Non-linear Hopf manifolds are locally conformally Kahler

TL;DR

This paper proves that non-linear Hopf manifolds can be embedded into linear ones and admit locally conformally Kahler metrics, expanding understanding of their geometric structures.

Contribution

It demonstrates that non-linear Hopf manifolds admit LCK metrics and can be holomorphically embedded into linear Hopf manifolds, extending known results from linear cases.

Findings

Non-linear Hopf manifolds admit LCK metrics.

Non-linear Hopf manifolds can be embedded into linear Hopf manifolds.

All Hopf manifolds defined by holomorphic contractions have LCK structures.

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.