Cohomological and Birational Invariants of Algebraic Varieties

该奖项支持起源于丢番图方程理论的代数几何、算术几何和数论之间的接口研究。主要的研究对象是代数族，由多个变量的多项式方程组定义。这样的方程式系统存在于数学、科学和工程领域。将离散或线性不变量与代数簇联系起来的思想在代数几何中得到了广泛而成功的应用，以了解代数簇的性质并对其进行分类。这个项目旨在利用变种的几何性质来更充分地研究这些不变量，这可能会导致朝着解决几个长期存在的问题取得决定性进展。主要目标之一是了解一个代数簇可以在多大程度上被独立参数参数化。在这个方向上，即使是立方曲线的情况--由四个或更多变量中的一个3次方程定义的变种--也远未完全被理解。该项目解决了四个问题。第一部分是关于代数簇的二元性。研究人员计划将基于零圈Chow群的性质的专门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Chow圈群和圈类到上同调群的映射：Hodge和Tate猜想的积分方面。这些问题可以通过计算未分支上同调群来解决。该项目将研究这些和相关的几何性质，例如簇上有理曲线的空间和R-等价。第三个问题涉及曲线函数域上的Galois理论不变量和局部-整体原理。特别重要的情况是有限域或阿贝尔簇上的三重。最后一个问题是关于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.