In this paper we show that if a number field K contains a nonreciprocal unit u of degree s+2t with s positive conjugates and 2t complex conjugates of equal moduli, where t ≧ 2, then s=(2t+2m)q-2t for some integers m ≧ 0 and q ≧ 2. On the other hand, for any s and t ≧ 2 related as above we construct a number field K with s real and 2t complex embeddings that contains a nonreciprocal unit u of degree s+2t with s positive conjugates and 2t complex conjugates of equal moduli. From this, for any pair of integers s ≧ 1, t ≧ 2 satisfying s ≠ (2t+2m)q-2t we deduce that the rank of the subgroup of units U whose 2t complex conjugates have equal moduli is smaller than s and, therefore, for any choice of an admissible subgroup A of K the corresponding Oeljeklaus-Toma manifold X(K,A) admits no locally conformal Kähler metric.

This research was supported by the Research Council of Lithuania grant No. MIP-068/2013/LSS-110000-740