Singularities and Poisson geometry of certain representation spaces

Huebschmann, Johannes

doi:10.1007/978-3-0348-8364-1_6

Cited by 17 publications

(23 citation statements)

References 42 publications

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“…lifting the embedding ρ : Γ ֒→ SO (3, R). The image of the center Z(Γ) of Γ would then be in the center of α(Γ), hence, as we noted above, in the center of SU (2). Then, the composition ρ = p • α would send Z(Γ) to 1, which is a contradiction.…”

Section: Representations Of Elementary Coxeter Groupsmentioning

confidence: 84%

“…In this paper we will prove that there are no "local" restrictions on geometry of representation schemes of 3-manifold groups to P O(3) and SL (2). Note that both groups H = P O(3) and H = SL(2) are affine algebraic group schemes defined over Q, thus, for every finitely-generated group Γ, the representation schemes Hom(Γ, H) and character schemes X(Γ, H) = Hom(Γ, H)//H are affine algebraic schemes over Q.…”

Section: Introductionmentioning

confidence: 99%

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On representation varieties of 3–manifold groups

Kapovich

Millson

2017

Geom. Topol.

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Section: Representations Of Elementary Coxeter Groupsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

On representation varieties of 3–manifold groups

Kapovich

Millson

2017

Geom. Topol.

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“…When G = SU(2), as already pointed out in the introduction, the space N is an exotic CP 3 , and the complement of the top stratum is a Kummer surface which, in turn, is the singular locus, in the sense of complex analytic stratified Kähler spaces, of this exotic CP 3 . For various special cases, the local structure of the space N near any of its points has been examined in [32]; see also [38][39][40].…”

Section: Reduction and Stratified Kähler Spacesmentioning

confidence: 99%

Kähler spaces, nilpotent orbits, and singular reduction

Huebschmann¹

2004

Memoirs of the AMS

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For a stratified symplectic space, a suitable concept of stratified Kähler polarization encapsulates Kähler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kähler space which establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces: The closure of a holomorphic nilpotent orbit or, equivalently, the closure of the stratum of the associated pre-homogeneous space of parabolic type carries a (positive) normal Kähler structure. In the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS's, and certain pre-homogeneous spaces appear as different incarnations of the same structure. The closure of the principal holomorphic nilpotent orbit arises from a semisimple holomorphic orbit by contraction. Symplectic reduction carries a positive Kähler manifold to a positive normal Kähler space in such a way that the sheaf of germs of polarized functions coincides with the ordinary sheaf of germs of holomorphic functions. Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups. Projectivization of holomorphic nilpotent orbits yields exotic (positive) stratified Kähler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics. The space of (in general twisted) representations of the fundamental group of a closed surface in a compact Lie group or, equivalently, a moduli space of central Yang-Mills connections on a principal bundle over a surface, inherits a (positive) normal (stratified) Kähler structure. Physical examples are provided by certain reduced spaces arising from angular momentum zero. , holomorphic nilpotent orbit, normal complex analytic space, constrained system, J(ordan) T(riple) S(ystem), reductive dual pair, invariant theory, real Lie algebra of hermitian type, prehomogeneous space, contraction of semisimple holomorphic orbits, moduli space of central Yang-Mills connections, moduli space of semistable holomorphic vector bundles.

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“…given by Kirwan [13, p.159]) where the geometric quotient has no singularities but the symplectic quotient has singularities in the sense that the Kähler structure coming from the reduction process is singular at certain points. Further examples of the relation between the structure of the singularities of the two type of quotients are given in the contribution of J. Huebschmann [9] to this volume.…”

Section: 2mentioning

confidence: 99%

Singular projective varieties and quantization

Schlichenmaier

2001

Quantization of Singular Symplectic Quotients

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By the quantization condition compact quantizable Kähler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.

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Singularities and Poisson geometry of certain representation spaces

Cited by 17 publications

References 42 publications

On representation varieties of 3–manifold groups

On representation varieties of 3–manifold groups

Kähler spaces, nilpotent orbits, and singular reduction

Singular projective varieties and quantization

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