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

代数变体是由多项式方程的解产生的几何形状。代数簇研究中的一个中心问题，称为模问题，是寻找代数簇的分类空间。解决模问题的一个突出策略是研究代数簇上的规范几何结构，称为爱因斯坦度量。本项目主要是利用爱因斯坦度量研究代数簇的模问题。这些问题是基础性的，并且与数学的其他领域有着深刻的联系，例如交换代数，微分几何和数学物理。该项目包括研究和学习的机会，为学生和早期职业研究人员在代数几何，通过研讨会，讲习班，暑期学校，和其他活动。更具体地说，PI将调查K-稳定性和K-模空间的Fano品种，连接到其他良好的研究模空间，以及相关问题的奇点。首先，PI将通过Kaehler-Ricci孤子构造K-不稳定Fano簇的紧模空间。其次，PI将通过壁交叉的框架研究Fano三重和log Fano对的显式模空间，并建立与曲线和K3曲面的模空间的联系。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将使用归一化体积理论研究奇点的有界性和模量问题。该奖项反映了NSF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.