Orbifold Uniformization of Complex Algebraic Variety by Stable Parabolic Higgs Bundle

Abstract: 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) is a compact divisor carrying rational parabolic weights (q_i/p_i). We study the parabolic Higgs bundle [ E_=(ΩX^1(\log D^p)\oplus\mathcal O_X) ] whose only non-zero compact weights occur on the conormal lines of the components of (D^c). The equality case of the parabolic Bogomolov--Gieseker inequality is formulated intrinsically on the root stack (X[\sqrt[p_i]{D_i^c}]). We prove that equality produces a flat trace-free adjoint harmonic bundle, a principal (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 [ ξ_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). Conversely, a branched complex-hyperbolic structure with this local normal form gives a mixed Poincare--cone current representing (c_1(K_X+Δ)); this class is automatically big, nef, and (Δ)-admissible. The induced Hodge metric gives parabolic polystability with respect to every admissible big and nef class, and the parabolic Chern equality.