# Positivity of LCK potential

## TL;DR

This paper proves that if a complex manifold with a flat line bundle admits a global LCK potential, then it also admits a positive one, enabling a holomorphic embedding into a Hopf manifold.

## Contribution

It establishes the existence of a global positive LCK potential on manifolds with a global potential, extending local results to a global setting.

## Key findings

- Existence of a global positive LCK potential under certain conditions
- Manifold admits a holomorphic embedding into a Hopf manifold
- Extension of local LCK potential results to global context

## Abstract

Let $M$ be a complex manifold and $L$ an oriented real line bundle on M equipped with a flat connection. An LCK ("locally conformally Kahler") form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian of a function called LCK potential. We consider a manifold $M$ with an LCK form admitting a global LCK potential, and prove that M admits a global, positive LCK potential. Then M admits a holomorphic embedding to a Hopf manifold.