Analytic Methods in Complex Algebraic Geometry

奖项：DMS 1707661，首席研究员：Mihai Paun代数簇是由多项式方程定义的对象。它们的分类是该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.Riemann的一维开创性工作之后，高维代数和解析簇理论在微分几何和整体分析方法的推动下得到了令人印象深刻的快速发展。正性的概念及其双重体现--以代数形式(除数和代数圈的正性)，或以分析形式(多次谐和性，正曲率)--在这种发展中发挥了关键作用。这些研究将通过分析技术来处理代数几何中出现的问题，从而进一步探索这一思想圈。主要的动机是表明，在分析中发展的方法可以提供一个非常强大的补充纯粹的代数方法。这个项目的成功实施将导致在代数和解析几何的分类问题方面取得重要进展。这些项目分为三个主要部分。第一部分是关于解析延拓的Ohsawa-Takegoshi定理的推广，它与代数几何的最小模型程序中出现的问题有关。第二条线是关于退化的Monge-Ampere方程的正则性以及对某些相对正则丛的有计划的应用。给定一个代数纤维空间，其对应的相对标准丛是其主要研究对象之一。在许多情况下，这个捆绑包带有一些积极的东西，项目是使用Monge-Ampere理论来量化这个捆绑包具有的积极程度。第三个研究方向是分析定义在射影簇对应的射影簇上的秩一全纯叶丛的正规丛的正性性质。这与小林双曲性的概念有关，在小林双曲性中，这种叶状结构是自然出现的。

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.