Nakajima--Grojnowski operators and derived autoequivalences of Hilbert schemes of points on surfaces

Nakajima--Grojnowski算子和导出的曲面上点的希尔伯特方案的自等价性

• 批准号: 257237495
• 负责人: Dr. Andreas Krug
• 金额: --
• 依托单位: Mathematics Institute
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/257237495?language=en
• 关键词: Nakajima; Grojnowski; operators; derived; autoequivalences

For every smooth algebraic surface X there is a series of higher-dimensional smooth varieties, namely the Hilbert schemes of points on X. A classical invariant of varieties is given by their cohomology. A central result in the theory of Hilbert schemes of points on surfaces is the description of their cohomology by means of the Nakajima--Grojnowski operators. The derived category is a more modern invariant of a variety. These categories and in particular their groups of autoequivalences have been object to intensive research in the area of algebraic geometry for about the last 15 years. One reason for the interest lies in conjectured connections to physical string theory by means of the homological mirror symmetry. The goal of the first part of the project is the construction of analogues of the Nakajima--Grojnowski operators as P-functors on the level of the derived categories. A P-functor, as defined by Addington, is a Fourier--Mukai transform of a special shape. Every P-functor automatically yields an autoequivalence of its target category. There is reason to hope that these new autoequivalences allow to give a description of the full group of autoequivalences of the Hilbert schemes in easier cases. It is the main goal of the second part of the project to achieve such descriptions.

对于每个光滑的代数表面X，都有一系列高维的平滑品种，即X上点的希尔伯特方案。X上的hilbert方案。一种经典的品种由其共同体学给出。希尔伯特（Hilbert）在表面上的希尔伯特（Hilbert）方案理论的核心结果是通过中岛（Nakajima） - grojnowski操作员对它们的共同体学描述。派生的类别是一种更现代的多样性。这些类别，尤其是它们的自动等量组是过去15年来代数几何学领域进行深入研究的反对。兴趣的原因之一是通过同源镜对称性猜想与物理弦理论的联系。该项目第一部分的目的是构建Nakajima-Grojnowski操作员的类似物，作为派生类别级别的P函数。 Addington定义的P函数是一种特殊形状的傅立叶 - mukai变换。每个P函数都会自动产生其目标类别的自动等效性。有理由希望这些新的自动等同允许在更容易的情况下对希尔伯特计划的全部自动等量进行描述。实现此类描述是项目第二部分的主要目标。

项目成果

On the derived category of the Hilbert scheme of points on an Enriques surface

关于恩里克斯曲面上点的希尔伯特方案的派生范畴

• DOI: 10.1007/s00029-015-0178-x
• 发表时间: 2015
• 期刊: Selecta Mathematica
• 作者: Andreas Krug (with P. Sosna)

Equivalences of equivariant derived categories

• DOI: 10.1112/jlms/jdv014
• 发表时间: 2014-11
• 期刊: Journal of the London Mathematical Society
• 作者: Andreas Krug;P. Sosna