Stability of the tangent bundle through conifold transitions

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2023-07-28 DOI:10.1002/cpa.22135

Tristan Collins, Sebastien Picard, Shing-Tung Yau

{"title":"Stability of the tangent bundle through conifold transitions","authors":"Tristan Collins, Sebastien Picard, Shing-Tung Yau","doi":"10.1002/cpa.22135","DOIUrl":null,"url":null,"abstract":"Let X be a compact, Kähler, Calabi-Yau threefold and suppose <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>↦</mo>\n <munder>\n <mi>X</mi>\n <mo>̲</mo>\n </munder>\n <mo>⇝</mo>\n <msub>\n <mi>X</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n <annotation>$X\\mapsto \\underline{X}\\leadsto X_t$</annotation>\n </semantics></math> , for <math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>∈</mo>\n <mi>Δ</mi>\n </mrow>\n <annotation>$t\\in \\Delta$</annotation>\n </semantics></math>, is a conifold transition obtained by contracting finitely many disjoint <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mo>−</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(-1,-1)$</annotation>\n </semantics></math> curves in X and then smoothing the resulting ordinary double point singularities. We show that, for <math>\n <semantics>\n <mrow>\n <mo>|</mo>\n <mi>t</mi>\n <mo>|</mo>\n <mo>≪</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$|t|\\ll 1$</annotation>\n </semantics></math> sufficiently small, the tangent bundle <math>\n <semantics>\n <mrow>\n <msup>\n <mi>T</mi>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mn>0</mn>\n </mrow>\n </msup>\n <msub>\n <mi>X</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n <annotation>$T^{1,0}X_{t}$</annotation>\n </semantics></math> admits a Hermitian-Yang-Mills metric <math>\n <semantics>\n <msub>\n <mi>H</mi>\n <mi>t</mi>\n </msub>\n <annotation>$H_t$</annotation>\n </semantics></math> with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of <math>\n <semantics>\n <msub>\n <mi>H</mi>\n <mi>t</mi>\n </msub>\n <annotation>$H_t$</annotation>\n </semantics></math> near the vanishing cycles of <math>\n <semantics>\n <msub>\n <mi>X</mi>\n <mi>t</mi>\n </msub>\n <annotation>$X_t$</annotation>\n </semantics></math> as <math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>→</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$t\\rightarrow 0$</annotation>\n </semantics></math>.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22135","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 14

Abstract

Let X be a compact, Kähler, Calabi-Yau threefold and suppose $X \mapsto \underset{̲}{X} ⇝ X_{t}$ , for $t \in Δ$ , is a conifold transition obtained by contracting finitely many disjoint $(- 1, - 1)$ curves in X and then smoothing the resulting ordinary double point singularities. We show that, for $| t | ≪ 1$ sufficiently small, the tangent bundle $T^{1, 0} X_{t}$ admits a Hermitian-Yang-Mills metric $H_{t}$ with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of $H_{t}$ near the vanishing cycles of $X_{t}$ as $t \to 0$ .

查看原文本刊更多论文

通过针叶树跃迁的切丛的稳定性

设$X$是一个紧致的K\“ahler，Calabi-Yau三重，并假设$X\mapsto\aunderline｛X｝\leadstoX_t$，对于$t\in\Delta$，是一个通过收缩$X$中的有限多个不相交的$（-1，-1）$曲线，然后对得到的普通双点奇点进行光滑化而得到的针叶树跃迁^{1,0}X_{t} $给出了Fu-Li-Yau构造的保形平衡度量的HermitianYang-Mills度量$H_t$。此外，我们将$X_t$在消失周期附近的行为描述为$t\rightarrow0$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.