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Hopf surfaces in locally conformally Kähler manifolds with potential. (English) Zbl 1443.53043

Authors’ abstract: An LCK manifold with potential is a quotient \(M\) of a Kähler manifold \(X\) equipped with a positive plurisubharmonic function \(f\), such that the monodromy group acts on \(X\) by holomorphic homotheties and maps \(f\) to a function proportional to \(f\). It is known that a compact \(M\) admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman, compact LCK manifold with potential contains a complex surface (possibly singular) with normalization biholomorphic to a Hopf surface \(H\). Moreover, \(H\) can be chosen non-diagonal, hence, also not admitting a Vaisman structure.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C40 Global submanifolds
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