Orbifold Uniformization of Complex Algebraic Variety via Polystable Parabolic Higgs Bundle

Let \(X\) be a smooth complex projective variety and let \(D=D^p+D^c\) be a simple normal crossing divisor, where \(D^p\) is a cusp divisor and \(D^c=\sum_{i\in I}D_i^c\) is a compact divisor whose components carry rational parabolic weights \(q_i/p_i\). We study the parabolic Higgs bundle \[ E_*=(\Omega_X^1(\log D^p)\oplus\mathcal O_X)_* \] whose only non-zero compact weights occur on the conormal lines of the components \(D_i^c\). The equality case of the parabolic Bogomolov--Gieseker inequality is formulated on the iterated root stack \(X[\sqrt[p_i]{D_i^c}]_{i\in I}\), obtained by taking the \(p_i\)-th root along every \(D_i^c\). We prove that equality produces a flat trace-free adjoint harmonic bundle, a principal \(\mathrm{PU}(n,1)\)-variation, and a period map to the unit ball. In root coordinates \(z_i=w_i^{p_i}\), its normal form is \[ \xi_i=u_i(w)w_i^{p_i-q_i}, \qquad u_i(0)\neq0. \] Thus the general equality case gives a branched complex-hyperbolic structure; it is an unramified orbifold ball uniformization exactly in the standard case \(q_i=p_i-1\) for every \(i\). Conversely, a branched complex-hyperbolic structure with this local normal form gives a mixed Poincar\'e--cone current representing \(c_1(K_X+\Delta)\); this class is big, nef, and \(\Delta\)-admissible. The induced Hodge metric gives parabolic polystability with respect to every admissible big and nef class, and the parabolic Chern equality.