Fano cone singularities and their links

在三维空间中，球体与所有其他曲面的区别在于它的曲率是均匀的，或者说，在包围相同体积的所有曲面中，球体的面积最小。这两个条件的高维类似分别称为爱因斯坦条件和K-稳定性。它们的等价性直到最近才被证明为Fano流形。我们将通过更代数的K-稳定性概念来研究Fano流形上的圆锥和它们之间的联系的爱因斯坦条件的几何含义。特别是，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，通过该点的所有曲线都弯曲到切线平面的同一侧，就像球体的情况一样，该点的正曲率是其特征。与球体上的情况不同，鞍点允许曲线弯曲到切线平面的对边。这种行为的特点是负曲率。在代数几何中，任何地方的正曲线物体都称为Fano簇。作为其他种类的“积木”，它们在代数几何中扮演着重要的角色。最近在Fano簇的研究中取得的突破是Birkar著名的有界性定理和Chen-Donaldson-Sun证明K-稳定性与爱因斯坦条件的等价性。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Fano簇上锥体的顶点。此外，我们还对某些关联对象，即所谓的链接感兴趣。这种链接上的佐佐木-爱因斯坦结构在理论物理中扮演着特殊的角色，物理学家们对寻找这种度量的新的明确例子很感兴趣。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。拟正则例子早已为人所知，并已通过射影代数几何进行了研究。不规则的例子是相对较新发现的，必须通过新的技术来处理。

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.