超环面流形与超Kähler矩映射
Hypertoric manifolds and hyperKähler moment maps
摘要：
我们讨论了各个方面的矩映射几何辛和hyperKähler几何。特别地，我们对维数为$4n$的完备超Kähler流形进行了分类，其中对维数为$n$的环面具有三Hamilton作用，而对贝蒂数的有限性没有任何假设.因此，我们发现，在这些情况下，超凯勒矩有连接纤维，辛矩映射的属性是真实的，是满射的。给出了无限拓扑型的超环面流形的新例子。我们给出了完备超凯勒流形上连通群的非阿贝尔三Hamilton群作用的例子，使得超凯勒矩映射不是满射的，并且有一些不连通的纤维。我们还讨论了辛切割，hyperKähler修改和内爆结构的关系。
英文摘要：
We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperKähler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connected groups on complete hyperKähler manifolds such that the hyperKähler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperKähler modifications and implosion constructions.
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