Research in higher dimensional algebraic geometry

这位研究人员将研究高维代数几何中的几个问题。在与Alexeev、Hassett和Kolár合作的一个项目中，PI计划完成稳定对数曲面和高维簇的粗模空间的存在证明，类似于稳定曲线的模空间。这是一个悬而未决的问题，其解决方案对该领域的许多应用至关重要。Kolár和PI共同实施的另一个项目将为上述项目提供重要成果。该项目的主要主题是了解出现在稳定原木品种上的奇点的变形。在一个相关的项目中，PI计划证明经典Kodaira消失定理的各种推广。在另一个项目中，PI将致力于关于典型极化光滑射影簇的模叠的子簇的精化Viehweg猜想。这个猜想是由Shafarevich的一个里程碑式的猜想及其由Arakelov和Parshin提出的解发展而来的，这个猜想在Faltings证明Mordell猜想的过程中发挥了重要作用。该项目的一部分是与Kebekus的联合工作。在另一个项目中，PI和Hacon将研究无处不在的微分形式的存在对潜在簇几何的影响。他们的目标是证明该地区几个突出的猜想。在另一个项目中，PI计划证明射影空间和二次超曲面的一个刻画，它将给出Mori定理(早期称为HartShorne猜想)和Beauville猜想的一个深远的共同推广。后者最近由阿劳霍、德鲁尔和PI解决。在另一个项目中，PI希望证明一个强大的有理分解定理。这一结果的一个应用是，具有有理奇点的变种允许只有有理奇点本身的紧致化。这项研究是在代数几何领域进行的，代数几何是现代数学中最古老的部分之一，但它发展到了解决几个世纪以来一直存在的问题的地步。最初，它仍然是最简单的形式，它处理由多项式在平面上定义的图形。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法，相反，它在这些领域被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。代数几何中的一个中心问题是对所有几何对象的分类。反过来，分类理论的一个重要部分就是模理论。后者的核心思想是，人们不仅想要了解这些物体，而且还想了解它们可以变形的方式。模空间在理论物理中起着非常重要的作用。研究模空间上的曲线提供了关于物体在时空中如何变化的信息。这个项目的焦点之一是紧模空间。这些通常是模空间的扩展，它们给出了关于奇异变形的附加信息，这些变形与其他变形本质上是不同的。该项目的其他目标包括更好地了解某些更高维度的品种。

The investigator will work on several problems in higher dimensional algebraic geometry. In a project, joint with Alexeev, Hassett, and Kollár, the PI plans to complete the proof of existence of a coarse moduli space of stable log surfaces and higher dimensional varieties, an analog of the moduli space of stable curves. This an open problem whose solution is crucial to many applications in the field. Another project, pursued jointly by Kollár and the PI, will provide crucial results for the above projects. The main theme of the project is to understand deformations of singularities that appear on stable log varieties. In a related project the PI is planning to prove various generalizations of the classical Kodaira vanishing theorem. In another project the PI is going to work on the refined Viehweg conjecture regarding subvarieties of moduli stacks of canonically polarized smooth projective varieties. This conjecture evolved from a landmark conjecture of Shafarevich, and its solution by Arakelov and Parshin, which played an important role in Faltings' proof of the Mordell Conjecture. Part of this project is joint work with Kebekus. In another project the PI and Hacon are going to study the impact of the existence of nowhere vanishing differential forms on the geometry of the underlying variety. Their goal is to prove several outstanding conjectures in the area. In another project the PI is planning to prove a a characterization of the projective space and quadric hypersurfaces that will give a far reaching common generalization of Mori's theorem (earlier known as Hartshorne's conjecture) and Beauville's conjecture. The latter was settled by Araujo, Druel and the PI recently. In yet another project the PI hopes to prove a strong rational resolution theorem. One application of this result would be that varieties with rational singularities admit a compactification with only rational singularities themselves.This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one that blossomed to the point where it has solved problems that have stood for centuries. Originally, and still in its simplest form it treats figures defined in the plane by polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one does not only want to understand these objects, but also understand the way they can be deformed. Moduli spaces play a very important role in theoretical physics. Studying curves on moduli spaces provides information on how an object is changing in space-time. One of the focuses of this project is on compact moduli spaces. Those are extensions of moduli spaces in general and they give additional information about singular deformations, ones that are essentially different from others. Other goals of the project involve a better understanding of certain higher dimensional varieties.