Compact pluricanonical manifolds are Vaisman

Abstract

A locally conformally Kahler manifold is a Hermitian manifold $(M,I,\omega)$ satisfying $d\omega=\theta\wedge \omega$, where $\theta$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nabla\theta$ is of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and Vaisman if $\nabla\theta=0$. We show that a compact LCK manifold is pluricanonical if and only if the Lee form has constant length and the Kahler form of its covering admits an automorphic potential. Using a degenerate Monge-Ampere equation and the classification of surfaces of Kahler rank one, due to Brunella, Chiose and Toma, we show that any pluricanonical metric on a compact manifold is Vaisman. Several errata to our previous work are given in the last Section.