K-Stability, Moduli, and Birational Geometry

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This is a project in the area of algebraic geometry, which is the study of shapes defined by polynomial equations. A fundamental topic in the field is moduli theory, which aims to construct and study spaces that parametrize all such shapes. For low dimensional shapes such moduli spaces are well understood, but less is known in higher dimensions. Recently, new perspectives from differential geometry have led to the construction of moduli spaces of certain Fano varieties, which are a class of positively curved shapes. The aim of this project is to apply newly developed techniques from the latter construction to the moduli theory of other classes of shapes. The project offers training opportunities for graduate students.K-stabilty is an algebraic notion introduced by differential geometers to characterize the existence of certain canonical metrics on complex projective varieties. While the notion can be defined for any projective variety, recent research has focused on the case of Fano varieties and culminated in an application to algebraic geometry: the construction of projective moduli spaces parametrizing K-stable Fano varieties. The goal of this project is to use tools from birational geometry and K-stability to construct moduli spaces of other classes of varieties. In particular, the PI will construct certain moduli of K-unstable Fano varieties, motivated by the study of Kahler-Ricci solitons in differential geometry. Next, the PI will develop new approaches for studying moduli of K-trivial varieties. Finally, the PI will apply tools from birational geometry to the study of K-stability of polarized varieties.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.