Orbifold Uniformization of Complex Algebraic Variety via Polystable Parabolic Higgs Bundle

Let $X$ be a smooth complex projective variety of dimension $n\ge 2$, and let $D=D^p+D^c$, $D^c={\sum }_i{D}_i^c$, be a simple normal crossing divisor. We regard $D^p$ as the cusp divisor and the components $D_i^c$ as compact orbifold divisors with standard weights ${\alpha }_i=1-1\mathrm{/}{p}_i$. Let $\mathcal{X}=X[\sqrt[p_i]{D_i^c}{]}_i$ be the root stack along the compact components. We study the canonical parabolic Higgs bundle $E_{\ast }=({\mathrm{\Omega }}_X^1(\log D^p)\oplus {\mathcal{O}}_X{)}_{\ast }$, whose compact weights are the ${\alpha }_i$ on the conormal lines of $D_i^c$, while the parabolic structure along $D^p$ is trivial. Assume that $(E_{\ast },\theta )$ is polystable with respect to some ample line bundle and that equality holds in the parabolic Bogomolov–Gieseker inequality. We prove that the trace-free adjoint Higgs bundle is flat. The associated principal $\mathrm{P}\mathrm{U}(n,1)$-variation gives a faithful monodromy representation $\rho :{\pi }_1^{\mathrm{o}\mathrm{r}\mathrm{b}}({\mathcal{X}}^o)\to \mathrm{P}\mathrm{U}(n,1)$ and a period map to the complex ball ${\mathbb{B}}^n$. The period map is unramified in the orbifold sense and identifies $(\mathcal{X},D^p)$ with the canonical orbifold toroidal compactification $({\mathcal{T}}_{\mathrm{\Gamma }},D_{\mathrm{\Gamma }}^{\mathrm{t}\mathrm{o}\mathrm{r}})$ of a finite-volume ball quotient, where $\mathrm{\Gamma }=\rho ({\pi }_1^{\mathrm{o}\mathrm{r}\mathrm{b}}({\mathcal{X}}^o))$ is a finite-volume lattice satisfying the regular log-root condition. We also formulate this standard-weight uniformization as an equivalence of categories: the compactified quotient construction and the monodromy construction define quasi-inverse functors between the regular log-root lattice category ${\mathsf{L}\mathsf{a}\mathsf{t}}_n^{\mathrm{r}\mathrm{e}\mathrm{g}}$ and the standard Bogomolov–Gieseker category ${\mathsf{B}\mathsf{G}}_n^{\mathrm{s}\mathrm{t}\mathrm{d}}$.