Boundedness and volume of generalised pairs

Abstract

In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models.Fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs (X, B + M ) under the given data, satisfies the DCC. Futhermore, we will show that in the klt case, the set of such pairs with ample KX + B + M and fixed volume, forms a bounded family.We prove a result about descent of nef divisors to bounded families. This is the key to proving the above and various other results.We will then apply the above to study projective lc pairs (X, B) with abundant KX +B of arbitrary Kodaira dimension. In particular, we show that the set of Iitaka volumes of such pairs satisfies DCC under some natural boundedness assumptions on the fibres of the Iitaka fibration.We define stable log minimal models which consist of a projective lc pair (X, B) with semi-ample KX + B together with a divisor A ≥ 0 so that KX + B + A is ample and A does not contain any non-klt centre of (X, B). This is a generalisation of both usual stable pairs of general type and stable log Calabi-Yau pairs. Fixing appropriate invariants we show that stable log minimal models form a bounded family. Then we discuss connection with moduli spaces. Caucher Birkar3.4. Bounded coefficients on generalised lc modifications 18 3.6. Bounded crepant models 21 3.8. Finiteness of generalised lc thresholds on bounded families 24 3.10. Construction of towers of crepant models 25 3.11. Boundedness of length of towers of crepant models 26 3.13. Descent of nef divisors to bounded models 27 4. DCC of volume of generalised pairs 30 4.1. DCC of volumes of pairs with fixed birational model 31 4.3. DCC of volumes for G glc (d, Φ) 34 5. Bounded birational models for G glc (d, Φ, v) 36 6. Boundedness of generalised pairs 39 6.1. Decrease of volume 39 6.4. Log discrepancies of F glc (d, Φ, v) 41 6.6. Boundedness of F gklt (d, Φ, v) 46 7. DCC of Iitaka volumes 48 7.1. Adjunction formula for I lc (d, Φ, u) 48 7.5. DCC of Iitaka volumes 52 8. Boundedness of stable log minimal models 53 8.1. Log discrepancies on stable log minimal models 53 8.4. Boundedness of S lc (d, Φ, u, v, σ) and S klt (d, Φ, u, v, <w) 56 References 57Acknowledgements. This work was supported by a grant of the Royal Society.

Iitaka fibrations for dlt pairs polarized by a nef and log big divisor

“…• p(K X + ∆ + M X ) is ample, and • vol(X, K X + ∆ + M X ) ≤ v. Then m(K X + ∆ + M X ) is very ample. In particular, (X, ∆, M) belongs to a bounded family F depending only on d, p, and v in the sense of Birkar [5].…”

Variations of generalised pairs

“…Boundedness in the generalised klt case is known, that is, F gklt (d, Φ, v) is bounded, see [B21,Theorem 1.4]. Surprisingly, we will show that F glc (d, Φ, v) is not bounded in general by giving a counter-example in dimension 3.…”

“…This follows from the proof of[B21, Lemma 8.2]. In [B21, Lemma 8.2], in addition to our assumptions, it is assumed that (X, B), A is a stable pair meaning that K X + B + A is ample and that (X, B + tA) is lc for some t > 0.…”

Boundedness of polarised Calabi-Yau fibrations

“…Because the general fiber (X g , ∆ g ) of (X, ∆) → Z is in a bounded family, there is a rational number α 1 > 0 such that a(E, X, ∆) ≥ −1 + a 1 for any divisor E on X whose center dominates Z. Since K Z + B Z + M Z is ample and (Z, B Z + M Z ) is generalised klt, by the main theorem of [Bir21b] and [Jia21], there is a rational number a 2 > 0 such that (Z, B Z + M Z ) is generalised a 2 -lc, then by Lemma 2.8. (d), a(E, X, ∆) ≥ −1 + a 2 for any divisor E on X whose center does not dominates Z.…”