Surface singularities and their deformations via principal bundles on elliptic curves

Abstract: It is well known that du Val (aka simple, Kleinian, ADE, ...) singularities of algebraic surfaces are classified by Dynkin diagrams of type ADE. A geometric link between the singularity and the Lie algebra of the same type was given by Brieskorn in the 70s, who showed that the singularity can be recovered by intersecting the nilpotent cone inside the Lie algebra with a transversal slice through a subregular nilpotent element. Brieskorn's construction also realises the entire transversal slice as the total space of a miniversal deformation of the singularity. In this talk, I will discuss an elliptic version of this story, where the Lie algebra is replaced with the stack of principal bundles on an elliptic curve. There is still a notion of subregular slice in this stack, and one gets a singular surface by intersecting such a thing with the locus of unstable bundles. I will explain which surfaces arise in this way, and in what sense the subregular slice is still the total space of a miniversal deformation. Time permitting, I will also touch on how the BCFG types are related to the ADE ones (in a different way to the story for Lie algebras!), and on some questions about Poisson structures and their quantisations.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

Free Mathematics Seminar

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