Eva Elduque, C. Geske, Moisés Herradón Cueto, L. Maxim, Botong Wang
{"title":"Mixed Hodge Structures on Alexander Modules","authors":"Eva Elduque, C. Geske, Moisés Herradón Cueto, L. Maxim, Botong Wang","doi":"10.1090/memo/1479","DOIUrl":null,"url":null,"abstract":"<p>Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\n <mml:semantics>\n <mml:mi>U</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a smooth connected complex algebraic variety and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper U right-arrow double-struck upper C Superscript asterisk\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:<!-- : --></mml:mo>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f\\colon U\\to \\mathbb {C}^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Superscript asterisk\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {C}^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\n <mml:semantics>\n <mml:mi>f</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> gives rise to an infinite cyclic cover <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Superscript f\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">U^f</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\n <mml:semantics>\n <mml:mi>U</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The action of the deck group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Superscript f\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">U^f</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> induces a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-bracket t Superscript plus-or-minus 1 Baseline right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:msup>\n <mml:mi>t</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>±<!-- ± --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}[t^{\\pm 1}]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-module structure on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript asterisk Baseline left-parenthesis upper U Superscript f Baseline semicolon double-struck upper Q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>H</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:mo>;</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H_*(U^f;\\mathbb {Q})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We show that the torsion parts <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript asterisk Baseline left-parenthesis upper U Superscript f Baseline semicolon double-struck upper Q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:mo>;</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_*(U^f;\\mathbb {Q})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the Alexander modules <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript asterisk Baseline left-parenthesis upper U Superscript f Baseline semicolon double-struck upper Q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>H</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:mo>;</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H_*(U^f;\\mathbb {Q})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> carry canonical <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-mixed Hodge structures. We also prove that the covering map <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Superscript f Baseline right-arrow upper U\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mi>U</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">U^f \\to U</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript asterisk Baseline left-parenthesis upper U Superscript f Baseline semicolon double-struck upper Q right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>U</mml:mi>\n <mml:mi>f</mml:mi>\n </mml:msup>\n <mml:mo>;</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_*(U^f;\\mathbb {Q})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper U right-arrow double-struck upper C Superscript asterisk\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:<!-- : --></mml:mo>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f\\colon U\\to \\mathbb {C}^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is proper, we prove the semisimplicity and purity of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript asterisk Baseline left-parenthesis upper U Superscript f Baseline semi","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1479","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let UU be a smooth connected complex algebraic variety and let f:U→C∗f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C∗\mathbb {C}^* by ff gives rise to an infinite cyclic cover UfU^f of UU. The action of the deck group Z\mathbb {Z} on UfU^f induces a Q[t±1]\mathbb {Q}[t^{\pm 1}]-module structure on H∗(Uf;Q)H_*(U^f;\mathbb {Q}). We show that the torsion parts A∗(Uf;Q)A_*(U^f;\mathbb {Q}) of the Alexander modules H∗(Uf;Q)H_*(U^f;\mathbb {Q}) carry canonical Q\mathbb {Q}-mixed Hodge structures. We also prove that the covering map Uf→UU^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A∗(Uf;Q)A_*(U^f;\mathbb {Q}), as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f:U→C∗f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of
受超曲面奇点胚芽的米尔诺纤维上的极限混合霍奇结构的启发,我们在光滑连通复代数簇的亚历山大模块的扭转部分上构造了一个自然的混合霍奇结构。更确切地说,设 U U 是光滑连通复代数簇,设 f : U → C ∗ fcolon U\to \mathbb {C}^* 是基本群中诱导外变形的代数映射。通过 f f 对 C ∗ \mathbb {C}^* 的普遍盖的拉回,可以得到 U U 的无限循环盖 U f U^f。甲板群 Z 在 U f U^f 上的作用在 H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) 上诱导出一个 Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] 模块结构。我们证明亚历山大模块 H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) 的扭转部分 A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) 带有典型的 Q \mathbb {Q} -mixed Hodge 结构。 -混合霍奇结构。我们还证明了覆盖映射 U f → U U^f \to U 在亚历山大模块的扭转部分上诱导了混合霍奇结构变形。作为应用,我们研究了 A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) 的半简单性,以及所构造的混合霍奇结构的可能权重。最后,在 f : U → C ∗ f\colon U\to \mathbb {C}^* 是适当的情况下,我们证明了 本文章由计算机程序翻译,如有差异,请以英文原文为准。