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.

对于每个光滑代数曲面X，存在一系列高维光滑簇，即X上点的希尔伯特格式。簇的一个经典不变量由簇的上同调给出。一个中心的结果在理论的希尔伯特计划的点在表面上是描述他们的上同调的Nakajima-Grojnowski运营商。派生范畴是一个簇的更现代的不变量。这些范畴，特别是它们的自等价群，在过去的15年里一直是代数几何领域深入研究的对象。这种兴趣的一个原因在于，通过同调镜像对称，它与物理弦理论有着明确的联系。该项目的第一部分的目标是建设类似物的Nakajima-Grojnowski运营商作为P-函子的水平上的派生类别。一个P-函子，由阿丁顿定义，是一个特殊形状的傅里叶-向井变换。每一个P-函子自动地产生它的目标范畴的一个自等价。有理由希望，这些新的自等价允许在更容易的情况下给出一个完整的自等价组的希尔伯特计划的描述。这是该项目的第二部分的主要目标，以实现这样的描述。

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