Higher Spin Structures on Orbifolds

这个项目是关于双曲Riemann和Klein 2-orbifolds上的m-自旋结构空间。一个黎曼2-orbifold是一个具有极大图集的2-orbifold，其过渡映射是全纯的。它可以被描述为双曲平面与Fuchsian群的商。一个Klein 2-orbifold是一个具有极大图集的2-orbifold，其过渡映射是全纯的或反全纯的。它可以被描述为一个双曲黎曼2-orbifold配备了一个反全纯对合。双曲Riemann 2-orbifold上的m-自旋结构是一个复线丛，使得这个线丛的m次张量幂同构于orbifold的余切丛。对于黎曼2-orbifold上的任何m-自旋结构，可以在简单轮廓的同伦类空间上分配一个唯一的函数，其值为Z/mZ，相关的m-Arf函数。双曲Klein轨道上的m-自旋结构是指在反全纯对合下不变的基础Riemann 2-轨道上的m-自旋结构，Natanzon和Pratoussevitch利用拓扑学方法研究了双曲Riemann和Klein 2-轨道上无锥点的m-自旋结构空间，确定了m-自旋结构空间的存在条件并描述了其所有连通分支的拓扑。Riley将这些结果推广到双曲Klein 2-orbifolds上的m-自旋结构，使得所有锥点都在蚁全纯对合的不动点集之外。研究了双曲Klein 2-orbifolds上的m-自旋结构空间，包括锥点在反全纯对合的不动点集上的2-orbifolds的情形.研究双曲2-orbifold（超椭圆，多边形等）上的一些特殊的m-自旋结构族以及它们与双曲2-orbifolds空间被m-自旋结构空间覆盖的分支点的关系.研究与拟齐次Gorenstein奇点族的联系。

This project is concerned with spaces of m-spin structures on hyperbolic Riemann and Klein 2-orbifolds. A Riemann 2-orbifold is a 2-orbifold with a maximal atlas whose transition maps are holomorphic. It can be described as a quotient of the hyperbolic plane by a Fuchsian group. A Klein 2-orbifold is a 2-orbifold with a maximal atlas whose transition maps are either holomorphic or anti-holomorphic. It can be described as a hyperbolic Riemann 2-orbifold equipped with an anti-holomorphic involution. An m-spin structure on a hyperbolic Riemann 2-orbifold is a complex line bundle such that the m-th tensor power of this line bundle is isomorphic to the cotangent bundle of the orbifolds. To any m-spin structure on a Riemann 2-orbifold, one can assign a unique function on the space of homotopy classes of simple contours with values in Z/mZ, the associated m-Arf function. An m-spin structure on a hyperbolic Klein orbifold is an m-spin structure on the underlying Riemann 2-orbifold which is invariant under the anti-holomorphic involution.Natanzon and Pratoussevitch used topological methods to study the space of m-spin structures on hyperbolic Riemann and Klein 2-orbifolds without cone points, to determine existence conditions and describe the topology of all connected components of the space of m-spin structures. Riley extended these results some cases of m-spin structures on hyperbolic Klein 2-orbifolds such that all cone points are outside of the fixed point set of the ant-holomorphic involution.The aims of this project:1. Study the spaces of m-spin structures on hyperbolic Klein 2-orbifolds, including the case of 2-orbifolds with cone points on the fixed point set of the anti-holomorphic involution.2. Study some special families of m-spin structures on hyperbolic 2-orbifolds (hyperelliptic, polygonal, etc.) and their connections with branching points of the covering of the space of hyperbolic 2-orbifolds by the space of m-spin structures.3. Investigate the connections with families of quasi-homogeneous Gorenstein singularities.