Proof. The assertion is local on the root chart. In the normal direction to \(D_i^c\), the bundle \(\mathcal{E}^{1,0}\) has frame \(\eta_i\), while \(\Omega^1_{\mathcal{X}}\) has frame \(dw_i\). By 3 , the morphism \(\iota\) sends \[\eta_i\longmapsto p_i w_i^{p_i-q_i-1}dw_i.\] Thus its vanishing order in this normal direction is \(p_i-q_i-1\). The remaining non-orbifold directions have weight zero, and along \(D^p\) the frame is the usual logarithmic frame. Therefore the determinant of \(\iota\) vanishes along \(\mathcal{D}_i^c\) with the asserted order. It is nowhere vanishing along \(\mathcal{D}_i^c\) precisely when \(p_i-q_i-1=0\), equivalently \(q_i=p_i-1\). ◻

Proof. The root-stack/parabolic correspondence identifies vector bundles on \(\mathcal{X}\) with locally abelian parabolic bundles on \((X,D^c)\) with the prescribed denominators; the corresponding Chern characters and degrees agree Iy-Si?. Adding \(D^p\) only records the logarithmic boundary with trivial parabolic filtration. Hence \((\mathcal{F},\varphi)\) is equivalently a regular filtered Higgs bundle on \((X,D^p+D^c)\).

Mochizuki’s Kobayashi–Hitchin correspondence for tame harmonic bundles Mochizuki2006? gives an adapted pluri-harmonic metric on \(F|_{X\setminus(D^p\cup D^c)}\), unique up to the usual ambiguity. In a root chart, if \(e_a\) has weight \(m_{a,i}/p_i\) along \(D_i^c\), the corresponding orbifold frame is \[\widetilde{e}_a=\prod_i w_i^{-m_{a,i}}\pi^*e_a.\] In these frames the singular powers prescribed by the parabolic weights are absorbed by the root coordinates. Thus the pulled-back metric and its inverse are locally bounded across the compact root divisors \(w_i=0\). The pulled-back Higgs field is smooth in the compact root directions. Logarithmic poles remain only along the inverse image of \(D^p\).

Next we prove that the pullback harmonic metric extends smoothly across the compact root divisors. Let \(U\) be a finite root chart with coordinates \((w,z,y)\), let \[B=\{w_1\cdots w_r=0\}\] be the compact root divisor, and let \(U^\dagger=U\setminus D^p\). It is enough to prove smoothness on every relatively compact open set \(U_0\Subset U^\dagger\); thus the logarithmic boundary \(D^p\) plays no role in the following local argument. Choose a holomorphic frame of \(\mathcal{F}\) on \(U_0\). The metric is represented on \(U_0^*:=U_0\setminus B\) by a positive Hermitian matrix \(H\), and the adaptedness just recalled gives \[C^{-1}I\le H\le CI.\] Write the Higgs field as \(\varphi=\sum_j A_j d\xi_j\) in the root coordinates \(\xi_j\). Since \(\varphi\) has no pole along \(B\), all matrices \(A_j\) are smooth and bounded on \(U_0\).

We now explain the removable regularity statement used below. In the frame chosen above the pluri-harmonic equation is the Hitchin–Simpson flatness equation. For local regularity it is enough to use its contraction with a fixed smooth background Hermitian metric on the coordinate ball; this contracted \((1,1)\)-equation can be written locally as \[\label{eq:local-harmonic-metric-equation} \sum_\mu \partial_{\bar\xi_\mu}\bigl(H^{-1}\partial_{\xi_\mu}H\bigr) + \sum_\mu \bigl[A_\mu,H^{-1}A_\mu^{*}H\bigr]=0,\tag{4}\] where \(A_\mu^{*}\) denotes the adjoint with respect to the fixed background Hermitian metric of the chosen frame. This formula is to be understood on \(U_0^*\). After choosing normal coordinates on the symmetric space \[\mathcal{P}_N:=\mathrm{GL}(N,\mathbb{C})/U(N)\] centered at a point in the compact ball containing the image of \(H\), it has the form \[\label{eq:forced-harmonic-map-system} L H^a+\Gamma^a_{bc}(H)\,\langle dH^b,dH^c\rangle_g=F^a(\xi,H).\tag{5}\] Here \(L\) is the scalar Laplace operator of a fixed smooth background Hermitian metric \(g\) on the coordinate ball, \(\Gamma^a_{bc}\) are the Christoffel symbols of \(\mathcal{P}_N\), and the term \(F\) is obtained from the Higgs commutator in 4 . Since the root powers have been absorbed into the orbifold frame, the matrices \(A_\mu\) are smooth up to \(B\). Since \(H\) and \(H^{-1}\) are bounded, \(F\) is smooth in \(\xi\), smooth in \(H\), and uniformly bounded on \(U_0\times K\), where \(K\Subset\mathcal{P}_N\) is the compact geodesic ball containing \(H(U_0^*)\). When \(F=0\), 5 is exactly the weak harmonic-map system considered by Hildebrandt–Kaul–Widman HildebrandtKaulWidman1977?. In our case the principal part is the same; the Higgs field contributes only the bounded zeroth-order forcing term \(F\).

We shall use the following local removable regularity fact, which is a minor forced variant of the proof of HildebrandtKaulWidman1977?.

Local removable regularity assertion. Let \(V\Subset U_0\) be a coordinate ball and let \(B\cap V\) be a normal crossing divisor. Let \[u:V\setminus B\longrightarrow \mathcal{P}_N\] be a \(C^2\) solution of 5 whose image lies in a compact regular geodesic ball \(K\Subset\mathcal{P}_N\). Assume that \(F\) is bounded on \(V\times K\). If \(u\) is bounded in \(K\), then \(u\) extends to a smooth solution on \(V\).

Indeed, first one obtains the Caccioppoli estimate. Let \(Q\in K\) be the center of a regular geodesic ball containing the image, let \(\mathcal{r}_Q^2\) be the squared distance to \(Q\), and test the weak form of 5 on \(V\setminus B\) by \(\chi^2\nabla\mathcal{r}_Q^2(u)\). The Hessian comparison estimate used in HildebrandtKaulWidman1977? is valid here because \(\mathcal{P}_N\) is simply connected with non-positive sectional curvature, hence geodesic balls are regular. It gives the usual positive term controlling \(\chi^2|du|^2\). The forcing term contributes \[\left|\int_{V\setminus B}\chi^2\langle F(\xi,u),\nabla\mathcal{r}_Q^2(u)\rangle\right| \le C\int_{V\setminus B}\chi^2,\] because \(F\) and \(\nabla\mathcal{r}_Q^2\) are bounded on \(V\times K\). The cross term is absorbed by Cauchy’s inequality. Therefore \[\label{eq:forced-caccioppoli} \int_{V\setminus B}\chi^2|du|^2 \le C\int_{V\setminus B}|d\chi|^2+C\int_{V\setminus B}\chi^2\tag{6}\] for all \(\chi\in C_c^\infty(V\setminus B)\).

Choose logarithmic cut-off functions \(\rho_\varepsilon\) for the divisor \(B\cap V\). For one component \(\{w=0\}\), \(\rho_\varepsilon=0\) on \(|w|\le\varepsilon^2\), \(\rho_\varepsilon=1\) on \(|w|\ge\varepsilon\), and \(\rho_\varepsilon\) is linear in \(\log |w|\) on the annulus \(\varepsilon^2<|w|<\varepsilon\); for a normal crossing divisor take the product over the components. Then \[0\le\rho_\varepsilon\le1, \qquad \rho_\varepsilon\to1 \text{ on } V\setminus B, \qquad \int_V |d\rho_\varepsilon|^2\to0 .\] Putting \(\chi=\psi\rho_\varepsilon\) in 6 and letting \(\varepsilon\to0\) gives \(u\in W^{1,2}_{\rm loc}(V)\). Testing the equation against \(\psi\rho_\varepsilon\Phi\), with \(\Phi\) an arbitrary compactly supported vector-valued test function in the normal coordinates of \(K\), and then letting \(\varepsilon\to0\), shows that \(u\) satisfies 5 weakly on all of \(V\). The terms involving \(d\rho_\varepsilon\) vanish by Cauchy–Schwarz, the bound just obtained for \(du\), and \(\int |d\rho_\varepsilon|^2\to0\).

It remains to prove continuity of this weak solution. The proof of HildebrandtKaulWidman1977? applies with only harmless changes. More explicitly, in the Green-function estimates of HildebrandtKaulWidman1977?, the weak harmonic-map identity is tested against the Green kernel multiplied by the normal-coordinate vector field. The new forcing term is bounded by \[C\int_{B_R(x_0)}G(x,x_0)\,dx,\] which is \(O(R^2)\) for real dimension at least three and \(O(R^2|\log R|)\) in real dimension two; in either case it tends to zero as \(R\to0\). The corresponding annular estimate used in their proof acquires the same vanishing error. Thus their oscillation estimate still gives \[\lim_{R\to0}\operatorname*{ess\,osc}_{B_R(x_0)}u=0\] for every \(x_0\in V\). Hence \(u\) is continuous. Once continuity is known, the equation is a uniformly elliptic system with natural quadratic growth \(|du|^2\) and an additional bounded right-hand side. The regularity step invoked after HildebrandtKaulWidman1977?, namely the local quasilinear elliptic regularity for such systems, gives Hölder continuity of \(u\); then the standard elliptic bootstrap in the smooth target coordinates of \(K\) gives \(u\in C^\infty(V)\). This proves the local assertion.

We now apply the assertion to the harmonic metric \(H\). The two-sided estimate \(C^{-1}I\le H\le CI\) says exactly that the map \(H:U_0^*\to\mathcal{P}_N\) has image in a compact geodesic ball. The symmetric space \(\mathcal{P}_N\) has non-positive sectional curvature, so this ball is regular. The preceding local assertion therefore shows that \(H\) extends smoothly across \(B\) on every relatively compact coordinate ball \(V\Subset U_0\). Since \(V\) was arbitrary, the pulled-back metric extends smoothly across the compact root divisor on the root chart. The extension is unique, hence is invariant under the finite stabilizer group, and therefore defines a smooth orbifold harmonic metric on \(\mathcal{X}^o\). The local gauge change by the root powers conjugates the Hitchin–Simpson connection on the complement, so harmonicity and flatness are unchanged by passing between the parabolic and root-stack descriptions.

Finally, uniqueness is inherited from Mochizuki’s uniqueness statement. Indeed, two orbifold adapted harmonic metrics give, in root frames, two invariant adapted metrics for the same regular filtered Higgs bundle on the coarse complement. Mochizuki’s uniqueness identifies them on stable summands up to positive constants, and in the polystable case up to constant Hermitian metrics on multiplicity spaces. Pulling this statement back to the root charts gives exactly the asserted orbifold uniqueness. ◻

Proof. The identity follows from \[\operatorname{ch}(\operatorname{End}F)=\operatorname{ch}(F)\operatorname{ch}(F^\vee).\] Writing \[\operatorname{ch}(F)=r+c_1(F)+\operatorname{ch}_2(F)+\cdots, \qquad \operatorname{ch}(F^\vee)=r-c_1(F)+\operatorname{ch}_2(F)+\cdots,\] the degree-one term cancels, and the degree-two term is \[r\operatorname{ch}_2(F)+r\operatorname{ch}_2(F)-c_1(F)^2.\] This gives 7 . The equality \[\left(\frac{c_1(F)^2}{2r}-\operatorname{ch}_2(F)\right)c_1(L)^{n-2}=0\] is equivalent to \[\bigl(2r\operatorname{ch}_2(F)-c_1(F)^2\bigr)c_1(L)^{n-2}=0.\] Taking \(F=\mathcal{E}\) gives 8 . Finally, \[\operatorname{End}(\mathcal{E})=\operatorname{End}_0(\mathcal{E})\oplus\mathcal{O}_{\mathcal{X}},\] and the trivial summand contributes no first or second Chern character. ◻

Proof. Polystability is preserved by duals, tensor products and direct summands for locally abelian parabolic Higgs bundles. Hence \(\operatorname{End}(\mathcal{E})=\mathcal{E}\otimes\mathcal{E}^\vee\) and its direct summand \(\operatorname{End}_0(\mathcal{E})\) are polystable.

By Lemma 1, \[\operatorname{orb\!-ch}_1(\operatorname{End}_0(\mathcal{E}))=0, \qquad \operatorname{orb\!-ch}_2(\operatorname{End}_0(\mathcal{E}))c_1(L)^{n-2}=0.\] Therefore the associated regular filtered Higgs bundle has trivial parabolic characteristic numbers. Proposition 6 gives the existence of an adapted pluri-harmonic metric on \(\operatorname{End}_0(\mathcal{E})|_{\mathcal{X}^o}\).

Let \(h\) be any adapted Hermitian–Einstein metric on this adjoint Higgs bundle, with the determinant normalized on the stable summands, and let \[\nabla_h=\partial_h+\bar\partial+\theta_{\operatorname{End}_0}+\theta_{\operatorname{End}_0,h}^{\dagger}\] be its Hitchin–Simpson connection. The parabolic Chern–Weil identity expresses the Bogomolov–Gieseker defect as a positive constant times the \(L^2\)-norm of the trace-free Hitchin–Simpson curvature. Since the defect vanishes and the bundle is trace-free adjoint, there is no residual central curvature term. Hence \(\nabla_h\) is flat. The resulting local system is semisimple by the harmonic-metric correspondence. ◻

Proof. Write \(W=V^{1,0}\) and \(L=V^{0,1}\). Since \(\dim L=1\), a point of \(K\) is represented by a pair \((A,\lambda)\in \mathrm{GL}(W)\times\mathrm{GL}(L)\), modulo simultaneous scalar multiplication. A pair of positive Hermitian metrics \((h_W,h_L)\) on \((W,L)\), again modulo simultaneous scalar multiplication, determines a positive Hodge metric on \(\mathfrak g\) as follows. On \[\mathfrak g^{-1,1}=\operatorname{Hom}(W,L),\qquad \mathfrak g^{1,-1}=\operatorname{Hom}(L,W),\] it is the usual Hom metric induced by \(h_W\) and \(h_L\), and the two summands are dual through the Killing form. On \[\mathfrak g^{0,0}=\bigl(\operatorname{End}(W)\oplus\operatorname{End}(L)\bigr)/\mathbb{C}\cdot \operatorname{Id}_V\] it is the metric induced by the Hermitian metric on \(\operatorname{End}(W)\) defined by \(h_W\); the \(\operatorname{End}(L)\)-factor is scalar and disappears after quotienting by the common scalar. This construction is exactly the \(K\)-orbit of \(h^0_{\mathfrak g}\), and the stabilizer of the model metric is \(K_0\). Hence the orbit is \(K/K_0\).

It remains to see that the conditions in the statement force a metric to be of this form. The restriction of \(H\) to \(\mathfrak g^{-1,1}=W^\vee\otimes L\) is a positive Hermitian metric on \(W^\vee\otimes L\). Choose a positive metric \(h_L\) on the one-dimensional space \(L\). Then the formula \[h_W^\vee(\alpha,\beta) :=\frac{H_{-1,1}(\alpha\otimes\ell,\beta\otimes\ell)}{h_L(\ell,\ell)}\] for any non-zero \(\ell\in L\) defines a positive Hermitian metric \(h_W^\vee\) on \(W^\vee\), hence a metric \(h_W\) on \(W\). Changing \(h_L\) by a positive scalar changes \(h_W\) by the inverse scalar, so the pair \((h_W,h_L)\) is well-defined modulo simultaneous scalar multiplication. Condition (ii) forces the metric on \(\mathfrak g^{1,-1}\) to be the Hom metric induced by the same pair. Condition (iii) then forces the metric on \(\mathfrak g^{0,0}\) to be the one induced by \(h_W\), because the bracket identifies \(\mathfrak g^{0,0}\) with the trace-free endomorphisms acting on \(W\). Hence \(H\) is the metric associated with \((h_W,h_L)\), and therefore belongs to \(K\cdot h^0_{\mathfrak g}\). ◻

Proof. The metric \(h\) is smooth on finite root charts away from the inverse image of \(D^p\), by Proposition 6, and remains tame along \(D^p\). We write its flat Hitchin–Simpson connection as \[\nabla_h=\partial_h+\bar\partial+\theta_{\mathcal{A}}+\theta_{\mathcal{A},h}^{\dagger}.\]

We first record the functoriality statement used below. Let \((F,\theta_F,h_F)\) and \((G,\theta_G,h_G)\) be two adapted pluri-harmonic bundles on \(\mathcal{X}^o\) arising from polystable logarithmic orbifold Higgs bundles on \((\mathcal{X},D^p)\) with trivial characteristic numbers. Let \[\phi:F\longrightarrow G\] be a holomorphic Higgs morphism which extends across \(D^p\) as a morphism of the corresponding logarithmic orbifold Higgs bundles. Then \(\phi\) is parallel for the induced flat connection on \(\operatorname{Hom}(F,G)\). Indeed, on \[H:=\operatorname{Hom}(F,G), \qquad \theta_H(u)=\theta_G\circ u-(u\otimes 1)\circ\theta_F,\] the induced metric \(h_H\) is pluri-harmonic, and the section \(\phi\) satisfies \[\bar\partial_H\phi=0, \qquad \theta_H(\phi)=0.\] The Bochner identity for a harmonic bundle gives, on \(\mathcal{X}^o\), \[\sqrt{-1}\Lambda\partial\bar\partial |\phi|_{h_H}^2 = |(\partial_{h_H}+\theta_{H,h_H}^{\dagger})\phi|_{h_H}^2.\] Choose a complete Poincaré type Kähler metric on \(\mathcal{X}^o\) and the standard cut-off functions near \(D^p\). Since \(\phi\) extends as a logarithmic orbifold morphism and the metrics are adapted, \(|\phi|_{h_H}\) is bounded; for the tensorial morphisms used below this boundedness is immediate from the fact that the tensors have weight zero in adapted frames. Multiplying the preceding identity by the square of a cut-off function \(\chi_\varepsilon\) and integrating by parts gives \[\int_{\mathcal{X}^o}\chi_\varepsilon^2 |(\partial_{h_H}+\theta_{H,h_H}^{\dagger})\phi|_{h_H}^2 \leq C\int_{\operatorname{supp} d\chi_\varepsilon}|d\chi_\varepsilon|^2 |\phi|_{h_H}^2.\] The right hand side tends to zero by the usual Poincaré cut-off estimate. Hence \[(\partial_{h_H}+\theta_{H,h_H}^{\dagger})\phi=0.\] Together with \(\bar\partial_H\phi=0\) and \(\theta_H(\phi)=0\), this says precisely that \(\phi\) is flat. The same proof applies to morphisms between any tensor constructions of such bundles, equipped with the tensor product and dual metrics.

We first show that \(h\) may be chosen to be a Hodge metric. The adjoint Higgs bundle is a system of Hodge bundles \[\overline{\mathcal{A}} =\overline{\mathcal{A}}^{-1,1}\oplus \overline{\mathcal{A}}^{0,0}\oplus \overline{\mathcal{A}}^{1,-1}, \qquad \theta_{\mathcal{A}}:\mathcal{A}^{p,-p}\longrightarrow \mathcal{A}^{p-1,-p+1}\otimes\Omega^1_{\mathcal{X}}(\log D^p)|_{\mathcal{X}^o}.\] For \(t\in U(1)\), let \(u_t\) act on \(\overline{\mathcal{A}}^{p,-p}\) by multiplication by \(t^{-p}\). Then \(u_t\) is an isomorphism from \((\overline{\mathcal{A}},\theta_{\mathcal{A}})\) to \((\overline{\mathcal{A}},t\theta_{\mathcal{A}})\). Since \(|t|=1\), a pluri-harmonic metric for \(\theta_{\mathcal{A}}\) is also a pluri-harmonic metric for \(t\theta_{\mathcal{A}}\). Thus \(u_t^*h\) is again an adapted pluri-harmonic metric for \((\mathcal{A},\theta_{\mathcal{A}})\). By the uniqueness assertion in Proposition 6, on each stable summand the metric is unique up to a positive constant, and on an isotypical summand \(S\otimes M\) the ambiguity is exactly a constant Hermitian metric on the multiplicity space \(M\). Averaging these finite-dimensional multiplicity metrics over the compact group \(U(1)\), and leaving the stable metrics fixed, gives another adapted pluri-harmonic metric. Replacing \(h\) by this normalized metric, we may assume that \(u_t\) is unitary for every \(t\in U(1)\). Therefore the three Hodge summands are mutually orthogonal for \(h\).

We now impose the algebraic tensors defining the adjoint group. We first fix the notation. If \(W\) is a finite-dimensional vector space, by a tensor construction in \(W\) we mean a finite direct sum of spaces of the form \[W^{\otimes a}\otimes (W^\vee)^{\otimes b}.\] Thus the Lie bracket of \(\mathfrak g\) is a point of \[\operatorname{Hom}(\wedge^2\mathfrak g,\mathfrak g) \subset \mathfrak g\otimes(\mathfrak g^\vee)^{\otimes 2},\] and the Killing form is a point of \(\operatorname{Sym}^2\mathfrak g^\vee\). We choose finitely many tensor constructions \(T_\alpha(\mathfrak g)\) and tensors \[\mathbf{t}_\alpha\in T_\alpha(\mathfrak g)\] with the following property: \[\operatorname{Ad}(G)=\{A\in\mathrm{GL}(\mathfrak g)\mid A\cdot \mathbf{t}_\alpha=\mathbf{t}_\alpha \text{ for every }\alpha\}, \qquad G=\mathrm{PGL}(V).\] The list is chosen to contain the Lie bracket and the Killing form. The existence of such a finite list follows from Chevalley’s theorem in its standard Tannakian form: for an algebraic subgroup \(H\subset\mathrm{GL}(W)\), there is a tensor construction \(T(W)\) and a line \(\ell\subset T(W)\) such that \(H\) is the stabilizer of \(\ell\). Applied to \(H=\operatorname{Ad}(G)\subset\mathrm{GL}(\mathfrak g)\), and replacing the line by a finite set of defining tensors, this gives the displayed description of \(\operatorname{Ad}(G)\); see, for example, DeligneMilne1982?.

Since the tensors \(\mathbf{t}_\alpha\) are \(G\)-invariant, they define global tensor sections \[\mathbf{t}_{\alpha,\mathcal{A}}\in H^0\bigl(\mathcal{X}^o,T_\alpha(\mathcal{A})\bigr).\] These sections are holomorphic Higgs tensors in the following sense. Locally the Higgs field on \(T_\alpha(\mathcal{A})\) is induced by the infinitesimal action of \(\tau\in\mathfrak g^{-1,1}\) on the tensor construction \(T_\alpha(\mathfrak g)\). Since \(\mathbf{t}_\alpha\) is fixed by \(G\), every element of \(\mathfrak g\) acts trivially on it. Hence \[\theta_{T_\alpha(\mathcal{A})}(\mathbf{t}_{\alpha,\mathcal{A}})=0.\] Equivalently, \(1\mapsto \mathbf{t}_{\alpha,\mathcal{A}}\) is a Higgs morphism from the trivial harmonic bundle to \(T_\alpha(\mathcal{A})\). The functoriality statement above therefore gives \[\nabla_{T_\alpha(h)}\mathbf{t}_{\alpha,\mathcal{A}}=0.\] Thus the parallel transport of \(\nabla_h\) preserves all defining tensors. In model frames it lies in the common stabilizer of the \(\mathbf{t}_\alpha\), namely in \(\operatorname{Ad}(G)\). Consequently the flat frame bundle of \((\mathcal{A},\nabla_h)\) reduces to \(\operatorname{Ad}(G)\). Together with the Hodge decomposition of \(\mathcal{A}\), this recovers the original holomorphic \(K\)-reduction \(P|_{\mathcal{X}^o}\).

We now pass from the adjoint metric to a \(K_0\)-reduction of \(P|_{\mathcal{X}^o}\). Choose the model Hermitian form \(S_0\) and the model adjoint Hodge metric \(h^0_{\mathfrak g}\) as in Lemma 2. That lemma identifies the relevant finite-dimensional space of adjoint Hodge metrics with \[K\cdot h^0_{\mathfrak g}\simeq K/K_0.\]

For a point \(x\in\mathcal{X}^o\) and a Hodge frame \(p\in P_x\), pull back the adjoint metric \(h_x\) to the model fiber: \[H_p:=p^*h_x\in \operatorname{Herm}^+(\mathfrak g).\] If \(p\) is replaced by \(p\cdot k\), \(k\in K\), then \(H_p\) is replaced by \(k^*H_p\). The tensors \(\mathbf{t}_{\alpha,\mathcal{A}}\) have Hodge degree zero. Since they are parallel for \(\nabla_h=(\partial_h+\bar\partial)+\theta_{\mathcal{A}}+\theta_{\mathcal{A},h}^{\dagger}\), and since \(h\) is Hodge, the different Hodge-degree components of \(\nabla_h\mathbf{t}_{\alpha,\mathcal{A}}=0\) vanish separately. In particular, the Chern connection \(\partial_h+\bar\partial\) preserves the Hodge decomposition and the defining tensor sections. Thus, in each Hodge frame, the pulled-back metric \(H_p\) is a positive adjoint Hodge metric compatible with the defining tensors, hence belongs to the finite-dimensional orbit \(K\cdot h^0_{\mathfrak g}\simeq K/K_0\). Consequently the rule \[x\longmapsto [H_p]\in K/K_0\] defines, independently of the local choice of \(p\), a smooth section of the associated bundle \[P|_{\mathcal{X}^o}\times_K (K/K_0).\] The standard equivalence between reductions of structure group and sections of associated homogeneous bundles gives a smooth \(K_0\)-reduction \[P_H\subset P|_{\mathcal{X}^o}.\] Equivalently, its fiber is \[(P_H)_x=\bigl\{p\in P_x\mid p^*h_x=h^0_{\mathfrak g}\bigr\}.\] The preceding pointwise transitivity says that this set is non-empty; the ambiguity is precisely right multiplication by \(K_0\). Since \(h\) is smooth, these fibers glue to a smooth adapted principal \(K_0\)-bundle. By construction, the adjoint metric induced by \(P_H\) is exactly \(h\).

The model Hodge metric also fixes the conjugation of \(\mathfrak g\) whose fixed Lie algebra is \(\mathfrak{pu}(n,1)\). Since \(P_H\) consists of frames preserving the model metric and the Hodge decomposition, this conjugation is transported to the bundle, and \(\tau+\tau^{\dagger}_{P_H}\) is \(\mathfrak g_0\)-valued.

Let \(\nabla^c_{P_H}\) be the Chern connection of the holomorphic \(K\)-bundle \(P|_{\mathcal{X}^o}\) with respect to the reduction \(P_H\). On the associated adjoint bundle, this is the Chern connection \(\partial_h+\bar\partial\). The adjoint \(\tau^{\dagger}_{P_H}\) is the \(\mathfrak g^{1,-1}\)-part determined by the same Hodge metric. Hence the adjoint connection of \[\nabla^{P_H}:=\nabla^c_{P_H}+\tau+\tau^{\dagger}_{P_H}\] on \(P_H\times_{K_0}G_0\), where \(G_0=\mathrm{PU}(n,1)\), is \[\partial_h+\bar\partial+\theta_{\mathcal{A}}+\theta_{\mathcal{A},h}^{\dagger}=\nabla_h.\] This proves the assertion. ◻

Proof. The principal bundle \(P\) and the principal Higgs field \(\tau\) were constructed above from the Hodge decomposition of \(\mathcal{E}\). The identification \[P\times_K\mathfrak g^{-1,1}\simeq \operatorname{Hom}(\mathcal{E}^{1,0},\mathcal{E}^{0,1})\] identifies the Higgs field with contraction against the image of \(\iota\). Thus \(\tau\) is dual to \(\iota\).

Under the adjoint representation of \(G\) on \(\mathfrak g\), the principal Higgs field \(\tau\) induces the adjoint Higgs bundle \[\bigl(P\times_K\mathfrak g,\operatorname{ad}(\tau)\bigr), \qquad \operatorname{ad}(\tau)(A)=[\tau,A].\] The natural identification \[P\times_K\mathfrak g\simeq \operatorname{End}_0(\mathcal{E})\] identifies \(\operatorname{ad}(\tau)\) with the commutator Higgs field \[\theta_{\operatorname{End}_0}(A)=[\theta_{\mathcal{E}},A].\] Thus \((P\times_K\mathfrak g,\operatorname{ad}(\tau))\) is exactly the adjoint Higgs bundle considered in Proposition 7.

By Proposition 7, the adjoint Higgs bundle just identified has an adapted pluri-harmonic metric \(h\) on \(\mathcal{X}^o\) whose Hitchin–Simpson connection is flat. Applying Lemma 3 to this metric gives an adapted \(K_0\)-reduction \(P_H\subset P|_{\mathcal{X}^o}\), and the principal connection on \(P_H\times_{K_0}\mathrm{PU}(n,1)\) has adjoint connection equal to the flat Hitchin–Simpson connection of \((P\times_K\mathfrak g,\operatorname{ad}(\tau),h)\). Since \(\mathrm{PU}(n,1)\) is an adjoint group, the adjoint representation is faithful; therefore the principal connection itself is flat. Hence \((P,\tau,P_H)\) is a principal \(\mathrm{PU}(n,1)\)-variation of Hodge structure on \(\mathcal{X}^o\). ◻

Proof. A principal \(G_0\)-variation of Hodge structure with Hodge reduction to \(K_0\) defines a horizontal period map from the orbifold universal cover to the symmetric space \(G_0/K_0\). The differential of this map is the Higgs field, equivalently the Kodaira–Spencer morphism \[\tau:T_{\mathcal{X}}(-\log D^p)\longrightarrow P\times_K\mathfrak g^{-1,1}.\] By Proposition 8, \(\tau\) is dual to \(\iota\). Proposition 5 shows that \(\iota\) is generically an isomorphism and that its only divisorial degeneracy is along \(\mathcal{D}^c\). Therefore \(\mathcal{P}\) is generically locally biholomorphic and is ramified precisely along the corresponding lifts of \(\mathcal{D}_i^c\), with the order computed in Section 4.2. ◻

Proof. Assume, for contradiction, that two components \(P_1,P_2\subset D^p\) meet transversely at a point \(x\). Choose local coordinates \[(z_1,z_2,z_3,\dots,z_n)\] centered at \(x\) such that \[P_1=\{z_1=0\},\qquad P_2=\{z_2=0\}.\] It is enough to work on the fiber over \(x\). Let \[e_1=\frac{dz_1}{z_1},\qquad e_2=\frac{dz_2}{z_2},\qquad e_a=dz_a\;(3\le a\le n),\qquad e_0=1\] be the standard local frame of \(E=\Omega_X^1(\log D^p)\oplus\mathcal{O}_X\). Let \(N_k=\operatorname{Res}_{P_k}(\theta)\) for \(k=1,2\). Since \(\theta\) is the tautological map from \(E^{1,0}\) to \(E^{0,1}\otimes\Omega_X^1(\log D)\), we have \[N_1(e_1)=e_0, \qquad N_1(e_j)=0\quad(j\ne1),\] and \[N_2(e_2)=e_0, \qquad N_2(e_j)=0\quad(j\ne2).\] Thus \[N_1^2=N_2^2=N_1N_2=N_2N_1=0.\] For \(t>0\), put \[N_t=N_1+tN_2, \qquad L_t=\operatorname{ad}_{N_t}:\operatorname{End}_0(E_x)\longrightarrow\operatorname{End}_0(E_x).\] Then \(N_t^2=0\), and hence \(L_t^3=0\).

By Proposition 7, \(\operatorname{End}_0(\mathcal{E})\) carries an adapted tame harmonic metric. Along \(D^p\), the parabolic structure is trivial. The residues \(\operatorname{ad}_{N_1}\) and \(\operatorname{ad}_{N_2}\) are nilpotent, so the harmonic bundle is tame nilpotent with trivial parabolic structure near \(x\). Applying Theorem 10 gives that the weight filtration \[W(L_t)=W\bigl(\operatorname{ad}_{N_1+tN_2}\bigr)\] is independent of \(t\in\mathbb{R}_{>0}\).

We compute \(W_{-2}(L_t)\). For \(A\in\operatorname{End}_0(E_x)\), \[\begin{align} L_t^2(A) &=[N_t,[N_t,A]]\\ &=N_t^2A-2N_t A N_t+AN_t^2\\ &=-2N_t A N_t. \end{align}\] Since \(N_t\) has rank one and image \(\mathbb{C}e_0\), the operator \(N_t A N_t\) is always a scalar multiple of \(N_t\). Conversely, define \(A_0\in\operatorname{End}_0(E_x)\) by \[A_0(e_0)=e_1, \qquad A_0(e_j)=0\quad(j\ne0).\] Then \(A_0\) has trace zero and \(N_t A_0 N_t=N_t\). Hence \[\label{eq:imageLt2} \operatorname{Im}(L_t^2)=\mathbb{C}\cdot N_t=\mathbb{C}\cdot(N_1+tN_2).\tag{9}\] By the elementary calculation above, \[W_{-2}(L_t)=\operatorname{Im}(L_t^2)=\mathbb{C}\cdot(N_1+tN_2).\] The lines \(\mathbb{C}(N_1+N_2)\) and \(\mathbb{C}(N_1+2N_2)\) are distinct because \(N_1\) and \(N_2\) are linearly independent. This contradicts Mochizuki’s positive-cone constancy. Therefore two components of \(D^p\) cannot meet. ◻

Proof. By Convention 4, the orbifold conormal frame is \[\eta_i=w_i^{-q_i}dz_i.\] Since \(z_i=w_i^{p_i}\), \[\eta_i=p_i\,w_i^{p_i-q_i-1}dw_i.\] The differential of the period map is the Kodaira–Spencer map \(\tau\), which is dual to the morphism \(\iota:\mathcal{E}^{1,0}\to\Omega^1_{\mathcal{X}}(\log D^p)\). Therefore its normal component is represented, up to a non-vanishing holomorphic factor, by the one-form \[w_i^{p_i-q_i-1}dw_i.\] Integrating the normal differential gives the local normal form \[\mathcal{P}_i(w)=v(w)w_i^{p_i-q_i},\qquad v(0)\ne0.\] Hence the ramification index is \(p_i-q_i\). ◻

Proof. Put \(m_i=p_i-q_i\). Let \(G_x=\prod_{i\in S}\mu_{p_i}\) be the local stabilizer. The local period map is equivariant with respect to the finite group \(\rho(\lambda_x(G_x))\), and this finite group fixes \(\mathcal{P}(x)\). By the holomorphic linearization theorem for finite groups acting near a fixed point, we may choose ball coordinates centered at \(\mathcal{P}(x)\) in which this action is linear and unitary.

In the root frame the morphism \(\iota:\mathcal{E}^{1,0}\to\Omega^1_{\mathcal{X}}(\log D^p)\) is diagonal in the compact normal directions: \[\eta_i\longmapsto p_i\,w_i^{m_i-1}dw_i, \qquad i\in S,\] while the remaining directions are multiplied by units. Since \(d\mathcal{P}\) is the Kodaira–Spencer morphism dual to \(\iota\), the first non-zero term of \(d\mathcal{P}\) in the \(i\)-th compact normal direction is a non-zero multiple of \(w_i^{m_i-1}dw_i\). Choose a linear target coordinate \(\xi_i\) on the corresponding eigenspace for the character \[\chi_i((\zeta_j)_{j\in S})=\zeta_i^{m_i}.\] Then \(\xi_i\circ\mathcal{P}\) is a \(\chi_i\)-eigenfunction for the stabilizer action. Its Taylor expansion contains only monomials whose \(G_x\)-character is \(\chi_i\). The preceding differential computation shows that the coefficient of \(w_i^{m_i}\) is non-zero. Equivariance rules out a term independent of \(w_i\), because such a term has trivial \(\mu_{p_i}\)-character. Thus \[\xi_i\circ\mathcal{P}_U=w_i^{m_i}u_i(w)\] with \(u_i(0)\ne0\). This proves the simultaneous normal form for all \(i\in S\). ◻

Proof. It is enough to show that every local stabilizer of \(\mathcal{X}^o\) injects into \(\pi_1^{\rm orb}(\mathcal{X}^o)\). Indeed, for an orbifold covering the stabilizer of a point in the universal cover is the kernel of the natural map from the corresponding local stabilizer downstairs to the orbifold fundamental group.

Let \(x\in\mathcal{X}^o\), and suppose that \(x\) lies on the compact stacky components \(\mathcal{D}_i^c\), \(i\in S\). In a root chart centered at \(x\), the local stabilizer is \[G_x=\prod_{i\in S}\mu_{p_i}, \qquad (\zeta_i)_{i\in S}\cdot w_i=\zeta_i w_i.\] Let \(\lambda_x:G_x\to\pi_1^{\rm orb}(\mathcal{X}^o)\) be the natural local stabilizer homomorphism. We prove that \(\lambda_x\) is injective by showing that \(\rho\circ\lambda_x\) is injective.

On the root chart, the local period map is equivariant: \[\mathcal{P}_U(g\cdot w)=\rho(\lambda_x(g))\,\mathcal{P}_U(w), \qquad g\in G_x.\] By Lemma 4, after shrinking the root chart and choosing simultaneous normal target coordinates \((\xi_i)_{i\in S}\), one has \[\xi_i\circ\mathcal{P}_U=u_i(w)w_i^{p_i-q_i}, \qquad u_i(0)\neq0\] for every \(i\in S\). If \(g=(\zeta_i)_{i\in S}\) lies in the kernel of \(\rho\circ\lambda_x\), then \(\mathcal{P}_U(g\cdot w)=\mathcal{P}_U(w)\). Comparing the first non-zero normal term in the \(i\)-th coordinate gives \[\zeta_i^{p_i-q_i}=1\] for every \(i\in S\). Since \[\gcd(p_i,p_i-q_i)=\gcd(p_i,q_i)=1,\] we get \(\zeta_i=1\) for every \(i\in S\). Hence \(g=1\), so \(\rho\circ\lambda_x\), and therefore \(\lambda_x\), is injective.

The only local stabilizers of \(\mathcal{X}^o\) occur along the compact root divisors \(\mathcal{D}_i^c\). Thus all stabilizers inject into the orbifold fundamental group, and the universal orbifold cover has no residual stabilizers. Hence it is represented by a simply connected smooth complex manifold. ◻

Proof. The source coarse coordinate is \(z_i=w_i^{p_i}\). The local period coordinate on the ball is \[\xi_i=w_i^{p_i-q_i}.\] The local stabilizer \(\mu_{p_i}\) acts on \(w_i\) by multiplication and hence acts on \(\xi_i\) by the character \(\zeta\mapsto \zeta^{p_i-q_i}\). Since \(\gcd(p_i,q_i)=1\), this character has order \(p_i\). Therefore the target coarse coordinate is, up to a unit, \[t_i=\xi_i^{p_i}=w_i^{p_i(p_i-q_i)}=z_i^{p_i-q_i}.\] Thus the induced map on coarse spaces is locally \[z_i\longmapsto z_i^{p_i-q_i},\] which has branch index \(p_i-q_i\). ◻

Proof. The principal reduction is projective, so only the metric on \(\operatorname{Hom}(\mathcal{E}^{1,0},\mathcal{E}^{0,1})\) is intrinsic. The displayed normalization is a local choice of lift; multiplying the lift by a common scalar changes the metrics on \(\mathcal{E}^{1,0}\) and \(\mathcal{E}^{0,1}\) simultaneously and leaves the Hom metric unchanged.

Along \(D^p\), the parabolic structure is trivial and the residues of the harmonic bundle are nilpotent. At a generic point of the cusp divisor the only non-zero residue of the Higgs field on \(E\) is \[N(e_1)=e_0, \qquad N(e_j)=0\quad(j\ne1).\] The associated nilpotent-orbit model puts \(e_1\) and \(e_0\) in opposite weights, while the transverse frames \(e_j\), \(j\ge2\), have weight zero. Mochizuki’s norm estimates for tame nilpotent harmonic bundles with trivial parabolic structure identify the harmonic metric, up to mutual boundedness, with this model. After the above local normalization one obtains \[|e_1|^2\sim s, \qquad |e_0|^2\sim s^{-1}, \qquad |e_j|^2\sim1\quad(j\ge2),\] where \(s=-\log|z_1|^2\).

The Kodaira–Spencer map sends \[z_1\frac{\partial}{\partial z_1}\longmapsto e_1^\vee\otimes e_0, \qquad \frac{\partial}{\partial z_j}\longmapsto e_j^\vee\otimes e_0\quad(j\ge2).\] The induced Hom metric is independent of the scalar normalization and satisfies \[|e_1^\vee\otimes e_0|^2\sim s^{-2}, \qquad |e_j^\vee\otimes e_0|^2\sim s^{-1}\quad(j\ge2).\] Thus \[\left|\frac{\partial}{\partial z_1}\right|^2\sim \frac{1}{|z_1|^2s^2}, \qquad \left|\frac{\partial}{\partial z_j}\right|^2\sim \frac{1}{s}\quad(j\ge2),\] which is 10 . ◻

Proof. Completeness follows from the normal Poincaré factor. Along a path with \(|z_1|=r\to0\), the length is bounded below by a constant multiple of \[\int_0^\varepsilon \frac{dr}{r|\log r|}=+\infty.\] For finite volume, Lemma 5 gives a local volume density mutually bounded by \[\frac{dV_{\rm eucl}}{|z_1|^2|\log |z_1|^2|^{n+1}}.\] In polar coordinates \(z_1=re^{\sqrt{-1}\theta}\), the normal integral is \[\int_0^\varepsilon\frac{r\,dr}{r^2|\log r^2|^{n+1}} = \frac{1}{2}\int_{-\log\varepsilon^2}^{+\infty}s^{-(n+1)}\,ds<\infty.\] ◻

Proof. By Proposition 12, the local normal component of the period map is \(\xi_i=w_i^{m_i}\), up to multiplication by a unit and up to higher-order tangential terms. The smooth ball metric in the \(\xi_i\)-direction pulls back as \[|d\xi_i|^2=m_i^2|w_i|^{2(m_i-1)}|dw_i|^2,\] up to mutual boundedness by positive functions. The volume contribution is locally integrable because \[\int_0^\varepsilon r^{2(m_i-1)}r\,dr<\infty.\] The metric is non-degenerate at \(w_i=0\) precisely when \(m_i=1\). ◻

Proof. We first fix the three levels involved in the construction. The stack \(\mathcal{X}=X[\sqrt[p_i]{D_i^c}]_{i\in I}\) has coarse moduli space \(X\), and \[\mathcal{X}^o=\mathcal{X}\setminus\pi^{-1}(D^p),\qquad X^o=X\setminus D^p.\] The divisor \(D^c\) is not removed; it is encoded by stabilizers on \(\mathcal{X}^o\). The auxiliary finite cover below is a cover of the quotient stack \(\mathcal{X}^o\), while its compactification is taken over the coarse variety \(X\).

Choose a finite-index neat normal subgroup \(\Gamma'\triangleleft\Gamma\), put \(F=\Gamma/\Gamma'\), and set \[U':=\mathbb{B}^n/\Gamma'.\] Then \(U'\) is smooth, its parabolic elements are unipotent, and the natural map \[U'=[\mathbb{B}^n/\Gamma']\longrightarrow [\mathbb{B}^n/\Gamma]\simeq\mathcal{X}^o\] is finite, representable, Galois, and étale as a morphism of orbifolds, with group \(F\). Let \(Y'\) be the normalization of the coarse projective variety \(X\) in the function field \(\mathbb{C}(U')\). Thus there is a finite morphism of normal projective varieties \[\nu_X:Y'\longrightarrow X,\] and \(\nu_X^{-1}(X^o)=U'\). Notice that this is a morphism to the coarse space \(X\), not a morphism to the root stack \(\mathcal{X}\). Over \(X^o\) the map is the coarse map induced by the representable orbifold cover \(U'\to\mathcal{X}^o\). Put \[E':=(\nu_X^{-1}D^p)_{\rm red}.\] Since \(D^c\subset X^o\), its preimage is contained in the open manifold \(U'\); it is not part of the boundary \(Y'\setminus U'\). Hence \(Y'\setminus U'=E'\).

We next verify the singularity hypothesis needed in the rigidity theorem of Deng–Cadorel DengCadorel2022?. Let \(y\in Y'\), and write \(x=\nu_X(y)\). Choose analytic coordinates on \(X\) near \(x\) such that \[D^p=\{z_1\cdots z_r=0\},\qquad D^c=\{x_1\cdots x_s=0\},\] with the compact component \(x_\beta=0\) carrying order \(p_\beta\). Pulling back to the root chart \[x_\beta=w_\beta^{p_\beta}\qquad(1\le \beta\le s)\] removes the compact stabilizers. Since \(U'\to\mathcal{X}^o\) is representable étale, its pull-back to this root chart is an ordinary finite étale cover away from the cusp boundary \(z_1\cdots z_r=0\). Along the cusp variables, the cover of the punctured polydisc is, after decomposing into connected components, given by finite-index subgroups of \(\mathbb{Z}^r\). By the analytic form of Abhyankar’s lemma, after cyclic base changes \[z_\alpha=t_\alpha^{m_\alpha}\qquad(1\le\alpha\le r)\] the normalization is finite étale over the smooth polydisc with coordinates \((t,w,\text{transverse})\). Therefore, locally analytically, the pair \((Y',E')\) is the quotient of a smooth simple-normal-crossing pair by a finite group preserving the boundary. In particular \((Y',E')\) has algebraic quotient singularities in the sense of DengCadorel2022?; hence \(Y'\) has quotient singularities and is klt.

We recall the precise form of Deng–Cadorel’s rigidity theorem used here. Let \(U=\mathbb{B}^n/\Lambda\) with \(\Lambda\subset\mathrm{PU}(n,1)\) torsion-free. If \(\overline{U}\) is a klt compactification of \(U\), if \(B^{(1)}\) is the divisorial part of \(\overline{U}\setminus U\), and if the Kähler–Einstein metric on \(U\), regarded as a metric on \(T_{\overline{U}}(-\log B^{(1)})|_U\), is adapted to log order near the generic point of every component of \(B^{(1)}\), then \((\overline{U},\overline{U}\setminus U)\) is the toroidal compactification of \(U\) DengCadorel2022?. In our situation \(U=U'\), \(\Lambda=\Gamma'\), \(\overline{U}=Y'\), and \(B^{(1)}=E'\).

It remains only to check the log-order metric hypothesis. The metric on \(U'\) obtained by pulling back the period metric is the Bergman metric of \(\mathbb{B}^n/\Gamma'\). Near a generic point of a component of \(D^p\), Lemma 5 gives the model \[\frac{\sqrt{-1}\,dz\wedge d\bar z}{|z|^2(\log |z|^2)^2} +\sum_{j=2}^n\frac{\sqrt{-1}\,du_j\wedge d\bar u_j}{-\log |z|^2}.\] After the finite normal cover \(z=t^m\), the normal Poincaré factor is preserved: \[\frac{\sqrt{-1}\,dz\wedge d\bar z}{|z|^2(\log |z|^2)^2} = \frac{\sqrt{-1}\,dt\wedge d\bar t}{|t|^2(\log |t|^2)^2},\] up to the harmless constant convention in the logarithm. The tangential factors remain mutually bounded after the finite étale change in the transverse variables. Thus the Bergman metric is adapted to log order for \(T_{Y'}(-\log E')|_{U'}\) near the generic point of every component of \(E'\). The compact divisors \(D_i^c\) do not enter \(E'\), and on their preimages the cover is an ordinary smooth interior cover in root coordinates.

All hypotheses of Deng–Cadorel’s theorem are therefore satisfied, and we obtain an isomorphism of compactifications \[(Y',E')\simeq (\overline{U}_{\Gamma'}^{\rm tor},D_{\Gamma'}^{\rm tor}).\] The finite group \(F\) acts on \(U'\), and this action extends to \(Y'\) by the universal property of normalization over the fixed coarse compactification \(X\). Under the preceding identification, it is the standard extension of the \(F\)-action to the toroidal compactification, since the two extensions agree on the dense open set \(U'\) and the compactifications are normal and separated.

The finite morphism \(\nu_X:Y'\to X\) is \(F\)-invariant, hence descends to a finite morphism \[Y'/F\longrightarrow X.\] It is an isomorphism over the dense open subset \[U'/F\simeq \mathbb{B}^n/\Gamma\simeq X^o.\] Both \(Y'/F\) and \(X\) are normal, so the finite birational morphism \(Y'/F\to X\) is an isomorphism. It carries \(E'/F\) to \(D^p\). Consequently \[(X,D^p)\simeq (\overline{U}_{\Gamma'}^{\rm tor}/F,D_{\Gamma'}^{\rm tor}/F) = (\overline{U}_{\Gamma}^{\rm tor},D_{\Gamma}^{\rm tor})\] as coarse toroidal pairs.

Finally the stack structure is exactly the root structure prescribed in Definition 1. The open stack is \([\mathbb{B}^n/\Gamma]\), the generic stabilizer along \(D_i^c\) is \(\mu_{p_i}\), and no root structure is imposed along the cusp boundary \(D^p\). Hence \[\mathcal{X}=X\Bigl[\sqrt[p_i]{D_i^c}\Bigr]_{i\in I}\] is the canonical orbifold toroidal compactification of \([\mathbb{B}^n/\Gamma]\). Since \(D_{\Gamma'}^{\rm tor}\) is a disjoint union of abelian varieties, each connected component of \(D^p\) is a finite quotient of an abelian variety. ◻

Proof of Corollary 1. If \(q_i=p_i-1\) for every \(i\), Proposition 5 gives \[\mathcal{E}^{1,0}=\Omega^1_{\mathcal{X}}(\log D^p),\] and the Kodaira–Spencer map \(\tau\) is an isomorphism everywhere on \(\mathcal{X}^o\). Therefore the period map \[\mathcal{P}:\widetilde{\mathcal{X}^o}\to\mathbb{B}^n\] is a local biholomorphism and a local isometry for the pull-back of the invariant metric of \(\mathbb{B}^n\).

By Lemma 6, the metric is complete near \(D^p\). Away from \(D^p\), the space is covered by finitely many compact orbifold charts after shrinking near the boundary. In the standard case the compact root divisors are not branch divisors, so the metric is non-degenerate across them in the orbifold sense. Hence no additional incomplete end occurs, and \(\widetilde{\mathcal{X}^o}\) is complete for the pull-back metric. Since \(\mathbb{B}^n\) is complete, connected and simply connected, the standard complete local-isometry argument gives \[\widetilde{\mathcal{X}^o}\simeq\mathbb{B}^n.\] The \(\rho\)-equivariance of \(\mathcal{P}\) identifies the orbifold deck group with the discrete subgroup \[\Gamma=\rho\bigl(\pi_1^{\rm orb}(\mathcal{X}^o)\bigr)\subset \mathrm{PU}(n,1),\] so \(\rho\) is faithful and \(\Gamma\) is discrete. The volume of \(\mathbb{B}^n/\Gamma\) equals the volume of \(\mathcal{X}^o\) for the Hodge metric. Lemma 6 gives finite volume near \(D^p\), and compactness away from \(D^p\) gives finite total volume. Hence \(\Gamma\) is a lattice.

Proposition 15 identifies \((\mathcal{X},D^p)\) with the canonical orbifold toroidal compactification of \([\mathbb{B}^n/\Gamma]\). This proves the corollary. ◻

Proof. Along \(D^p\), the harmonic bundle is tame nilpotent with trivial parabolic structure. Hence the local monodromy around a component of \(D^p\) is unipotent. The residue is non-zero because the Higgs residue sends the logarithmic frame \(d\log z_1\) to \(1\). Therefore the corresponding element of \(\mathrm{PU}(n,1)\) is a non-trivial unipotent isometry of complex hyperbolic space, hence parabolic.

Along \(D_i^c\), the divisor is not removed. It is part of the orbifold structure. On the root chart \(z_i=w_i^{p_i}\), the local stabilizer \(\mu_{p_i}\) acts by \(w_i\mapsto\zeta w_i\). The period normal coordinate is \(\xi_i=w_i^{p_i-q_i}\), so the induced action on the target normal coordinate is \[\xi_i\longmapsto \zeta^{p_i-q_i}\xi_i.\] Because \(\gcd(p_i,p_i-q_i)=\gcd(p_i,q_i)=1\), this character has order \(p_i\). Hence the local monodromy is elliptic of order \(p_i\) in the normal direction. ◻

Proof of Theorem 1. Proposition 7 gives flatness of the adjoint harmonic bundle. Proposition 8 supplies the principal \(\mathrm{PU}(n,1)\)-system, and Proposition 9 gives the period map. Proposition 12 gives the ramification index, while Proposition 5 gives the unramified criterion. Proposition 13 proves that the orbifold universal cover is represented by a simply connected complex manifold and that compact local stabilizers act faithfully in the normal period coordinates. The assertions on \(D^p\) and on peripheral monodromy follow from Propositions 11 and 16. ◻

Proof. We first record the zero-Lelong-number estimate. Lelong numbers are monotone for positive currents, so it suffices to check the local model terms in Definition 2. The Poincaré term has potential \(-\log(-\log |z|^2)\), whose quotient by \(\log |z|^2\) tends to zero. For the cone term, write \(a=1-\beta\), \(\beta>0\); up to a positive constant, \[\sqrt{-1}\,\partial\bar\partial |x|^{2\beta} = \frac{\sqrt{-1}\,dx\wedge d\bar x}{|x|^{2a}}\] on the punctured disc, and the potential \(|x|^{2\beta}\) is continuous at the origin. Hence all model terms, and therefore \(T\), have zero Lelong number along \(D^p+D^c\). Since \(T\) is smooth on \(X^\circ\), it has zero Lelong number there as well. Thus \(T\) has zero Lelong number at every point of \(X\), and no component of \(D^p+D^c\) occurs in the Siu divisorial part of \(T\).

Fix a smooth Kähler form \(\omega_0\) on \(X\), and choose a smooth representative \(\theta\) of the class \([T]\). Write \[T=\theta+\sqrt{-1}\,\partial\bar\partial\varphi\] for a quasi-plurisubharmonic potential \(\varphi\).

By Demailly’s regularization theorem with control of Lelong numbers Demailly1992?, for every \(\varepsilon>0\) there exists a closed current \(T_\varepsilon\in[T]\) with analytic singularities such that \[T_\varepsilon\geq -\varepsilon\omega_0,\] and whose Lelong numbers are bounded by those of \(T\). Since all Lelong numbers of \(T\) vanish, the analytic singularities of \(T_\varepsilon\) are removable. Thus \(T_\varepsilon\) is represented by a smooth real \((1,1)\)-form in \([T]\) satisfying the same lower bound. Hence \([T]\) is nef.

It remains to prove bigness. Choose a coordinate ball \(B\Subset X^\circ\). Since \(T|_{X^\circ}\) is Kähler, after shrinking \(B\) there is a constant \(c>0\) such that \[T\geq c\omega_0\] on \(B\). The regularization may be chosen to converge to \(T\) smoothly on compact subsets of the locus where \(T\) is smooth; hence, for all sufficiently small \(\varepsilon\), \[T_\varepsilon+\varepsilon\omega_0\geq \frac{c}{2}\omega_0\] on \(B\), while \(T_\varepsilon+\varepsilon\omega_0\geq0\) on all of \(X\). Therefore \[([T]+\varepsilon[\omega_0])^n =\int_X (T_\varepsilon+\varepsilon\omega_0)^n \geq \int_B (T_\varepsilon+\varepsilon\omega_0)^n \geq \left(\frac{c}{2}\right)^n\int_B\omega_0^n>0.\] Letting \(\varepsilon\to0\) gives \([T]^n>0\). Since \([T]\) is nef and \(X\) is projective, the standard nef volume criterion implies that \([T]\) is big; see, for instance, DemaillyPaun2004?. The same current \(T\) satisfies Definition 2, so \([T]\) is \(\Delta\)-admissible by Definition 3. ◻

Proof of Theorem 2. Proof of (1). The Bergman form is \(\mathrm{PU}(n,1)\)-invariant, so \(\mathcal{P}^*\omega_{\mathbb{B}}\) is invariant under the orbifold deck group and descends to a smooth Kähler form on \(X^\circ\). Near \(D_i^c\), use \(x_i=w_i^{p_i}\) and the local normal form \(\xi_i=u_i(w)w_i^{m_i}\). In the normal direction, \[d\xi_i=(\text{unit})\,w_i^{m_i-1}dw_i.\] Since the Bergman metric is smooth and positive in the target coordinate, the pull-back normal part is mutually bounded by \[|w_i|^{2m_i-2}\sqrt{-1}\,dw_i\wedge d\bar w_i.\] Using \(m_i=p_i-q_i\) and \(x_i=w_i^{p_i}\), this is, up to a positive smooth factor, \[|x_i|^{-2q_i/p_i}\sqrt{-1}\,dx_i\wedge d\bar x_i.\] The cusp hypothesis gives the Poincaré bound along \(D^p\). This proves (1).

Proof of (2). Because the mixed model is locally integrable, \(\omega_{\rm br}\) has locally finite mass near \(|D|\). The form is closed and positive on \(X^\circ\); by the Skoda–El Mir extension theorem for closed positive currents DemaillyBook? its trivial extension across \(|D|\) is a closed positive current. We denote the extension again by \(\omega_{\rm br}\), and put \[\Theta:=\frac{n+1}{2\pi}\,\omega_{\rm br}.\]

It remains to identify the de Rham class of \(\Theta\). Put \[a_i:=\frac{q_i}{p_i},\qquad L:=K_X+D^p+\sum_{i\in I} a_i\,D_i^c.\] On \(X^\circ\), the complex-hyperbolic metric satisfies \[\operatorname{Ric}(\omega_{\rm br})=-(n+1)\omega_{\rm br}.\] Let \(h_K\) be the Hermitian metric on \(K_X|_{X^\circ}\) induced by the volume form \(\omega_{\rm br}^n\). The standard curvature formula gives \[c_1(K_X,h_K) = -\frac{1}{2\pi}\operatorname{Ric}(\omega_{\rm br}) = \Theta\] on \(X^\circ\).

Choose an integer \(N>0\) such that \(Na_i\in\mathbb{Z}\) for all \(i\), and set \[L_N:=N L.\] Then \(L_N\) is a holomorphic line bundle. It suffices to prove \([N\Theta]=c_1(L_N)\). Write \[D^p=\sum_{\ell\in J_p}D_\ell^p,\qquad D^c=\sum_{i\in I}D_i^c.\] Let \(s_\ell^p\) and \(s_i^c\) be the canonical sections of \(\mathcal{O}_X(D_\ell^p)\) and \(\mathcal{O}_X(D_i^c)\), respectively, and choose arbitrary smooth Hermitian metrics \(h_\ell^p\) and \(h_i^c\) on these divisor line bundles. On \(X^\circ\) define a singular Hermitian metric on \(L_N\) by \[H_N := h_K^{\otimes N} \otimes \bigotimes_{\ell\in J_p} \left(\frac{h_\ell^p}{|s_\ell^p|_{h_\ell^p}^2}\right)^{\otimes N} \otimes \bigotimes_{i\in I} \left(\frac{h_i^c}{|s_i^c|_{h_i^c}^2}\right)^{\otimes Na_i}.\] On \(X^\circ\) these divisor factors have zero curvature, since their canonical sections have constant norm \(1\). Hence \[c_1(L_N,H_N)=N\,c_1(K_X,h_K)=N\Theta\] on \(X^\circ\).

Now we check the singularities of \(H_N\). Choose local coordinates \[D^p=\{z_1\cdots z_r=0\},\qquad D^c=\{x_1\cdots x_s=0\}.\] In local frames with \(s_\alpha^p=z_\alpha e_\alpha^p\) and \(s_\beta^c=x_\beta e_\beta^c\), the divisor factors contribute \(|z_\alpha|^{-2N}\) and \(|x_\beta|^{-2Na_\beta}\). The mixed Poincaré–cone model gives the divisorial part of the determinant metric in the form \[|\sigma_K|_{h_K}^2 = \prod_{\alpha=1}^r |z_\alpha|^2 \prod_{\beta=1}^s |x_\beta|^{2a_\beta} \cdot \Lambda,\] where \(\sigma_K\) is a local frame of \(K_X\), and \(\log\Lambda\) has only log-log growth in the cusp variables and bounded, equivalently orbifold-bounded, growth in the compact root directions. Therefore, in the corresponding local frame of \(L_N\), \[|\sigma_{L_N}|_{H_N}^2=\Lambda^N.\] Thus \(H_N\) has locally integrable weights, with no remaining \(\log|z_\alpha|^2\) or \(\log|x_\beta|^2\) divisorial term. Its curvature current is well-defined and has no extra Siu divisorial mass along \(D^p+D^c\). The trivial extension of \(\Theta\) also has no divisorial mass by the mixed growth estimate and the zero-Lelong-number estimate in Lemma 8. Since the two currents agree on \(X^\circ\), they agree on all of \(X\): \[c_1(L_N,H_N)=N\Theta.\]

Finally, let \(H_0\) be any smooth Hermitian metric on \(L_N\). Since \(H_N=H_0e^{-\varphi}\) for a global \(L^1_{\rm loc}\) function \(\varphi\), \[c_1(L_N,H_N) = c_1(L_N,H_0) + \frac{\sqrt{-1}}{2\pi}\partial\bar\partial\varphi.\] The second term is exact as a current. Hence \[[N\Theta]=[c_1(L_N,H_N)]=c_1(L_N),\] and division by \(N\) gives \[[\Theta]=c_1(K_X+\Delta) =c_1\left(K_X+D^p+\sum_{i\in I}\frac{q_i}{p_i}D_i^c\right).\]

We have also proved that \(\Theta\) has at most mixed Poincaré–cone growth and is a smooth Kähler form on \(X^\circ\). Lemma 8 then shows that \(c_1(K_X+\Delta)\) is big and nef, and that it is \(\Delta\)-admissible. This proves (2).

Proof of (3). Equip \(E|_{X^\circ}=\Omega_X^1|_{X^\circ}\oplus\mathcal{O}_X\) with the Hodge metric \[h=\omega_{\rm br}^{-1}\oplus h_0,\] where \(h_0\) is the constant metric on \(\mathcal{O}_X\). This is the pull-back of the standard homogeneous Hodge metric on the system of Hodge bundles over \(\mathbb{B}^n\). Hence the projective Hitchin–Simpson connection is flat. Equivalently, if \[F_{h,\theta}:=F_h+[\theta,\theta_h^\dagger]\] denotes the Hitchin–Simpson curvature of \((E,\theta,h)\), then \[\label{eq:hs-flat-conv} F_{h,\theta}^{\perp}=0.\tag{11}\] Here \(\perp\) denotes the trace-free part of the Hitchin–Simpson curvature. The local model gives \[|dx_i|_h^2\sim |x_i|^{2q_i/p_i}\] in the conormal direction to \(D_i^c\), exactly the growth prescribed by the parabolic weight \(q_i/p_i\). Along \(D^p\) the metric is the usual tame nilpotent cusp model. Thus \(h\) is adapted to \(E_*\) and acceptable with respect to the mixed Poincaré–cone model.

Let \(\alpha\) be a \(\Delta\)-admissible big and nef class, and choose a current \(T\in\alpha\) as in Definition 3. Put \(\omega_T:=T|_{X^\circ}\), which is a smooth Kähler form on \(X^\circ\). Let \(S_*\subset E_*\) be a saturated Higgs subsheaf of rank \(s\), and put \(r=\operatorname{rk}E=n+1\). Let \[Z\subset X\] be the union of \(|D|\) and the locus where \(E/S\) is not locally free. Then \(Z\setminus |D|\) has codimension at least two, and on \[U:=X\setminus Z\] the sheaf \(S\) is a holomorphic subbundle of \(E\) preserved by \(\theta\). Denote by \(\Pi\) the \(h\)-orthogonal projection from \(E|_U\) to \(S|_U\), and by \(h_S\) the induced metric on \(S|_U\).

We use the following Chern–Weil formula for parabolic degrees.

Let \[b_j(S):=\sum_a a\,\operatorname{rk}\bigl(\operatorname{Gr}^j_a(S_*)\bigr),\] where \(\operatorname{Gr}^j_a(S_*)\) is the graded piece of the parabolic filtration of \(S_*\) along \(D_j^c\). Thus \[\operatorname{par\!-c}_1(S_*)=c_1(S)+\sum_j b_j(S)[D_j^c].\] Choose a finite Galois cover \(g:Y\to X\) adapted to the denominators of the weights, so that \(g^*D_j^c=m_j\,p_j\widetilde{D}_j^c\). By Biswas1997?, the pull-back of every locally abelian parabolic bundle is an equivariant orbifold bundle. Let \(\widetilde{S}\) be the equivariant bundle corresponding to \(S_*\), and let \(\widetilde{h}_S\) be the metric induced by \(h_S\) in the root frames. Then \[c_1(\widetilde{S})=g^*\operatorname{par\!-c}_1(S_*),\] and, on \(Y\setminus g^{-1}(D^p+D^c)\), \[\operatorname{tr}F_{\widetilde{h}_S}=g^*\operatorname{tr}F_{h_S}.\] Locally this is the change from the coarse adapted frame to the root frame. If \(x_j=w_j^{p_j}\) and the weight is \(a=q_j/p_j\), the factor \(|x_j|^{2a}\) in the coarse adapted metric becomes a smooth factor in the orbifold frame \(w_j^{-q_j}dx_j\). Hence the divisor contribution in \(\operatorname{par\!-c}_1(S_*)\) is exactly converted into the ordinary first Chern class of \(\widetilde{S}\) on the cover.

Put \(\widetilde{T}:=g^*T\). Since \(T\) is dominated by the mixed Poincaré–cone model, \(\widetilde{T}\) has at most Poincaré growth along \(g^{-1}D^p\) and locally finite mass along the inverse image of \(D^c\). A cut-off argument gives the required intersection formula.

Dividing by the degree of the cover and using the change-of-variables formula for the finite map \(g\), one obtains \[\label{eq:par-degree-cw-current} \operatorname{par\!-deg}_\alpha(S_*) = \frac{\sqrt{-1}}{2\pi} \int_U \operatorname{tr}F_{h_S}\wedge T^{n-1}, \qquad \operatorname{par\!-deg}_\alpha(E_*) = \frac{\sqrt{-1}}{2\pi} \int_U \operatorname{tr}F_h\wedge T^{n-1}.\tag{12}\]

Because \(S\) is \(\theta\)-invariant, we have \[\operatorname{tr}[\theta_S,\theta_{S,h_S}^\dagger]=0.\]

Thus, in the trace appearing in 12 , \(F_{h_S}\) may equivalently be replaced by the Hitchin–Simpson curvature \[F_{h_S,\theta_S}=F_{h_S}+[\theta_S,\theta_{S,h_S}^\dagger].\]

On \(U\), the standard second fundamental form identity for Higgs bundles gives \[\label{eq:second-fund-current} \begin{align} \frac{\sqrt{-1}}{2\pi}\operatorname{tr}F_{h_S,\theta_S} &-\frac{s}{r}\frac{\sqrt{-1}}{2\pi}\operatorname{tr}F_{h,\theta} \\ &=\frac{\sqrt{-1}}{2\pi}\operatorname{tr}\bigl(\Pi F_{h,\theta}^{\perp}\bigr) -\mathcal{Q}(\Pi). \end{align}\tag{13}\] Here \(\mathcal{Q}(\Pi)\) is the non-negative \((1,1)\)-form whose contraction with \(\omega_T:=T|_{X^\circ}\) is \[\Lambda_{\omega_T}\mathcal{Q}(\Pi) = c_n\Bigl(|\bar\partial_E\Pi|^2_{h,\omega_T}+|[\theta,\Pi]|^2_{h,\omega_T}\Bigr)\] for a positive dimensional constant \(c_n\). This formula is the usual Chern–Weil formula for the induced Higgs metric: the first norm is the ordinary second fundamental form, and the second norm measures the failure of the orthogonal splitting to be Higgs-invariant.

After integrating 13 against \(T^{n-1}\) on \(U\), the parabolic Chern–Weil formula 12 identifies the left-hand side with \(\operatorname{par\!-deg}_\alpha(S_*)-\frac{s}{r}\operatorname{par\!-deg}_\alpha(E_*)\). Lemma 9 accounts for the boundary and codimension-two contributions. Using 11 , we obtain \[\label{eq:slope-current-identity} \begin{align} \operatorname{par\!-deg}_\alpha(S_*)-\frac{s}{r}\operatorname{par\!-deg}_\alpha(E_*) &=-C_n\int_U \Bigl(|\bar\partial_E\Pi|^2_{h,\omega_T}+|[\theta,\Pi]|^2_{h,\omega_T}\Bigr) \frac{\omega_T^n}{n!} \\ &\le 0, \end{align}\tag{14}\] where \(C_n>0\). Dividing by the ranks gives \[\mu_\alpha(S_*)\le \mu_\alpha(E_*).\] Thus \((E_*,\theta)\) is \(\mu_\alpha\)-semistable.

If equality holds in 14 , both non-negative terms in the integral vanish. Since \(\omega_T\) is a Kähler metric on \(X^\circ\), one obtains \[\bar\partial_E\Pi=0, \qquad [\theta,\Pi]=0\] on \(U\). Therefore \(\Pi\) is a holomorphic Higgs projection on \(U\). Since \(S\) and \(E/S\) are saturated, the two summands extend uniquely across the codimension at least two set by reflexivity. The local growth of \(h\) along \(D^p+D^c\) shows that the extended summands are compatible with the induced parabolic filtrations; equivalently, the splitting is a splitting in the category of parabolic Higgs sheaves. Hence any saturated Higgs subsheaf with the same \(\alpha\)-slope is a direct summand. This proves (3).

Proof of (4). The parabolic Chern classes are computed by the same adapted Chern–Weil interpretation as in Lemma 9, applied on an adapted cover and then divided by the degree of the cover. On \(X^\circ\), equation 11 says that the Hitchin–Simpson curvature of \((E,\theta,h)\) is scalar. Therefore the induced Hitchin–Simpson curvature on \(\operatorname{End}_0(E)\), and hence also on \(\operatorname{End}(E)\), is zero. The adapted-cover Chern–Weil representative of \(\operatorname{par\!-ch}_2(\operatorname{End}E_*)\) is consequently zero on the complement of the boundary; the growth estimates used above show that no boundary current is produced. Thus \[\operatorname{par\!-ch}_2(\operatorname{End}E_*)=0.\] For a rank \(r=n+1\) bundle, \[\operatorname{par\!-ch}_2(\operatorname{End}E_*)=2r\,\operatorname{par\!-ch}_2(E_*)-\operatorname{par\!-ch}_1(E_*)^2,\] which gives the asserted equality. ◻

Proof of Corollary 2. The orbifold ball quotient structure gives the orbifold universal cover \[\mathbb{B}^n\longrightarrow \mathcal{X}^o\] and a period map which is the identity map on \(\mathbb{B}^n\), up to the action of \(\Gamma\). Near a compact orbifold divisor of stabilizer order \(p_i\), the standard orbifold local model has root coordinate \(z_i=w_i^{p_i}\), and the normal ball coordinate is a unit multiple of \(w_i\). Since \(q_i=p_i-1\), this is exactly the local normal form \[\xi_i=u_i(w)w_i^{p_i-q_i}=u_i(w)w_i, \qquad u_i(0)\ne0.\] At the cusp boundary \(D^p\), the toroidal compactification gives the usual Poincaré cusp growth. Hence the weighted pair \((X,D^p,D^c,\{(p_i-1)/p_i\})\) satisfies Definition 4 with \[\Delta=D^p+\sum_{i\in I}\left(1-\frac{1}{p_i}\right)D_i^c.\] Theorem 2 therefore gives that \(K_X+\Delta\) is big and nef, that \(c_1(K_X+\Delta)\) is \(\Delta\)-admissible, and that the asserted polystability and Chern equality hold for every \(\Delta\)-admissible big and nef class. ◻