光滑平面网的曲率整体性判据及其在 PC2$\mathbb {P}^{2}_\{mathbb {C}}$ 上同质叶状体对偶网的应用

Pub Date : 2024-09-01 DOI:10.1002/mana.202400150
Samir Bedrouni, David Marín
{"title":"光滑平面网的曲率整体性判据及其在 PC2$\\mathbb {P}^{2}_\\{mathbb {C}}$ 上同质叶状体对偶网的应用","authors":"Samir Bedrouni,&nbsp;David Marín","doi":"10.1002/mana.202400150","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\ge 3$</annotation>\n </semantics></math> be an integer. For a holomorphic <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-web <span></span><math>\n <semantics>\n <mi>W</mi>\n <annotation>$\\mathcal {W}$</annotation>\n </semantics></math> on a complex surface <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$M$</annotation>\n </semantics></math>, smooth along an irreducible component <span></span><math>\n <semantics>\n <mi>D</mi>\n <annotation>$D$</annotation>\n </semantics></math> of its discriminant <span></span><math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n <mo>(</mo>\n <mi>W</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\Delta (\\mathcal {W})$</annotation>\n </semantics></math>, we establish an effective criterion for the holomorphy of the curvature of <span></span><math>\n <semantics>\n <mi>W</mi>\n <annotation>$\\mathcal {W}$</annotation>\n </semantics></math> along <span></span><math>\n <semantics>\n <mi>D</mi>\n <annotation>$D$</annotation>\n </semantics></math>, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) <span></span><math>\n <semantics>\n <mrow>\n <mi>Leg</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$\\mathrm{Leg}\\mathcal {H}$</annotation>\n </semantics></math> of a homogeneous foliation <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> of degree <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <msubsup>\n <mi>P</mi>\n <mi>C</mi>\n <mn>2</mn>\n </msubsup>\n <annotation>$\\mathbb {P}^{2}_{\\mathbb {C}}$</annotation>\n </semantics></math>, generalizing some of our previous results. This then allows us to study the flatness of the <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-web <span></span><math>\n <semantics>\n <mrow>\n <mi>Leg</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$\\mathrm{Leg}\\mathcal {H}$</annotation>\n </semantics></math> in the particular case where the foliation <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> is Galois. When the Galois group of <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> is cyclic, we show that <span></span><math>\n <semantics>\n <mrow>\n <mi>Leg</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$\\mathrm{Leg}\\mathcal {H}$</annotation>\n </semantics></math> is flat if and only if <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> is given, up to linear conjugation, by one of the two 1-forms <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>ω</mi>\n <mn>1</mn>\n <mi>d</mi>\n </msubsup>\n <mo>=</mo>\n <msup>\n <mi>y</mi>\n <mi>d</mi>\n </msup>\n <mi>d</mi>\n <mi>x</mi>\n <mo>−</mo>\n <msup>\n <mi>x</mi>\n <mi>d</mi>\n </msup>\n <mi>d</mi>\n <mi>y</mi>\n </mrow>\n <annotation>$\\omega _1^{d}=y^d\\mathrm{d}x-x^d\\mathrm{d}y$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>ω</mi>\n <mn>2</mn>\n <mi>d</mi>\n </msubsup>\n <mo>=</mo>\n <msup>\n <mi>x</mi>\n <mi>d</mi>\n </msup>\n <mi>d</mi>\n <mi>x</mi>\n <mo>−</mo>\n <msup>\n <mi>y</mi>\n <mi>d</mi>\n </msup>\n <mi>d</mi>\n <mi>y</mi>\n </mrow>\n <annotation>$\\omega _2^{d}=x^d\\mathrm{d}x-y^d\\mathrm{d}y$</annotation>\n </semantics></math>. When the Galois group of <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> is noncyclic, we obtain that <span></span><math>\n <semantics>\n <mrow>\n <mi>Leg</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$\\mathrm{Leg}\\mathcal {H}$</annotation>\n </semantics></math> is always flat.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on \\n \\n \\n P\\n C\\n 2\\n \\n $\\\\mathbb {P}^{2}_{\\\\mathbb {C}}$\",\"authors\":\"Samir Bedrouni,&nbsp;David Marín\",\"doi\":\"10.1002/mana.202400150\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>≥</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d\\\\ge 3$</annotation>\\n </semantics></math> be an integer. For a holomorphic <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>-web <span></span><math>\\n <semantics>\\n <mi>W</mi>\\n <annotation>$\\\\mathcal {W}$</annotation>\\n </semantics></math> on a complex surface <span></span><math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$M$</annotation>\\n </semantics></math>, smooth along an irreducible component <span></span><math>\\n <semantics>\\n <mi>D</mi>\\n <annotation>$D$</annotation>\\n </semantics></math> of its discriminant <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Δ</mi>\\n <mo>(</mo>\\n <mi>W</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\Delta (\\\\mathcal {W})$</annotation>\\n </semantics></math>, we establish an effective criterion for the holomorphy of the curvature of <span></span><math>\\n <semantics>\\n <mi>W</mi>\\n <annotation>$\\\\mathcal {W}$</annotation>\\n </semantics></math> along <span></span><math>\\n <semantics>\\n <mi>D</mi>\\n <annotation>$D$</annotation>\\n </semantics></math>, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Leg</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$\\\\mathrm{Leg}\\\\mathcal {H}$</annotation>\\n </semantics></math> of a homogeneous foliation <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> of degree <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <msubsup>\\n <mi>P</mi>\\n <mi>C</mi>\\n <mn>2</mn>\\n </msubsup>\\n <annotation>$\\\\mathbb {P}^{2}_{\\\\mathbb {C}}$</annotation>\\n </semantics></math>, generalizing some of our previous results. This then allows us to study the flatness of the <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>-web <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Leg</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$\\\\mathrm{Leg}\\\\mathcal {H}$</annotation>\\n </semantics></math> in the particular case where the foliation <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> is Galois. When the Galois group of <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> is cyclic, we show that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Leg</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$\\\\mathrm{Leg}\\\\mathcal {H}$</annotation>\\n </semantics></math> is flat if and only if <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> is given, up to linear conjugation, by one of the two 1-forms <span></span><math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mi>ω</mi>\\n <mn>1</mn>\\n <mi>d</mi>\\n </msubsup>\\n <mo>=</mo>\\n <msup>\\n <mi>y</mi>\\n <mi>d</mi>\\n </msup>\\n <mi>d</mi>\\n <mi>x</mi>\\n <mo>−</mo>\\n <msup>\\n <mi>x</mi>\\n <mi>d</mi>\\n </msup>\\n <mi>d</mi>\\n <mi>y</mi>\\n </mrow>\\n <annotation>$\\\\omega _1^{d}=y^d\\\\mathrm{d}x-x^d\\\\mathrm{d}y$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mi>ω</mi>\\n <mn>2</mn>\\n <mi>d</mi>\\n </msubsup>\\n <mo>=</mo>\\n <msup>\\n <mi>x</mi>\\n <mi>d</mi>\\n </msup>\\n <mi>d</mi>\\n <mi>x</mi>\\n <mo>−</mo>\\n <msup>\\n <mi>y</mi>\\n <mi>d</mi>\\n </msup>\\n <mi>d</mi>\\n <mi>y</mi>\\n </mrow>\\n <annotation>$\\\\omega _2^{d}=x^d\\\\mathrm{d}x-y^d\\\\mathrm{d}y$</annotation>\\n </semantics></math>. When the Galois group of <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> is noncyclic, we obtain that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Leg</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$\\\\mathrm{Leg}\\\\mathcal {H}$</annotation>\\n </semantics></math> is always flat.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202400150\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202400150","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

设为整数。对于复曲面上的全形网 ,沿其判别式的不可还原分量光滑,我们建立了沿其判别式的曲率全态的有效判据,推广了马林、佩雷拉和皮里奥关于可分解网的结果。作为一个应用,我们推导出了一个完整的特征,即沿Ⅳ的阶均质叶幅的 Legendre 变换(对偶网)曲率的全态性,并推广了我们之前的一些结果。这样,我们就可以研究褶为伽罗瓦的特殊情况下的-网的平坦性。当伽罗华群为循环群时,我们证明,当且仅当由两个 1-forms 之一给出,且不计线性共轭时,-web 是平坦的。当伽罗华群为非循环群时,我们会得到总是平坦的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
分享
查看原文
A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on P C 2 $\mathbb {P}^{2}_{\mathbb {C}}$

Let d 3 $d\ge 3$ be an integer. For a holomorphic d $d$ -web W $\mathcal {W}$ on a complex surface M $M$ , smooth along an irreducible component D $D$ of its discriminant Δ ( W ) $\Delta (\mathcal {W})$ , we establish an effective criterion for the holomorphy of the curvature of W $\mathcal {W}$ along D $D$ , generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) Leg H $\mathrm{Leg}\mathcal {H}$ of a homogeneous foliation H $\mathcal {H}$ of degree d $d$ on P C 2 $\mathbb {P}^{2}_{\mathbb {C}}$ , generalizing some of our previous results. This then allows us to study the flatness of the d $d$ -web Leg H $\mathrm{Leg}\mathcal {H}$ in the particular case where the foliation H $\mathcal {H}$ is Galois. When the Galois group of H $\mathcal {H}$ is cyclic, we show that Leg H $\mathrm{Leg}\mathcal {H}$ is flat if and only if H $\mathcal {H}$ is given, up to linear conjugation, by one of the two 1-forms ω 1 d = y d d x x d d y $\omega _1^{d}=y^d\mathrm{d}x-x^d\mathrm{d}y$ , ω 2 d = x d d x y d d y $\omega _2^{d}=x^d\mathrm{d}x-y^d\mathrm{d}y$ . When the Galois group of H $\mathcal {H}$ is noncyclic, we obtain that Leg H $\mathrm{Leg}\mathcal {H}$ is always flat.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
×
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
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
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 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信