Orbifolds and Birational Geometry

轨道折叠和双有理几何

基本信息

批准号：
EP/H023267/1
负责人：
Miles Reid
金额：
$ 45.12万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH023267%2F1
关键词：
Orbifolds Birational Geometry

项目摘要

Our main object of study are orbifolds - roughly speaking, spaces which are locally the quotient M/G of a complex manifold M by the action of a finite group G.Subgroups of linear groups play a prominent role in geometry and algebra at all levels: finite subgroups of rotations of 3-space appear as the symmetry groups of regular solids, that is, the regular cylinders and the famous Platonic solids. The finite subgroups of SL(2,\C) are classified as an ADE scheme, with cyclic groups, binary groups and groups obtained as spinor double covers of the Platonic groups. The quotients of complex 2-space \C^2 by these groups are a famous family of surface singularities studied by Felix Klein around 1870, and by Coxeter and Du Val in the 1930s; each of these singularities has a resolution by a surface containing a bunch of algebraic curves (spheres) with configuration graph the same ADE diagram. As a notable example, the binary icosahedral group gives the singular hypersurface (x^2+y^3+z^5 = 0) in complex 3-space \C^3, that appears in many guises in topology and algebra, and whose resolution is the Dynkin diagram E8.John McKay observed in the 1980s that the ADE resolution graph of the Klein singularities is given by the McKay quiver in the representation theory of the abstract group G. This observation was translated into geometry by Gonzales-Sprinberg and Verdier in 1978: an irreducible representation of G gives a tautological vector bundle on the resolved orbifold, and the classes of these bundles base the K theory. Conjectures of Reid from the early 1990s generalized this McKay correspondence to higher dimensions, and the subject has grown from then into a whole field of study, exploring the rich and intricate relations between the equivariant geometry of M (that is, primarily, the representation theory of G) and the geometry of the resolved orbifold. Reid's 1999 Bourbaki seminar includes a colloquial summary of these matters; the correspondence takes place on many levels - cohomology, K-theory, derived categories, motivic integration, as moduli spaces, as stacks, and so on.Orbifold geometry appears in many quite different contexts, including the 3-fold terminal and canonical singularities of minimal model theory, moduli space problems, the geometry of varieties in weighted projective spaces or other toric ambient spaces and the representation theory surrounding the McKay correspondence, as exploited notably by Mark Haiman. More generally, one need not insist that the local cover M is nonsingular, leading for example to the hyperquotient singularities (hypersurface singularity divided by a group action) that play an important role in Mori and Reid's study of 3-fold terminal singularities.

我们的主要研究对象是轨道折叠 - 粗略地说，空间是有限群 G 作用下的复流形 M 的局部商 M/G。线性群的子群在各级几何和代数中发挥着重要作用： 3-空间旋转的有限子群表现为正多面体的对称群，即正则圆柱体和著名的柏拉图多面体。 SL(2,\C) 的有限子群被分类为 ADE 方案，其中有循环群、二元群和作为柏拉图群的旋量双覆盖获得的群。这些群的复数 2-空间 \C^2 的商是 Felix Klein 于 1870 年左右以及 Coxeter 和 Du Val 在 1930 年代研究的著名表面奇点族。这些奇点中的每一个都有一个包含一堆代数曲线（球体）的表面的分辨率，其配置图与 ADE 图相同。作为一个值得注意的例子，二元二十面体群给出了复数 3 空间 \C^3 中的奇异超曲面 (x^2+y^3+z^5 = 0)，它以多种形式出现在拓扑和代数中，并且其分辨率是Dynkin图E8。John McKay在20世纪80年代观察到克莱因奇点的ADE分辨率图是由抽象群表示论中的McKay颤动给出的G. 这个观察结果被 Gonzales-Sprinberg 和 Verdier 于 1978 年转化为几何学：G 的不可约表示给出了解析轨道折叠上的同义反复向量丛，并且这些向量丛的类别基于 K 理论。里德从 20 世纪 90 年代初的猜想将这种麦凯对应关系推广到了更高的维度，从那时起，这个学科就发展成为一个完整的研究领域，探索 M 的等变几何（即主要是表示论）之间丰富而复杂的关系。 G）和解析的轨道折叠的几何形状。里德 1999 年布尔巴基研讨会包括对这些问题的口语总结；对应关系发生在许多层面上——上同调、K 理论、派生范畴、动机积分、模空间、堆栈等等。轨道几何出现在许多完全不同的上下文中，包括 3 重终端和规范奇点最小模型理论、模空间问题、加权射影空间或其他环面环境空间中的簇几何以及围绕麦凯对应的表示理论，马克·海曼（Mark Haiman）特别利用了这一点。更一般地，无需坚持局部覆盖 M 是非奇异的，例如导致超商奇点（超曲面奇点除以群作用），这在 Mori 和 Reid 的 3 重终端奇点研究中发挥着重要作用。