Logarithmic and Non-Archimedean Gromov-Witten invariants

对数和非阿基米德 Gromov-Witten 不变量

• 批准号: 2275887
• 金额: --
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2275887
• 关键词: Logarithmic; Non; Archimedean; Gromov; Witten

Since the early 1990s, curve-counting invariants have played an increasingly important role in algebraic geometry. Typically, these invariants give meaningful answers to questions such as "How manycurves of a fixed degree and genus are contained in a given algebraic variety?" There has been recently a great deal of development in logarithmic Gromov-Witten theory, which allows the imposition of tangency conditions with divisors. This is allows a good theory of Gromov-Witten invariants for degenerations of algebraic varieties.More recently, Tony Yue Yu has developed a theory of non-Archimedean Gromov-Witten invariants, which also provides a good theory of invariants for degenerations of algebraic varieties. This leads to the first fundamental question: what is the relationship between these two kinds of invariants. Johnston will work towards a comparison result for these invariants. Initially, he will also explore the recent work of Keel-Yu on constructions of mirror pairs using non-Archimedean algebraic geometry, where a direct comparison with the Gross-Siebert approach is likely to be more readily achieved. Once this is done, he will prove a general comparison result. This will allow the two theories to be used interchangeably where appropriate.

自 20 世纪 90 年代初以来，曲线计数不变量在代数几何中发挥着越来越重要的作用。通常，这些不变量可以对诸如“给定代数簇中包含多少条固定度和属的曲线？”之类的问题提供有意义的答案。最近，对数 Gromov-Witten 理论有了很大的发展，它允许施加与除数的相切条件。这为代数簇的退化提供了一个很好的Gromov-Witten不变量理论。最近，Tony Yue Yu发展了一种非阿基米德Gromov-Witten不变量理论，这也为代数簇的退化提供了一个很好的不变量理论。这就引出了第一个基本问题：这两种不变量之间的关系是什么。约翰斯顿将致力于对这些不变量进行比较结果。首先，他还将探索 Keel-Yu 最近使用非阿基米德代数几何构造镜像对的工作，其中与 Gross-Siebert 方法的直接比较可能更容易实现。一旦完成，他将证明一个一般的比较结果。这将使这两种理论可以在适当的情况下互换使用。