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Regular version of the site

Working paper

Compact pluricanonical manifolds are Vaisman

Ornea L., Verbitsky M.
A locally conformally Kahler manifold is a Hermitian manifold (M,I,ω) satisfying dω=θ∧ω, where θ is a closed 1-form, called the Lee form of M. It is called pluricanonical if ∇θ is of Hodge type (2,0)+(0,2), where ∇ is the Levi-Civita connection, and Vaisman if ∇θ=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.