On effective log Iitaka fibrations and existence of complements

TL;DR

This paper explores the connection between Iitaka fibrations and the existence of complements, proving the conjecture in dimension 3 and establishing new bounds for Calabi-Yau type varieties.

Contribution

It demonstrates that the existence of complements implies the effective log Iitaka fibration conjecture and introduces the decomposable Iitaka fibration conjecture.

Findings

Effective log Iitaka fibration conjecture holds in dimension 3.

Calabi-Yau type varieties have n-complements with n depending only on dimension and invariants.

The decomposable Iitaka fibration conjecture relates to the structure of ample models.

Abstract

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 effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_X$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a general fiber of an Iitaka fibration of $-K_X$. We also formulate the decomposable Iitaka fibration conjecture, a variation of the effective log Iitaka fibration conjecture which is closely related to the structure of ample models of pairs with non-rational coefficients, and study its relationship with the forestated conjectures.