K-Stability, Moduli Spaces, and Singularities

簇是多变量多项式方程常见解的几何对象。品种可以通过改变多项式的系数而变化，何时它们的几何形状将保持良好状态是一个重要的问题。当变种是被称为Fano变种的正曲线时，一种自然的方法是观察那些具有爱因斯坦结构的变种，这是一个源于广义相对论的概念。爱因斯坦结构与K-稳定性密切相关，K-稳定性是代数几何中的稳定性理论。PI将研究具有爱因斯坦结构的Fano变种，以及它们如何退化和发展奇点。这些问题是基础性的，在不同的数学领域，如代数几何、微分几何、交换代数和数学物理之间有着深刻的联系。PI将研究Fano簇的K-稳定性，它们的模空间，以及奇点几何。PI的目的是证明K-多稳定Fano簇的模空间是紧的，这与找到K-不稳定Fano簇上的Hard-Narasimhan滤子有关。PI将研究对数Fano对及其壁交的显式模空间，并建立与曲线和K3曲面的模空间的联系。PI将研究Kawamata对数终端奇点的局部体积分布，他的目的是证明0是唯一的聚集点。以上提出的项目将使用最小模型程序和几何不变理论中的经典技术，以及K-稳定性代数理论、K-模空间和PI及其合作者开发的归一化体积的新工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.