Poncelet Polygons and the Painleve Equations
Poncelet Polygons and the Painleve Equations
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Poncelet 多边形和 Painleve 方程
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发表时间：
2004
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通讯作者：
N. Hitchin
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作者：
N. Hitchin
Narasimhan和Seshadri的著名定理将曲线上的稳定向量束与其基本群的酉表示联系起来，这一定理已经成为最近大量代数几何和拓扑学相互交织的结果的模型。在所有这些推广中，在这两个领域之间起中介作用的对象是联系的概念，而各种联系的存在性定理则提供了建立这些定理的手段。从某种意义上说，本文的动机是超越存在，要求更明确。它们之间的联系是怎样的？我们能把它们写下来吗？这个问题是我们的出发点。我们这次演讲的新奇之处在于，答案涉及到一段时间的旅行，从1965年Narasimhan和Seshadri定理的证明开始，将我们带回到200多年前。为简单起见，我们不考虑高属曲线上的稳定束，而是考虑复射影线CP上的抛物稳定束的类似情况，即Mehta和Seshadri[11]意义上的稳定束。这样的稳定束由在n个标记点a1，…处具有加权标志结构的矢量束组成。,一个。与它相关的幺正连接是平的，并且在点上有奇点。在一般情况下，向量束本身是平凡的，我们正在寻找的平面连接可以写成亚纯m×m矩阵值1形式，在每个点ai上有一个简单的极点。抛物线结构可以很容易地从形式的残馀中读出。方程的另一边是基本群π1（CP \{a1，…）的表示。， U(m)中的an})，连接的完整性，这带来了更多的问题。这些问题引起了福克斯、克莱因等人的注意
The celebrated theorem of Narasimhan and Seshadri [13] relating stable vector bundles on a curve to unitary representations of its fundamental group has been the model for an enormous range of recent results intertwining algebraic geometry and topology. The object which mediates between the two areas in all of these generalizations is the notion of a connection, and existence theorems for various types of connection provide the means of establishing the theorems. In one sense, the motivation for this paper is to pass beyond the existence and demand more explicitness. What do the connections look like? Can we write them down? This question is our point of departure. The novelty of our presentation here is that the answer involves a journey which takes us backwards in time over two hundred years from the proof of Narasimhan and Seshadri’s theorem in 1965. For simplicity, instead of considering stable bundles on curves of higher genus we consider the analogous case of parabolically stable bundles, in the sense of Mehta and Seshadri [11], on the complex projective line CP. Such a bundle consists of a vector bundle with a weighted flag structure at n marked points a1, . . . , an. The unitary connection that is associated with it is flat and has singularities at the points. In the generic case, the vector bundle itself is trivial, and the flat connection we are looking for can be written as a meromorphic m×m matrix-valued 1-form with a simple pole at each point ai. The parabolic structure can easily be read off from the residues of the form. The other side of the equation is a representation of the fundamental group π1(CP \{a1, . . . , an}) in U(m), the holonomy of the connection, and this presents more problems. Such questions occupied the attention of Fuchs, Klein and others in