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Aspects of the moduli theory of sheaves and varieties

滑轮和变体模量理论的各个方面

基本信息

批准号：
1303389
负责人：
Alina Marian
金额：
$ 17.9万
依托单位：
Northeastern University
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：
2013-08-15 至 2016-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303389&HistoricalAwards=false
关键词：
Aspects moduli theory sheaves varieties

项目摘要

The PI proposes to study various spaces of sheaves on curves and surfaces, often in relation to the geometry of the moduli space of curves and of some moduli spaces of surfaces. In particular, the PI will investigate how the geometry of the Verlinde sheaves over the moduli space of smooth curves of genus g is reflected in the structure of the kappa ring. The PI also plans to further the current understanding of the geometry of determinant line bundles over moduli spaces of sheaves on a surface. For abelian or K3 surfaces, when the surface is allowed to vary, interesting questions arise about the cohomology of the moduli space of polarized abelian or K3 surfaces. In a different direction, the PI will interpret geometrically a q-defomation of two-dimensional Yang-Mills theory. In the mathematical study of important geometric objects, there is considerable interest in describing geometrically the set itself of all objects of a certain type. Such a set is called a moduli space. In particular, moduli spaces of sheaves on varieties are important in both mathematics and theoretical physics. Spaces of sheaves over a variety often reflect geometric properties of the underlying variety, while in physics, moduli spaces of vector bundles arise naturally in the fundamental study of gauge theories. The PI plans to analyze moduli spaces of sheaves in various settings, with the potential to shed light on a range of interesting geometries.

PI建议研究曲线和曲面上的层的各种空间，通常与曲线的模空间和曲面的某些模空间的几何有关。特别是，PI将研究如何在亏格g的光滑曲线的模空间上的Verlinde层的几何结构反映在卡帕环的结构中。PI还计划进一步了解目前的几何行列式线丛的模空间上的一个表面上的层。对于阿贝尔曲面或K3曲面，当允许曲面变化时，极化阿贝尔曲面或K3曲面的模空间的上同调性会产生有趣的问题。在另一个方向上，PI将从几何上解释二维杨-米尔斯理论的q-变形。在重要几何对象的数学研究中，人们对用几何方法描述某种类型的所有对象的集合本身非常感兴趣。这样的集合称为模空间。特别是，簇上层的模空间在数学和理论物理中都很重要。簇上的层空间通常反映了基础簇的几何性质，而在物理学中，向量丛的模空间在规范理论的基础研究中自然出现。PI计划在各种环境中分析层的模空间，并有可能揭示一系列有趣的几何形状。