Orbifold Uniformization of Complex Algebraic Variety via Polystable Parabolic Higgs Bundle

Abstract

Let \(X\) be a smooth complex projective variety of dimension \(n\geq 2\), 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 \(\alpha_i=1-1/p_i\). 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 \(E_*=(\Omega_X^1(\log D^p)\oplus\mathcal O_X)_*\), 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 \((E_*,\theta)\) 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 \(\mathrm{PU}(n,1)\)-variation gives a faithful monodromy representation \(\rho:\pi_1^{\mathrm{orb}}(\mathcal X^o)\to\mathrm{PU}(n,1)\) 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 \((\mathcal T_\Gamma,D_\Gamma^{\mathrm{tor}})\) of a finite-volume ball quotient, where \(\Gamma=\rho(\pi_1^{\mathrm{orb}}(\mathcal X^o))\) 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 \(\mathsf{Lat}_n^{\mathrm{reg}}\) and the standard Bogomolov--Gieseker category \(\mathsf{BG}_n^{\mathrm{std}}\).