Symplectic geometry of the moduli space of projective structures in homological coordinates
Symplectic geometry of the moduli space of projective structures in homological coordinates
复制标题
同调坐标下射影结构模空间的辛几何
DOI：
10.1007/s00222-017-0739-z
复制
发表时间：
2017
影响因子：
3.1
通讯作者：
C. Norton
中科院分区：
文献类型：
--
作者：
M. Bertola;D. Korotkin;C. Norton
我们研究全纯系数为 φ′′-uφ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} 的线性微分方程空间的辛几何\setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi ''-u\varphi =0$$\end{document} 在属 g 的黎曼曲面上。该空间与射影连接的模空间重合，射影连接是仿射丛，建模于余切丛 T∗Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$T^*{\mathcal {M}}_g$$\end{文档}。我们证明，对于原点或基全纯射影连接（例如 Bergman、Wirtinger 或 Schottky）的几种选择，T*Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} 上的规范泊松结构\usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*{\mathcal {M}}_g$$\end{document} 在单一字符变体上引入 Goldman 括号。这些不同的选择产生了射影连接空间上的等效辛结构，但辛极化不同；我们找到相应的生成函数。结合 Kawai 的先验定理，我们的结果显示了 T*Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} 的嵌入之间的辛等价性\setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*{\mathcal {M}}_g$$\end{document} 由 Bers 和 Bergman 射影连接引入射影结构空间。主要技术工具是全纯二次微分空间上同调达布坐标的变分公式。特别是，我们得到了一个新的微分方程组，用于由二次微分定义的黎曼曲面的规范双片覆盖的 Prym 矩阵。
We study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form φ′′-uφ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi ''-u\varphi =0$$\end{document} on Riemann surfaces of genus g. This space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle T∗Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*{\mathcal {M}}_g$$\end{document}. We show that for several choices of the origin, or base, holomorphic projective connection (such as Bergman, Wirtinger or Schottky) the canonical Poisson structure on T∗Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*{\mathcal {M}}_g$$\end{document} induces the Goldman bracket on the monodromy character variety. These different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations; we find the corresponding generating functions. Combined with a prior theorem of Kawai, our results show the symplectic equivalence between the embeddings of T∗Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*{\mathcal {M}}_g$$\end{document} induced by the Bers and Bergman projective connections into the space of projective structures. The main technical tools are variational formulas with respect to homological Darboux coordinates on the space of holomorphic quadratic differentials. In particular, we get a new system of differential equations for the Prym matrix of the canonical two-sheeted covering of a Riemann surface defined by a quadratic differential.