Calabi-Yau Manifolds & Algebraic Geometry

The main objectives of this project are:
1.to study the existence of certain special canonical metrics on Fano manifolds by algebro-geometric methods,
2.to apply these results to the study of families and degenerations of Fano manifolds in dimension 3.

Fano manifolds play a major role in birational geometry and are one of the building blocks of pure geometric type used to build all algebraic varieties. Roughly speaking, they are positively curved, and have spherical geometry.
A key problem in complex geometry studies the existence of canonical metrics. A few years ago, Donaldson and his collaborators showed that for Fano manifolds, the existence of such metrics is equivalent to a stability criterion in Algebraic geometry (K-stability). This is an exciting development, but our understanding of this criterion is still very limited, and we only understand a few explicit cases in dimension 3.
This project will apply techniques from birational geometry to study valuative criteria for K-stability on explicit families of Fano 3-folds. The results obtained will enhance the current understanding of K-stability and provide examples in dimension 3. The second main goal of the project is to study the moduli theory of certain Fano 3-folds. In other words, I want to understand the geometry of degenerations of explicit Fano 3-folds such as quartic hypersurfaces in 4-dimensional projective space, and answer natural questions such as which singularities can appear on degenerations
This research is of interest to the algebraic and/or complex geometer; but also potentially to scientists that use Fano varieties. These appear in domains ranging from Theoretical Physics to Phylogenetic trees.