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Algebraic Groups Whose Orbit Closures Contain Only Finitely Many Orbits

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We explore connected affine algebraic groups G, which enjoy the following finiteness property (F): for every algebraic action of G, the closure of every G-orbit contains only finitely many G-orbits. We obtain two main results. First, we classify such groups. Namely, we prove that a connected affine algebraic group G enjoys property (F) if and only if G is either a torus or a product of a torus and a one-dimensional connected unipotent algebraic group. Secondly, we obtain a characterization of such groups in terms of the modality of action in the sense of V. Arnol’d. Namely, we prove that a connected affine algebraic group G enjoys property (F) if and only if for every irreducible algebraic variety X endowed with an algebraic action of G, the modality of X is equal to dim X − maxxϵX dim G∙x.

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  1. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

    VLADIMIR L. POPOV

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To the memory of E. B. Vinberg

This work was done under a grant from the Russian Science Foundation (project no. 19-11-00237).

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POPOV, V.L. Algebraic Groups Whose Orbit Closures Contain Only Finitely Many Orbits. Transformation Groups 26, 671–689 (2021). https://doi.org/10.1007/s00031-020-09633-w

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