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

对于每一个光滑代数曲面X，都存在一系列高维光滑变型，即X上点的Hilbert格式。变种的经典不变量由它们的上同调给出。表面上点的希尔伯特格式理论的一个中心结果是用Nakajima—Grojnowski算子描述它们的上同调。派生范畴是一个变种的更现代的不变量。这些类别，特别是它们的自等价组，在近15年来一直是代数几何领域深入研究的对象。产生这种兴趣的一个原因在于通过同调镜像对称来推测与物理弦理论的联系。项目第一部分的目标是在派生范畴的水平上构建作为p函子的Nakajima- Grojnowski算子的类似物。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函子自动地产生其目标范畴的自等价。我们有理由希望这些新的自等价可以在更简单的情况下给出Hilbert格式的全部自等价组的描述。实现这样的描述是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.