scholarly journals Pushing down the Rumin complex to conformally symplectic quotients

2014 ◽  
Vol 35 ◽  
pp. 255-265 ◽  
Author(s):  
Andreas Čap ◽  
Tomáš Salač
Keyword(s):  
Symplectic Quotients ◽  
Rumin Complex

scholarly journals Corrigendum: Gromov-Witten invariants of symplectic quotients and adiabatic limits

2009 ◽  
Vol 7 (3) ◽  
pp. 377-379
Author(s):  
A. Rita Gaio ◽  
A. Rita Gaio ◽  
Dietmar A. Salamon ◽  
Dietmar A. Salamon
Keyword(s):  
Adiabatic Limits ◽  
Symplectic Quotients

scholarly journals POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

2020 ◽  
pp. 1-52
Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU
Keyword(s):  
Differential Form ◽  
Scale Invariance ◽  
Differential Forms ◽  
Sobolev Inequalities ◽  
Exact Form ◽  
Heisenberg Groups ◽  
Contact Manifolds ◽  
Bounded Geometry ◽  
Exterior Differential ◽  
Rumin Complex

Abstract In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

THE KIRWAN MAP FOR SINGULAR SYMPLECTIC QUOTIENTS

2006 ◽  
Vol 73 (01) ◽  
pp. 209-230 ◽  
Author(s):  
YOUNG-HOON KIEM ◽  
JONATHAN WOOLF
Keyword(s):  
Symplectic Quotients

scholarly journals Symplectic quotients have symplectic singularities

2020 ◽  
Vol 156 (3) ◽  
pp. 613-646 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Gerald W. Schwarz ◽  
Christopher Seaton
Keyword(s):  
Lie Group ◽  
Moment Map ◽  
Compact Lie Group ◽  
The Real ◽  
Level Zero ◽  
Symplectic Quotients ◽  
Complex Symplectic

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.

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