Hopf surfaces in locally conformally Kahler manifolds with potential

TL;DR

This paper explores the structure of locally conformally Kahler (LCK) manifolds with potential, showing that non-Vaisman cases contain complex surfaces biholomorphic to non-diagonal Hopf surfaces, expanding understanding of their geometric properties.

Contribution

It proves that non-Vaisman LCK manifolds with potential contain non-diagonal Hopf surfaces, revealing new geometric structures within these manifolds.

Findings

Non-Vaisman LCK manifolds with potential contain Hopf surfaces.

Such Hopf surfaces can be chosen to be non-diagonal.

Non-diagonal Hopf surfaces do not admit Vaisman structures.

Abstract

An LCK manifold with potential is a compact quotient M of a Kahler 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 M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman LCK manifold with potential contains a complex surface with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.