Birational Geometry of Moduli Spaces and Applications

模空间双有理几何及其应用

• 批准号: 0968968
• 负责人: Radu Laza
• 金额: $ 10.06万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968968&HistoricalAwards=false
• 关键词: Birational; Geometry; Moduli; Spaces; Applications

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator is interested in the study of the geometry of moduli spaces, especially those that are birational to modular varieties of orthogonal or unitary type (examples include the moduli space of K3 surfaces or low genus curves). Laza's work exploits the existence of multiple birational models for a moduli space (e.g. obtained by using different compactification methods, such as Geometric Invariant Theory (GIT) or Hodge theory) to extract useful geometric information about a given moduli space. The principal investigator plans to apply this type of ideas to various projects involving moduli spaces. A first question that the PI proposes to investigate is the problem of finding a geometric compactification for the moduli of polarized K3 surfaces. The methods of the variation of GIT quotients and ideas coming from the minimal model program offer a promising approach to the low degree cases. A second project addresses various questions about the moduli of compact hyperkaehler manifolds, higher dimensional analogues of the K3 surfaces. In particular, the PI plans to investigate from an arithmetic and geometric point of view the case of moduli space of double EPW sextics (introduced by O'Grady). A third project is concerned with some concrete questions about the birational geometry of moduli spaces of genus 4 curves and cubic threefolds. The general area of the proposal is algebraic geometry, the branch of mathematics that is concerned with the geometric properties of algebraic varieties (geometric objects defined by polynomial equations). A simple example of algebraic variety is the complex torus, that has the shape of a doughnut. While in other branches of mathematics (such as topology) all doughnuts have the same shape, in algebraic geometry the precise shape (in this case the ratio between the diameter and width) is very important. In fact, the precise quantification of the shape of the geometric objects within a given topological class is the subject of the moduli theory. Moduli theory is a central field of study in algebraic geometry and has numerous applications in mathematics and modern physics.

该奖项是根据 2009 年美国复苏和再投资法案（公法 111-5）资助的。主要研究者对模空间的几何研究感兴趣，特别是那些与正交或酉类型的模变体双有理的几何空间（示例包括 K3 曲面或低亏格曲线的模空间）。 Laza 的工作利用模空间的多个双有理模型的存在（例如通过使用不同的紧化方法获得，例如几何不变理论（GIT）或霍奇理论）来提取有关给定模空间的有用几何信息。首席研究员计划将这种类型的想法应用到涉及模空间的各种项目中。 PI 建议研究的第一个问题是找到偏振 K3 表面模量的几何紧化问题。 GIT 商的变化方法和来自最小模型程序的想法为低度案例提供了一种有前途的方法。第二个项目解决了有关紧致超凯勒流形（K3 表面的高维类似物）模量的各种问题。特别是，PI 计划从算术和几何的角度研究双 EPW 六次幂的模空间的情况（由 O'Grady 引入）。 第三个项目涉及有关 4 格曲线和三次三次的模空间双有理几何的一些具体问题。该提案的总体领域是代数几何，这是数学的一个分支，涉及代数簇（由多项式方程定义的几何对象）的几何性质。代数簇的一个简单例子是复杂的环面，其形状为甜甜圈。虽然在数学的其他分支（例如拓扑）中，所有甜甜圈都具有相同的形状，但在代数几何中，精确的形状（在这种情况下是直径与宽度之间的比率）非常重要。事实上，给定拓扑类内几何对象形状的精确量化是模理论的主题。模理论是代数几何的核心研究领域，在数学和现代物理学中有广泛的应用。