Weak Zariski decompositions and log terminal models for generalized pairs

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

“…By the argument of Shokurov polytope (cf. [25,Subsection 3.3]) and [12], we see that 𝐾 𝑋 + Δ + 𝑙𝑀 is semiample. We fix an effective R-divisor…”

“…We may carry out [22, Proof of Lemma 5.3] in the framework of generalized pairs. It is because we may use the argument of the length of extremal rays [25,Proposition 3.17 Proof. Replacing (𝑌, Γ, M) by a Q-factorial generalized dlt model, we may assume that (𝑌, Γ, M) is a Q-factorial generalized dlt pair.…”

“…Definition 2.10 (Models, cf. [25]). Let (𝑋, Δ, M)/𝑍 be a generalized lc pair, and let (𝑋 , Δ , M )/𝑍 be a generalized pair.…”

“…The goal of this appendix is to prove that the definition of generalized dlt pairs by Birkar [4] and that by Han-Li [25] are equivalent when the nef part of a generalized pair is a finite…”

“…The proof in [14] heavily depends on a vanishing theorem for quasi-log schemes (see [17,Theorem 5.7.3]). On the other hand, for the termination of the MMP of Theorem 1.1, we need to discuss the MMP in the framework of generalized pairs (see, for example, [25] by Han-Li) and we need the techniques developed in [31]. By Theorem 3.15, we also know the existence of a minimal model for generalized lc pairs with a polarization (Theorem 3.17), which was mentioned in [32].…”