Meromorphic Limits of Automorphisms

Biliotti, Leonardo; Ghigi, Alessandro

doi:10.1007/s00031-020-09551-x

Cited by 4 publications

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References 39 publications

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“…Denote by g n = exp(t n ξ). By Theorem 2 in [10], up to passing to a subsequence, there exist a proper analytic subset U of Z such that.…”

Section: Convexity Properties Of Gradient Mapmentioning

confidence: 99%

Convexity properties of gradient maps associated to real reductive representations

Biliotti

2020

Journal of Geometry and Physics

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Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson [38]. A similar result is proved for the gradient map associated to the natural G action on P(V ). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result.2010 Mathematics Subject Classification. 22E45,53D20; 14L24.

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“…Denote by g n = exp(t n ξ). By Theorem 2 in [10], up to passing to a subsequence, there exist a proper analytic subset U of Z such that.…”

Section: Convexity Properties Of Gradient Mapmentioning

confidence: 99%

Convexity properties of gradient maps associated to real reductive representations

Biliotti

2020

Journal of Geometry and Physics

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“…Remark 81. In a recent paper [16], the authors studied the compactification of the connected component of the automorphism group of a complex manifold X in the Fujiki class C given by its closure in Barlet cycle space, that we denote by B(X). The boundary points of B(X) give rise to non-dominant meromorphic self-maps of X and convergence in cycle space yields convergence of the corresponding meromorphic maps.…”

Section: Proof By Lemma 33 We Can Find Uniquementioning

confidence: 99%

Satake-Furstenberg compactifications and gradient map

Biliotti¹

2020

Preprint

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Let G be a real semisimple Lie group with finite center and let g = k⊕p be a Cartan decomposition of its Lie algebra. Let K be a maximal compact subgroup of G with Lie algebra k and let τ be an irreducible representation of G on a complex vector space V . We denote by µp : P(V ) −→ p the G-gradient map and by O the unique closed orbit of G in P(V ) which is a K-orbit [33,40]. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of G induced by τ are completely determined by the facial structure of the polar orbitope E = conv(µp(O)). Moreover, any parabolic subgroup of G admits a unique closed orbit which is well-adapted to O and µp respectively. These results are new also in the complex reductive case. The connection between E and τ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure γ on O, we construct a map Ψγ from the Satake compactification of G/K associated to τ and E . If γ is a K-invariant measure then Ψγ is an homeomorphism of the Satake compactification and E . Finally, we prove that for a large class of measures the map Ψγ is surjective.

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Hessenberg varieties and Poisson slices

Crooks,

Röser

2023

Compactifications, Configurations, and Cohomology

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This expository article considers a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian G G -variety μ : G × S ⟶ g \mu :G\times \mathcal {S}\longrightarrow \mathfrak {g} to each complex semisimple Lie algebra g \mathfrak {g} with adjoint group G G and fixed Kostant section S ⊆ g \mathcal {S}\subseteq \mathfrak {g} . This variety is one of Bielawski’s hyperkähler slices, and it is central to Moore and Tachikawa’s work on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian G G -varieties μ ¯ S : G × S ¯ ⟶ g \overline {\mu }_{\mathcal {S}}:\overline {G\times \mathcal {S}}\longrightarrow \mathfrak {g} and ν : H e s s ⟶ g \nu :\mathrm {Hess}\longrightarrow \mathfrak {g} . The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification G ¯ \overline {G} , while the latter is the standard family of Hessenberg varieties. Each of μ ¯ \overline {\mu } and ν \nu is known to be a fibrewise compactification of μ \mu . We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our work is shown to be compatible with a Poisson isomorphism obtained by Bălibanu.

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Meromorphic Limits of Automorphisms

Cited by 4 publications

References 39 publications

Convexity properties of gradient maps associated to real reductive representations

Convexity properties of gradient maps associated to real reductive representations

Satake-Furstenberg compactifications and gradient map

Hessenberg varieties and Poisson slices

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