Fano cone singularities and their links

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，或者在所有包围相同体积的表面中具有最小的面积。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。直到最近才在法诺流形上证明了它们的等价性。我们将通过更代数的K-稳定性概念来研究Fano流形上锥的爱因斯坦条件及其联系的几何含义。特别地，我们将利用代数工具证明高维球面上爱因斯坦度量的正则性。曲率是几何对象的一个重要特征。对于曲面，在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，就像球面的情况一样。与球面上的情况相反，鞍点允许曲线向切平面的相对侧弯曲。这种行为表现为负曲率。在代数几何中，任何正弯曲的物体都被称为法诺簇。作为“积木”的其他品种，他们发挥了重要作用，代数几何。Fano簇研究的最新突破是Birkar著名的有界性定理和Chen-Donaldson-Sun关于K-稳定性与Einstein条件等价的证明。事实上，这种奇点的一个典型例子是法诺簇上的锥的顶点。此外，我们还感兴趣的某些关联对象，所谓的链接。这种联系上的Sasaki-Einstein结构在理论物理学中起着杰出的作用，物理学家有兴趣找到这种度量的新的明确例子。佐佐木-爱因斯坦结构有两种：准规则结构和不规则结构。准正则的例子已经有一段时间了，它们已经通过射影代数几何进行了研究。不规则的例子是最近才发现的，必须通过新技术来处理。

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.