Cohomological and Birational Invariants of Algebraic Varieties

该奖项支持在代数几何、算术几何和起源于丢番图方程理论的数论方面的研究。研究的主要对象是由多个变量的多项式方程组定义的代数变量。这样的方程组在数学、科学和工程中随处可见。将离散不变量或线性不变量与代数变量联系起来的思想已经在代数几何中得到了广泛而成功的应用，以理解代数变量的性质并对它们进行分类。该项目旨在采用现代技术，利用该品种的几何特性来更充分地研究这些不变量，这可能会在解决几个长期存在的问题方面取得决定性进展。其中一个主要目标是了解在多大程度上代数变量可以被独立参数参数化。在这个方向上，即使是立方体的情况——由四个或更多变量的单个3次方程定义的变量——也远没有被完全理解。该项目解决了四个问题。第一部分是关于代数变量的双族性质。研究人员计划应用专业化技术，基于零循环的周氏群的性质，有理表面上的二次振动。第二个问题是关于代数变体上的周群和到上同调群的循环类映射：霍奇和塔特猜想的积分方面。这些问题可以通过计算未分叉上同调群来解决。该项目将研究这些和相关的几何性质，如有理曲线的空间上的变化和r等价。第三个问题涉及曲线函数域上的伽罗瓦理论不变量和局部-全局原理。特别重要的情况是在有限域或阿贝尔变异上的三倍。最后一个问题研究代数闭域上代数群的分类空间的性质。

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.