Analytic Methods in Complex Algebraic Geometry

获奖：DMS 1707661，首席研究员：潘宓海。代数变量是由多项式方程定义的对象。它们的分类是该领域影响深远的问题，本项目旨在回答这个框架中的几个核心问题。继B. Riemann在一维上的开创性工作之后，高维代数变分理论和解析变分理论在微分几何和全局分析方法的推动下迅速发展。正性的概念，其双重体现——在代数术语中（除数和代数循环的正性），或在分析术语中（多次调和性，正曲率）——在这种发展中发挥了关键作用。这些研究将通过解析技术处理代数几何中产生的问题，进一步探索这一思想循环。主要的动机是表明分析中发展的方法可以为纯代数方法提供非常有力的补充。这个项目的成功实施将导致代数和解析几何分类问题的重要进展。这些项目主要分为三个部分。第一部分是关于Ohsawa-Takegoshi定理在解析扩展上的推广，并与代数几何极小模型规划中出现的问题相联系。第二方面的研究涉及退化蒙日-安培方程的正则性以及在某些相对规范束上的计划应用。给定一个代数纤维空间，主要研究对象之一是其对应的相对规范束。在许多情况下，这个束带有一些积极性，这个项目是使用蒙日-安培理论来量化这个束所具有的积极性的数量。第三个研究方向是分析定义在射影变体对应的射流空间上的秩一全纯叶的正则束的正性质。这与小林双曲的概念有关，在小林双曲中，这样的叶子自然出现。

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.