Singularities on the base of a Fano type fibration

TL;DR

This paper proves Shokurov's conjecture on the boundedness of singularities in Fano type fibrations when the general fiber belongs to a bounded family, advancing understanding of singularity behavior in algebraic geometry.

Contribution

The paper establishes the boundedness of singularities on the base of a Fano type fibration under specific boundedness conditions on the fibers, confirming a key conjecture in the field.

Findings

Proves Shokurov's conjecture for bounded fibers.

Demonstrates boundedness of singularities on the base.

Advances the theory of Fano type fibrations.

Abstract

Let $f\colon X\to Z$ be a Mori fibre space. McKernan conjectured that the singularities of $Z$ are bounded in terms of the singularities of $X$. Shokurov generalised this to pairs: let $(X,B)$ be a klt pair and $f\colon X\to Z$ a contraction such that $K_X+B\sim_\R 0/Z$ and that the general fibres of $f$ are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor $B_Z$ and a moduli divisor $M_Z$ on $Z$. it is then conjectured that the singularities of $(Z,B_Z+M_Z)$ are bounded in terms of the singularities of $(X,B)$. We prove Shokurov conjecture when $(F,\Supp B_F)$ belongs to a bounded family where $F$ is a general fibre of $f$ and $K_F+B_F=(K_X+B)|_F$.