  In this paper we investigate the exactness of the Grassmannian BGG complexes
introduced in a previous work (arXiv:1211.2486), and obtain some inequalities
between some Hodge numbers of some irregular varieties. In particular, we
obtain sharp lower bounds for the Hodge numbers of smooth subvarieties of
Abelian varieties, as well as some improvements of results of Lazarsfeld and
Popa and Lombardi concerning threefolds and fourfolds.
