On classification of compact complex solvmanifolds

In this paper, we complete Nakamuraʼs classification of compact complex parallelizable solvmanifolds up to the complex dimension five. We find that the holomorphic symplectic ones are either nilpotent or pseudo-kähler-like, i.e., with a complex solvable Lie group as that of a compact complex solvable pseudo-kähler space in Guan (2010) [Gu1]. We also found that, for any even complex dimension, all the compact complex pseudo-kähler solvmanifolds are hypersymplectic. Therefore, for compact complex solvmanifolds, hypersymplectic is as general as pseudo-kähler.