{"title":"Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs","authors":"Tim Graefnitz","doi":"10.1090/jag/794","DOIUrl":null,"url":null,"abstract":"<p>Consider a log Calabi-Yau pair <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> consisting of a smooth del Pezzo 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> of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"greater-than-or-equal-to 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\geq 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and a smooth anticanonical divisor <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\n <mml:semantics>\n <mml:mi>D</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of <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> intersecting <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\n <mml:semantics>\n <mml:mi>D</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"21","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/794","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: 21
Abstract
Consider a log Calabi-Yau pair (X,D)(X,D) consisting of a smooth del Pezzo surface XX of degree ≥3\geq 3 and a smooth anticanonical divisor DD. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of XX intersecting DD in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of (X,D)(X,D) from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.
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.