Moduli of surfaces, vector bundles, and mirror symmetry
曲面模、向量丛和镜像对称
基本信息
- 批准号:1201439
- 负责人:
- 金额:$ 17.13万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-09-01 至 2016-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The PI has established a connection between the classification of stable vector bundles on an algebraic surface and the compactification of the moduli space of deformations of the surface. The PI will study this correspondence in detail for surfaces of general type, in particular, its relation to the Donaldson theory of instanton invariants of smooth 4-manifolds. Jointly with Mark Gross and Sean Keel, the PI has described an explicit construction of the mirror partner to a non-compact Calabi--Yau manifold of complex dimension 2. The PI will pursue two related projects: an algebraic description of the symplectic cohomology ring of a non-compact Calabi-Yau manifold, and the construction of a canonical basis of global sections of an ample line bundle on a K3 surface, analogous to theta functions for polarized abelian varieties.The PI will study the classification of certain 4-dimensional geometric spaces and the ways in which such a space can be continuously deformed or undergo a "degeneration" given by a subset of the space collapsing to a point. In prior work the PI related such degenerations to the classification of bundles of linear spaces over the given geometric space. He will study this correspondence in detail, in particular its connection with work of Donaldson motivated by theoretical physics. Mirror symmetry is a mysterious correspondence between pairs of geometric spaces called Calabi-Yau manifolds arising in string theory. The PI will pursue two projects inspired by mirror symmetry. The first is a description in explicit terms of an algebraic structure built from counts of area-minimizing surfaces inside a Calabi-Yau manifold. The second is the construction of natural functions on Calabi-Yau manifolds. In the simplest case of a torus (the surface of a donut) these functions were known classically and are important in many areas of mathematics.
PI建立了代数曲面上稳定向量丛的分类与曲面变形模空间的紧化之间的联系。PI将详细研究一般类型表面的这种对应关系,特别是它与光滑4-流形的瞬子不变量的唐纳森理论的关系。PI与Mark Gross和Sean Keel一起,描述了一个非紧的复维数为2的卡拉比-丘流形的镜像伙伴的显式构造。PI将开展两个相关项目:非紧Calabi-Yau流形的辛上同调环的代数描述,以及K3曲面上的样本线丛的整体截面的典范基的构造,类似于极化阿贝尔变种的θ函数。PI将研究某些4-维几何空间和这样的空间可以连续变形或经历由空间的子集坍缩到一个点给出的“退化”的方式。在以前的工作中,PI将这种退化与给定几何空间上的线性空间束的分类联系起来。他将详细研究这一对应关系,特别是它与工作的唐纳森动机的理论物理。镜像对称是弦理论中出现的称为卡-丘流形的几何空间对之间的一种神秘对应。PI将追求两个受镜像对称启发的项目。第一个是一个明确的代数结构的描述建立在计数的面积最小化表面内的卡-丘流形。第二部分是Calabi-Yau流形上自然函数的构造。在最简单的情况下,一个环面(表面的甜甜圈),这些职能是已知的经典和重要的许多领域的数学。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Paul Hacking其他文献
Paul Hacking的其他文献
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{{ truncateString('Paul Hacking', 18)}}的其他基金
Mirror Symmetry, Birational Geometry, and Moduli.
镜像对称、双有理几何和模。
- 批准号:
2200875 - 财政年份:2022
- 资助金额:
$ 17.13万 - 项目类别:
Standard Grant
Collaborative Research: AGNES: Algebraic Geometry NorthEastern Series
合作研究:AGNES:代数几何东北系列
- 批准号:
1937705 - 财政年份:2019
- 资助金额:
$ 17.13万 - 项目类别:
Continuing Grant
Collaborative Research: AGNES: Algebraic Geometry NorthEastern Series
合作研究:AGNES:代数几何东北系列
- 批准号:
1650256 - 财政年份:2017
- 资助金额:
$ 17.13万 - 项目类别:
Continuing Grant
Holomorphic Symplectic Varieties, Mirror Symmetry, and Cluster Algebras
全纯辛簇、镜像对称和簇代数
- 批准号:
1601065 - 财政年份:2016
- 资助金额:
$ 17.13万 - 项目类别:
Standard Grant
Collaborative Research: AGNES: Algebraic Geometry NorthEastern Series, April 25-27, 2014
合作研究:AGNES:代数几何东北系列,2014 年 4 月 25-27 日
- 批准号:
1360543 - 财政年份:2014
- 资助金额:
$ 17.13万 - 项目类别:
Continuing Grant
Collaborative Research: AGNES. Algebraic Geometry North Eastern Series.
合作研究:AGNES。
- 批准号:
1064426 - 财政年份:2011
- 资助金额:
$ 17.13万 - 项目类别:
Continuing Grant
Exceptional vector bundles and degenerations of surfaces
特殊的向量束和表面的退化
- 批准号:
0968824 - 财政年份:2009
- 资助金额:
$ 17.13万 - 项目类别:
Standard Grant
Exceptional vector bundles and degenerations of surfaces
特殊的向量束和表面的退化
- 批准号:
0855760 - 财政年份:2009
- 资助金额:
$ 17.13万 - 项目类别:
Standard Grant
Moduli problems in algebraic geometry
代数几何中的模问题
- 批准号:
0650052 - 财政年份:2006
- 资助金额:
$ 17.13万 - 项目类别:
Standard Grant
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