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February 2022 Visible actions of compact Lie groups on complex spherical varieties
Yuichiro Tanaka
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J. Differential Geom. 120(2): 375-388 (February 2022). DOI: 10.4310/jdg/1645207534


With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.

Funding Statement

This work was supported by a JSPS Kakenhi (17K14155).


Download Citation

Yuichiro Tanaka. "Visible actions of compact Lie groups on complex spherical varieties." J. Differential Geom. 120 (2) 375 - 388, February 2022.


Received: 30 March 2019; Accepted: 19 September 2019; Published: February 2022
First available in Project Euclid: 23 February 2022

Digital Object Identifier: 10.4310/jdg/1645207534

Primary: 22E46
Secondary: 32A37 , 53C30

Keywords: compact Lie group , spherical variety , visible action

Rights: Copyright © 2022 Lehigh University


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Vol.120 • No. 2 • February 2022
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