Numerical analogues of the Kodaira dimension and the Abundance Conjecture

TL;DR

This paper introduces new numerical invariants related to the Kodaira dimension, establishes their equivalences, and demonstrates the equivalence of the Abundance Conjecture and its generalized form for non-uniruled varieties.

Contribution

It extends Lehmann's list of numerical analogues to the Kodaira dimension, proves their equivalence, and links the Abundance Conjecture to its generalized form using these invariants.

Findings

Most numerical analogues to the Kodaira dimension are equal.

The Abundance Conjecture and the Generalized Abundance Conjecture are equivalent for non-uniruled varieties.

Fills a gap in Lehmann's arguments to establish these results.

Abstract

We add further notions to Lehmann's list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann's arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.