LCK rank of locally conformally Kahler manifolds with potential

An LCK manifold with potential is a quotient of a kahler manifold X equipped with a positive Kahler potential f, such that the monodromy group acts on X by holomorphic homotheties and multiplies f by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and b(1)(M). Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last section. (C) 2016 Elsevier B.V. All rights reserved.