复曲面超Kahler簇上剩余圆作用的性质
Properties of the residual circle action on a toric hyperkahler variety
摘要：
我们考虑由C^n的Kahler约化得到的流形X，并将其超Kahler类似物M定义为由同一群T^*C^n = H^n的超Kahler约化。在群是阿贝尔群且X是光滑环面簇的情况下，M是环面超卡勒流形，如Bielawski-Dancer所定义，并由Konno和Hausel-Sturmfels进一步研究。流形M带有S^1的自然作用量，这是由S^1在T^*C^n的纤维上的标量作用量引起的。本文研究了这类作用，计算了它的不动点和等变上同调。作为一个应用程序，我们使用相关的Z/2行动的真正的轨迹M计算变形的Orlik-Solomon代数的光滑，通用，真正的超平面安排，依赖于非平凡的仿射结构的安排。这种变形由复化的补的Z/2-等变上同调给出，其中Z/2通过复共轭作用。
英文摘要：
We consider a manifold X obtained by a Kahler reduction of C^n, and we define its hyperkahler analogue M as a hyperkahler reduction of T^*C^n = H^n by the same group. In the case where the group is abelian and X is a smooth toric variety, M is a toric hyperkahler manifold, as defined by Bielawski-Dancer, and further studied by Konno and Hausel-Sturmfels. The manifold M carries a natural action of S^1, induced by the scalar action of S^1 on the fibers of T^*C^n. In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z/2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, generic, real hyperplane arrangement, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z/2-equivariant cohomology of the complement of the complexification, where Z/2 acts by complex conjugation.
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