Volume-type functions for numerical cycle classes

TL;DR

This paper introduces a volume-type function for numerical cycle classes that helps distinguish big classes from boundary classes, extending the concept of volume from divisors to higher cycles.

Contribution

It constructs an analogue of the volume function for arbitrary cycle classes, providing a new tool to identify big classes in the cone of effective classes.

Findings

Defines a volume-type function for cycle classes.

Provides criteria to distinguish big classes from boundary classes.

Extends the concept of volume from divisors to higher codimension cycles.

Abstract

A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We construct analogues for arbitrary cycle classes of the volume function for divisors which distinguishes big classes from boundary classes.