Toric vector bundles: Stability, Cohomology, and Applications.

这项拨款的主题是代数几何，研究几何对象，定义为有限多个多项式方程的消失轨迹，称为代数变量。一个基本问题是代数变量的分类。向量束是与代数变量相关的几何对象，它们可以组合在一起形成一个新的变量，称为模空间。这些模空间是由旧模空间构造新模空间和揭示底层模空间几何性质的重要工具。它们有一个几何输入数据，即Chern类，对于这些模空间存在的输入数据，只有几种类型的变体是已知的。我建议研究一类叫做环面品种的矢量束。虽然这些品种非常特殊，但它们表现出额外的组合结构，这使得它们可以使用一套全新的工具进行研究。他们是一个成功的故事，作为猜想和发展新理论的例子。环面品种有一类特殊的矢量束，称为环面矢量束，可用于研究环面品种上一般矢量束的性质。这些环向向量束有组合学和线性代数的描述，本提案将代数几何中的问题与组合学和线性代数中的问题联系起来，从而在这些领域之间建立进一步的桥梁。通过提供更大的工具集和向两个领域引入新的研究问题，这将为这些领域之间新的交叉受精打开大门。本提案的目标是系统地发展环向向量束理论，以研究与代数几何及其邻近领域相关的问题。该方案的主要目标之一是识别环变上存在模空间的输入数据。另一个主要目标是揭示几何和代数之间的基本关系内在的代数变量的定义，在环变的情况下，通过研究，对于一个给定的环变的嵌入，一个给定的次的最小定义方程的数量，和这些给定的次的定义方程之间的最小高级代数关系（合）的数量，在嵌入的几何。

The topic of this grant is in algebraic geometry, the study of geometric objects defined as the vanishing locus of finitely many polynomial equations, called algebraic varieties. One basic question is the classification of algebraic varieties. Vector bundles are geometric objects associated to algebraic varieties that can be put together to form a new variety, called a moduli space. These moduli spaces are an important tool to construct new varieties from old ones, and to reveal geometric properties of the underlying variety. They have a geometric input data, the Chern class, and it is known only for a few types of varieties for what input data these moduli spaces exist. I propose to study vector bundles on a class of varieties called toric varieties. While these varieties are very special, they exhibit additional combinatorial structure, that allows their study with a completely new set of tools. They have been a success story serving as examples for conjectures and to develop new theories. Toric varieties carry a special class of vector bundles called toric vector bundles that can be used to study properties of general vector bundles on toric varieties. These toric vector bundles have descriptions in terms of combinatorics and linear algebra, and this proposal will build further bridges between these fields by relating questions originating in algebraic geometry to questions in combinatorics and linear algebra. This will open the door to new cross-fertilization between these fields, by giving access to a much larger toolset and by introducing new research questions to both fields. The goal of this proposal is to systematically develop the theory of toric vector bundles in order to study questions that are of relevance to algebraic geometry and neighboring fields. One of the main objectives of the proposal is to identify the input data for the existence of moduli spaces on toric varieties. Another main objective is to reveal the fundamental relationship between geometry and algebra intrinsic in the definition of algebraic varieties in the case of toric varieties, by studying, for a given embedding of a toric variety, the numbers of minimal defining equations of a given degree, and the number of minimal higher algebraic relations (syzygies) between these defining equations of a given degree in terms of the geometry of the embedding.