Wednesday, March 20, 18:30, room 306

Evgeny Goncharov

Symplectic resolutions of certain hypersurfaces in A^5

The hypersurfaces we are going to consider are obtained by the general procedure of constructing Coulomb branches (an object of interest to physicists that was rigorously defined mathematically in a series of papers by A. Braverman, M. Finkelberg and H. Nakajima). The general procedure is quite involved but for the case we will consider (quiver with one vertex and l loops with dim V = 2, dim W = 1) there is a theorem that describes the generators (and gives a way to find relations between them) of the coordinate ring of the Coulomb branch (it is an algebraic variety). We will apply the theorem to find the Coulomb branch which turns out to be a hypersurface in A^5.
Then I will explain how to (naturally) define a symplectic form on the smooth locus of this hypersurface. For l=1 the Coulomb branch turns out to be Sym^2 A^2 and its symplectic resolution (resolution of singularities such that the pullback of the symplectic form on the smooth locus extends to define a symplectic form on the resolution) has been known for a long time (it is given by the Hilbert scheme Hilb^2 A^2). I will show that a naive attempt to construct a symplectic resolution for a general l does not work and discuss the general properties of our hypersurface (compute the Hilbert polynomial, etc).
We will then note that a certain construction in a Lie algebra sp_{2l} (Slodowy slice to the subsubregular orbit of the nilpotent of type (2l-2, 1, 1)) has precisely the same properties and compute the Slodowy slice explicitly to see that the resulting variety is the same. The symplectic resolution of the Slodowy slice, however, is easy to describe. This argument gives a new family of symplectic hypersurfaces (apart from the known Kleinian singularities in A^3). Time permitting we will discuss what happens if one varies the dimensions of the vector spaces V and W in the quiver we started with.