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Singular Metrics in Kahler Geometry

卡勒几何中的奇异度量

基本信息

批准号：
1906216
负责人：
Nicholas Edelen
金额：
$ 29.64万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906216&HistoricalAwards=false
关键词：
Singular Metrics Kahler Geometry

项目摘要

A central theme in the PI's research is the connections between partial differential equations (PDEs) such as Einstein's equations in general relativity, and the algebraic or geometric properties of the spaces on which these equations are studied. Much research has been focused on the study of PDEs on smooth spaces, where the local geometry has a very simple model, however singularities can arise naturally in nature, the most well known being black holes. The current research project focuses on the study of PDEs on such singular spaces. This study will have applications in algebraic geometry, where singular spaces play a central role in modern research, as well as in the theory of partial differential equations. At the same time, singular spaces occur naturally as limits of families of smooth spaces, which could collapse in certain directions for instance. In this way the proposed research will shed new light on the behavior of such families. Aside from pursuing these research projects, the PI will continue training PhD students and postdoctoral researchers. In addition the PI will also co-organize a yearly summer workshop forundergraduate students aimed at conveying ideas in geometry to them which do not typically appear in the undergraduate curriculum. The PI will also co-organize a yearly bridge program aimed at helping incoming graduate students with diverse backgrounds get up to speed, to ensure their success.The objective of the project is to investigate singularities in Kahler geometry from different points of view. On the one hand, singularities can arise in the limit of a sequence of smooth spaces, and it is important to understand the structure of such limit spaces. The PI will study the question of when the limit space can be identified with a singular Kahler space, building on work of Donaldson-Sun as well as Liu and the PI. Of particular interest are non-collapsed limit spaces of Kahler manifolds with Ricci curvature bounds, however the PI will also investigate the much less understood collapsed limit spaces. In a related direction, the PI will study canonical metrics, such as Kahler-Einstein metrics, on singular complex varieties, in particular the behavior of such metrics near the singular set. A good understanding of the geometry of such singular metrics would lead to applications of differential geometric techniques to the algebraic geometry of singular varieties. Finally, in relation to the collapsing theory of Kahler manifolds, the PI will study the construction of canonical metrics on fibrations that are almost collapsed.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.

PI 研究的中心主题是偏微分方程 (PDE)（例如广义相对论中的爱因斯坦方程）与研究这些方程的空间的代数或几何性质之间的联系。许多研究都集中在光滑空间上的偏微分方程的研究，其中局部几何有一个非常简单的模型，但奇点在自然界中可能自然出现，最著名的是黑洞。当前的研究项目重点研究此类奇异空间上的偏微分方程。这项研究将在代数几何以及偏微分方程理论中得到应用，其中奇异空间在现代研究中发挥着核心作用。与此同时，奇异空间自然地作为平滑空间族的限制而出现，例如，它可能会在某些方向上崩溃。通过这种方式，拟议的研究将为此类家庭的行为提供新的线索。除了开展这些研究项目外，PI还将继续培训博士生和博士后研究人员。此外，PI 还将与本科生共同组织每年一次的夏季研讨会，旨在向他们传达本科课程中通常不会出现的几何思想。 PI 还将共同组织一个年度桥梁项目，旨在帮助具有不同背景的研究生加快步伐，以确保他们的成功。该项目的目标是从不同的角度研究卡勒几何中的奇点。一方面，奇点可能出现在一系列光滑空间的极限中，理解这种极限空间的结构很重要。 PI 将在 Donaldson-Sun 以及 Liu 和 PI 的工作基础上，研究何时可以用奇异卡勒空间来识别极限空间的问题。特别令人感兴趣的是具有里奇曲率界限的卡勒流形的非塌陷极限空间，然而 PI 还将研究不太了解的塌陷极限空间。在相关方向上，PI 将研究奇异复杂变体的规范度量，例如卡勒-爱因斯坦度量，特别是此类度量在奇异集附近的行为。对此类奇异度量几何的良好理解将导致微分几何技术在奇异簇代数几何中的应用。最后，关于卡勒流形的塌陷理论，PI 将研究几乎塌陷的纤维的规范度量的构建。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。