Singularities on toric fibrations

Abstract: In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M$^\rm{c}$Kernan) which roughly says that if $(X,B)\to Z$ is an $\epsilon$-lc Fano type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $\epsilon$ and $\dim X$ where $B_Z,M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $\epsilon$-lc, then the multiplicities of the fibres over codimension one points are bounded depending only on $\epsilon$ and $\dim X$.