The Albanese map for manifolds without meromorphic functions

In this paper, the investigation of compact complex manifolds on which there are no nonconstant meromorphic functions, begun in [1], is continued. Here, in the case when such a manifold is kähler, it is proved that the kodaira dimension of the general fiber of the Albanese map $\leqslant0$; the case when the irregularity is equal to the dimension minus one is studied in more detail.