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: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2022-05-01
|
| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022135?viewType=HTML |
| _version_ | 1811247644783149056 |
|---|---|
| author | Xin Lu Jinbang Yang Kang Zuo |
| author_facet | Xin Lu Jinbang Yang Kang Zuo |
| author_sort | Xin Lu |
| collection | DOAJ |
| description | 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>. |
| first_indexed | 2024-04-12T15:13:26Z |
| format | Article |
| id | doaj.art-a84e132a68054f6db279c727219657d1 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-12T15:13:26Z |
| publishDate | 2022-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-a84e132a68054f6db279c727219657d12022-12-22T03:27:42ZengAIMS PressElectronic Research Archive2688-15942022-05-013072643266210.3934/era.2022135Strict Arakelov inequality for a family of varieties of general typeXin Lu0Jinbang Yang1Kang Zuo21. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500 Dongchuan Road, Shanghai 200241, China2. Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China3. Institut für Mathematik, Universität Mainz, Mainz 55099, GermanyLet $ 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>.https://www.aimspress.com/article/doi/10.3934/era.2022135?viewType=HTMLarakelov inequalityfamily |
| spellingShingle | Xin Lu Jinbang Yang Kang Zuo Strict Arakelov inequality for a family of varieties of general type Electronic Research Archive arakelov inequality family |
| title | Strict Arakelov inequality for a family of varieties of general type |
| title_full | Strict Arakelov inequality for a family of varieties of general type |
| title_fullStr | Strict Arakelov inequality for a family of varieties of general type |
| title_full_unstemmed | Strict Arakelov inequality for a family of varieties of general type |
| title_short | Strict Arakelov inequality for a family of varieties of general type |
| title_sort | strict arakelov inequality for a family of varieties of general type |
| topic | arakelov inequality family |
| url | https://www.aimspress.com/article/doi/10.3934/era.2022135?viewType=HTML |
| work_keys_str_mv | AT xinlu strictarakelovinequalityforafamilyofvarietiesofgeneraltype AT jinbangyang strictarakelovinequalityforafamilyofvarietiesofgeneraltype AT kangzuo strictarakelovinequalityforafamilyofvarietiesofgeneraltype |