Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds

Abstract

The cohomology algebra of the canonical bundle of a compact Kähler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order to understand the regularity properties of this module. We also relate it to the infinitesimal theory of the canonical linear series inside paracanonical space. Finally, we apply vector bundle methods on the polynomial ring side to obtain inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kähler manifolds without irregular fibrations.