Locally conformally Kähler metrics obtained from pseudoconvex shells

Abstract:

A locally conformally Kähler (LCK) manifold is a complex manifold $M$ admitting a Kähler covering $\tilde {M}$, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if its covering admits an automorphic Kähler potential. It is known that in this case $\tilde {M}$ is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone $C$, and show that the set of Kähler metrics with potential which could arise from an LCK structure is in bijective correspondence with the set of pseudoconvex shells, that is, pseudoconvex hypersurfaces in $C$ meeting each orbit of the associated $\mathbb {R}^{>0}$-action exactly once and transversally. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea, Belgun and Kamishima-Ornea.