Toward a classification of real compact solvmanifolds with real symplectic structures

In this paper, we deal with the problem of classifying compact complex solvmanifolds with holomorphic symplectic structures and obtain some structure results which make the classification possible. In particular, we reduce the classification to the nilpotent case with the same dimension, which we call the nilpotent reduction. The same method also works for the real compact solvmanifolds with real symplectic structures. The real six-dimensional case was treated completely. This is one of the major steps to obtain further examples of compact holomorphic symplectic manifolds. For example, the Kodaira–Thurston surface is NOT a complex homogeneous manifold with a transitive Lie group action which keeps the complex structure invariant, but a real solvmanifold with a complex structure.