Logarithmic geometry and its applications to moduli and birational geometry

这一提议的工作是来自物理学的弦理论和代数几何的交集。在弦理论中，粒子被在时空中运动的弦的小环路所取代。据推测，时空是十维的。在我们所知的四维时空中，有一种六维纤维，它的纤维被称为卡拉比-尤流形。Gromov-Witten不变量是研究Calabi-Yau流形和弦理论的重要组成部分。自从Gromov-Witten不变量的发现以来，新的见解正在迅速发展，这也为代数几何中的经典问题提供了许多有趣的新方法。在Calabi-Yau流形上计算Gromov-Witten不变量是非常困难的。该项目旨在开发一种新的Gromov-Witten不变量的计算方法，并将其应用于代数几何中产生新的理解。该项目的主要重点是研究Gromov-Witten理论的退化。这将从与丹·阿布拉莫维奇共同开发的稳定对数图的角度进行，并由马克·格罗斯和贝恩德·西伯特独立开发。特别地，PI建议证明对数Gromov-Witten不变量的一般粘合公式。稳定对数映射理论也为研究拟投射簇上的有理曲线提供了一个有用的工具。本课题的另一部分工作是将Mori的理论推广到非真情形，并对拟投射簇的双射几何有进一步的理解。

The work of this proposal lies at the intersection of string theory from physics and algebraic geometry. In string theory, particles are replaced by small loops of strings moving in space-time. Conjecturally, the space-time is ten-dimensional. Over the four-dimensional space-time we are aware of, there is a six-dimensional fibration whose fiber is known as the Calabi-Yau manifold. The Gromov-Witten invariants serve as important ingredients in the study of Calabi-Yau manifolds and string theory. Since the discovery of the Gromov-Witten invariants, new insights are developing rapidly, which also provide many interesting new approaches to classical problems from algebraic geometry. The calculation of Gromov-Witten invariants is extremely difficult to perform on Calabi-Yau manifolds. This project aims to develop a new calculation method of Gromov-Witten invariants, and to apply the developments to produce new understandings in algebraic geometry.The major focus of this project is to study the degeneration of Gromov-Witten theory. This will be carried out from the perspective of stable logarithmic maps developed jointly with Dan Abramovich, and independently by Mark Gross and Bernd Siebert. In particular, the PI proposes to prove a general gluing formula of logarithmic Gromov-Witten invariants. The theory of stable logarithmic maps also provides a useful tool for the study of rational curves on quasi-projective varieties. Another part of this project is to generalize Mori's theory in the non-proper cases, and to have a further understanding of the birational geometry of quasi-projective varieties.