Grassmannian BGG complexes and Hodge numbers of irregular varieties

In this paper, we investigate the exactness of the Grassmannian Bernstein-Gel'fand- Gel'fand complexes introduced in a previous work of the author, 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-Popa and Lombardi concerning threefolds and fourfolds.