Toric vector bundles: Stability, Cohomology, and Applications.

这个补助金的主题是在代数几何，几何对象的研究定义为消失轨迹的许多多项式方程，称为代数簇。一个基本问题是代数簇的分类。向量丛是与代数簇相关联的几何对象，这些代数簇可以放在一起形成一个新的簇，称为模空间。这些模空间是从旧簇构造新簇和揭示簇的几何性质的重要工具。他们有一个几何输入数据，陈类，它是已知的，只有少数类型的品种，这些模空间存在的输入数据。 我建议研究一类称为环面簇的向量丛。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，作为榜样，为企业家和发展新的理论。复曲面簇带有一类特殊的向量丛，称为复曲面向量丛，可以用来研究复曲面簇上一般向量丛的性质。这些环面向量束有描述的组合学和线性代数，这一建议将建立进一步的桥梁，这些领域之间的相关问题，起源于代数几何的问题，在组合学和线性代数。这将为这两个领域之间的新的交叉施肥打开大门，通过提供更大的工具集并为这两个领域引入新的研究问题。这个提议的目标是系统地发展环面向量丛的理论，以便研究与代数几何和邻近领域有关的问题。该提案的主要目标之一是确定输入数据的存在模空间的复曲面品种。另一个主要目的是揭示几何和代数之间的基本关系内在的定义代数簇的情况下，环面品种，通过研究，为一个给定的嵌入环面品种，数量的最小定义方程的一个给定的程度，最小高等代数关系的个数（合）之间的这些定义方程的一个给定的程度方面的几何嵌入。

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.