On cohomological decomposition of generalized-complex structures

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the ${\mathcal{C}}^{\mathrm{\infty }}$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwasawa manifold are investigated.