EPSRC Reference: 
EP/V013270/1 
Title: 
Fano cone singularities and their links 
Principal Investigator: 
Suess, Dr H 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Faculty of Mathematics and Computer Sci 
Organisation: 
Friedrich Schiller University Jena 
Scheme: 
Standard Research 
Starts: 
01 August 2022 
Ends: 
31 July 2025 
Value (£): 
145,960

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
In threedimensional 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. Higherdimensional analogues of these two conditions are called Einstein condition and Kstability, 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 Kstability. In particular, we will prove regularity properties of Einstein metrics on higherdimensional 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 ChenDonaldsonSun's proof of the equivalence of Kstability with the Einstein condition.
In this project the main objects of our interest are socalled 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, socalled links. SasakiEinstein structures on such links play a distinguished role in theoretical physics and physicists are interested in finding new explicit examples of such metrics. SasakiEinstein structures come in two flavours: quasiregular and irregular ones. Quasiregular 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.

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