Abstract
We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group G on a real submanifold X of a Kähler manifold Z. More precisely, we suppose that the action of a compact Lie group U with Lie algebra \({\mathfrak {u}}\) extends holomorphically to an action of the complexified group \(U^\mathbb {C}\) and that the U-action on Z is Hamiltonian. If \(G\subset U^\mathbb {C}\) is compatible, there is a corresponding gradient map \(\mu _{\mathfrak {p}} : X\rightarrow {\mathfrak {p}}\), where \({\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}\) is a Cartan decomposition of the Lie algebra of G. The concept of energy complete action of G on X is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a G-equivariant function called maximal weight. We also prove the classical Hilbert–Mumford criteria for semistability and polystability conditions.
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Acknowledgements
We wish to thank Alessandro Ghigi and Peter Heinzner for interesting discussions. We would also like to thank the anonymous referee for carefully reading our paper and for giving such constructive comments which substantially helped improving the quality of the paper.
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The first author was partially supported by PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics ” and “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INDAM).
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Biliotti, L., Windare, O.J. Stability, Analytic Stability for Real Reductive Lie Groups. J Geom Anal 33, 92 (2023). https://doi.org/10.1007/s12220-022-01146-0
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DOI: https://doi.org/10.1007/s12220-022-01146-0