权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Fano cone singularities and their links

法诺锥奇点及其联系

基本信息

批准号：
EP/V013270/1
负责人：
Hendrik Suess
金额：
$ 18.6万
依托单位：
Friedrich Schiller University Jena
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV013270%2F1
关键词：
Fano cone singularities their links

项目摘要

In three-dimensional space the sphere is distinguished from all other surfaces by the uniformity of its curvature or alternatively by having the smallest area among all surfaces enclosing the same volume. Higher-dimensional analogues of these two conditions are called Einstein condition and K-stability, respectively. Their equivalence was proved for Fano manifolds only recently. We will study the geometric implications of the Einstein condition for cones over Fano manifolds and their links via the more algebraic notion of K-stability. In particular, we will prove regularity properties of Einstein metrics on higher-dimensional spheres by using algebraic tools.Curvature is an important feature of geometric objects. For surfaces positive curvature at a point is characterised by the fact that all curves through this point bend to the same side of a tangent plane, as it is the case for a the sphere. In contrast to the situation on the sphere, a saddle point admits curves which bend to opposite sides of a tangent plane. This behaviour characterises negative curvature. In algebraic geometry everywhere positively curved objects are called Fano varieties. As "building blocks" of other varieties they play an important role within algebraic geometry. Recent breakthroughs in the study of Fano varieties have been Birkar's celebrated Boundedness Theorem and Chen-Donaldson-Sun's proof of the equivalence of K-stability with the Einstein condition.In this project the main objects of our interest are so-called klt singularities, which can be seen as local analogues of Fano varieties. Indeed, a prototypical example of such a singularity is the vertex of the cone over a Fano variety. Moreover, we are also interested certain associated objects, so-called links. Sasaki-Einstein structures on such links play a distinguished role in theoretical physics and physicists are interested in finding new explicit examples of such metrics. Sasaki-Einstein structures come in two flavours: quasi-regular and irregular ones. Quasi-regular examples are known for a while and they have been studied via projective algebraic geometry. Irregular examples were discovered relatively recently and they have to be approached via new techniques.

在三维空间中，球体与所有其他表面的区别在于其曲率的均匀性，或者通过具有包围相同体积的所有表面中的最小面积。这两个条件的高维类似物分别称为爱因斯坦条件和 K 稳定性。直到最近，它们的等价性才在 Fano 流形上得到证明。我们将通过更代数的 K 稳定性概念来研究 Fano 流形上锥体的爱因斯坦条件的几何含义及其联系。特别是，我们将利用代数工具证明爱因斯坦度量在高维球体上的正则性。曲率是几何物体的一个重要特征。对于曲面，一点处的正曲率的特征在于，通过该点的所有曲线都弯曲到切平面的同一侧，就像球体的情况一样。与球体上的情况相反，鞍点允许弯曲到切平面的相对两侧的曲线。这种行为的特征是负曲率。在代数几何中，凡是正弯曲的物体都被称为法诺簇。作为其他类型的“构建块”，它们在代数几何中发挥着重要作用。 Fano簇研究的最新突破是Birkar著名的有界定理和Chen-Donaldson-Sun对K稳定性与爱因斯坦条件等价性的证明。在这个项目中，我们感兴趣的主要对象是所谓的klt奇点，它可以被视为Fano簇的局部类似物。事实上，这种奇点的一个典型例子是法诺簇上的圆锥体的顶点。此外，我们还对某些关联对象感兴趣，即所谓的链接。这种链接上的佐佐木-爱因斯坦结构在理论物理学中发挥着杰出的作用，物理学家有兴趣寻找此类度量的新的明确例子。佐佐木-爱因斯坦结构有两种类型：准规则结构和不规则结构。拟正则例子已经为人所知一段时间了，并且已经通过射影代数几何对它们进行了研究。不规则的例子是最近才发现的，必须通过新技术来处理它们。