On generalised abundance. I
The starting point of the paper are three well-known conjectures in minimal model theory: the Non-vanishing, Abundance and Semiampleness Conjecture. The authors generalize these conjectures into the so-called Generalized Non-vanishing (GN) and Generalized Abundance (GA) Conjectures. Then they study the relations between these conjectures and the Termination-of-Flips Conjecture. The main results are as follows: Theorem A: Termination of flips, Abundance and Semiampleness (all in dimension \(\le n\)) imply GN in dimension \(n\). Theorem B: Termination of flips, Abundance and Semiampleness (all in dimension \(\le n\)) imply GA in dimension \(n\). Corollary C: GN and GA hold in dimension \(2\). Corollary D: GN and GA hold in dimension \(3\), except possibly in some cases where \(\nu(K_X + \Delta) = 0\). Theorem E: Assume termination of flips in dimension \(n\). Then in order to prove GN in dimension \(n\), it is sufficient to do so under the additional assumption that \(K_X + \Delta + L\) is nef. The main technical result of the paper is Theorem 5.3, which says that if termination of flips in dimension \(n\), Abundance and Semiampleness in dimension \(\le n\) and GN in dimension \(\le n-1\) hold, then the positive part of a certain Nakayama-Zariski decomposition is numerically equivalent to a semiample divisor.
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