Inequalities for the Hodge numbers of irregular compact Kaehler manifolds

TL;DR

This paper establishes inequalities for Hodge numbers of irregular compact Kähler manifolds using derivative complexes, providing asymptotic bounds for 3- and 4-folds and bounds on cohomology modules.

Contribution

It introduces new inequalities and bounds for Hodge numbers and cohomology modules of irregular compact Kähler manifolds, extending previous work.

Findings

Derived inequalities for all Hodge numbers of certain Kähler manifolds.

Provided asymptotic bounds for Hodge numbers of 3- and 4-folds.

Bounded the regularity of exterior cohomology modules.

Abstract

Based on work of R. Lazarsfeld and M. Popa, we use the derivative complex associated to the bundle of the holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kaehler manifolds. For 3-folds and 4-folds we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the BGG correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms.