Lefschetz theorems, Hodge-Riemann relations and Ample vector bundles

TL;DR

This paper develops a new Hermitian metric framework on the cohomology of compact Kähler manifolds, generalizing classical Hodge theory and applying it to prove results about the Lefschetz property and Chern classes of ample vector bundles.

Contribution

It introduces a novel Hermitian metric on cohomology rings and exterior algebras, extending Hodge theory and providing new proofs and applications in algebraic geometry.

Findings

New Hermitian metric on cohomology rings of Kähler manifolds

Generalization of classical Hodge theory to this setting

Applications to Lefschetz property and Chern classes of ample vector bundles

Abstract

We introduce a new Hermitian metric on the cohomology ring of compact K\"ahlerian manifolds with a pair $(v,w)$ satisfying certain Hodge-Riemann relations. An Hermitian metric on the exterior algebra of the cotangent bundle is also defined and we establish the corresponding theory of harmonic forms, relating the global metric and local metric. This generalizes the classical Hodge theory. As an immediate application we give a new proof of Dinh-Nguyen's theorem on the Hodge-Riemann relations for mixed K\"ahler classes. We give several other applications to the Lefschetz property and Hodge-Riemann relations of Chern classes of ample vector bundles.