Canonical Maps from Spaces of Higher Complex Structures to Hitchin Components

Abstract: 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 Teichm\"uller 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.