Modular Galois covers associated to symplectic resolutions of singularities

TL;DR

This paper studies the Galois properties of deformation spaces of symplectic resolutions of singularities, revealing a modular Galois group structure linked to Weyl groups and Dynkin diagrams, with broader implications for related geometric contexts.

Contribution

It proves that the deformation map is Galois and explicitly calculates the Galois group as a product of Weyl groups, extending to affine and Calabi-Yau cases.

Findings

The deformation map is Galois with a well-defined Galois group.

The Galois group is a product of Weyl groups associated with singularity components.

Generalizations include affine symplectic and Calabi-Yau threefold cases.

Abstract

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.