Embedding of LCK Manifolds with Potential into Hopf Manifolds Using Riesz-Schauder Theorem
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.
KeywordsComplex surface Contraction Holomorphic embedding Locally conformally Kähler Potential Riesz-Schauder Spherical shell
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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