A Kollár-type vanishing theorem for k-positive vector bundles

IF 1 3区 数学 Q1 MATHEMATICS Forum Mathematicum Pub Date : 2024-03-25 DOI:10.1515/forum-2023-0332
Chen Zhao
{"title":"A Kollár-type vanishing theorem for k-positive vector bundles","authors":"Chen Zhao","doi":"10.1515/forum-2023-0332","DOIUrl":null,"url":null,"abstract":"Given a proper holomorphic surjective morphism <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0235.png\" /> <jats:tex-math>{f:X\\rightarrow Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle <jats:italic>E</jats:italic> on <jats:italic>X</jats:italic>, we prove Kollár-type vanishing theorems on cohomologies with coefficients in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊗</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0130.png\" /> <jats:tex-math>{R^{q}f_{\\ast}(\\omega_{X}(E))\\otimes F}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>F</jats:italic> is a <jats:italic>k</jats:italic>-positive vector bundle on <jats:italic>Y</jats:italic>. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0114.png\" /> <jats:tex-math>{L^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Dolbeault resolution of the higher direct image sheaf <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0332_eq_0132.png\" /> <jats:tex-math>{R^{q}f_{\\ast}(\\omega_{X}(E))}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is of interest in itself.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0332","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Given a proper holomorphic surjective morphism f : X Y {f:X\rightarrow Y} between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle E on X, we prove Kollár-type vanishing theorems on cohomologies with coefficients in R q f ( ω X ( E ) ) F {R^{q}f_{\ast}(\omega_{X}(E))\otimes F} , where F is a k-positive vector bundle on Y. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane–Takayama, and an L 2 {L^{2}} -Dolbeault resolution of the higher direct image sheaf R q f ( ω X ( E ) ) {R^{q}f_{\ast}(\omega_{X}(E))} , which is of interest in itself.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
k 正向向量束的科拉型消失定理
给定紧凑 Kähler 流形之间的适当全态投射态 f : X → Y {f:X\rightarrow Y} 和 X 上的中野半正全态向量束 E,我们证明了 R q f ∗ ( ω X ( E ) ) 中系数的同调上的 Kollár 型消失定理。 ⊗ F {R^{q}f_{\ast}(\omega_{X}(E))\otimes F} 。 证明的主要输入是 Berndtsson 和 Mourougane-Takayama 关于高直达像的中野半实在性的深入结果,以及一个 L 2 {L^{2}} -Dolbeault 解析。 高直映像 Sheaf R q f ∗ ( ω X ( E ) ) 的 L 2 {L^{2}} -Dolbeault 解析。 {R^{q}f_{\ast}(\omega_{X}(E))} {R^{q}f_{\ast}(\omega_{X}(E))} ,这本身就很有趣。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Forum Mathematicum
Forum Mathematicum 数学-数学
CiteScore
1.60
自引率
0.00%
发文量
78
审稿时长
6-12 weeks
期刊介绍: Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.
期刊最新文献
Is addition definable from multiplication and successor? The stable category of monomorphisms between (Gorenstein) projective modules with applications Big pure projective modules over commutative noetherian rings: Comparison with the completion Discrete Ω-results for the Riemann zeta function Any Sasakian structure is approximated by embeddings into spheres
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1