CAREER: K-stability and moduli spaces of higher dimensional varieties

Algebraic varieties are geometric shapes arising from solutions to polynomial equations. A central problem in the study of algebraic varieties, known as the moduli problem, is to find a classifying space of algebraic varieties. One prominent strategy to solve the moduli problem is to study canonical geometric structures on algebraic varieties, known as Einstein metrics. This project is to study the moduli problem for algebraic varieties with the help of Einstein metrics. The problems are foundational and have deep connections to other areas of mathematics, such as commutative algebra, differential geometry, and mathematical physics. The project includes research and learning opportunities for students and early-career researchers in algebraic geometry, through seminars, workshops, summer schools, and other activities. More specifically, the PI will investigate K-stability and K-moduli spaces of Fano varieties, connections to other well-studied moduli spaces, and related problems for singularities. Firstly, the PI will construct compact moduli spaces of K-unstable Fano varieties through Kaehler-Ricci solitons. Secondly, the PI will study explicit moduli spaces of Fano threefolds and log Fano pairs via the framework of wall crossings, and establish connections to moduli spaces of curves and K3 surfaces. Thirdly, The PI will construct moduli spaces of log Calabi-Yau pairs connecting K-moduli spaces and KSBA moduli spaces. Finally, the PI will study boundedness and moduli problems for singularities using the theory of normalized volumes. 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.