Locally conformally Kahler metrics obtained from pseudoconvex shells

TL;DR

This paper explores the relationship between algebraic cones, pseudoconvex shells, and locally conformally Kahler (LCK) metrics, providing explicit constructions of LCK and Vaisman metrics on Hopf manifolds.

Contribution

It establishes a bijective correspondence between LCK metrics with potential and pseudoconvex shells in algebraic cones, enabling explicit metric constructions.

Findings

Characterization of LCK metrics via pseudoconvex shells

Explicit LCK and Vaisman metrics on Hopf manifolds

Generalization of previous constructions by Gauduchon-Ornea and Kamishima-Ornea

Abstract

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it 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 Kahler 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 R-action exactly once. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea and Kamishima-Ornea.