Cohomological and Birational Invariants of Algebraic Varieties

该奖项支持源自丢番图方程理论的代数几何、算术几何和数论的交叉研究。主要研究对象是代数簇，由多个变量的多项式方程组定义。这样的方程组存在于数学、科学和工程中。将离散或线性不变量与代数簇相关联的思想已在代数几何中得到广泛而成功的应用，以理解代数簇的性质并对它们进行分类。该项目旨在采用现代技术，利用品种的几何特性来更全面地研究这些不变量，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。 主要目标之一是了解代数簇可以在多大程度上通过独立参数进行参数化。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 次方程定义的变体——也远没有被完全理解。该项目解决了四个问题。第一个是关于代数簇的双有理性质。研究人员计划将基于零循环 Chow 群特性的专业化技术应用于有理曲面上的二次纤维。第二个问题涉及代数簇上的循环群以及上同调群的循环类映射：霍奇和泰特猜想的积分方面。这些问题可以通过计算无分支上同调群来解决。该项目将研究这些以及相关的几何特性，例如品种上的有理曲线空间和 R 等价性。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。特别重要的情况在有限域或阿贝尔簇上有三重。最后一个问题集中于代数闭域上代数群空间的分类性质。

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.