Numerical analogues of the Kodaira dimension and the Abundance Conjecture

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.