Related papers: Iitaka fibrations for dlt pairs polarized by a nef…

In this paper we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if $X$ is a projective variety of…

We study the Iitaka-Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general…

We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.

In this paper, we investigate the boundedness of log pairs with log Calabi--Yau fibration structures. We prove that total spaces of log Calabi--Yau fibrations are bounded modulo crepant birational equivalence when the Iitaka volumes of log…

For good minimal models with klt singularities, polarized by Weil divisors that are relatively nef and big over the bases of the Iitaka fibration, we show that, after fixing appropriate numerical invariants, they form a bounded family. As…

In this paper we investigate various properties of generalised pairs in families, especially boundedness of several kinds. We show that many statements for usual pairs do not hold for generalised pairs. In particular, we construct an…

We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a…

For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the very general fibres of an Iitaka…

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the…

We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. The result follows from a theorem on the restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms, which allows…

We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension$\le 4$. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log…

We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the…

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt…

In this paper, we establish a structure theorem for projective klt pairs $(X,\Delta)$ with nef anti-log canonical divisor; specifically, we prove that, up to replacing $X$ with a finite quasi-\'etale cover, $X$ admits a locally trivial…

We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at…

We prove a boundedness result for klt pairs $(X,B)$ such that $K_X+B\equiv 0$ and $B$ is big. As a consequence we obtain a positive answer to the Effective Iitaka Fibration Conjecture for klt pairs with big boundary.

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…

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally…

In this paper, we investigate the explicit birational geometry for projective $\epsilon$-lc varieties polarised by nef and big Weil divisors. We show that if $X$ is a projective $\epsilon$-lc variety, $H$ is a nef and big Weil divisor with…

Under the assumption of the minimal model theory for projective klt pairs of dimension $n$, we establish the minimal model theory for lc pairs $(X/Z,\Delta)$ such that the log canonical divisor is relatively log abundant and its restriction…