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On Generalised Abundance, II ON GENERALISED ABUNDANCE, II VLADIMIR LAZIC´ AND THOMAS PETERNELL Abstract. In our previous work, we introduced the Generalised Non- vanishing Conjecture, which generalises several central conjectures in algebraic geometry. In this paper, we derive some surprising nonvanish- ing results for pluricanonical bundles which were not predicted by the Minimal Model Program, by making progress towards the Generalised Nonvanishing Conjecture in every dimension. The main step is to estab- lish that a somewhat stronger version of the Generalised Nonvanishing Conjecture holds almost always in the presence of metrics with gener- alised algebraic singularities, assuming the Minimal Model Program in lower dimensions. Contents 1. Introduction 1 2. Preliminaries 5 3. Reduction to the Weak Generalised Nonvanishing Conjecture 12 4. A criterion for Nonvanishing 22 5. Metrics with generalised algebraic singularities, I 28 6. Metrics with generalised algebraic singularities, II 31 7. Numerical dimension 1 39 8. Maps to abelian varieties 40 References 41 1. Introduction arXiv:1809.02500v4 [math.AG] 7 May 2019 In this paper we study the Generalised Nonvanishing Conjecture in every dimension, and derive some surprising nonvanishing results for pluricanoni- cal bundles which were not predicted by the Minimal Model Program. Generalised Nonvanishing Conjecture. Let (X, ∆) be a klt pair such that KX +∆ is pseudoeffective. Let L be a nef Q-divisor on X. Then for Lazi´cwas supported by the DFG-Emmy-Noether-Nachwuchsgruppe “Gute Strukturen in der h¨oherdimensionalen birationalen Geometrie”. Peternell was supported by the DFG grant “Zur Positivit¨at in der komplexen Geometrie”. We would like to thank S. Schreieder for asking a question about the paper [LP18a] at a seminar in Munich in December 2017 which motivated this paper, and to J.-P. Demailly for useful discussions. 2010 Mathematics Subject Classification: 14E30. Keywords: abundance conjecture, Minimal Model Program. 1 2 VLADIMIR LAZIC´ AND THOMAS PETERNELL every t ≥ 0 the numerical class of the divisor KX +∆+ tL belongs to the effective cone. If the numerical class of a divisor belongs to the effective cone, then we call the divisor num-effective, see Section 2. The Generalised Nonvanishing Conjecture as well as the Generalised Abundance Conjecture were intro- duced in our paper [LP18b] as generalisations of several central conjectures in algebraic geometry. This conjecture contains as special cases the Nonvanishing Conjecture – often viewed as part of the Abundance Conjecture – and the nonvanishing part of the Semiampleness Conjecture (for nef line bundle on varieties with numerically trivial canonical class). 1.1. Numerical versus linear equivalence. Constructing many non-tri- vial effective or basepoint free divisors on a projective variety is one of the main goals of the Minimal Model Program. Assume, for instance, that (X, ∆) is a projective klt pair such that KX + ∆ is semiample. It is a basic question to determine which Q-divisors D in the numerical class of KX +∆ are effective (that is, κ(X, D) ≥ 0) or semiample. If the numerical and Q- 1 linear equivalence on X coincide, i.e. if H (X, OX ) = 0, then all such D are trivially effective, respectively semiample, and one would expect that this is the only case where such behaviour happens. Our first main result is that, surprisingly, such behaviour happens al- most always. More precisely, whether each such D is effective, respectively semiample, depends not (only) on the vanishing of the cohomology group 1 H (X, OX ), but on the behaviour of the Euler-Poincar´echaracteristic of the structure sheaf of X. We obtain the following unconditional results in dimensions at most 3, which were previously not even conjectured. Theorem A. Let (X, ∆) be a projective klt pair of dimension at most 3 such that KX +∆ is pseudoeffective. Assume that χ(X, OX ) 6= 0. (i) Then for every Q-divisor G with KX +∆ ≡ G we have κ(X, G) ≥ 0. (ii) If KX +∆ is semiample, then every Q-divisor G with KX +∆ ≡ G is semiample. Theorem A is a special case of Corollaries 6.7 and 6.8 below. Another instance where our results hold unconditionally in every dimen- sion is the case where the numerical dimension is 1; this is the first highly nontrivial case beyond those when the numerical dimension is 0 or maximal. Theorem B. Let (X, ∆) be a projective terminal pair of dimension n such that KX +∆ is nef and let L be a nef Q-Cartier divisor on X. Assume that ν(X, KX +∆+ L) = 1 and χ(X, OX ) 6= 0. Then KX +∆+ tL is num-effective for all t ≥ 0. This is proved in Section 7. ONGENERALISEDABUNDANCE,II 3 As an illustration, consider a terminal Calabi-Yau pair (X, ∆), i.e. a ter- minal pair such that KX +∆ ≡ 0. Let L be a nef divisor on X of numerical dimension 1. If χ(X, OX ) 6= 0, then L is num-effective. 1.2. Metrics with algebraic singularities. Theorem A above is a special instance of more general results which work in all dimensions but require inductive hypotheses. Let (X, ∆) be a klt pair such that KX +∆ is nef and X is not uniruled, and let L be a nef Q-divisor on X. Our main technical result, Theorem 4.1 below, gives a general criterion for the existence of sections of some multiple of KX +∆+ L. The criterion generalises the main results of [LP18a]: the proof uses the birational stability of the cotangent bundle from [CP15] (this is where the non-uniruledness of X is necessary) together with the very carefully chosen MMP techniques in Theorem 4.2 (this is where the nefness of KX + ∆ is used). We then apply Theorem 4.1 to line bundles possessing metrics with gen- eralised algebraic singularities; the importance of this class of metrics is explained in Section 2. We expect these methods to be crucial in the reso- lution of the Generalised Nonvanishing Conjecture. In Section 6 we prove our central inductive statement. Theorem C. Assume the termination of flips in dimensions at most n − 1, and the Abundance Conjecture in dimensions at most n − 1. Let (X, ∆) be a projective klt pair of dimension n such that KX +∆ is pseudoeffective, and let L be a nef Q-divisor on X. Suppose that KX +∆ and KX +∆+ L have singular metrics with generalised algebraic singularities and semipositive curvature currents, and that χ(X, OX ) 6= 0. (i) Then for every Q-divisor L′ with L ≡ L′ there exists a positive ra- tional number t0 such that ′ κ(X, KX +∆+ tL ) ≥ 0 for every 0 ≤ t ≤ t0. (ii) Assume additionally the semiampleness part of the Abundance Con- jecture in dimension n. Then KX +∆+ tL is num-effective for every t ≥ 0. With notation from Theorem C, we note that by [LP18b, Theorem A], all divisors of the form KX +∆+ L should possess singular metrics with generalised algebraic singularities and semipositive curvature currents. We mention a special case, where the conclusion of Theorem C is much stronger: when KX +∆ and KX +∆+ L additionally have smooth metrics with semipositive curvature, or, more generally, singular metrics with semi- positive curvature currents and vanishing Lelong numbers, then KX +∆+L is actually semiample. This is Theorem 5.2, and the proof uses crucially the main result of [GM17]. As a corollary of Theorem C, we obtain the following: 4 VLADIMIR LAZIC´ AND THOMAS PETERNELL Corollary D. Let (X, ∆) be a projective klt pair of dimension 4 such that KX +∆ is pseudoeffective and let L be a nef Q-divisor on X. Suppose that KX +∆ and KX +∆+ L have singular metrics with generalised algebraic singularities and semipositive curvature currents, and that χ(X, OX ) 6= 0. Then there exists a positive rational number t0 such that κ(X, KX +∆+ tL) ≥ 0 for every 0 ≤ t ≤ t0. Theorem C has an unexpected consequence for the Minimal Model Pro- gram in terms of the existence of sections of divisors numerically equivalent to adjoint divisors, generalising Theorem A to every dimension. The follow- ing result is a special case of Corollary 6.5 and Theorem 6.6. Theorem E. Assume the termination of flips in dimensions at most n − 1, and the existence of good models in dimensions at most n. Let (X, ∆) be a projective klt pair of dimension n such that KX +∆ is pseudoeffective. Assume that χ(X, OX ) 6= 0. (i) Then for every Q-divisor G with KX +∆ ≡ G we have κ(X, G) ≥ 0. (ii) If KX +∆ is semiample, then every Q-divisor G with KX +∆ ≡ G is semiample. 1.3. The Weak Generalised Nonvanishing Conjecture. As an impor- tant part of this paper on which the previous results depend, we show that the assumptions of the Generalised Nonvanishing Conjecture can be weak- ened considerably. In fact, we reduce the Generalised Nonvanishing Conjec- ture to the following. Weak Generalised Nonvanishing Conjecture. Let (X, ∆) be a Q-facto- rial projective klt pair of dimension n such that X is not uniruled and KX +∆ is nef. Let L be a nef Q-divisor on X such that n(X, KX +∆+ L) = n. Then for every t ≥ 0 the numerical class of the divisor KX +∆+ tL belongs to the effective cone. Here, for any Q-Cartier divisor D on X, we denote by n(X, D) the nef dimension of D, see Section 2. The condition n(X, D) = dim X is equivalent to saying that X cannot be covered by curves C such that D · C = 0. Conjecturally, when n(X, KX +∆+ L) = dim X as above, then KX +∆+ L is big. This conjecture is weaker than the Generalised Nonvanishing Conjecture in three different ways: the variety X is not uniruled; the pair (X, ∆) is a minimal model; and the nef dimension of KX +∆+ L is maximal; see Section 2 for details.
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