Hopf surfaces in locally conformally Kahler manifolds with potential

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.

11 pages, version 3.0, minor changes

References in corpus (5)

• LCK rank of locally conformally Kahler manifolds with potential
• Vanishing theorems for locally conformal hyperkaehler manifolds
• On pluricanonical locally conformally Kähler manifolds
• On compact complex surfaces of Kähler rank one
• Locally conformally symplectic structures on compact non-Kähler complex surfaces

Cited by in corpus (1)

• Positivity of LCK potential