Fano cone singularities and their links

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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.