Classification of orbit closures in the variety of 4-dimensional symplectic Lie algebras

The aim of this paper is to study the natural action of the real symplectic group, $\mathrm{Sp}(4,\mathbb{R})$, on the algebraic set of $4$-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on four dimensional Lie groups.