# Complex symplectic quotients with symplectic singularities

Expositor: Christopher W. Seaton
11/07/2017 , 17:00 | Sala 236

Let $K$ be a compact Lie group and $V$ a unitary $K$-module. Considering $V$ as a real symplectic manifold and choosing the homogeneous quadratic moment map $J$, let $X$ denote the singular symplectic quotient $X = M/K$    where $M=J^{-1}(0)$    is the zero fiber of the moment map. The space $X$ has several structures, including that of a symplectic stratified space and a semialgebraic set, and its algebra of smooth functions has a graded Poisson subalgebra $mathbb{R}[X]$  of regular functions defined as the quotient of the real polynomial invariants of the $K$-action on $V$ by the invariant part of the vanishing ideal of $M$. Let $G$ denote the complexification of $K$, and then the corresponding action of $G$ on $V+V^*$   is Hamiltonian with moment map given by tensoring $J$ with $mathbb{C}$ . The complex symplectic quotient $Y$ is given by the affine quotient of the variety of $mu$ by $G$.

We will show that, under mild hypotheses, the complex symplectic quotient $Y$ has symplectic singularities as defined by Beauville. We will discuss consequences of this result for the ring $mathbb{R}[X]$  of regular functions on the real symplectic quotient.

Joint work with Hans-Christian Herbig and Gerald Schwarz