广义非线性系统的仿射不变量
The affine invariant of generalized semitoric systems
摘要：
4-辛流形上的广义广义对偶系统F：=（J，H）：M --> R^2是一个可积系统，它的基本性质是F是一个真映射，它的正则值集是连通的，J生成一个S^1-作用且不一定是真的.这些系统可以表现出焦点-焦点奇点，这对应于F的纤维拓扑多捏环面。图像F（M）是一个奇异仿射流形，它在其内部包含一组独特的孤立点：F的焦点-焦点值{（x_i，y_i）}。通过沿直线{x：=x_i}进行垂直切割，构造了一个同胚f：F（M）--> f（F（M）），它限制于远离这些垂直线的仿射同胚，推广了Vu Ngoc的一个构造. R^2中的集合Δ：=f（F（M））是（M，ω，F）的辛不变量，它编码了F的仿射结构。此外，Δ可以被描述为四种不同类型的平面区域的可数并集，其中每种类型被定义为具有各种属性（分段线性，连续，凸等）的两个函数的图之间的边界区域。如果F是一个复曲面系统，Δ是一个凸多边形（如Atiyah和Guillemin-Sternberg证明的），f是恒等式。
英文摘要：
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which correspond to fibers of F which are topologically multipinched tori. The image F(M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(x_i,y_i)} of F. By performing a vertical cutting procedure along the lines {x:=x_i}, we construct a homeomorphism f : F(M) --> f(F(M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vu Ngoc. The set Δ:=f(F(M)) in R^2 is a symplectic invariant of (M,ω,F), which encodes the affine structure of F. Moreover, Δmay be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, Δis a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity.
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