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.
Highlights
During the last decade a plethora of numerical analogues to the Kodaira dimension for pseudoeffective divisors on complex projective varieties was introduced, by Nakayama [18], Boucksom et al [2], Siu [19] and Lehmann [16]
Lehmann clarified lots of relations between these numerical dimensions, adding some new notions, ordering them by the way how they are constructed and showing that most of them are at least related by an inequality
We show that the Abundance Conjecture as formulated in the context of the Minimal Model Program is equivalent to a Generalised Abundance Conjecture introduced in [2]
Summary
During the last decade a plethora of numerical analogues to the Kodaira dimension for pseudoeffective divisors on (smooth) complex projective varieties was introduced, by Nakayama [18], Boucksom et al [2], Siu [19] and Lehmann [16]. W (defined in 1.9); (10) κν,Leh(D) := min dim W |∀ > 0 : φW∗ D − EW not pseudoeffective , where φW : X → X is any birational morphism of smooth varieties such that OX (EW ) = φ−1IW · OX For attributions of these definitions see Sect. All the notions of numerical dimension listed in Definition 0.1 are equal, except κν,Leh(D) which may be smaller This theorem is a consequence of the following net of equalities and inequalities: νVol ( D ). 4 we show that the numerical dimension of a pseudoeffective divisor behaves well under birational morphisms, following the ideas of Nakayama but explicitly using Theorem 0.2: Proposition 0.4. (Abundance Conjecture, MMP version [17, Conj.3-3-4]) Let S be a minimal model of a non-uniruled smooth projective complex variety X. The author still thinks that it is worth presenting the argument for Theorem 4.5, emphasizing in particular that not all the possible definitions of numerical dimension are shown to be birationally invariant
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