Proof. The identities are the ordinary planar Green identities with \(\Delta=4\partial_z\partial_{\bar z}\). For the limiting assertion, \(\log(1/|z|)\) belongs to every finite \(L^b(\mathbb D)\), so Hölder’s inequality and dominated convergence apply to the right-hand side. Continuity gives convergence of the circle means. ◻

Proof. We give Brody’s rescaling argument in the normalization needed here (Brody 1978). Suppose first that there are maps \(f_j\colon\mathbb D\to X\) with \(|f_j'(0)|_k\to\infty\). Choose \(z_j\) at which the continuous function \[z\longmapsto (1/2-|z|)|f_j'(z)|_k\] attains its maximum on \(\{|z|\leq1/2\}\). For all sufficiently large \(j\) this maximum is positive, so \(|z_j|<1/2\). Put \[\lambda_j=|f_j'(z_j)|_k, \qquad R_j=(1/2-|z_j|)\lambda_j, \qquad g_j(w)=f_j(z_j+w/\lambda_j).\] The map \(g_j\) is defined on \(\{|w|<R_j\}\), and \[R_j\geq\tfrac12|f_j'(0)|_k\longrightarrow\infty, \qquad |g_j'(0)|_k=1.\] If \(|w|<R_j/2\), then \[1/2-|z_j+w/\lambda_j| \geq\tfrac12(1/2-|z_j|).\] The maximizing property of \(z_j\) therefore gives \(|g_j'(w)|_k\leq2\) on this smaller disc.

For every fixed radius these maps are consequently equicontinuous and take values in the compact metric space \(X\). The Arzelà–Ascoli theorem and a diagonal subsequence give a locally uniform limit \(g\colon\mathbb C\to X\). To verify holomorphy, choose a point of the source and a target coordinate chart containing its image under \(g\). On a sufficiently small source neighborhood, the limit and all sufficiently late maps take values in that chart. Their coordinate representations converge uniformly on smaller neighborhoods. The usual theorem on uniform limits of holomorphic functions, followed by the Cauchy integral formula for derivatives, shows that \(g\) is holomorphic and \(|g'(0)|_k=1\). This contradicts the hypothesis on entire maps. The central derivative bound follows.

For \(z\in\mathbb D\), apply this bound to \(w\mapsto f(z+(1-|z|)w)\). Its central derivative is \((1-|z|)f'(z)\), proving Equation (6). The estimate gives equicontinuity on every compact source disc. A second application of Arzelà–Ascoli and the same coordinate argument proves normality and convergence of derivatives. Increasing \(D\) if necessary makes it strictly positive. ◻

Proof. Choose \(C_q\geq0\) such that \[|q(v,v)|\leq C_q h(v,v)\] for every tangent vector \(v\). Select \(r_0\in[1/2,1)\) so close to \(1\) that \(2C_q\log(1/r_0)\leq\delta/2\). For a disc \(f\) write \[a_f(z)=h(f'(z),f'(z)),\qquad b_f(z)=q(f'(z),f'(z)).\] On the outer annulus \(r_0<|z|<1\), the function \[T_f(z)=\delta a_f(z)-2\log(1/|z|)b_f(z)\] satisfies \[ T_f(z)\geq\frac\delta2 a_f(z)\geq0. \tag{8}\] On the inner disc, Lemma 3 and comparison of \(h\) with \(k\) give a uniform bound for \(a_f\) and \(|b_f|\). Since \(\log(1/|z|)\) is integrable there, there is a constant \(K\) independent of \(f\) such that \[\left|\frac1\pi\int_{|z|<r_0} \bigl(2\log(1/|z|)b_f(z)-\delta a_f(z)\bigr)\,dS\right| \leq K.\] Consequently, \[ J_{h,q,\delta}(f) \leq D+K-\frac\delta{2\pi}\int_{r_0<|z|<1}a_f\,dS. \tag{9}\] In particular, the supremum is finite. It is nonnegative because the constant disc has value zero.

Choose a maximizing sequence \((f_j)\) with \(J_{h,q,\delta}(f_j)\geq-1\). Equation (9) bounds its outer-annulus areas uniformly, and the compact-interior derivative estimate bounds its inner areas. Thus \[\sup_j A_h(f_j)<\infty.\] By Lemma 3, after passing to a subsequence the maps converge locally uniformly to a holomorphic disc \(f\) centered at \(x\). Derivative convergence on compact subsets gives pointwise convergence of \(a_{f_j}\) and \(b_{f_j}\). Fatou’s lemma gives \(A_h(f)\leq\liminf_j A_h(f_j)<\infty\).

We verify the required upper semicontinuity of \(J\). Its central term converges by derivative convergence. Its inner-disc integral converges by dominated convergence, using the uniform bound above together with the integrable logarithmic weight. On the outer annulus, the nonnegative functions \(T_{f_j}\) converge pointwise to \(T_f\), so Fatou’s lemma yields \[-\frac1\pi\int_{r_0<|z|<1}T_f\,dS \geq\limsup_j \left(-\frac1\pi\int_{r_0<|z|<1}T_{f_j}\,dS\right).\] All these integrals are finite. Indeed, finite area controls the unweighted integrals and the logarithmic weight is bounded on the outer annulus. Combining the three terms proves \[J_{h,q,\delta}(f)\geq\limsup_jJ_{h,q,\delta}(f_j),\] which proves attainment. ◻

Proof. Let \(f_r(z)=f(rz)\). A change of variables gives \[d(f_r)=r^2d(f),\qquad A_h(f_r)=A_{h,r}(f),\qquad A_{q,s}(f_r)=A_{q,rs}(f).\] The identity \(\log(1/|z|)=\int_{|z|}^1ds/s\) and Fubini’s theorem give \[P_q(f)=2\int_0^1 A_{q,s}(f)\,\frac{ds}{s}.\] Fubini is valid also for this signed integrand: near the origin the coordinate derivatives of \(f\) are bounded, while away from the origin the logarithmic weight is bounded and \(|q(f',f')|\) is controlled by the finite \(h\)-area. Applying the same identity to \(f_r\) gives \[ P_q(f)-P_q(f_r) =2\int_r^1 A_{q,s}(f)\,\frac{ds}{s}. \tag{12}\] Since \(f_r\) has the same center and finite area, maximality gives \(J_{h,q,\delta}(f)\geq J_{h,q,\delta}(f_r)\). Substituting the three identities proves Equation (10); its leftmost inequality uses the positivity of \(h\).

Divide the rightmost expression in Equation (10) by \(1-r\). As \(r\uparrow1\), it tends to \(2d(f)+2A_q(f)\). It is nonnegative for every \(r\), and hence \(A_q(f)\geq-d(f)\). For \(r\geq1/2\), the original right-hand side of Equation (10) is at most \[2d(f)(1-r)+4C_qA_h(f)(1-r).\] This bounds the \(h\)-area tail by a constant times \(1-r\). Uniform comparison of \(k\) and \(h\) then gives the second assertion in Equation (11). ◻

Proof. We first make precise the uniform choice of target charts for small source discs. Choose finitely many nested coordinate neighborhoods \[V_\alpha\Subset W_\alpha\Subset U_\alpha\] such that the \(V_\alpha\) cover \(X\) and each \(U_\alpha\) is a holomorphic coordinate chart. By compactness, there is a number \(\rho>0\) such that if \(y\in V_\alpha\), its metric ball of radius \(\rho\) is contained in \(W_\alpha\). On each \(\overline{W_\alpha}\), the metric \(k\) and the Euclidean coordinate metric are uniformly comparable; one comparison constant suffices for this finite collection.

For \(z\in\mathbb D\) put \(t=1-|z|\). Let \(0<c<1/4\), to be fixed independently of \(z\) and \(f\). For \(w\in B(z,ct)\), the straight segment from \(z\) to \(w\) stays at source distance at least \((1-c)t\) from \(\partial\mathbb D\). Equation (6) therefore bounds the length of its image by \[C\sqrt D\,\frac{c}{1-c},\] where \(C\) accounts only for the convention relating real and complex tangent norms. Choose \(c\) small enough that this bound is less than \(\rho\). Selecting \(\alpha\) with \(f(z)\in V_\alpha\), we obtain \[f\bigl(B(z,ct)\bigr)\subset W_\alpha.\]

In these coordinates the derivative of \(f\) is a vector of holomorphic functions. Its squared Euclidean norm is subharmonic. The mean-value inequality and the uniform metric comparisons give \[|f'(z)|_k^2 \leq\frac{C_1}{t^2} \int_{B(z,ct)}|f'(w)|_k^2\,dS(w).\] Every point of \(B(z,ct)\) has modulus greater than \(1-(1+c)t\). For \(t\) sufficiently small, the area-tail estimate in Equation (11) consequently gives \[\int_{B(z,ct)}|f'(w)|_k^2\,dS(w) \leq\pi\bigl(A_k(f)-A_{k,1-(1+c)t}(f)\bigr) \leq C_2t.\] This proves Equation (13).

For \(r<s<1\) sufficiently close to \(1\) and \(\zeta\in\partial\mathbb D\), integration along the radial segment gives \[\operatorname{dist}_k\bigl(f(r\zeta),f(s\zeta)\bigr) \leq C_3\int_r^s(1-u)^{-1/2}\,du \leq2C_3\sqrt{1-r}.\] The estimate is uniform in \(\zeta\). Since \(X\) is complete, each radial path has a limit, and the continuous maps \(\zeta\mapsto f(r\zeta)\) converge uniformly as \(r\uparrow1\). Their limit is therefore continuous on \(\partial\mathbb D\). The same uniform estimate, together with continuity of this boundary map, proves that the resulting extension of \(f\) is continuous on all of \(\overline\mathbb D\).

It remains to improve the integrability of the derivative. For all sufficiently large integers \(j\), let \[E_j=\{z\in\mathbb D:2^{-j-1}<1-|z|<2^{-j}\}.\] The area-tail estimate and Equation (13) imply \[\int_{E_j}|f'|_k^2\,dS\leq C_4\,2^{-j}, \qquad \sup_{E_j}|f'|_k\leq C_5\,2^{j/2}.\] Thus, for \(2<p<4\), \[\begin{split} \int_{E_j}|f'|_k^p\,dS &\leq\bigl(\sup_{E_j}|f'|_k\bigr)^{p-2} \int_{E_j}|f'|_k^2\,dS\\ &\leq C_p\,2^{-j(4-p)/2}. \end{split}\] This is summable in \(j\). On the remaining compact interior disc the derivative is bounded, so \(|f'|_k\in L^p(\mathbb D)\).

For completeness, the asserted Sobolev regularity is understood in local target coordinates. Continuity on \(\overline\mathbb D\) allows a finite cover of the closed source disc by small neighborhoods whose images lie in relatively compact target charts. In each such chart, the coordinate functions are bounded, are smooth in the source interior, and have first derivatives in \(L^p\) by the estimate just proved and metric comparison. These classical derivatives are their weak derivatives, as is seen by testing against compactly supported smooth functions in the source interior. Hence the coordinate functions lie in \(W^{1,p}\) up to the boundary, with the continuous boundary values already constructed. This proves the proposition. ◻

Proof. In a holomorphic target chart, the matrix of \(B\) is holomorphic and nonsingular. Thus \[\partial_z\partial_{\bar z}\log\det N =\partial_z\partial_{\bar z}\log\det h(f(z)) =-R_h(f',f').\] The first equality follows because \(\log|\det B|^2\) is harmonic; the formula is independent of the target chart. The right-hand side lies in \(L^{p_0/2}\). Apply Lemma 2 and use \(\log\det N=0\) on the boundary. ◻

Proof. The central term and differentiation under the integrals. Since \(F(t,0)=x\), the vector \(\partial_z F(t,0)\) is holomorphic in \(t\) and lies in the fixed vector space \(T_x^{1,0}X\). Its \(t_j\) derivative at zero is \(b_j(0)\). Differentiating its squared \(k(x)\) norm gives Equation (38). The holomorphic dependence of this central derivative also follows directly from Cauchy’s formula on a smaller source circle.

We next justify the variations under the area and Poisson integrals. Choose finitely many fixed source patches and target charts containing the image of \(F\) for sufficiently small \(t\). In each chart the map \(t\mapsto F(t,\cdot)\) is holomorphic into \(W^{1,p_0}\). The smooth metric coefficients and the product rule imply that \[t\longmapsto q(F_z(t,\cdot),F_z(t,\cdot))\] is twice continuously differentiable into \(L^{p_0/2}\); the same holds with \(h\). Indeed, each parameter derivative is a sum of products of two first source derivatives in \(L^{p_0}\), with uniformly bounded Sobolev coefficient factors. A fixed partition of unity in the source gives the global assertion. Since \(p_0/2>1\), integration against \(1\) or against \(\log(1/|z|)\) is a continuous functional on this space. Differentiation under both integrals is therefore valid.

Proof. Every sufficiently small parameter value gives an admissible finite-area disc with the same center. Hence the real-valued twice differentiable function \(J(F_t)\) has a local maximum at zero, and \(\mathcal LJ(F_t)\le0\). Proposition 13 gives the displayed inequality. The remaining assertions are Lemma 12 and the dilation inequality of Lemma 5. ◻

Proof. For \(0<\delta\leq 1\) and a prescribed center \(x\), let \(f\) maximize \[J_\delta(f)=d(f)+P_{R_k}(f)-\delta A_k(f).\] Such a maximizer exists by Proposition 4. Use the boundary-normalized holomorphic frame \(B\) associated to this maximizer and put \(N(z)=B(z)^*k(f(z))B(z)\). Set \[M=\sup_X|\mathop{\mathrm{tr}}_k R_k|.\] The dilation inequality, Equation (10), gives \(A_{R_k}(f)\geq-d(f)\geq-D\). Consequently, Equation (45), with \(h=k\) and \(q=R_k\), implies \[\mathop{\mathrm{tr}}N(0) \leq \delta\bigl(n-A_{R_k}(f)\bigr)+M \leq n+D+M.\] The determinant identity, Equation (37), and the arithmetic-geometric mean inequality yield \[P_{R_k}(f)=\log\det N(0) \leq n\log\!\left(\frac{n+D+M}{n}\right)=:C_1.\] Since \(A_k(f)\geq0\) and \(d(f)\leq D\), the maximal value of \(J_\delta\) is at most \(D+C_1\), independently of both \(x\) and \(\delta\). Now fix any finite-area disc \(g\), use its center in this maximization, and obtain \[d(g)+P_{R_k}(g)-\delta A_k(g)\leq D+C_1.\] Letting \(\delta\downarrow0\) for this fixed disc and using \(d(g)\geq0\) proves the assertion with \(C=D+C_1\). This limiting argument requires no area bound uniform in \(\delta\) for the maximizing discs. ◻

Proof. Fix \(x\in X\). Use the maximizer from Proposition 4 with \(q=-h\) and \(\delta=1\), so that its functional is \[J(f)=d(f)-P_h(f)-A_h(f).\] The constant disc centered at \(x\) has value zero. Consequently a maximizing disc \(f\) satisfies \[ P_h(f)+A_h(f)\le d(f)\le D. \tag{50}\] Both terms on the left are nonnegative. The assumed Ricci inequality therefore gives \[A_{R_h}(f)\ge-A_h(f)\ge-D, \qquad P_{R_h}(f)\ge-P_h(f)\ge-D.\]

Let \(B\) be the holomorphic frame along \(f\) normalized to be \(h\)-orthonormal on the boundary. Since \(\mathop{\mathrm{tr}}_h(-h)=-n\), the maximum inequality, Equation (45), yields \[\mathop{\mathrm{tr}}\bigl(B(0)^*k(x)B(0)\bigr) \le 2n-A_{R_h}(f)\le2n+D.\] Meanwhile, the determinant identity, Equation (37), gives \[\det\bigl(B(0)^*h(x)B(0)\bigr) =\exp\bigl(P_{R_h}(f)\bigr)\ge e^{-D}.\] The arithmetic–geometric mean inequality for the eigenvalues of the positive Hermitian matrix \(B(0)^*k(x)B(0)\) gives \[\det\bigl(B(0)^*k(x)B(0)\bigr) \le\left(\frac{2n+D}{n}\right)^n.\] Taking the quotient cancels the common factor \(|\det B(0)|^2\): \[\frac{h^n}{k^n}(x) =\frac{\det\bigl(B(0)^*h(x)B(0)\bigr)} {\det\bigl(B(0)^*k(x)B(0)\bigr)} \ge e^{-D}\left(\frac{n}{2n+D}\right)^n.\] The point \(x\) was arbitrary, proving the assertion. ◻

Proof. Write \[\alpha=[-R_k]=2\pi c_1(K_X).\] The nefness proved in Proposition 18 means that, for each \(t>0\), there is a smooth real function \(a_t\) for which \[-R_k+i\partial\bar\partial a_t\ge-\frac t2 k.\] It follows that \[\gamma_t=-R_k+tk+i\partial\bar\partial a_t\ge\frac t2 k>0\] is a Kähler form. Apply Theorem 20 with this form and with the smooth positive volume form \(\Omega_t=e^{a_t}k^n\). The resulting Kähler form \[h_t=\gamma_t+i\partial\bar\partial w_t\] satisfies \[ h_t^n=e^{a_t+w_t}k^n. \tag{51}\] Taking \(-i\partial\bar\partial\) of the logarithm of the ratio of the volume forms in Equation (51) shows, with our Ricci convention, that \[ R_{h_t}=R_k-i\partial\bar\partial(a_t+w_t)=-h_t+tk\ge-h_t. \tag{52}\] Proposition 19 therefore applies, with the same constant for every \(t>0\): \[\int_Xh_t^n\ge c_{n,D}\int_Xk^n.\] On the other hand \([h_t]=\alpha+t[k]\), so closedness and Stokes’ theorem give \[\int_Xh_t^n=\int_X(\alpha+t[k])^n.\] The expression on the right is a polynomial in \(t\). Passing to its value at zero yields \[ (2\pi)^n\int_Xc_1(K_X)^n =\int_X\alpha^n \ge c_{n,D}\int_Xk^n>0. \tag{53}\] This passage uses only cohomology. No convergence of the metrics \(h_t\), or uniform bounds on their potentials, is required. ◻

Proof. Put \(\beta=2\pi c_1(L)\) and \(V=\int_X\beta^n>0\). For \(0<\varepsilon\le1\), choose a smooth metric on \(L\) with real curvature \(\Theta_\varepsilon\ge-\varepsilon k\). Thus \[P_\varepsilon=\Theta_\varepsilon+2\varepsilon k\] is positive definite. At a point where \(\Theta_\varepsilon\) has a negative eigenvalue, write its eigenvalues relative to \(k\) as \(\lambda_1,\ldots,\lambda_n\) and choose \(j\) with \(\lambda_j<0\). Then \[|\lambda_j|\le\varepsilon, \qquad |\lambda_i|\le\lambda_i+2\varepsilon\quad(i\ne j).\] Consequently \[\left|\prod_{i=1}^n\lambda_i\right| \le\varepsilon\prod_{i\ne j}(\lambda_i+2\varepsilon) \le\varepsilon\sum_{j=1}^n\prod_{i\ne j}(\lambda_i+2\varepsilon).\] In terms of volume forms this is \[ |\Theta_\varepsilon^n|\le n\varepsilon\,P_\varepsilon^{n-1}\wedge k \tag{55}\] on the entire locus with a negative eigenvalue. The right-hand side has integral \[n\varepsilon\int_X(\beta+2\varepsilon[k])^{n-1}\wedge[k]=O(\varepsilon).\] The implied constant is independent of the chosen metric: the remaining cohomological integral is a fixed polynomial in \(\varepsilon\), bounded for \(0\le\varepsilon\le1\).

The degenerate locus contributes zero to \(\Theta_\varepsilon^n\). Removing the loci of index at least two from its total integral therefore changes that integral by at most \(O(\varepsilon)\) in absolute value. Since \(\int_X\Theta_\varepsilon^n=V\), we obtain \[\int_{X(\Theta_\varepsilon,\le1)}\Theta_\varepsilon^n \ge V-O(\varepsilon)>0\] for a sufficiently small fixed \(\varepsilon\). The criterion in Equation (54) applies to this one smooth metric and proves the assertion. In dimension one the locus of index at least two is empty; the same argument, with \(P_\varepsilon^0=1\), applies without modification. ◻

Proof. Kodaira’s lemma gives an integer \(m>0\), an ample Cartier divisor \(H\), and an effective divisor \(E\) such that \[ mK_X\sim H+E. \tag{56}\] For the needed form of this lemma, choose a smooth very ample hyperplane section \(H\). Bigness gives \(h^0(X,mK_X)\ge cm^n\) along sufficiently divisible powers, whereas the weak holomorphic Morse inequality gives \(h^0(H,mK_X|_H)=O(m^{n-1})\) (Demailly 1985, Theorem 0.1(a)). The restriction sequence therefore gives a nonzero section of \(mK_X-H\) for some \(m\), producing \(E\). For \(n=1\), \(H\) is a finite set and the same estimate is \(O(1)\).

Choose a sufficiently small positive rational number \(\eta\) so that \((X,\eta E)\) is Kawamata log terminal. This choice is possible even when \(E\) is singular or nonreduced. Choose a log resolution \(\pi:Y\to X\) of \((X,E)\), whose existence follows from (Kollár 2007, Theorem 35), and write \[K_Y-\pi^*K_X=\sum_i a_iF_i, \qquad \pi^*E=\widetilde E+\sum_i b_iF_i,\] where \(F_i\) are exceptional prime divisors and \(\widetilde E\) is the strict transform with its multiplicities. The discrepancy criterion for a Kawamata log terminal pair is given in (Fujino 2011, sec. 4.4). The exceptional discrepancies have the form \(a_i-\eta b_i\), with \(a_i\ge0\) because \(X\) is smooth and \(b_i\ge0\) because \(E\) is effective. There are finitely many such coefficients. Taking \(\eta>0\) sufficiently small keeps each discrepancy greater than \(-1\) and each strict-transform boundary coefficient less than one, which is the required condition.

The log cone theorem says that a projective Kawamata log terminal pair \((X,\Delta)\) over \(\mathbb C\) with effective rational boundary has a rational curve of negative \((K_X+\Delta)\)-degree whenever \(K_X+\Delta\) is not nef. This follows, for example, from the cone decomposition and rational-curve assertion in (Fujino 2011, Theorem 1.1, especially part (5)); for a Kawamata log terminal pair the non-log-canonical locus in that statement is empty. The theorem has no restriction on the dimension.

Since \(X\) has no rational curves, it follows that \[N=K_X+\eta E\] is nef, in the numerical sense that its degree on every curve is nonnegative. Combining this with Equation (56) gives \[(1+\eta m)K_X\sim_{\mathbb Q}N+\eta H.\] The sum of a nef rational divisor and an ample rational divisor on a projective variety is ample by Kleiman’s ampleness criterion (Fujino 2011, Definition 4.9 and Theorem 4.10): it is positive on every nonzero element of the closed cone of curves. Hence \((1+\eta m)K_X\) is ample. Clearing the positive rational coefficient shows that a positive integral multiple of \(K_X\) is ample, and therefore \(K_X\) itself is ample. ◻

Completion of the proof of Theorem 1. A rational curve in \(X\) would give a nonconstant holomorphic map \(\mathbb P^1\to X\) through its normalization. Its restriction to \(\mathbb C\subset\mathbb P^1\) would still be nonconstant, by the identity theorem. The hypothesis therefore excludes rational curves. We have proved that \(X\) is projective and \(K_X\) is big, so Proposition 24 proves that \(K_X\) is ample. Equivalently, a sufficiently high positive tensor power of \(K_X\) is very ample and its global holomorphic sections define a holomorphic embedding into projective space. This is the assertion in every positive complex dimension. ◻

Proof. Theorem 1 makes \(X\) smooth projective with \(K_X\) ample. Apply Fujita freeness in its all-powers form (OpenAI 2026a, Corollary 6.3) with \(L=K_X\) and exponent \(m-1\ge n+1\); the resulting globally generated adjoint bundle is \(K_X\otimes L^{\otimes(m-1)}=K_X^{\otimes m}\). ◻

Proof. Theorem 1 makes \(X\) projective. The semialgebraic-cover classification (OpenAI 2026b, Theorem 1.1) gives \(\widetilde X\simeq D\times\mathbb C^m\times F\), with \(F\) simply connected, normal and projective. Smoothness of this product forces \(F\) to be smooth. Brody hyperbolicity passes to covers and factors, so \(m=0\) and \(F\) is Brody hyperbolic. If \(\dim F>0\), Theorem 1 makes \(K_F\) ample, and Kobayashi’s finite-automorphism theorem (Kobayashi 1959, Theorem, p. 184) shows that \(\operatorname{Aut}(F)\) is finite. This also holds when \(F\) is a point.

For a deck transformation of \(D\times F\), the first coordinate is constant on each compact connected fiber \(F\). Applying the same observation to its inverse shows that its base map is an automorphism of \(D\) and its fiber maps are automorphisms of \(F\). Since \(D\) is connected and \(\operatorname{Aut}(F)\) is finite, these fiber maps are independent of the base point. Thus the deck group \(\Gamma\) acts by products. The kernel \(\Gamma_0\) of its homomorphism to \(\operatorname{Aut}(F)\) has finite index and acts faithfully and freely on \(D\). Proper discontinuity on \(D\) follows by applying the deck-action property to \(K\times F\) for compact \(K\subset D\); in particular \(\Gamma_0\) is discrete. Hence \(\widetilde X/\Gamma_0\simeq(D/\Gamma_0)\times F\) is a connected finite unramified holomorphic cover of \(X\), and therefore a finite étale cover of the projective manifold \(X\). Its compactness implies that \(D/\Gamma_0\) is compact. ◻

Proof. First, \(t\mapsto S_x(e^t)\) is nondecreasing and convex. To see convexity, fix \(0<r_1<r_2\leq R\) and compare the restriction of \(\psi\) to each complex line through \(x\) with the harmonic function on its annulus \(r_1<|z-x|<r_2\) that is affine in \(\log|z-x|\) and takes the constant values \(S_x(r_1)\) and \(S_x(r_2)\) on the boundary circles. The subharmonic maximum principle bounds \(\psi\) by this function on each annulus. On the inner ball the bound follows from the monotonicity of the interpolation. Taking the supremum over an intermediate ball proves convexity.

Apply this convexity at the three logarithmic radii \(\log r\), \(\log r+\log A\), and \(\log R\). It gives \[S_x(Ar)-S_x(r) \leq\frac{\log A}{\log(R/r)}\bigl(S_x(R)-S_x(r)\bigr),\] which implies Equation (57).

For fixed \(r\), the supremum over the closed translation ball is upper semicontinuous. Indeed, for a convergent sequence of centers, choose maximizing translation vectors and pass to a convergent subsequence in \(\overline B(0,r)\); upper semicontinuity of \(\psi\) gives the required upper bound. It is also a locally bounded supremum of the plurisubharmonic translations \(x\mapsto\psi(x+\zeta)\), \(|\zeta|\leq r\). Its upper semicontinuity therefore makes it plurisubharmonic. ◻

Proof of Lemma 17. Local potentials and regularization. Choose finitely many coordinate patches \(U_i\), smooth real functions \(a_i\) with \(\alpha=i\partial\bar\partial a_i\) on neighborhoods of \(\overline U_i\), and open sets \[V_i\Subset W_i\Subset U_i\] such that the \(V_i\) cover \(M\). Such local potentials exist because \(\alpha\) is a smooth closed real \((1,1)\)-form. On overlaps, \(a_i-a_j\) is pluriharmonic.

The distributional positivity of \(\alpha+i\partial\bar\partial u\) means that \(a_i+u\) has a unique plurisubharmonic representative \(\psi_i\). These representatives are locally bounded: their almost-everywhere bounds follow from boundedness of \(u\) and smoothness of \(a_i\), and the submean inequality and upper semicontinuity extend these bounds to every point. Moreover, \[\psi_i-\psi_j=a_i-a_j\] pointwise on overlaps, since the two plurisubharmonic functions \(\psi_i\) and \(\psi_j+a_i-a_j\) agree almost everywhere. Thus, after replacing \(u\) on a set of measure zero, we may write \(\psi_i=a_i+u\) everywhere. The functions \(\psi_i\) are bounded above and below on \(U_i\), with bounds uniform over the finite collection. The positive distances between the compact sets \(\overline W_i\) and the boundaries of \(U_i\) allow all subsequent regularizations to be defined on neighborhoods of \(\overline W_i\).

In the coordinates of \(U_i\), let \[S_{i,b}(x)=\sup_{|y-x|\leq b}\psi_i(y).\] Fix a smooth nonnegative kernel of integral one supported in the unit ball, and let \(\rho_{b/4}\) be its rescaling to radius \(b/4\). For all sufficiently small \(b>0\), set \[\varphi_{i,b}=S_{i,b}*\rho_{b/4}\] near \(\overline W_i\). Lemma 27 and convolution show that \(\varphi_{i,b}\) is smooth and plurisubharmonic. If \(|y-x|\leq b/4\), then \[B(x,b/2)\subset B(y,b)\subset B(x,2b).\] Taking suprema and then averaging proves \[ S_{i,b/2}(x)\leq\varphi_{i,b}(x)\leq S_{i,2b}(x). \tag{58}\]