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2001
In the first part of this paper we study different blow-up constructions on symplectic orbifolds. Unlike the manifold case, we can define different blow-ups by using different circle actions. In the second part, we use some of these constructions to describe the behavior of reduced spaces of a Hamiltonian circle action on a symplectic orbifold, when passing a critical level of its Hamiltonian function. Using these descriptions, we generalize, in the manifold case, the wall-crossing theorem of Guillemin and Sternberg to the case of a Hamiltonian torus action not necessarily quasi-free and also the Duistermaat-Heckman theorem to intervals of values of the Hamiltonian function containing critical values.
Symmetry and Spaces, 2009
We study Hamiltonian actions of compact Lie groups K on Kähler manifolds which extend to a holomorphic action of the complexified group K C . For a closed normal subgroup L of K we show that the Kählerian reduction with respect to L is a stratified Hamiltonian Kähler K C /L C -space whose Kählerian reduction with respect to K/L is naturally isomorphic to the Kählerian reduction of the original manifold with respect to K. * The first author is supported by a Promotionsstipendium of the Studienstiftung des deutschen Volkes and by SFB/TR 12 of the DFG.
SYMPLECTIC QUOTIENTS AND COMPLEX GEOMETRY
In this memoire we present the theory of symplectic quotients and the interactions with the theory of kahler quotients. The starting point is the following natural question: in the presence of a symplectic action of a Lie group G on a symplectic manifold pM, ωq can one define in a coherent way the quotient of M by G (so that the result will have a natural symplectic structure)? Even if we assume that G is compact and the action is free, we see that in general M {G cannot be symplectic (for dimension reasons). In order to obtain a symplectic quotient, one needs a new ingredient: a moment map. The fundamental theorem of the theory states that, assuming that G acts freely and properly around the zero locus of the moment map, then the G-quotient of this zero locus has a natural structure of a symplectic manifold. We will solve in detail the existence and unicity problems for moment maps. The main tool needed here is Lie algebra cohomology. We will continue with explicit computations of moment maps and explicit descriptions of symplectic quotients. We will prove that many interesting manifolds (projective spaces, Grassman manifolds, flag manifolds) can be obtained as symplectic quotients. We will conclude with a general principle which emphasizes an interesting relation between symplectic geometry and complex geometry: Let X be a Kähler manifold, K be a compact Lie group and G " K CˆX Ñ X be a holomorphic action on X which restricts to a Hamiltonian symplectic action of K which is free around µ´1p0q. Then the corresponding symplectic quotient has a natural structure of complex manifold. We will illustrate this principle in the examples we study.
Singularities and semistable degenerations for symplectic topology
Comptes Rendus Mathematique, 2018
We overview our work [6, 7, 8, 9, 10] defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
Canonical transformations for hyperkahler structures and hyperhamiltonian dynamics
Journal of Mathematical Physics, 2014
We discuss generalizations of the well known concept of canonical transformations for symplectic structures to the case of hyperkahler structures. Different characterizations, which are equivalent in the symplectic case, give rise to non-equivalent notions in the hyperkahler framework; we will thus distinguish between hyperkahler and canonical transformations. We also discuss the properties of hyperhamiltonian dynamics in this respect.
Reduction of polysymplectic manifolds
Journal of Physics A: Mathematical and Theoretical, 2015
The aim of this paper is to generalize the classical Marsden-Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogous to the Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed.
Symplectic quotients have symplectic singularities
Compositio Mathematica, 2020
Let KKK be a compact Lie group with complexification GGG, and let VVV be a unitary KKK-module. We consider the real symplectic quotient M0M_{0}M0 at level zero of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of M0M_{0}M0. We show that if (V,G)(V,G)(V,G) is 333-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 KKK is a torus or operatornameSU2\operatorname{SU}_{2}operatornameSU2, we show that these results hold without the hypothesis that (V,G)(V,G)(V,G) is 333-large.