Positivity of LCK potential

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.