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Symplectic quotients have symplectic singularities

Part of: Linear algebraic groups and related topics Local rings and semilocal rings General commutative ring theory Symplectic geometry, contact geometry

Published online by Cambridge University Press:  31 January 2020

Hans-Christian Herbig ,
Gerald W. Schwarz  and
Christopher Seaton
Hans-Christian Herbig
Affiliation:
Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia – Bloco C, CEP: 21941-909 Rio de Janeiro, Brazil email herbighc@gmail.com
Gerald W. Schwarz
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA02454-9110, USA email schwarz@brandeis.edu
Christopher Seaton
Affiliation:
Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN38112, USA email seatonc@rhodes.edu
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Abstract

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_{0}$ at level zero of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_{0}$. We show that if $(V,G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This implies in particular that the real symplectic quotient is graded Gorenstein. In case $K$ is a torus or $\operatorname{SU}_{2}$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.

Keywords

singular symplectic reductionsymplectic singularitymoment mapHamiltonian actiongraded Gorenstein

MSC classification

Primary: 53D20: Momentum maps; symplectic reduction 13A50: Actions of groups on commutative rings; invariant theory
Secondary: 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 57S15: Compact Lie groups of differentiable transformations 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 3 , March 2020 , pp. 613 - 646
Copyright
© The Authors 2020

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Footnotes

C.S. was supported by the E.C. Ellett Professorship in Mathematics.

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