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建议研究曲线和表面上的各种束带，通常与曲线模量空间的几何形状和表面的某些模量空间有关。特别是，PI将研究如何在Kappa环的结构中反映在平滑曲线的模量曲线上的Verlinde滑轮的几何形状。 PI还计划进一步进一步了解行李束在表面上的模束空间上的几何形状的当前理解。对于Abelian或K3表面，当允许表面变化时，关于极化Abelian或K3表面的模量空间的共同学出现了有趣的问题。在不同的方向上，PI将以几何形式解释二维Yang-Mills理论的Q-Defomation。在对重要几何对象的数学研究中，有很大的兴趣在几何上描述某种类型的所有对象的集合本身。这样的集合称为模量空间。特别是，在数学和理论物理学中，束带的模束空间都很重要。在各种种类上的滑轮空间通常反映了基本品种的几何特性，而在物理学中，矢量束的模量空间自然出现在仪表理论的基本研究中。 PI计划在各种环境中分析带束带的模量空间，并有可能阐明一系列有趣的几何形状。