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发挥了越来越重要的作用。通常，这些不变量对于诸如“给定代数变量中包含多少固定次和属的曲线”这样的问题给出了有意义的答案。近年来，对数格罗莫夫-威腾理论有了很大的发展，它允许对除数施加相切条件。这提供了一个关于代数变元退化的Gromov-Witten不变量的良好理论。最近，余宇（Tony Yue Yu）发展了一个非阿基米德Gromov-Witten不变量理论，这也为代数变元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？Johnston将致力于这些不变量的比较结果。最初，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。一旦这样做了，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地方互换使用。

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.