论文标题
从较高复杂结构到hitchin组件的规范图
Canonical Maps from Spaces of Higher Complex Structures to Hitchin Components
论文作者
论文摘要
对于$ s $ a nus $ g \ geq2 $的封闭表面，我们从$ 3 $ fock-thomas space $ \ mathcal {t}^3（s）$构建了一个规范的差异性，高于$ \ \ \ text {SL}（3，\ Mathbb {r}）$ Hitchin Components $ Hitchin Component。关于映射类组$ \ text {mod}（s）$的自然动作，我们的构造是均等的。对于所有$ n \ geq 3 $，我们表明fock-thomas space $ \ mathcal {t}^n（s）$在Teichmüller空间上具有规范的矢量束结构。然后，我们构造一个$ \ text {mod}（s）$ - equivariant捆绑套件，从$ \ mathcal {t}^n（s）$到$ \ text {psl}（n，\ n，\ nathbb {r}）$ hitchian loc的$ \ text {psl}（n，psl}的切线捆绑的限制。结果，我们证明，复杂结构的高度模量空间是Riemann表面的模量空间的捆绑，并且$ \ text {mod}（s）$在$ \ Mathcal {t}^n（s）上的动作是对既适用于Canonical Instrument a Canonical consonical consonical complemplemphist的适当动作''。我们方法的核心是对更高程度的差异组的仔细分析。
For $S$ a closed surface of genus $g\geq2$, we construct a canonical diffeomorphism from the degree $3$ Fock-Thomas space $\mathcal{T}^3(S)$ of higher complex structures to the $\text{SL}(3,\mathbb{R})$ Hitchin component. Our construction is equivariant with respect to natural actions of the mapping class group $\text{Mod}(S)$. For all $n \geq 3$, we show that the Fock-Thomas space $\mathcal{T}^n(S)$ has a canonical vector bundle structure over Teichmüller space. We then construct a $\text{Mod}(S)$-equivariant bundle isomorphism from $\mathcal{T}^n(S)$ to a sub-bundle of the restriction of the tangent bundle of the $\text{PSL}(n, \mathbb{R})$ Hitchin component to the Fuchsian locus. As consequences, we prove that the higher degree moduli space of complex structures is a bundle over the moduli space of Riemann surfaces and that the action of $\text{Mod}(S)$ on $\mathcal{T}^n(S)$ is a proper action by holomorphic automorphisms with respect to a canonical complex structure. The core of our approach is a careful analysis of higher degree diffeomorphism groups.
