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Momentum polytopes of projective spherical varieties and related Kähler geometry

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Abstract

We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be Kähler and classify the invariant compatible complex structures of a given Kähler multiplicity free compact and connected Hamiltonian manifold.

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Notes

  1. From now on we implicitly fix a choice of such a facet for all \(\alpha \in \Sigma _{\mathbb {Q}}(\Xi ,Q)\); see Lemma 4.20-(2).

  2. Simple roots are numbered here according to the usual Bourbaki notation [6].

  3. From now on we implicitly fix a choice of such a face for all \(\alpha \in \Sigma _{\mathbb {R}}(\Xi ,{\mathcal {P}})\).

  4. As recalled in Sect. 3.2, it is a general fact, due to Brion [11], that the set of spherical roots of a spherical variety is linearly independent.

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Acknowledgements

The authors thank the referee for several useful suggestions, which improved the paper. S. C.-F. is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics.” B. V. S. received support from the (US) N.S.F. through Grant DMS 1407394 and from the City University of New York PSC-CUNY Research Award Program. He thanks Friedrich Knop and the Department of Mathematics at the FAU for hosting him in 2016–2017 and Medgar Evers College for his 2016–2017 Fellowship Award.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

    Stéphanie Cupit-Foutou

  2. Dipartimento di Matematica “Guido Castelnuovo”, “Sapienza” Università di Roma, Rome, Italy

    Guido Pezzini

  3. Department Mathematik, FAU Erlangen-Nürnberg, Erlangen, Germany

    Bart Van Steirteghem

  4. Department of Mathematics, Medgar Evers College & The Graduate Center, City University of New York, New York, USA

    Bart Van Steirteghem

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  1. Stéphanie Cupit-Foutou
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  3. Bart Van Steirteghem
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Correspondence to Bart Van Steirteghem.

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Cupit-Foutou, S., Pezzini, G. & Van Steirteghem, B. Momentum polytopes of projective spherical varieties and related Kähler geometry. Sel. Math. New Ser. 26, 27 (2020). https://doi.org/10.1007/s00029-020-0549-9

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