Logarithmic and Non-Archimedean Gromov-Witten invariants

自20世纪90年代初以来，曲线计数不变量在代数几何中扮演着越来越重要的角色。通常，这些不变量给出了一些有意义的答案，例如“在一个给定的代数簇中包含多少固定次数和亏格的曲线？”最近，对数Gromov-Witten理论有了很大的发展，它允许施加带有因子的相切条件。最近，Tony Yue Yu发展了非阿基米德的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：这两种不变量之间有什么关系。约翰斯顿将致力于得出这些不变量的比较结果。首先，他还将探索Keel-Yu最近关于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。一旦做到这一点，他将证明一个一般的比较结果。这将允许这两种理论在适当的情况下互换使用。

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.