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A Calabi--Matsushima decomposition for real reductive group actions

  • Leonardo Biliotti
  • Alessandro Minuzzo
  • Fabio Podestà
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Abstract

Let Z be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group U, and let G⊂ Uℂ be a real compatible subgroup. Using the gradient map μp associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer Gz at a critical point of f∘μp, where f is the restriction to p of a suitable AdU-invariant strictly convex function on iu. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of Uℂ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of f∘μp is a maximal compact subgroup of the identity component of the G-stabilizer.

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Cite

BibTeXRIS
BibTeX
@misc{biliotti2026calabimatsushima,
  title = {{A Calabi--Matsushima decomposition for real reductive group actions}},
  author = {Leonardo Biliotti and Alessandro Minuzzo and Fabio Podestà},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39860},
  eprint = {2609.39860},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39860},
}
RIS
TY  - UNPB
TI  - A Calabi--Matsushima decomposition for real reductive group actions
AU  - Leonardo Biliotti
AU  - Alessandro Minuzzo
AU  - Fabio Podestà
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39860
UR  - https://arxiv.org/abs/2609.39860
AB  - Let Z be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group U, and let G⊂ U^ℂ be a real compatible subgroup. Using the gradient map μ_p associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer G_z at a critical point of f∘μ_p, where f is the restriction to p of a suitable Ad_U-invariant strictly convex function on iu. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of U^ℂ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of f∘μ_p is a maximal compact subgroup of the identity component of the G-stabilizer.
ER  -