CAREER: Birational Geometry and K-stability of Algebraic Varieties

Algebraic geometry studies algebraic varieties which are geometric objects defined by polynomial equations. It plays an important role in neighboring fields such as differential geometry. A natural and important question connecting these areas is to find and classify algebraic varieties with nice geometric structures, such as metrics with constant curvature. In recent years, the development of higher dimensional algebraic geometry has led to a number of breakthroughs in the search for such metrics when the algebraic varieties are positively curved. The goal of this project is to advance these ideas to further understand the geometric structure of general algebraic varieties. In addition the project provides training and research opportunities for students and early-career researchers in related areas, through seminars, summer schools, and other activities. In more detail, the project is motivated by the influential Yau-Tian-Donaldson Conjecture that existence of canonical metrics should be equivalent to some algebraic stability condition known as K-stability. The PI will to develop a local K-stability theory for general Kawamata log terminal singularities, with a view towards the understanding of their birational geometry and moduli. The PI will also investigate some interesting new questions in birational geometry that are inspired by recent progress in K-stability. Finally the project will further advance the algebraic theory of K-stability from the Fano case to the general polarized case, and attack some open problems related to the Yau-Tian-Donaldson conjecture. 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.