K-stability and Higher Dimensional Geometry

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The Principal Investigator (PI) will study varieties. Varieties are defined as the set of solutions of systems of polynomial equations. They are fairly easy to compute and Nash proved every space can be well approximated by varieties. The main aim of the project is to understand how varieties vary if we change the coefficients of the defining polynomial equations, especially for the ones which are positively curved. Such varieties are called Fano varieties. In particular the research will try to understand situations when a family of Fano varieties degenerates to one with singularities. The PI intends to prove that among all Fano varieties, the K-polystable ones can be parametrised by a universal space, called moduli space. As part of the this project, the PI aims to show the moduli space is Hausdorff and compact. The PI aims to understand which Fano varieties are K-semistable by understanding concrete examples as well as some general phenomena. The PI aims to understand the degeneration of Calabi-Yau manifolds, through the interplay between birational and non-archimedean geometry.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.