Birational Geometry of Moduli Spaces and Applications

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。主要研究人员对模空间的几何研究感兴趣，特别是那些与正交或酉型模簇(例如，K3曲面或低亏格曲线的模空间)具有对偶关系的模空间。Laza的工作利用了模空间的多个双态模型的存在(例如，通过使用不同的紧化方法，如几何不变理论(GIT)或Hodge理论)来提取关于给定的模空间的有用的几何信息。首席研究员计划将这种类型的想法应用于涉及模空间的各种项目。PI建议研究的第一个问题是找到极化K3曲面的模数的几何紧化问题。Git商的变分方法和最小模型规划的思想为解决低阶次问题提供了一种很有前途的途径。第二个项目解决了关于紧致超Kaehler流形的各种问题，紧致超Kaehler流形是K3曲面的高维类似物。特别是，PI计划从算术和几何的角度研究双EPW六次元的模空间的情况(由O‘Grady引入)。第三个项目涉及亏格为4的曲线和三次三重的模空间的二次几何的一些具体问题。该提案的一般领域是代数几何学，这是数学的一个分支，涉及代数族(由多项式方程定义的几何对象)的几何性质。代数变化的一个简单例子是具有甜甜圈形状的复杂环面。虽然在其他数学分支(如拓扑学)中，所有甜甜圈的形状都相同，但在代数几何中，精确的形状(在这种情况下是直径和宽度的比率)是非常重要的。事实上，给定拓扑类内几何对象的形状的精确量化是模理论的主题。模理论是代数几何的中心研究领域，在数学和现代物理中有着广泛的应用。

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.