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Embedding of LCK Manifolds with Potential into Hopf Manifolds Using Riesz-Schauder Theorem

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Complex and Symplectic Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 21))

Abstract

A locally conformally Kähler (LCK) manifold with potential is a complex manifold with a cover which admits a positive automorphic Kähler potential. A compact LCK manifold with potential can be embedded into a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We provide an alternative argument for compact complex surfaces, deducing the embedding theorem from the Spherical Shell Conjecture.

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Notes

  1. 1.

    Recall that for a ring A with a proper ideal \(\mathfrak{m}\) , the \(\mathfrak{m}\) -adic topology on A is given by the base of open sets formed by \(\mathfrak{m}^{k}\) and their translates.

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Acknowledgements

The authors thank Georges Dloussky for his kind advice and for a bibliographical information, and the anonymous referee for very useful remarks. The author “Liviu Ornea” was partially supported by University of Bucharest grant 1/2012. The author “Misha Verbitsky” was partially supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Bucharest, 14 Academiei Str., 70109, Bucharest, Romania

    Liviu Ornea

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str., Bucharest, 010702, Romania

    Liviu Ornea

  3. Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University HSE, 6, Usacheva Str., Moscow, Russia

    Misha Verbitsky

  4. Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine, C.P. 218/01, Boulevard du Triomphe, B-1050, Brussels, Belgium

    Misha Verbitsky

Authors
  1. Liviu Ornea
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  2. Misha Verbitsky
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Correspondence to Liviu Ornea .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica ‘Ulisse Dini’, University of Florence, Firenze, Italy

    Daniele Angella

  2. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, University of Parma, Parma, Italy

    Costantino Medori

  3. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, University of Parma, Parma, Italy

    Adriano Tomassini

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Ornea, L., Verbitsky, M. (2017). Embedding of LCK Manifolds with Potential into Hopf Manifolds Using Riesz-Schauder Theorem. In: Angella, D., Medori, C., Tomassini, A. (eds) Complex and Symplectic Geometry. Springer INdAM Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-62914-8_11

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