Holomorphic Symplectic Varieties, Mirror Symmetry, and Cluster Algebras

Hyper-Kähler流形是一类具有丰富内部结构的几何空间，是代数几何和微分几何中的重要几何对象。这些空间在数学和理论物理中都扮演着重要的角色。这个研究项目的主要目的是构建新的例子，或者证明它们不存在。这项研究将研究超Kähler流形如何退化或分解为更简单的部分。研究人员将首先研究已知的例子如何退化，推断出一般模式，然后努力逆转这一过程来构建新的例子。这项工作将利用镜像对称性，这是物理学家发现的两对几何空间之间的一种非凡的对应。对于Hyper-Kähler流形，镜像对称性可以非常明确地理解。它提供了关于超Kähler流形退化的信息，例如，不同部分组合在一起的方式。该奖项支持在超Kähler流形上代数几何和微分几何之间的界面的研究，这在几何和理论物理中是重要的。只有几种类型的例子是已知的，以前的构造都是基于经典的代数几何。研究人员将使用来自镜像对称性、团簇代数和形变理论的新技术来产生新的例子，或者发现它们存在的有形障碍。在之前的合作工作中，他用环面几何和双曲面几何对簇代数进行了几何解释。这些被称为簇簇的几何空间有望构成超Kähler流形退化的基础。在随后的合作工作中，研究人员研究了集群品种的镜像对称性。这涉及到分段线性几何结构，可以用来理解如何将集群品种粘合在一起形成超级Kähler流形。

Hyper-Kähler manifolds, a type of geometric space with rich internal structure, are key geometric objects in algebraic geometry and differential geometry. These spaces play important roles in both mathematics and theoretical physics. The main goal of this research project is to construct new examples, or to prove that they do not exist. The research will investigate how hyper-Kähler manifolds degenerate, or break into simpler pieces. The investigator will first study how the known examples degenerate, to infer general patterns, and will then work to reverse the process to construct new examples. The work will make use of mirror symmetry, a remarkable correspondence between pairs of geometric spaces discovered by physicists. For hyper-Kähler manifolds, mirror symmetry can be understood quite explicitly. This provides information about degenerations of hyper-Kähler manifolds, for example, the combinatorial way in which the different pieces fit together.This award supports research at the interface between algebraic geometry and differential geometry on hyper-Kähler manifolds, which are important in geometry and theoretical physics. Only a few types of examples are known, and previous constructions have been based on classical algebraic geometry. The investigator will use new techniques coming from mirror symmetry, cluster algebras, and deformation theory to produce new examples, or alternatively, to discover tangible obstructions to their existence. In prior collaborative work, he developed a geometric interpretation of cluster algebras in terms of toric and birational geometry. These geometric spaces, called cluster varieties, are expected to form the building blocks of degenerations of hyper-Kähler manifolds. In subsequent collaborative work, the investigator studied mirror symmetry for cluster varieties. This involves piecewise linear geometric structures that can be used to understand how to glue cluster varieties together to form a hyper-Kähler manifold.