Complex Hyperbolic Geometry, Arithmetic, and Commutative Algebra

复杂的双曲几何、算术和交换代数

• 批准号: 0200877
• 负责人: Domingo Toledo
• 金额: $ 22.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200877&HistoricalAwards=false
• 关键词: Complex; Hyperbolic; Geometry; Arithmetic; Commutative

The principal investigators study the complex hyperbolic geometry of the moduli spaces of cubic surfaces and cubic threefolds. For stable cubic surfaces over the complex numbers, previous work of D. Allcock and the investigators gives a period map from the moduli space to the quotient of the four-ball by a lattice defined over the Eisenstein integers, which is an isomorphism between these two spaces and which takes the subspace of moduli of singular surfaces to a configuration of hyperplanes in the ball. The investigators now study the arithmetic properties of cubic surfaces whose periods are Eisenstein rational points. For stable cubic threefolds over the complex numbers, there is a period map from their moduli space to the quotient of the ten-ball by a lattice also defined over the Eisenstein integers. The investigators study detailed properties of this period map in order to prove that this map is an isomorphism between the two spaces (after some blowing up and down) which carries the subspace of moduli of singular threefolds to a hyperplane configuration in the ball. The main technical point under study is the computation of the differential of the restriction of the period map to each stratum that parametrizes equisingular varieties. The extension of the Griffiths residue calculus to this situation turns out to be quite involved and to require new techniques in commutative algebra.The understanding of cubic equations has been a central theme in mathematics, whith deep implications for mathematics, science and engineering. In the sixteenth century cubic equations in one variable were solved by finding a formula for its solutions analogous to the well-known formula for the solutions of quadratic equations. This formula led to the introduction of complex numbers, which are now a standard tool in science and engineering. In the eighteenth and nineteenth centuries it was realized that quadratic equations in any number of variables could be understood, and that, for a fixed number of variables, all quadratic equations are essentially equivalent by a change of variables. It was also realized that cubic equations in two or more variables are not all equivalent by change of variables. The different equivalence classes are now called the moduli space. The moduli space of cubic equations in two variables is the hyperbolic plane of non-Euclidean geometry. This connection has led to manydiscoveries, from special functions used to solve problems in physics andengineering, to advances in number theory and cryptography. The aim of thisproject is to study the special non-Euclidean geometry that has recently beendiscovered on the moduli spaces of cubic equations in three and fourvariables. The reason for carrying this study is not only the solution ofthe problems posed in the proposal, but also the expectation that this studyof cubics in more variables will continue to be a point of departure for newideas that will affect other areas of mathematics, science and engineering.

主要研究人员研究三次曲面和三次三次曲面的模空间的复双曲几何。对于复数上稳定的三次曲面，D.Allcock和研究者以前的工作通过定义在Eisenstein整数上的格给出了从模空间到四球的商的周期映射，它是这两个空间之间的同构，它将奇异曲面的模空间的子空间带到球中的超平面的构形。研究人员现在研究周期为艾森斯坦有理点的三次曲面的算术性质。对于复数上稳定的三次三重数，存在从它们的模空间到十球的商的周期映射，该映射也定义在Eisenstein整数上。研究人员研究了这一周期映射的详细性质，以证明这一映射是两个空间(在一些爆破后)之间的同构，它带有奇异三重模子空间子空间到球中的超平面构形。研究的主要技术要点是周期映射对等距变异参数化层的约束微分的计算。格里菲斯剩余演算在这种情况下的推广是相当复杂的，需要交换代数中的新技巧。理解三次方程一直是数学的中心主题，对数学、科学和工程都有深远的意义。在16世纪，人们通过找到一个类似于著名的二次方程解的公式来求解一元三次方程。这个公式导致了复数的引入，复数现在是科学和工程中的标准工具。在18世纪和19世纪，人们认识到，可以理解任意数量的变量的二次方程，并且对于固定数量的变量，所有的二次方程通过变量的改变基本上是等价的。还认识到通过变量的变换，两个或多个变量的三次方程并不都是等价的。不同的等价类现在称为模空间。二元三次方程的模空间是非欧几里德几何的双曲平面。这种联系导致了许多发现，从用于解决物理和工程问题的特殊函数，到数论和密码学的进步。这个项目的目的是研究最近在三次和四次方程的模空间上发现的特殊的非欧几里德几何。进行这项研究的原因不仅是为了解决提案中提出的问题，而且还希望这项关于更多变量的立方曲线的研究将继续成为新的起点，这将影响到数学、科学和工程的其他领域。