We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has $U$-rank different from Morley rank. We also give a sufficient condition for a Kähler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are $K3$ surfaces which satisfy these conditions.
"Compact complex manifolds with the DOP and other properties." J. Symbolic Logic 67 (2) 737 - 743, June 2002. https://doi.org/10.2178/jsl/1190150107