Orbifolds and Birational Geometry

我们的主要研究对象是旋涡--粗略地说，空间是复流形M在有限群G的作用下局部的商M/G。线性群的子群在几何和代数中都扮演着重要的角色：3-空间的旋转的有限子群表现为正则固体的对称群，即正则柱面和著名的柏拉图固体。SL(2，C)的有限子群被归类为ADE格式，得到的循环群、二元群和群是柏拉图群的旋量双覆盖。由这些群构成的复2-空间C^2的商是由Felix Klein在1870年左右以及由Coxeter和Du Val在20世纪30年代研究的一族著名的曲面奇点，每个奇点都有一个曲面的分辨率，该曲面包含一组构形图相同的ADE图的代数曲线(球面)。作为一个明显的例子，二元二十面体群给出了复3-空间\C^3中的奇异超曲面(x^2+y^3+z^5=0)，它在拓扑学和代数中以多种伪装出现，其分辨率是dykin图E8。John McKay在20世纪80年代观察到Klein奇点的ADE分解图是由抽象群G的表示理论中的McKay箭图给出的。这一观察在1978年由Gonzales-Sprint berg和Verdier转化为几何：G的一个不可约表示给出了已分解的或分叉上的重言式向量丛，而这些丛的类基于K理论。20世纪90年代初Reid的猜想将这种McKay对应推广到了更高的维度，从那时起，这个主题已经发展成为一个完整的研究领域，探索M的等变几何(主要是G的表示理论)和所分解的奥比福尔德的几何之间丰富而复杂的关系。里德1999年的Bourbaki研讨会包括对这些问题的口语化总结；对应发生在许多层面上-上同调、K-理论、派生范畴、动机积分、作为模空间、作为堆栈等等。奥比博尔德几何出现在许多完全不同的背景下，包括极小模型理论的三重终端和正则奇点、模空间问题、加权射影空间或其他环面环境空间中的簇的几何以及围绕McKay对应的表示理论，正如Mark Haiman特别利用的那样。更一般地，人们不需要坚持局部覆盖M是非奇异的，例如，导致在Mori和Reid的三重终端奇点研究中扮演重要角色的超商奇点(超曲面奇点被群作用除以)。

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.