Positivity of LCK Potential
Let M be a complex manifold and L an oriented real line bundle on M equipped with a flat connection. A “locally conformally Kähler” (LCK) 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 an LCK potential (globally on M), and prove that M admits a positive LCK potential. Then M admits a holomorphic embedding to a Hopf manifold, as shown in Ornea and Verbitsky (Math Ann 348:25–33, 2010).