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

Abstract

We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.

A note on lc‐trivial fibrations

“…By combining Theorem 1.1 with a result from [20], we will prove an effective base point free theorem for lc-trivial fibrations with log big moduli parts. Theorem 1.4 (Theorems 4.1 and 4.2).…”

“…By the cone and contraction theorem [26, Theorem 3.7], the divisor $$n^{\prime }(K_{W}+{\bf B}_{W}+{\bf M}_{W})$$ is Cartier and it is the pullback of $$K_{Z}+{\bf B}_{Z}+{\bf M}_{Z}$$ to $$W$$. By [20, Theorem 5.8], there exists $$n^{\prime \prime }$$, depending only on $$\dim Z$$ and $$n^{\prime }$$, such that $$n^{\prime \prime }(K_{W}+{\bf B}_{W}+{\bf M}_{W})$$ is base point free. We denote $$n^{\prime \prime }$$ by $$n^{\prime \prime }(\dim Z,\,n^{\prime })$$, and we define $$$$\begin{equation*} n:=mn^{\prime }\prod _{i=0}^{d}n^{\prime \prime }(i,\,n^{\prime }).$$…”

“…We may write $$$$\begin{equation*} K_{Z^{\prime }}+B^{\prime }+{\bf M}_{Z^{\prime }}=\phi ^{*}(K_{Z}+{\bf B}_{Z}+{\bf M}_{Z})+E^{\prime }, \end{equation*}$$$$where $$B^{\prime } \geqslant 0$$ and $$E^{\prime } \geqslant 0$$ have no common components. By [20, Theorem 1.1], the generalized lc pair $$(Z^{\prime },B^{\prime }+{\bf M})$$ has a good minimal model. By [11, Section 4], $$K_{Z}+{\bf B}_{Z}+{\bf M}_{Z}$$ birationally has a Nakayama–Zariski decomposition (for definition of the Nakayama–Zariski decomposition, see [28, III, 1.12.…”

“…By combining Theorem 1.1 with a result from [20], we will prove an effective base point free theorem for lc‐trivial fibrations with log big moduli parts. Theorem For every $$d$$, $$m \in \mathbb {Z}_{>0}$$, and $$v \in \mathbb {R}_{>0}$$, there exists $$n \in \mathbb {Z}_{>0}$$, depending only on $$d$$, $$m$$, and $$v$$, satisfying the following.…”