Remark on nefness in higher codimension

TL;DR

This paper extends the concept of nef cones to higher codimension in complex manifolds, providing explicit examples, vanishing theorems, and insights into Zariski decompositions, advancing the understanding of positivity in complex geometry.

Contribution

It constructs higher codimension nef cones, presents vanishing theorems in this context, and demonstrates the optimality of Boucksom's divisorial Zariski decomposition.

Findings

Explicit examples of higher codimension nef cones

Two versions of Kawamata-Viehweg vanishing theorems

Optimality of Boucksom's divisorial Zariski decomposition

Abstract

In this work, following the fundamental work of Boucksom we construct the nef cone of a compact complex manifold in higher codimension and give explicit examples where these cones are different. In the third section, we give two versions of Kawamata-Viehweg vanishing theorems in terms of nefness in higher codimension and numerical dimensions. We also show by examples the optimality of the divisoral Zariski decomposition defined by Boucksom.