Hard Lefschetz properties, complete intersections and numerical dimensions

TL;DR

This paper characterizes when complete intersection classes of nef classes have the hard Lefschetz property on compact complex tori and Kähler manifolds, linking it to numerical dimensions and revealing a polymatroid structure.

Contribution

It provides new necessary and sufficient conditions for the hard Lefschetz property of nef classes on complex tori and Kähler manifolds, based on numerical dimensions.

Findings

Characterization of classes with hard Lefschetz property on complex tori.

Identification of conditions for non-vanishing of intersection classes.

Discovery of a polymatroid structure on nef classes via numerical dimensions.

Abstract

We study the positivity of complete intersections of nef classes. We first give a sufficient and necessary characterization on the complete intersection classes which have hard Lefschetz property on a compact complex torus, equivalently, in the linear case. In turn, this provides us new kinds of cohomology classes which have Hodge-Riemann property or hard Lefschetz property on an arbitrary compact K\"ahler manifold. We also give a complete characterization on when the complete intersection classes are non-vanishing on an arbitrary compact K\"ahler manifold. Both characterizations are given by the numerical dimensions of various partial summations of the given nef classes. As an interesting byproduct, we show that the numerical dimension endows any finite set of nef classes with a loopless polymatroid structure.