K-Stability and Birational Geometry of Fano Varieties

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Algebraic geometry is the study of the geometric objects – called algebraic varieties – defined by systems of polynomial equations. A fundamental problem is to find and classify algebraic varieties with nice geometric structures. When the varieties are positively curved, it is natural to search for metrics that satisfy Einstein's equation from general relativity. The main theme of this project is to study such Einstein metrics using an algebro-geometric stability theory called K-stability, and to investigate their interaction with other structures that lie at the crossroads of algebraic geometry, differential geometry, commutative algebra and mathematical physics. The Principal Investigator plans to organize activities to enhance communication between different groups of researchers (including under-represented groups).The project has three closely related parts. The first is to prove K-stability of explicit Fano varieties, such as complete intersections and their mildly singular degenerations. With the help of birational geometry, the PI aims to show that most of these Fano varieties are K-stable and explicitly describe some of their corresponding compact K-moduli. The second part is to study the moduli theory of Fano varieties and singularities using K-stability. The goal is to construct compact moduli spaces that parametrize certain optimal degenerations of the varieties or singularities. The last part is to explore the connection of K-stability with other geometric properties of Fano varieties, especially their birational rigidity and rationality.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.