Reduction of complex HamiltonianG-spaces

Heinzner, Peter; Loose, Frank

doi:10.1007/bf01896243

Cited by 65 publications

(87 citation statements)

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“…This paper makes essential use of the holomorphic slice theorem and the existence of a potential ρ only in the case that U is commutative. One can give short proofs of the required results in this case, thus circumventing the use of the results in [HeLo94]. Then our results in this paper generalize those of [HeLo94] and are independent of the results there.…”

Section: Commutative Groupsmentioning

confidence: 52%

Cartan decomposition of the moment map

Heinzner

Schwarz

2006

Math. Ann.

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Section: Commutative Groupsmentioning

confidence: 52%

Cartan decomposition of the moment map

Heinzner

Schwarz

2006

Math. Ann.

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“…We collect the main results about Hamiltonian G-spaces in Theorem 3.1 ([HL94], [HHL94]). Let X be a Hamiltonian G-space.…”

Section: Kähler Structuresmentioning

confidence: 99%

“…* for the action of a maximal compact subgroup K of G with respect to a K-invariant Kähler structure on X, the fundamental link between symplectic and complex geometry is given by the concept of semistability: generalising fundamental work of Kirwan [Kir84], it was shown by Heinzner and Loose [HL94] that the set of µ-semistable points …”

Section: Introductionmentioning

confidence: 99%

Projectivity of analytic Hilbert and Kähler quotients

Greb¹

2010

Trans. Amer. Math. Soc.

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Abstract. We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.

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“…Though we are not interested here in the results for Kähler reduction (The reader is referred to [5,12], for example, for Marsden-Weinstein reduction on Kähler manifolds.) we notice that the almost complex structure can be dropped to the quotient space P/G.…”

Section: Abstract Mechanical Connectionmentioning

confidence: 99%

Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Pekarsky

Marsden

2001

Journal of Applied Mathematics

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When the phase space P of a Hamiltonian G-system (P, ω, G, J, H) has an almost Kähler structure, a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (due to the work of A. Blaom), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.

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Reduction of complex HamiltonianG-spaces

Cited by 65 publications

References 4 publications

Cartan decomposition of the moment map

Cartan decomposition of the moment map

Projectivity of analytic Hilbert and Kähler quotients

Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

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