Modular Galois covers associated to symplectic resolutions of singularities

Abstract

Let Y be a normal projective variety and π : X → Y a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def (X) and Def (Y) are both smooth, of the same dimension, and π induces a finite branched cover ƒ : Def (X) → Def (Y). We prove that ƒ is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Namikawa's work. The modular Galois group G is the product of Weyl groups of finite type, indexed by such irreducible components B. Each Weyl group factor W_B is that of a Dynkin diagram, obtained as a quotient of the Dynkin diagram of the singularity-type of B, by a group of Dynkin diagram automorphisms.

Finally we consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities. We prove that ƒ : Def (X) → Def (Y) is a Galois cover of its image. This explains the analogy between the above results and related work of Nakajima, on quiver varieties, and of Szendrői on enhanced gauge symmetries for Calabi-Yau threefolds.