Strict Arakelov inequality for a family of varieties of general type

TL;DR

This paper proves a strict Arakelov inequality for families of varieties of general type over curves, establishing a bound on the degree of certain sheaves related to pluricanonical systems, answering a previously open question.

Contribution

It establishes a new strict inequality for families of varieties of general type, extending Arakelov inequalities to pluricanonical sheaves under birationality conditions.

Findings

Proves the strict Arakelov inequality for all u with birational pluricanonical systems.

Answers a question posed by M"oller, Viehweg, and the third author.

Provides bounds relating degrees of sheaves to the geometry of the base curve.

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 \[\frac{\mathrm{deg}\, f_*\omega_{X/Y}^\nu}{\mathrm{rank}\, f_*\omega_{X/Y}^\nu} < {n\nu\over 2}\cdot\mathrm{deg}\,\Omega^1_Y(\log S),\] for all $\nu\in \mathbb N$ such that the $\nu$-th pluricanonical linear system $|\omega^\nu_F|$ is birational. This answers a question asked by M\"oller, Viehweg and the third named author.