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

TL;DR

This paper explores the algebraic structure of the cohomology of the canonical bundle in compact K"ahler manifolds using advanced algebraic tools, deriving inequalities for key topological invariants.

Contribution

It introduces a novel approach combining BGG correspondence and Generic Vanishing to analyze the cohomology module and derive inequalities for Hodge numbers.

Findings

Established regularity properties of the cohomology module.

Derived new inequalities for the holomorphic Euler characteristic.

Linked algebraic structures to geometric properties of K"ahler manifolds.

Abstract

The cohomology algebra of the canonical bundle of a compact K\"ahler 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 K\"ahler manifolds without irregular fibrations.