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Analytic Methods in Complex Algebraic Geometry

复杂代数几何中的解析方法

基本信息

批准号：
1707661
负责人：
Julius Ross
金额：
$ 24.7万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707661&HistoricalAwards=false
关键词：
Analytic Methods Complex Algebraic Geometry

项目摘要

Award: DMS 1707661, Principal Investigator: Mihai Paun Algebraic varieties are objects defined by polynomial equations. Their classification is the far-reaching problem in this field, and this project aims to answer a few central questions in this framework. After the ground-breaking work of B. Riemann in dimension one, the theory of higher dimensional algebraic and analytic varieties developed impressively fast under the impetus of differential geometry and global analysis methods. The concept of positivity, with its twofold incarnations - in algebraic terms (positivity of divisors and algebraic cycles), or in analytic terms (plurisubharmonicity, positive curvature) - played a key role in such development. These investigations will pursue further this circle of ideas by treating problems arising from algebraic geometry via analytic techniques. The main motivation is to show that the methods developed in analysis can offer a very powerful complement to the purely algebraic methods. A successful implementation of this project would lead to important progress towards classification problems in both algebraic and analytic geometry.These projects are structured in three main parts. The first concerns generalizations of the Ohsawa-Takegoshi theorem on analytic extensions, with connections to questions arising in the minimal model program of algebraic geometry. The second line of investigation concerns regularity properties of degenerate Monge-Ampere equations and planned applications to certain relative canonical bundles. Given an algebraic fiber space, one of the main objects of study is its corresponding relative canonical bundle. In many instances, this bundle carries some positivity, and the project is to use the Monge-Ampere theory in order to quantify the amount of positivity this bundle has. The third research direction is an analysis of the positivity properties of the canonical bundle of rank one holomorphic foliations defined on the jet spaces corresponding to a projective variety. This is in connection to the notion of Kobayashi hyperbolicity, where such foliations appear naturally.

奖项：DMS 1707661，首席研究员：Mihai Paun代数变种是由多项式方程定义的对象。它们的分类是这一领域中影响深远的问题，本项目旨在回答这一框架中的几个核心问题。在B的开创性工作之后。在微分几何和整体分析方法的推动下，高维代数簇和解析簇的理论得到了令人印象深刻的快速发展。积极性的概念，以其双重体现-在代数术语（因子和代数圈的积极性），或在分析术语（plurisubharmonicity，正曲率）-在这种发展中发挥了关键作用。这些调查将进一步追求这一圈的想法所产生的问题，通过处理代数几何分析技术。主要的动机是表明，在分析中开发的方法可以提供一个非常强大的补充，纯代数方法。这个项目的成功实施将导致在代数和解析几何分类问题的重要进展。这些项目的结构分为三个主要部分。第一个关注的推广大泽武越定理的解析扩展，与连接的问题所产生的最小模型计划的代数几何。第二条线的调查关注的正则性退化的蒙赫-安培方程和计划的应用程序，某些相对规范束。给定一个代数纤维空间，其对应的相对标准丛是研究的主要对象之一。在许多情况下，这个捆绑包带有一些积极性，该项目将使用Monge-Ampere理论来量化这个捆绑包的积极性。第三个研究方向是分析定义在对应于射影簇的jet空间上的秩一全纯叶理的标准丛的正性性质。这与小林双曲面的概念有关，在这种双曲面中，叶理是自然出现的。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

奇异克勒爱因斯坦度量的变体：正小平维度

DOI：
10.1515/crelle-2021-0028
发表时间：
2021
期刊：
Journal für die reine und angewandte Mathematik (Crelles Journal
影响因子：
0
作者：
Cao, Junyan;Guenancia, Henri;Păun, Mihai
通讯作者：
Păun, Mihai

Covers of rational double points in mixed characteristic

混合特征中有理双点的覆盖

DOI：
10.5427/jsing.2021.23h
发表时间：
2021
期刊：
Journal of Singularities
影响因子：
0.4
作者：
Carvajal-Rojas, Javier;Ma, Linquan;Polstra, Thomas;Schwede, Karl;Tucker, Kevin
通讯作者：
Tucker, Kevin

Twisted Kähler–Einstein metrics

扭曲的克勒爱因斯坦度量