论文标题
更高的复杂结构和更高的Teichmüller理论
Higher Complex Structures and Higher Teichmüller Theory
论文作者
论文摘要
在本博士学位论文中，我们为更高的Teichmüller理论提供了一种新的几何方法。特别是我们在表面上构建了几何结构，概括了复杂的结构，并探索了其与Hitchin组件的联系。 该结构的构建称为较高的复杂结构，采用了平面的守时希尔伯特方案。它的模量空间允许与希钦（Hitchin）的组件相似。给定较高的复杂结构，我们尝试将其变形为平坦的连接。通过模仿Atiyah-Bott还原，可以获得称为“抛物线”的这种连接的空间。它是通勤差分运算符的一个空间。在某些猜想下，我们在模量空间和Hitchin的组件之间建立了规范的差异。 最后，我们将某些构造（例如守时希尔伯特方案和更高的复杂结构）概括为简单的代数。
In this PhD thesis, we give a new geometric approach to higher Teichmüller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The construction of this structure, called higher complex structure, uses the punctual Hilbert scheme of the plane. Its moduli space admits similar properties to Hitchin's component. Given a higher complex structure, we try to canonically deform it to a flat connection. The space of such connections, called "parabolic", is obtained by imitating the Atiyah--Bott reduction. It is a space of pairs of commuting differential operators. Under some conjecture, we establish a canonical diffeomorphism between our moduli space and Hitchin's component. Finally, we generalize certain constructions, like the punctual Hilbert scheme and the higher complex structure, to the case of a simple Lie algebra.
