{"title":"Global rigidity of the period mapping","authors":"B. Farb","doi":"10.1090/jag/809","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper M Subscript g comma n\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal M}_{g,n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denote the moduli space of smooth, genus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">g\\geq 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> curves with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n\\geq 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> marked points. Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A Subscript h\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mi>h</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal A}_h</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denote the moduli space of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\">\n <mml:semantics>\n <mml:mi>h</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">h</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-dimensional, principally polarized abelian varieties. Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g greater-than-or-equal-to 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">g\\geq 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h less-than-or-equal-to g\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>h</mml:mi>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">h\\leq g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. If <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F colon script upper M Subscript g comma n Baseline right-arrow script upper A Subscript upper H Baseline\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>F</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mi>H</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">F:{\\mathcal M}_{g,n} \\to {\\mathcal A}_H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a nonholomorphic map, then <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h equals g\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>h</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">h=g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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> is the classical period mapping, assigning to a Riemann surface <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> its Jacobian.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/809","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 5
Abstract
Let Mg,n{\mathcal M}_{g,n} denote the moduli space of smooth, genus g≥1g\geq 1 curves with n≥0n\geq 0 marked points. Let Ah{\mathcal A}_h denote the moduli space of hh-dimensional, principally polarized abelian varieties. Let g≥3g\geq 3 and h≤gh\leq g. If F:Mg,n→AHF:{\mathcal M}_{g,n} \to {\mathcal A}_H is a nonholomorphic map, then h=gh=g and FF is the classical period mapping, assigning to a Riemann surface XX its Jacobian.
让 M g , n {\ 公元mathcal} {g, n} denote的平滑,属moduli太空》 用g≥1 g \ geq曲线 n≥0 \ geq 0标记分。把h和h的维空间分开，父异母变种的分布空间分开。让 g≥3 g \ geq和 g h≤h \ leq g。如果 F : M g , n → A H F: {\ mathcal M} {g的,n} \ {\ mathcal百万}_H是个nonholomorphic文件夹,然后 h = F g h = g和F是古典期《绘图,assigning to a是一个类比地面X X Jacobian。
期刊介绍：
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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