Quantum cohomology of homogeneous spaces

Gromov-Witten理论起源于弦理论的物理模型。Kontsevich和Manin关于稳定映射的模空间的开创性工作为该理论奠定了严格的数学基础，并为解决代数几何中突出的枚举问题做出了贡献。因此，他们开创了一个新的研究分支，在代数几何和辛几何以及理论物理中有着深远的应用。粗略地说，Gromov-Witten不变量计数光滑射影变化上的固定次和属曲线。考虑曲线必须满足特定的偶然性，从而使所得到的曲线数量有限。Gromov-Witten不变量可以概括为所谓的量子上同环。这是一个梯度代数，其量子积结构是普通杯积的变形。这个乘积的结合律产生了枚举解之间的非平凡关系。本文研究了齐次空间上的Gromov-Witten理论——一类具有传递群作用的光滑射影变体。在这种情况下，稳定映射的模空间具有特别理想的性质，它允许直观地接近Gromov-Witten理论。特别是，可以对量子产物中的最小度作出更明确的表述。首先，在这个项目中，我们将研究这些最小度的分布，并尝试计算所有最小度的最小上界。此外，该项目的目的是证明比稳定映射模空间的不可约性更精细的性质。模空间的性质，如拟齐性，简化了Gromov-Witten不变量的描述，并为其结构的理论理解铺平了道路。因此，这个项目最终将是关于计算Gromov-Witten不变量的新方法。

Gromov-Witten theory has its origins in physical models of string theory. Groundbraking work of Kontsevich and Manin on the moduli space of stable maps gave the theory a rigorous mathematical foundation and contributed to solve outstanding enumerative problems in algebraic geometry. Thereby, they initiated a new branch of research with far-reaching applications in algebraic and symplectic geometry and theoretical physics.Roughly speaking, Gromov-Witten invariants count curves of fixed degree and genus on a smooth projective variety. The considerer curves have to satisfy specific incidences which force the resulting number of curves to be finite. Gromov-Witten invariants can be summarized in the so-called quantum cohomology ring. This is a graded algebra whose quantum product structure is a deformation of the ordinary cup product. The associativity of this product yields non-trivial relations among the enumerative solutions.The present project is about Gromov-Witten theory on homogeneous spaces - a class of smooth projective varieties which carry a transitive group action. In this case, the moduli space of stable maps has particularly desirable properties which allow an intuitive approach to Gromov-Witten theory. In particular, it is possible to make sharper statements about the minimal degrees in quantum products. First of all, we will study in this project the distribution of these minimal degrees and try to compute a minimal upper bound of all minimal degrees.Moreover, the aim of the project is to prove finer properties than irreducibility of the moduli space of stable maps. Properties of the moduli space such as quasi-homogeneity simplify the description of Gromov-Witten invariants and prepare the path for a theoretical understanding of their structure. Thus, the project will be finally about new methods for computing Gromov-Witten invariants.