Non-linear Hopf manifolds are locally conformally Kahler

Abstract

A Hopf manifold is a quotient of $C^n\backslash 0$ 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 Kahler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kahler (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.

11 pages, Latex (no change in the article, but the title in the submission was wrong)

Cited by in corpus (4)

• Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
• Bimeromorphic geometry of LCK manifolds
• Algebraic cones of LCK manifolds with potential
• Do products of compact complex manifolds admit LCK metrics?