Birational geometry and K-stability of Fano varieties

这个博士项目是一个更广泛的项目的一部分，该项目涉及到双曲面几何和Fano品种的K稳定性之间的相互作用。例如，代数簇是由多项式公式定义的几何形状。普通球面是一种代数族。这种形状出现在科学家研究由多项式方程描述的现象的所有领域，从计算机辅助图形设计到密码学和数学生物学。转动几何学简化了对任何几何形状的研究，以理解一些纯粹几何类型的积木。这些纯类型对应于Fano品种(正弯曲)、Calabi-Yau品种(平坦)和一般类型的变种(负弯曲)。近年来，在Fano簇的情况下，保证某些规范度量存在的微分几何性质与称为K-稳定性的代数性质等价，这一点已变得很清楚。这些都是微妙的属性，我们对这些属性的几何直觉仍然太少。此外，我们现在明白，K-稳定性是理解Fano变种家族的关键--或这些变种是如何退化的。这个项目将集中在显式Fano 3折叠的K稳定性，并将考虑稳定性性质如何在Fano 3折叠的变形族中变化。

This PhD project is part of a wider project concerned with the interplay between birational geometry and K-stability of Fano varieties. Algebraic varieties are geometric shapes that are defined by polynomial equations; for example. an ordinary sphere is an algebraic variety. Such shapes appear in all areas where scientists study phenomena described by polynomial equations from computer-aided graphic design to cryptography and mathematical biology. Birational geometry reduces the study of any geometric shape to understanding some building blocks which are of pure geometric type. These pure types correspond to Fano varieties (positively curved), Calabi-Yau varieties (flat) and varietes of general type (negatively curved). In recent years, it has become clear that a differential geometric property guaranteeing the existence of certain canonical metrics was equivalent, in the case of Fano varieties, to an algebraic property called K-stability. These are subtle properties, about which we still have too little geometric intuition. Further, we now understand that K-stability holds the key to understanding families of Fano varieties - or how these degenerate. This project will focus on K-stability of explicit Fano 3-folds, and will consider how stability properties vary in deformation families of Fano 3-folds.