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A Partial Comparison of Stability Notions in Kähler Geometry

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Moduli of K-stable Varieties

Part of the book series: Springer INdAM Series ((SINDAMS,volume 31))

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Abstract

In this follow up work to Dyrefelt (J Geom Anal, 2017. https://doi.org/10.1007/s12220-017-9942-9), Dervan and Ross (Math Res Lett 24, 2017), Dervan (Math Ann, 2017. https://doi.org/10.1007/s00208-017-1592-5), and Sjöström Dyrefelt (Int Math Res Not 2018. https://doi.org/10.1093/imrn/rny094) we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and transcendental K-polystability, equivariant K-polystability, and the geodesic K-polystability notion introduced by the author in Sjöström Dyrefelt (Int Math Res Not 2018. https://doi.org/10.1093/imrn/rny094). We provide a partial comparison of the above notions, and show equivalence of some of these notions provided that the underlying manifold satisfies a certain ‘weak cscK’ condition. As an application this proves K-polystability of a new family of cscK manifolds with irrational polarization.

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Acknowledgements

It is a pleasure to thank the referee for helpful remarks and comments.

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Authors and Affiliations

  1. The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, TS, Italy

    Zakarias Sjöström Dyrefelt

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  1. Zakarias Sjöström Dyrefelt
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Correspondence to Zakarias Sjöström Dyrefelt .

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Editors and Affiliations

  1. Dept. of Mathematics and Physics, Roma Tre University, Rome, Italy

    Giulio Codogni

  2. DPMMS, University of Cambridge, Cambridge, UK

    Ruadhaí Dervan

  3. Dept. of Mathematics and Physics, Roma Tre University, Rome, Italy

    Filippo Viviani

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Dyrefelt, Z.S. (2019). A Partial Comparison of Stability Notions in Kähler Geometry. In: Codogni, G., Dervan, R., Viviani, F. (eds) Moduli of K-stable Varieties. Springer INdAM Series, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-13158-6_6

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