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 fiber $ F $ of general type. We show the strict Arakelov inequality $ {\deg f_*\omega_{X/Y}^\nu \over {{{\rm{rank\,}}}} f_*\omega_{X/Y}^\nu} < {n\nu\over 2}\cdot\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 Möller, Viehweg and the third named author [1].

Keywords

• arakelov inequality
• family