Abstract
A Hopf manifold is a quotient of \({\mathbb C}^n\backslash 0\) by the cyclic group generated by a holomorphic contraction. Hopf manifolds are diffeomorphic to \(S^1\times S^{2n-1}\) and hence do not admit Kähler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kähler (LCK) metrics. In this paper, we prove that the Hopf manifolds defined by non-linear holomorphic contractions admit holomorphic embeddings into linear Hopf manifolds, and moreover, they admit LCK metrics.
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Notes
In the sequel, a differential form which is multiplied by a constant factor by the action of the deck group is called automorphic.
A set which has compact closure is called precompact.
Here, as elsewhere, the notation \(A \Subset B\) means that A is relatively compact in B, that is, A is a subset of B and its closure is compact.
Compare with the proof of [17, Theorem 2.14].
A Schauder basis in a Banach space W is a set \(\{x_i\}\) of vectors such that the closure of the space generated by \(x_i\) is W, and the closure of the space generated by all of \(\{x_i\}\) except one of them does not contain the last one.
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Acknowledgements
The authors are grateful to Florin Belgun for insightful discussions about the subject of this paper and to Victor Vuletescu for a careful reading of a first draft. They also thank the referee for her or his very useful suggestions.
Funding
Liviu Ornea is partially supported by Romanian Ministry of Education and Research, Program PN-III, Project Number PN-III-P4-ID-PCE-2020-0025, Contract 30/04.02.2021. Misha Verbitsky is partially supported by the HSE University Basic Research Program, FAPERJ E-26/202.912/2018 and CNPq—Process 313608/2017-2.
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Ornea, L., Verbitsky, M. Non-linear Hopf Manifolds are Locally Conformally Kähler. J Geom Anal 33, 201 (2023). https://doi.org/10.1007/s12220-023-01273-2
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DOI: https://doi.org/10.1007/s12220-023-01273-2