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

Chapter
Part of the Springer INdAM Series book 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.

Keywords

Complex surface Contraction Holomorphic embedding Locally conformally Kähler Potential Riesz-Schauder Spherical shell 

Notes

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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Copyright information

© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Laboratory of Algebraic GeometryFaculty of Mathematics, National Research University HSEMoscowRussia
  4. 4.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium

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