Inequalities for the Hodge Numbers of Irregular Compact Kähler Manifolds

Based on the work of Lazarsfeld and Popa, we use the derivative complex associated to the bundle of holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kahler manifolds. For three-folds and four-folds, we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the Bernstein-Gel'fand-Gel'fand correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms.