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On the Automorphisms of Hyperplane Sections of Generalized Grassmannians

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Abstract

Given a smooth hyperplane section H of a rational homogeneous space G/P with Picard number one, we address the question whether it is always possible to lift an automorphism of H to the Lie group G, or more precisely to Aut(G/P). Using linear spaces and quadrics in H, we show that the answer is positive up to a few well understood exceptions related to Jordan algebras. When G/P is an adjoint variety, we show how to describe Aut(H) completely, extending results obtained by Prokhorov and Zaidenberg when G is the exceptional group G2.

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Acknowledgements

We would like to thank Mikhail Zaidenberg for his useful comments on a preliminary version of the paper. We would also like to express our gratitude to Alexander Kuznetsov for his careful and insightful reading, which led to significant improvements.

Funding

This work received support from the ANR project FanoHK, grant ANR-20-CE40-0023. The first author is partially supported by the EIPHI Graduate School (contract ANR-17-EURE-0002).

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, F-31062, Toulouse Cedex 9, France

    Laurent Manivel

  2. Institut de Mathématiques de Bourgogne, UMR CNRS 5584, Université de Bourgogne et Franche-Comté, 9 Avenue Alain Savary, BP 47870, 21078, Dijon Cedex, France

    Vladimiro Benedetti

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Correspondence to Laurent Manivel.

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Manivel, L., Benedetti, V. On the Automorphisms of Hyperplane Sections of Generalized Grassmannians. Transformation Groups (2022). https://doi.org/10.1007/s00031-022-09757-1

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