Paper detail

Modular Galois covers associated to symplectic resolutions of singularities

Let Y be a normal projective variety and p a morphism from X to Y, which is 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 p induces a finite branched cover f from Def(X) to Def(Y). We prove that f 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 the morphism f from Def(X) to 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 Szendroi on enhanced gauge symmetries for Calabi-Yau threefolds.