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On generalised abundance. II. (English) Zbl 1444.14045

The non-vanishing conjecture states the following: let \((X,\Delta)\) be a projective \(k\)lt pair; if \(K_X+\Delta\) is pseudoeffective, then some positive multiple of \(K_X+\Delta\) is effective. In their previous work [Publ. Res. Inst. Math. Sci. 56, No. 2, 353–389 (2020; Zbl 1466.14019)], the authors conjectured a more general form: in the above setting, let \(L\) be a nef \(\mathbb{Q}\)-divisor on X; then, for any \(t\geqslant 0\), the numerical class of \(K_X+\Delta + tL\) belongs to the effective cone. In this paper the authors make progress towards the generalized non-vanishing conjecture and related questions by assuming some standard conjectures of the Minimal Model Program, such as termination of flips. In some instances the results do not need any additional assumptions. For example, since the MMP in known up to dimension three, the authors prove the following.
Theorem A. Let \((X,\Delta)\) be a projective klt pair of dimension at most three such that \(K_X + \Delta\) is pseudoeffective. Assume that \(\chi (X, \mathcal{O}_X)\neq 0\).
(i) Then for every \(\mathbb{Q}\)-divisor \(G\) with \(K_X+\Delta \equiv G\) we have \(\kappa (X,G)\geqslant 0\).
(ii) If \(K_X+\Delta\) is semiample, then every \(\mathbb{Q}\)-divisor \(G\) with \(K_X+\Delta \equiv G\) is semiample.

MSC:

14E30 Minimal model program (Mori theory, extremal rays)

Citations:

Zbl 1466.14019
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References:

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