CAREER: K-stability and moduli spaces of higher dimensional varieties

代数变量是由多项式方程的解产生的几何形状。模问题是代数变量研究中的一个中心问题，即如何找到代数变量的分类空间。解决模问题的一个重要策略是研究代数变量上的正则几何结构，即爱因斯坦度量。本课题是在爱因斯坦度量的帮助下研究代数变量的模问题。这些问题是基础的，并且与数学的其他领域有很深的联系，比如交换代数、微分几何和数学物理。该项目包括通过研讨会、讲习班、暑期学校和其他活动，为代数几何领域的学生和早期职业研究人员提供研究和学习机会。更具体地说，PI将研究Fano变体的k -稳定性和k -模空间，与其他研究得很好的模空间的连接，以及奇点的相关问题。首先，PI将通过Kaehler-Ricci孤子构造k -不稳定Fano变体的紧模空间。其次，PI将通过墙交的框架研究Fano三倍和对数Fano对的显式模空间，并建立与曲线和K3曲面模空间的联系。第三，PI将构造连接k模空间和KSBA模空间的对数Calabi-Yau对的模空间。最后，PI将利用归一化体积理论研究奇点的有界性和模性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic varieties are geometric shapes arising from solutions to polynomial equations. A central problem in the study of algebraic varieties, known as the moduli problem, is to find a classifying space of algebraic varieties. One prominent strategy to solve the moduli problem is to study canonical geometric structures on algebraic varieties, known as Einstein metrics. This project is to study the moduli problem for algebraic varieties with the help of Einstein metrics. The problems are foundational and have deep connections to other areas of mathematics, such as commutative algebra, differential geometry, and mathematical physics. The project includes research and learning opportunities for students and early-career researchers in algebraic geometry, through seminars, workshops, summer schools, and other activities.More specifically, the PI will investigate K-stability and K-moduli spaces of Fano varieties, connections to other well-studied moduli spaces, and related problems for singularities. Firstly, the PI will construct compact moduli spaces of K-unstable Fano varieties through Kaehler-Ricci solitons. Secondly, the PI will study explicit moduli spaces of Fano threefolds and log Fano pairs via the framework of wall crossings, and establish connections to moduli spaces of curves and K3 surfaces. Thirdly, The PI will construct moduli spaces of log Calabi-Yau pairs connecting K-moduli spaces and KSBA moduli spaces. Finally, the PI will study boundedness and moduli problems for singularities using the theory of normalized volumes.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.