K-Stability, Moduli Spaces, and Singularities

变量是作为多变量多项式方程公解的几何对象。变量可以通过改变多项式的系数而变化，而多项式的几何形状在什么时候能保持良好的状态是一个重要的问题。当这些变种是正弯曲的，被称为法诺变种时，一种自然的方法是观察那些具有爱因斯坦结构的变种，这是一个起源于广义相对论的概念。爱因斯坦结构与代数几何中的稳定性理论k -稳定性密切相关。PI将研究具有爱因斯坦结构的法诺变种，以及它们如何退化和发展奇点。这些问题是基础的，并且在数学的不同领域之间有着深刻的联系，例如代数几何、微分几何、交换代数和数学物理。PI将研究Fano变体的k稳定性，它们的模空间，以及奇点的几何。该PI旨在证明k -多稳定型Fano的模空间是紧致的，这与在k -不稳定型Fano上找到一个Harder-Narasimhan过滤有关。PI将研究log Fano对的显式模空间及其壁交叉，并建立与曲线和K3曲面的模空间的连接。PI将研究Kawamata日志终端奇点的局部体积分布，他的目标是证明0是唯一的积累点。上述建议的项目将使用最小模型规划和几何不变理论中的经典技术，以及PI和合作者开发的k -稳定性，k模空间和归一化体积的代数理论中的新工具。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。

Varieties are geometric objects as common solutions of multivariable polynomial equations. Varieties can vary by changing the coefficients of polynomials, and it is an important question on when their geometric shapes will remain in a nice way. When the varieties are positively curved called Fano varieties, a natural approach is to look at those with Einstein structures, a notion originated from general relativity. The Einstein structures are deeply related to K-stability, a stability theory in algebraic geometry. The PI will study Fano varieties with Einstein structures, and how they degenerate and develop singularities. The problems are foundational and have deep connections among different areas of mathematics, such as algebraic geometry, differential geometry, commutative algebra, and mathematical physics.The PI will investigate K-stability of Fano varieties, their moduli spaces, and geometry of singularities. The PI aims to show that moduli spaces of K-polystable Fano varieties are compact, which is related to finding a Harder-Narasimhan filtration on K-unstable Fano varieties. The PI will study explicit moduli spaces of log Fano pairs and their wall crossings, and estabilish connections to moduli spaces of curves and K3 surfaces. The PI will study distribution of local volumes of Kawamata log terminal singularities, where he aims to show that 0 is the only accumulation point. The proposed projects above will use classical techniques from the Minimal Model Program and Geometric Invariant Theory, as well as new tools from algebraic theory of K-stability, K-moduli spaces, and normalized volumes developed by the PI and collaborators.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.