Calabi-Yau Manifolds & Algebraic Geometry

这个项目的主要目标是：1.to通过代数几何方法研究Fano流形上某些特殊正则度量的存在性，2.to将这些结果应用于三维Fano流形的族和退化的研究。法诺流形在双有理几何中起着重要的作用，并且是用于构建所有代数簇的纯几何类型的构建块之一。粗略地说，它们是正弯曲的，并且具有球形几何形状。复几何中的一个关键问题是研究正则度量的存在性。几年前，唐纳森和他的合作者证明，对于Fano流形，这样的度量的存在性等价于代数几何中的稳定性判据（K-稳定性）。这是一个令人兴奋的发展，但我们对这一标准的理解仍然非常有限，我们只了解第三维度中的几个明确案例。本计画将应用双有理几何的技巧来研究显式Fano 3-folds族的K-稳定性的评价准则。所获得的结果将提高目前的K-稳定性的理解，并提供在3维的例子。该项目的第二个主要目标是研究某些Fano 3-折叠的模量理论。换句话说，我想了解显式Fano 3-folds（如四维射影空间中的四次超曲面）退化的几何学，并回答自然问题，如退化上可能出现的奇点。这项研究对代数和/或复几何学感兴趣;但也可能对使用Fano品种的科学家感兴趣。这些出现在从理论物理到系统发育树的领域。

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 degenerationsThis 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.