Logarithmic geometry and its applications to moduli and birational geometry

这个建议的工作在于弦理论从物理学和代数几何的交叉点。在弦理论中，粒子被在时空中运动的小弦环所取代。从理论上讲，时空是十维的。在我们所知道的四维时空中，有一个六维纤维化，其纤维被称为卡-丘流形。Gromov-Witten不变量是卡-丘流形和弦理论研究中的重要组成部分。自从Gromov-Witten不变量被发现以来，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。Gromov-Witten不变量的计算在Calabi-Yau流形上是非常困难的。本计画旨在发展一种新的Gromov-Witten不变量的计算方法，并应用其发展来产生代数几何的新认识，主要研究Gromov-Witten理论的退化。这将从与Dan Abramovich共同开发的稳定对数图的角度进行，并由Mark Gross和Bernd Siebert独立开发。特别是，PI提出证明对数Gromov-Witten不变量的一般胶合公式。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。本项目的另一部分是在非真情形下推广Mori的理论，并进一步理解拟投射簇的双有理几何。

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.