Tyurin Degenerations of K3 Surfaces

这个项目的目的是研究一种叫做K3表面的表面的性质。K3曲面是一类称为Calabi-Yau流形的二维情况，其特殊性质意味着它们出现在纯数学的许多领域：从代数几何分类问题的中心位置，到数论中一维Calabi-Yau流形（又名椭圆曲线）的使用，以及K3曲面和三维Calabi-Yau流形在数学物理和弦理论中发挥的特殊作用。该项目涉及到K3表面退化的构建和研究。这些退化包括在数学上对K3曲面进行变形，直到它们分解或退化成一组更简单的碎片。该项目将研究一类被称为“秋林退化”的退化，这种退化在镜像对称的数学中起着特殊的作用，但目前人们对其知之甚少。该项目的目的是构建新的Tyurin退化的例子，研究它们的性质，并为它们建立一个更广泛的分类理论。

The aim of this project is to investigate the properties of a certain type of surface, called a K3 surface. K3 surfaces are the two-dimensional case of a class of manifold called Calabi-Yau manifolds, whose special properties mean that they appear in many areas of pure mathematics: from a central position in classification problems in algebraic geometry, to the use of 1-dimensional Calabi-Yau manifolds (a.k.a. elliptic curves) in number theory, and the special role played by K3 surfaces and 3-dimensional Calabi-Yau manifolds in mathematical physics and string theory.The proposed project involves the construction and study of degenerations of K3 surfaces. These degenerations involve mathematically deforming K3 surfaces until they break up, or degenerate, into a collection of simpler pieces. The project will study a certain class of degenerations, called Tyurin degenerations, which play a special role in the mathematics of mirror symmetry and yet are currently poorly understood. The aim of the project is to construct new examples of Tyurin degenerations, study their properties, and work towards a broader classification theory for them.