling a concrete mirror construction for marked irreducible symplectic orbifolds endowed with a Kähler class. The results unify and extend known smooth-case theorems to orbifolds, provide new examples like Nikulin-type orbifolds, and lay groundwork for explicit Kähler/movable-cone descriptions and mirror pairs in this broader context.
Abstract
We generalize the Hodge version of the global Torelli theorem in the framework of irreducible symplectic orbifolds. We also propose a generalization of several results related to the Kähler cone and the notion of wall divisors introduced in the smooth case by Mongardi. As an application we propose a definition of the mirror symmetry as an involution on a moduli space; this construction is also new in the smooth case.