In symplectic geometry, the quotient of a symplectic manifold by a Lie group acting as symplectomorphisms does not usually inherit a symplectic structure. Rather, one forms the \emph{symplectic quotient}, the quotient by of a subset of defined as a fiber of the \emph{moment map} where is the Lie algebra of and its dual; this subset of is called the \emph{shell}. At a regular value of the moment map, the shell is a submanifold of and its quotient is a symplectic manifold. Otherwise, the shell and symplectic quotient are singular varieties. To consider the kinds of singularities that arise, one can restrict to the case when for a finite-dimensional linear -representation . In this case, there is a natural homogeneous quadratic moment map such that is a singular value, hence the shell and symplectic quotient are singular.
For many cases of and , it has been demonstrated by Bellamy--Schedler, Becker, Terpereau, and others that has \emph{symplectic singularities}; equivalently, is Gorenstein with rational singularities and its smooth locus admits a holomorphic symplectic form. In some of these cases, the variety already has rational singularities. We will present recent results demonstrating that for a given , the shell has rational singularities and the symplectic quotient has symplectic singularities in the case of ``most" -modules . This includes the important case where is semisimple and with , optimizing results of Aizenbud--Avni and generalizing a theorem of Budur. We will also discuss applications indicating that the representation varieties of surface groups have rational singularities and the corresponding character varieties have symplectic singularities.
Joint work with Hans-Christian Herbig and Gerald Schwarz
Rational singularities of the zero fiber of the moment map Sponsored by the Meyer Fund