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滑轮的各种空间，通常与曲线模空间和曲面的某些模空间的几何相关。特别是，PI 将研究 Verlinde 滑轮在 genus g 平滑曲线模空间上的几何形状如何反映在 kappa 环的结构中。 PI 还计划进一步加深对表面滑轮模空间上行列式线束几何形状的理解。对于阿贝尔或 K3 曲面，当允许曲面变化时，会出现有关极化阿贝尔或 K3 曲面的模空间上同调的有趣问题。在不同的方向上，PI 将从几何角度解释二维 Yang-Mills 理论的 q 变形。在重要几何对象的数学研究中，人们对以几何方式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，簇上滑轮的模空间在数学和理论物理中都很重要。簇上的滑轮空间通常反映了底层簇的几何特性，而在物理学中，矢量丛的模空间自然出现在规范理论的基础研究中。 PI 计划分析各种设置下滑轮的模量空间，有可能揭示一系列有趣的几何形状。