Calabi-Yau Manifolds: Families, Fibrations, and Degenerations

该项目的目的是研究某种类型的流形的性质，称为Calabi-Yau流形。卡-丘流形的特殊性质意味着它们出现在纯数学的许多领域，从代数几何中分类问题的中心位置，到一维卡-丘流形的使用。椭圆曲线），以及三维卡-丘流形在数学物理和弦理论中的特殊作用。该项目本身将涉及卡-丘流形的构造和研究。该项目的第一部分将涉及对二维卡-丘流形的研究，这些流形通常被称为K3曲面。特别是，该项目将集中在K3曲面的行为，因为他们在家庭中的变化，重点是如何采取限制导致这些家庭退化。然后，这个理论将被应用于构建三维卡-丘流形，承认纤维化结构的K3曲面研究的第一部分。最后，我们将详细研究这些K_3阶的Calabi-Yau三折叠的性质。

The aim of this project is to investigate the properties of a certain type of manifold, called a Calabi-Yau manifold. The special properties of Calabi-Yau manifolds 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 3-dimensional Calabi-Yau manifolds in mathematical physics and string theory.The project itself will involve the construction and study of Calabi-Yau manifolds. The first part of the project will involve a study of 2-dimensional Calabi-Yau manifolds, which are more commonly known as K3 surfaces. In particular, the project will focus on the behaviour of K3 surfaces as they vary in families, with a focus on how taking limits causes these families to degenerate.This theory will then be applied to construct 3-dimensional Calabi-Yau manifolds which admit fibration structures by the K3 surfaces studied in the first part. Finally, the properties of these K3-fibred Calabi-Yau threefolds will be explicitly studied.