Basepoint-free theorems: saturation, b-divisors, and canonical bundle formula

Abstract

We reformulate basepoint-free theorems using notions introduced by Shokurov, such as b-divisors and saturation of linear systems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the basepoint-free theorem in Fukuda's paper "On numerically effective log canonical divisors".

On minimal model theory for log abundant lc pairs

“…The class is very useful for inductive arguments on dimension of varieties. For example, the abundance conjecture for abundant klt pairs and log abundant lc pairs are studied by a lot of people ( [28], [37], [10], [19], [13], [17], [23]), and the conjecture is currently known in full generality ( [23], and [17] for projective case). In klt case, the existence of good minimal models or Mori fiber spaces is known ( [32] and [21]).…”

Existence of log canonical closures

“…In Section 4, we observe that the result in Section 3 and a result of Fujino (cf. [Fujino05]), which generalizes the main theorem of Kawamata in [Kawamata85], immediately imply that any minimal model of (X, ∆) is indeed a good model. Therefore, the proof of (1.1) is reduced to proving the existence of a minimal model for the dlt pair (X, ∆).…”

“…In this section, we recall a version of Kawamata's theorem (cf. [Kawamata85], [Ambro05], [Fujino05], [Fujino11] and [FG11]) on good minimal models. (1) the image of any strata S I of S = ⌊∆⌋ intersects U 0 ,…”

Remarks on the non-vanishing conjecture

“…Then, by Remark 2.2, the abundance conjecture (1.1) in dimension ≤ n − 1, and [FG2,Theorem 4.12] (cf. [F4,Corollary 6.7]), there exists a good minimal model f ′ : (W ′ , Γ ′ ) → Z of (W, Γ) in the sense of Birkar-Shokurov over Z. If some S k contracts by the birational map W W ′ (may not be contracting), then K W + Γ − δ S k is pseudo-effective for some δ > 0 from the positivity property of the definition of minimal models (cf.…”