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The cohomology algebra of the canonical bundle of a compact K\"ahler manifoldis naturally viewed as a module over an exterior algebra. We use theBernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishingtheory, in order to understand the regularity properties of this module. Wealso relate it to the infinitesimal theory of the canonical linear seriesinside paracanonical space. Finally, we apply vector bundle methods on thepolynomial ring side to obtain inequalities for the holomorphic Eulercharacteristic and the Hodge numbers of compact K\"ahler manifolds withoutirregular fibrations.

Derivative complex, BGG correspondence, and numerical inequalities for compact K\"ahler manifolds