Strict Arakelov inequality for a family of varieties of general type

Xin Lu; Jinbang Yang; Kang Zuo; Xin Lu; Jinbang Yang; Kang Zuo

doi:10.3934/era.2022135

Electronic Research Archive

2022, Volume 30, Issue 7: 2643-2662. doi: 10.3934/era.2022135

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Strict Arakelov inequality for a family of varieties of general type

1.
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500 Dongchuan Road, Shanghai 200241, China
2.
Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
3.
Institut für Mathematik, Universität Mainz, Mainz 55099, Germany

Received: 13 April 2021 Revised: 26 December 2021 Accepted: 03 January 2022 Published: 16 May 2022

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].
- Arakelov inequality,
- family
Citation: Xin Lu, Jinbang Yang, Kang Zuo. Strict Arakelov inequality for a family of varieties of general type[J]. Electronic Research Archive, 2022, 30(7): 2643-2662. doi: 10.3934/era.2022135

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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].

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