Hodge theory and algebraic cycles

霍奇理论和代数圈

• 批准号: 1303105
• 负责人: Burt Totaro
• 金额: $ 34.62万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303105&HistoricalAwards=false
• 关键词: Hodge; theory; algebraic; cycles

The Hodge conjecture is one of the central problems of algebraic geometry. If true, it would give a strong connection between topology and algebraic geometry. The integral Hodge conjecture is a stronger statement which fails for some algebraic varieties, as shown by Atiyah and Hirzebruch. Nonetheless, the project aims to prove the integral Hodge conjecture for a large class of 3-dimensional varieties. In another direction, the integral Hodge conjecture often fails for quotients of varieties by finite groups. The project will compute the Chow ring of algebraic cycles for quotient varieties by many finite groups. This can be viewed as finding the correct replacement for the integral Hodge conjecture when it fails.Algebraic geometry is the study of shapes defined by polynomial equations. It is a central area of mathematics, at the border between arbitrary shapes (topology) and the theory of equations (algebra and computation). Users of mathematics have always had to solve systems of polynomial equations, and that remains a computationally challenging problem today. The project deals with the question of which shapes can be moved continuously to become defined by polynomial equations. There is a plausible answer, but a proof seems exceedingly hard to find. The project aims to solve the problem in the most important low-dimensional cases.

霍奇猜想是代数几何的中心问题之一。如果这是真的，它将在拓扑学和代数几何学之间建立一种强有力的联系。积分霍奇猜想是一个更强的声明，失败的一些代数簇，如Atiyah和Hirzebruch。尽管如此，该项目的目的是证明积分霍奇猜想的一大类3维品种。在另一个方向上，积分霍奇猜想对于有限群簇的商经常失败。该项目将计算许多有限群的商簇的代数圈的Chow环。这可以被看作是找到正确的替代积分霍奇猜想时，它失败了。代数几何是研究形状定义的多项式方程。它是数学的中心领域，介于任意形状（拓扑学）和方程理论（代数和计算）之间。数学用户总是不得不解决多项式方程组，这在今天仍然是一个具有挑战性的计算问题。该项目涉及的问题是，哪些形状可以不断移动，成为由多项式方程定义。有一个看似合理的答案，但似乎很难找到证据。该项目旨在解决最重要的低维情况下的问题。