K-Stability in Higher Dimensional Geometry

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The project in algebraic geometry is a study of varieties, which are geometric spaces defined by polynomial equations. Among all geometric spaces, varieties have the advantage of being easier to study from a computational viewpoint. Moreover, since every space can be approximated by varieties, they are crucial construct for algebraic geometers. Inspired by the study of spaces with a metric satisfying certain Einstein field equations, the main aim of the proposal is to provide a new framework to understand varieties which are positively curved, in both global and local settings. One focus will be how these varieties vary in families, and degenerate to others with more special structures. There are several research thrusts to this project. First, the PI aims to solve the local higher rank finite generation conjecture, and thus, complete the local stability theory by establishing the stable degeneration conjecture for any Kawamata log terminal singularity. In addition, the project includes investigation into the moduli theory for general Fano varieties without a K-stability assumption and a study of K-stability for explicit examples of Fano varieties. There are also aims to combine birational and non-archimedean geometry together to understand degenerations of Calabi-Yau manifolds. Graduate students will participate in the research project.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.