On toric Fano fibrations

TL;DR

This paper extends Borisov's classification of toric singularities to toric Fano fibrations, including non--actorial cases, and verifies a conjecture on bounded complements in the toric setting.

Contribution

It generalizes Borisov's classification to a broader class of toric Fano fibrations and confirms Shokurov's conjecture within this framework.

Findings

Classification of toric Fano fibrations extended

Verification of Shokurov's conjecture in toric case

Bounded complements exist for these fibrations

Abstract

A. Borisov classified into finitely many series the set of isomorphism classes of germs of toric $\Q$-factorial singularities, of fixed dimension and with minimal log discrepancy over the special point bounded from below by a fixed real number. We extend this classification to germs of toric Fano fibrations, possibly not $\Q$-factorial. As an application, we verify in the toric setting a conjecture proposed by V. V. Shokurov on the existence of bounded complements.