Compact pluricanonical manifolds are Vaisman

TL;DR

This paper proves that compact pluricanonical locally conformally Kähler manifolds are necessarily Vaisman, establishing a strong link between these geometric structures using advanced techniques and classification results.

Contribution

It demonstrates that all compact pluricanonical LCK manifolds are Vaisman, connecting pluricanonicality with Vaisman structures through new analytical and classification methods.

Findings

A compact LCK manifold is pluricanonical iff the Lee form has constant length and admits an automorphic potential.

Any pluricanonical metric on a compact manifold is Vaisman.

The proof involves a degenerate Monge-Ampère equation and classification of Kähler rank one surfaces.

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.