Higher Dimensional Algebraic Varieties: Geometry and Arithmetic

代数几何主要研究多项式方程的解。在这一领域的指导问题是理解几何对象，所谓的代数簇，产生于多项式方程和分类等形状。这个项目还涉及数论的应用，希望了解这些形状的几何形状可以告诉我们这些多项式方程的有理数或整数解。这些问题也可以应用到其他数学领域（例如微分几何）和物理学（例如弦理论）。高维代数簇的模空间不像它们的曲线对应物那样被很好地理解。这个项目的首要问题是理解高维代数簇的模空间的紧化。所采用的方法将是由最小模型程序（MMP）以及K稳定性等技术产生的组合。特别是，PI计划了解特定情况下的显式紧化（例如K3曲面），并研究如何以模块化的方式在各种紧化之间进行插值。第二个首要问题，是调查几何的高维品种和他们的模量如何影响算术，研究代数几何，双曲性和算术几何之间的丰富的相互作用。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Algebraic geometry focuses on the study of solutions of polynomial equations. Guiding questions in this fields are to understand the geometric objects, called algebraic varieties, arising from polynomial equations and to classify such shapes. This project also concerns applications to number theory, with hopes to understand what the geometry of these shapes can tell us about the rational number or integral solutions to these polynomial equations. These problems have applications to other fields of math (e.g. differential geometry) as well as physics (e.g. string theory). Moduli spaces of higher dimensional algebraic varieties are not as well understood as their curve counterparts. The first overarching question in this project is to the understand compactifications of moduli spaces of higher dimensional algebraic varieties. The approaches taken will be a combination of techniques arising from, e.g. the minimal model program (MMP), as well as K-stability. In particular, the PI plans to understand explicit compactifications in specific cases (e.g. K3 surfaces), and to study how to interpolate between various compactifications in a modular way. The second overarching question, is to investigate how geometry of higher dimensional varieties and their moduli influences arithmetic, studying the rich interplay between algebraic geometry, hyperbolicity, and arithmetic geometry.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.