Comparing numerical dimensions

TL;DR

This paper proves the equivalence of different definitions of the numerical dimension of pseudo-effective divisors and introduces new properties and characterizations of this measure.

Contribution

It establishes the equality of various proposed definitions of the numerical dimension and provides new insights and properties of this invariant.

Findings

Proved the equality of Nakayama and Boucksom et al. definitions.

Provided several new characterizations of the numerical dimension.

Established new properties of the numerical dimension.

Abstract

The numerical dimension is a numerical measure of the positivity of a pseudo-effective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama (2004) and Boucksom et al. (2004). We prove the equality of these notions and give several additional characterizations. We also prove some new properties of the numerical dimension.