Singularities on Fano fibrations and beyond

TL;DR

This paper proves several conjectures related to singularities on Fano and Calabi-Yau fibrations, deriving new results on rationally connected varieties, boundedness, and fiber properties, advancing understanding in algebraic geometry.

Contribution

It establishes proofs for conjectures of McKernan, Shokurov, and the author on singularities in Fano and Calabi-Yau fibrations, introducing a general multiplicity control result.

Findings

Proved conjectures on singularities of Fano type fibrations.

Derived a variant of a conjecture on rationally connected varieties with nef anti-canonical divisor.

Established bounds on klt complements and properties of fibers in fibrations.

Abstract

In this paper, we investigate singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau fibrations. From these we derive a variant of a conjecture of McKernan and Prokhorov on rationally connected varieties with nef anti-canonical divisor. We present further applications to other problems including boundedness of klt complements for Fano fibrations over curves, torsion index of rationally connected Calabi-Yau pairs, and gonality of fibres of del Pezzo fibrations. We prove a general result on controlling multiplicities of fibres of certain fibrations (not necessarily of Fano type) which is the key ingredient of the proofs of the above results.