{"title":"算术Okounkov体与adelic Cartier除数的正性","authors":"François Ballaÿ","doi":"10.1090/jag/821","DOIUrl":null,"url":null,"abstract":"In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{\\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cartier divisor <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">\\overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{\\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">\\overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Arithmetic Okounkov bodies and positivity of adelic Cartier divisors\",\"authors\":\"François Ballaÿ\",\"doi\":\"10.1090/jag/821\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R\\\"> <mml:semantics> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">{\\\\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cartier divisor <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D overbar\\\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\\\"false\\\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R\\\"> <mml:semantics> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">{\\\\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D overbar\\\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\\\"false\\\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":\"48 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-10-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/jag/821\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/jag/821","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic R{\mathbb {R}}-Cartier divisor D¯\overline {D} are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized R{\mathbb {R}}-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of D¯\overline {D} coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.
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.