Spherical gradient manifolds
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, p. 2235-2260
We study the action of a real-reductive group G=Kexp(𝔭) on a real-analytic submanifold X of a Kähler manifold. We suppose that the action of G extends holomorphically to an action of the complexified group G on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map μ 𝔭 :X𝔭. We show that μ 𝔭 almost separates the K–orbits if and only if a minimal parabolic subgroup of G has an open orbit. This generalizes Brion’s characterization of spherical Kähler manifolds with moment maps.
Nous étudions l’action d’un groupe réel-réductif G=Kexp(𝔭) sur une sous-variété réel-analytique X d’une variété kählérienne. Nous supposons que l’action de G peut être prolongée en une action holomorphe du groupe complexifié G sur cette variété kählérienne telle que l’action d’un sous-groupe maximal compact de G soit hamiltonienne. L’application moment induit une application gradient μ 𝔭 :X𝔭. Nous montrons que μ 𝔭 sépare presque les orbites de K si et seulement si un sous-groupe minimal parabolique de G possède une orbite ouverte dans X. Ce résultat généralise la caractérisation de Brion des variétés kählériennes sphériques qui admettent une application moment.
DOI : https://doi.org/10.5802/aif.2582
Classification:  32M05,  22E46,  53D20
Keywords: Real-reductive Lie group, Hamiltonian action, gradient map, spherical variety
@article{AIF_2010__60_6_2235_0,
     author = {Miebach, Christian and St\"otzel, Henrik},
     title = {Spherical gradient manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     pages = {2235-2260},
     doi = {10.5802/aif.2582},
     zbl = {1214.32007},
     mrnumber = {2791656},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_6_2235_0}
}
Spherical gradient manifolds. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2235-2260. doi : 10.5802/aif.2582. https://aif.centre-mersenne.org/item/AIF_2010__60_6_2235_0/

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