Intersection theory of nef b-divisor classes

TL;DR

This paper develops an intersection theory for nef b-divisors on projective varieties, proving their approximation by Cartier classes and establishing variants of the Hodge index theorem, with applications to curve classes and Zariski decompositions.

Contribution

It introduces a new intersection theory for nef b-divisors, showing they can be approximated by Cartier classes and connecting to Zariski decompositions of curves.

Findings

Nef b-divisor classes are limits of nef Cartier classes.

Established variants of the Hodge index theorem for nef b-divisors.

Big and basepoint free curve classes are powers of nef b-divisors.

Abstract

We prove that any nef b-divisor class on a projective variety defined over an algebraically closed field of characteristic 0 is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to Zariski decompositions of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of b-divisors which were defined in our previous work.5