Strict Arakelov inequality for a family of varieties of general type
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-05-01
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| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022135?viewType=HTML |
| Summary: | 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 <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. |
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| ISSN: | 2688-1594 |