Cohomological and Birational Invariants of Algebraic Varieties

该奖项支持研究在接口的代数几何，算术几何，数论起源于丢番图方程的理论。主要的研究对象是代数簇，由多个变量的多项式方程系统定义。这样的方程组出现在数学、科学和工程学中。将离散或线性不变量与代数簇相关联的思想已被广泛地并成功地用于代数几何中，以理解代数簇的性质并对其进行分类。该项目旨在采用现代技术，利用品种的几何特性，更充分地研究这些不变量，这可能会导致对解决几个长期存在的问题取得决定性进展。 其中一个主要目标是要了解在何种程度上代数簇可以参数化的独立参数。在这个方向上，即使是三次函数--由四个或更多变量的三次方程定义的变量--也远未被完全理解。该项目涉及四个问题。第一个是关于代数簇的双有理性质。研究人员计划应用专业化技术，基于性质的周群零圈，二次纤维有理曲面。第二个问题涉及周群的循环代数簇和循环类映射到上同调群：积分方面的霍奇和泰特代数。这些问题可以通过计算非分歧上同调群来解决。该项目将调查这些和相关的几何性质，如空间的合理曲线品种和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-整体原理。特别重要的情况是三倍有限域或阿贝尔品种。最后一个问题是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.