Canonical metrics and stability in complex geometry

Differential geometers study geometric structures and their properties by using tools similar to those employed in Calculus. Algebraic geometry utilizes methods and tools that come from algebra to understand geometric entities. In this project, the PI aims to focus on a class of geometric objects known as projective manifolds and investigate questions related to a geometric entity called the scalar curvature. The PI will explore both the differential and algebraic properties of these structures as well as their underlying spaces and study their interactions with each other. This research connects the field of differential geometry and algebraic geometry and will lead to collaborations and training of researchers with diverse backgrounds. Broader impacts of the project include student and postdoctoral mentoring, creation of a vertically integrated research group, preparation and publishing lecture notes, as well as conference and seminar organization. A central problem in complex differential geometry is to construct canonical metric structures on Kahler manifolds. Over projective manifolds, the Yau-Tian-Donaldson (YTD) conjecture aims to find stability conditions to guarantee the existence of canonical Kahler metrics. The main aim of this project is to study the YTD conjecture for constant scalar curvature Kahler (cscK) metrics. The PI will continue to study uniform K-stability for models as a sufficient condition for the existence of cscK metrics and how this stability condition is related to a more standard K-stability for test configurations. To achieve this, the PI will bridge the Archimedean and non-Archimedean theory by relating slopes of energy functionals along geodesic rays in the space of Kahler metrics to non-Archimedean invariants of algebraic degenerations. The PI will further extend such study to more general weighted cscK metrics and explore new connections between weighted cscK metrics with other canonical Kahler metrics, including Kahler-Ricci solitons, extremal Kahler metrics and conformally Einstein-Maxwell Kahler metrics. The PI will also study un-stable polarized manifolds which do not carry cscK metrics. He will then find ways to construct their optimal degenerations as substitutes of cscK metrics. 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.