Complex Geometric Properties of Period Maps

代数几何研究作为多项式方程组解的集合的几何性质。许多熟悉的集合都是以这种方式出现的，包括线和面，圆和球/肥皂泡。代数几何在数学之外也有许多应用，包括统计学、机器人、纠错码、系统发育学、博弈论和整数规划。“模问题”是代数几何的一个分支，研究族中的这些集合。例如，可以考虑所有经过固定点的线。这些家庭，以及相关的问题，可能相当复杂。他们研究中的一个重要工具是“时期图”，这是一个与家庭相关的功能。周期图允许人们从其他数学领域引入技术，包括复杂几何（其中定义函数不是多项式，如代数几何，而是全纯函数）和表示理论（线性代数的复杂扩展）。该奖项支持：(i)对周期图属性的研究，这些研究是由模问题的应用驱动的，以及（ii）研究生和早期职业博士后研究人员的培训。PI的工作包括通过多个部门和大学项目积极指导本科生和研究生。一半的项目旨在构建周期图的完成，动机是预期应用于模空间的紧化。这包括一个Ohsawa-Takegoshi型扩展问题，它将完成一个推广Satake-Baily-Borel紧化的程序；一个特定科的葫芦属品种的这种结构的调查；并为不同时期的地图建造了加藤-臼井式的完井，包括与卡拉比-丘三叠特定家族相关的地图。其他项目继续由PI发起的一个项目，通过它们的特征形式（与周期图相关的无穷小、复杂几何不变量）来研究周期图（和霍奇结构的变化）。这个程序的灵感来自于一个非常成功的Hwang-Mok程序的密切类比，通过它们的最小有理切线的变化来研究Fano流形。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 geometry studies geometric properties of sets that arise as solutions to systems of polynomial equations. Many familiar sets arise in this way, including lines and planes, and circles and spheres/soap bubbles. Algebraic geometry also has many applications outside of mathemayics, including statistics, robotics, error-correcting codes, phylogenetics, game theory and integer programming. “Moduli problems” is a sub-branch of algebraic geometry that studies these sets in families. For example, one might consider all the lines passing through a fixed point. These families, and the related questions, can be quite complicated. An important tool in their study is the “period map”, a function that is associated with the family. The period map allows one to bring to bear techniques from other areas of mathematics, including complex geometry (where the defining functions are not polynomials, as in algebraic geometry, but holomorphic functions) and representation theory (a sophisticated extension of linear algebra). This award supports both: (i) research into properties of period maps that are motived by applications to moduli problems, and (ii) the training of graduate students and early career postdoctoral researchers. The PI’s work includes active mentorship of both undergraduate and graduate students through multiple departmental and university programs.Half of the projects aim to construct completions of period maps, motivated by anticipated applications to compactifications of moduli spaces. This includes an Ohsawa-Takegoshi type extension problem that will complete a program to generalize the Satake-Baily-Borel compactification; an investigation of this construction for a specific family of Calabi-Yau varieties; and the construction of a Kato-Usui type completions for various period maps, including that associated to a specific family of Calabi-Yau 3-folds. The other projects continue a program initiated by the PI to study period maps (and variations of Hodge structure) by their characteristic forms (which are infinitesimal, complex-geometric invariants associated with a period maps). This program is inspired by a close analogy with the very successful Hwang-Mok program to study Fano manifolds via their varieties of minimal rational tangents.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.