Cohomological and Birational Invariants of Algebraic Varieties

代数簇的上同调和双有理不变量

• 批准号: 1601680
• 负责人: Alena Pirutka
• 金额: $ 17.96万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-05-01 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601680&HistoricalAwards=false
• 关键词: Cohomological; Birational; Invariants; Algebraic; Varieties

This award supports research at the interface of algebraic geometry, arithmetic geometry, and number theory originating in the theory of Diophantine equations. The main objects of study are algebraic varieties, defined by systems of polynomial equations in several variables. Such systems of equations occur throughout mathematics, science, and engineering. The idea of associating discrete or linear invariants to algebraic varieties has been intensively and successfully used in algebraic geometry to understand the properties of algebraic varieties and to classify them. This project aims to employ modern techniques that make use of the geometric properties of the variety to more fully investigate these invariants, which may lead to decisive progress towards the solution of several long-standing problems. One of the main objectives is to understand to what extent an algebraic variety could be parametrized by independent parameters. In this direction, even the case of cubics -- varieties defined by a single equation of degree 3 in four or more variables -- is far from being completely understood. The project addresses four questions. The first is about birational properties of algebraic varieties. The investigator plans to apply specialization techniques, based on properties of Chow group of zero-cycles, to quadric fibrations over rational surfaces. The second problem concerns Chow groups of cycles on algebraic varieties and the cycle class maps to the cohomology groups: integral aspects of the Hodge and Tate conjectures. These questions can be approached by computing unramified cohomology groups. The project will investigate these and related geometric properties, such as spaces of rational curves on varieties and R-equivalence. The third problem concerns Galois-theoretic invariants and local-global principles over function fields of curves. The cases of particular importance are threefolds over finite fields or abelian varieties. The last problem focuses on properties of classifying spaces of algebraic groups over algebraically closed fields.

该奖项支持在代数几何形状，算术几何形状和数字理论中的研究，该理论源于二聚体方程理论。研究的主要对象是代数品种，由多个变量中的多项式方程式定义。这种方程式在整个数学，科学和工程中都发生。将离散或线性不变性与代数品种相关联的想法已被密切而成功地用于代数几何形状，以了解代数品种的特性并将其分类。该项目旨在采用利用该品种的几何特性的现代技术来更全面地研究这些不变性，这可能会导致果断的进步，以解决几个长期存在的问题。 主要目标之一是了解代数变体可以在多大程度上通过独立参数来参数。在这个方向上，即使是立方体的情况（由四个或多个变量中的第3级方程定义的品种）也远非完全理解。该项目解决了四个问题。首先是关于代数品种的生物元特性。研究人员计划根据零循环的Chow群的特性应用专业化技术，以在有理表面上进行四次振动。第二个问题涉及代数品种上的周期和周期类图的周期组：霍奇和泰特猜想的整体方面。这些问题可以通过计算未经塑造的共同体学组来解决。该项目将研究这些和相关的几何特性，例如在品种和R等效性上的理性曲线空间。第三个问题涉及曲线功能领域的Galois理论不变性和局部全球原则。特别重要的案例是有限田或阿伯利亚品种的三倍。最后一个问题集中在代数封闭字段上代数组分类空间的属性。