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

对于每个光滑代数曲面X，都有一系列高维光滑簇，即X上的点的Hilbert格式。由它们的上同调给出簇的一个经典不变量。曲面上点的Hilbert格式理论的一个中心结果是用Nakajima-Grojnowski算子刻画了它们的上同调。派生范畴是一种更现代的不变量。在过去的15年里，这些范畴，特别是它们的自等价群，一直是代数几何领域深入研究的对象。人们感兴趣的一个原因是通过同调镜像对称性推测了与物理弦理论的联系。该项目第一部分的目标是在派生范畴的水平上构造作为P-函子的Nakajima-Grojnowski算子的类似算子。根据Addington的定义，P函子是一种特殊形状的傅里叶-Mukai变换。每个P-函子自动生成其目标范畴的一个自等价。有理由希望这些新的自等价能够在更简单的情况下给出希尔伯特方案的全部自等价的描述。实现这样的描述是本项目第二部分的主要目标。

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.