LCK rank of locally conformally Kähler manifolds with potential

An LCK manifold with potential is a quotient of a Kähler manifold XX equipped with a positive Kähler potential ff, such that the monodromy group acts on XX by holomorphic homotheties and multiplies ff 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 b1(M)b1(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.