On the Lefschetz and Hodge-Riemann theorems

TL;DR

This paper presents an abstract formulation of fundamental theorems in Kähler geometry, including the hard Lefschetz theorem, Lefschetz decomposition, and Hodge-Riemann relations, applicable to compact Kähler manifolds.

Contribution

It introduces an abstract framework for key theorems in Kähler geometry, extending their applicability beyond classical settings.

Findings

Abstract version of the hard Lefschetz theorem established

Lefschetz decomposition generalized in an abstract setting

Hodge-Riemann relations extended to broader contexts

Abstract

We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge-Riemann theorem for compact Kaehler manifolds.