Toward a Classification of Compact Nilmanifolds with Symplectic Structures

Abstract

In this paper, we obtain an inductive process to classify all the compact nilmanifolds with symplectic structures. As a consequence, we give a counter-example to a many years question that any compact nilmanifold with a symplectic structure has a nilpotent Lie group as its universal covering such that as a solvable Lie group it has at most two steps.