群值内爆与抛物结构
Group-valued Implosion and Parabolic Structures
摘要：
本文件有两个目的。首先，我们将辛内爆的概念推广到拟Hamilton $K$-流形范畴，其中$K$是单连通紧李群。双$K\times K$的内爆截面在适当的意义上是普适的。它是一个奇异空间，但它的某些层具有非奇异闭包。这一观察导致了有趣的新的准哈密顿$K$-流形的例子，例如$K=\SU(n)$的"旋转$2n$-球"。其次，我们构造了一个标记Riemann曲面上抛物丛的泛模空间，其结构群为$K$。主模空间具有$K$的极大环面和环面不变分层到流形的自然作用，每个流形具有辛结构。宇宙的内爆是这个结构的一个基本要素。巧合的是，虽然普适内爆没有复杂的结构（它是$K=\SU(2)$的四球），但主模空间是一个复代数簇。
英文摘要：
The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian $K$-manifolds, where $K$ is a simply connected compact Lie group. The imploded cross-section of the double $K\times K$ turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian $K$-manifolds, such as the ``spinning $2n$-sphere'' for $K=\SU(n)$. Secondly we construct a universal (``master'') moduli space of parabolic bundles with structure group $K$ over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of $K$ and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for $K=\SU(2)$), the master moduli space turns out to be a complex algebraic variety.
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