Strict Arakelov inequality for a family of varieties of general type

Abstract

Let $f:X\to Y$ be a semistable non-isotrivial family of $n$-folds over a smooth projective curve with discriminant locus $S\subseteq Y$ and with general fibre $F$ of general type. We show the strict Arakelov inequality \[\frac{\mathrm{deg}\, f_*ω_{X/Y}^ν}{\mathrm{rank}\, f_*ω_{X/Y}^ν} < {nν\over 2}\cdot\mathrm{deg}\,Ω^1_Y(\log S),\] for all $\nu \in \mathbb{N}$ such that the $\nu$-th pluricanonical linear system $|{\omega }_F^{\nu }|$ is birational. This answers a question asked by Möller, Viehweg and the third named author.

17 pages