Canonical ampleness of compact hyperbolic Kähler manifolds

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(\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 [4]. Suppose first that there are maps f_j\colon\D\to X with |f_j'(0)|_k\to\infty. Choose z_j at which the continuous function

attains its maximum on \{|z|\leq1/2\}. For all sufficiently large j this maximum is positive, so |z_j|<1/2. Put

The map g_j is defined on \{|w|<R_j\}, and

If |w|<R_j/2, then

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\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\D, apply this bound to w\mapsto f(z+(1-|z|)w). Its central derivative is (1-|z|)f'(z), proving Equation (3.1). 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

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

On the outer annulus r_0<|z|<1, the function

satisfies

On the inner disc, Lemma 3.1 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

Consequently,

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 (3.4) bounds its outer-annulus areas uniformly, and the compact-interior derivative estimate bounds its inner areas. Thus

By Lemma 3.1, 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

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

which proves attainment.□

Proof. Let f_r(z)=f(rz). A change of variables gives

The identity \log(1/|z|)=\int_{|z|}^1ds/s and Fubini’s theorem give

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

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 (3.5); its leftmost inequality uses the positivity of h.

Divide the rightmost expression in Equation (3.5) 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 (3.5) is at most

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 (3.6).□

Proof. We first make precise the uniform choice of target charts for small source discs. Choose finitely many nested coordinate neighborhoods

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\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\D. Equation (3.1) therefore bounds the length of its image by

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

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

Every point of B(z,ct) has modulus greater than 1-(1+c)t. For t sufficiently small, the area-tail estimate in Equation (3.6) consequently gives

This proves Equation (3.8).

For r<s<1 sufficiently close to 1 and \zeta\in\partial\D, integration along the radial segment gives

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\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\D.

It remains to improve the integrability of the derivative. For all sufficiently large integers j, let

The area-tail estimate and Equation (3.8) imply

Thus, for 2<p<4,

This is summable in j. On the remaining compact interior disc the derivative is bounded, so |f'|_k\in L^p(\D).

For completeness, the asserted Sobolev regularity is understood in local target coordinates. Continuity on \overline\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. The continuous pullback bundle over \overline\D is topologically trivial. Its coordinate transition matrices are W^{1,p}: they are smooth functions of f with bounded derivatives on the compact subsets of the target charts in use. Start with a continuous frame, approximate its coefficients uniformly in finitely many local frames, and combine the approximations with a smooth partition of unity on the source. This gives a W^{1,p} frame v uniformly close to the original one; sufficiently close approximation preserves its pointwise independence.

In the frame v, the Dolbeault operator on the open disc has the form

where we identify a (0,1)-form with its coefficient. Extend a by zero to a disc \D_R=\{|z|<R\} with R>1. On a sufficiently small disc U\subset\D_R of radius r, solve

Here and below the same notation for a matrix equation is understood entrywise. For completeness, the solid Cauchy transform

satisfies \bar\partial T_Ug=g weakly in U. Hölder’s inequality for its kernel and the L^p estimate for the planar Cauchy singular integral [28, Section 4.1 of the authors’ preprint] give

Thus U_0=I-T_U(aU_0) is solved by contraction in the sup norm when r is small. The radius can also be chosen so that the solution stays within distance 1/2 of I. It is therefore invertible and, by Equation (4.2), belongs to W^{1,p}.

For any two such solutions U_i,U_j, the transition matrix U_i^{-1}U_j satisfies \bar\partial(U_i^{-1}U_j)=0 weakly on the overlap. It is consequently holomorphic there. These transitions define a holomorphic vector bundle on \D_R. A holomorphic vector bundle on a disc is holomorphically trivial, by the Oka–Grauert principle [15, 12]. Choose a global holomorphic frame of this bundle. Relative to its local frames U_i, its coefficients are holomorphic; relative to the original trivial bundle they are therefore W^{1,p} on compact subsets of \D_R. In particular the resulting matrix U is continuous, invertible, and W^{1,p} on \overline\D, and satisfies \bar\partial U=-aU on \D. The frame e=vU has all the required properties.□

Proof. Choose e by Lemma 4.1 and set

This is a continuous, positive definite W^{1,p} matrix on the closed disc. In particular, there are constants 0<m\le M_0<\infty such that

The Hardy-space factorization. Write H^2=H^2(\partial\D,\C^n) for the vector Hardy space with nonnegative Fourier modes, and put \dd\sigma=\dd\theta/(2\pi). Equip the same vector space with the equivalent Hilbert norm

Multiplication by z is an isometry, and its closed image zH^2 has codimension n. Let s_1,\ldots,s_n be an orthonormal basis of W=H^2\ominus_M zH^2 and form the matrix S=(s_1,\ldots,s_n). The vectors z^j s_a, with j\ge0 and 1\le a\le n, are an orthonormal basis. Indeed, iterating H^2=W\oplus_M zH^2 shows that the orthogonal complement of their span is contained in every z^jH^2; the intersection of these spaces is zero.

Their orthogonality identifies all Fourier coefficients of the L^1 matrix S^*MS, and gives

Equation (4.4) then bounds the boundary values of S in L^\infty. Its holomorphic Hardy extension is bounded on \D, as follows entrywise from the Poisson integral. It is invertible at every point \zeta\in\D: evaluation at \zeta is a continuous surjection from H^2 onto \C^n, whereas the evaluations of all z^j s_a lie in the range of S(\zeta). Their dense span forces that range to be \C^n.

Regularity of the boundary factor. The factor constructed so far has only almost-everywhere boundary values. We now obtain continuity and a Sobolev exponent above two, which will also give nonsingularity at every boundary point. We first record the fractional regularity supplied by M\in W^{1,p}. If M_\theta(e^{i\alpha})=M(e^{i(\alpha+\theta)}), then

One can see this directly, without a trace theorem. Set t=|\theta| and choose \rho\in[1-2t,1-t] such that the angular L^p norm of \partial_\alpha M(\rho e^{i\alpha}) is at most Ct^{-1/p}\|M\|_{W^{1,p}}. Radial integration and Hölder’s inequality bound \|M|_{\partial\D}-M(\rho\,\cdot)\|_{L^p} by Ct^{1-1/p}\|M\|_{W^{1,p}}. At radius \rho, angular integration gives the same bound for translation through \theta. The triangle inequality proves Equation (4.6); the argument for Sobolev functions follows by approximation, with the continuous boundary values identifying the trace.

Let H^s(S^1) now denote the fractional Sobolev space on the circle, defined by square summability of Fourier coefficients with weight 1+|l|^{2s}. This differs from the Hardy-space notation H^2 above. It follows that M|_{\partial\D}\in H^s(S^1) for every

Indeed, on a circle of finite measure the L^p bound controls the L^2 bound, and integration of the squared rotation differences against |\theta|^{-1-2s}\,\dd\theta is finite. Parseval’s identity identifies this condition with \sum_{l\in\mathbb Z}(1+|l|^{2s})|M_l|^2<\infty.

Let \Pi denote the unweighted orthogonal Hardy projection. The Toeplitz operator

is bounded, self-adjoint, and satisfies \langle T_Mv,v\rangle\ge m\|v\|_2^2. Consequently it has a bounded inverse: its range is closed by the lower bound and dense because its orthogonal complement is the kernel of its adjoint. For a column v of S, membership in W says precisely that T_Mv is a constant vector. Rotating this identity and subtracting it from the original one yields

The boundedness of v and T_M^{-1} gives

The same rotation characterization therefore gives v\in H^s.

Write v(z)=\sum_{l\ge0}v_lz^l. Since s>1/2, Cauchy–Schwarz shows that this series converges absolutely and uniformly on \overline\D. Thus v is continuous there. More precisely, for 1/2<r<1, Parseval’s identity and Cauchy–Schwarz give

For any p_0>2, interpolation of these two estimates bounds \|v'(r\,\cdot)\|_{p_0}^{p_0} by a constant times

This is integrable in r provided

The interval is nonempty because s>1/2, and we choose p_0 also at most p. It follows that S\in W^{1,p_0}(\D).

By continuity, Equation (4.5) now holds at every boundary point. It implies that S is invertible there as well as in the interior. Compactness gives uniform nonsingularity on \overline\D. Finally B=eS is a continuous, nonsingular, W^{1,p_0} holomorphic frame and satisfies Equation (4.3). Notice that uniform nonsingularity was obtained after boundary regularity; it was not assumed in the Hardy-space argument.□

Proof. Fiber coordinates. Choose a frame e from Lemma 4.1 and write B=eS. The matrix S=e^{-1}B is holomorphic on \D and belongs to W^{1,p_0} on \overline\D. We first construct fiber coordinates near the graph of f. Choose a finite cover of \overline\D by relatively open sets U_i and target charts \phi_i\colon V_i\longrightarrow\C^n so that f(\overline U_i) is a compact subset of V_i, after shrinking the source sets as necessary. Let \rho_i be a smooth partition of unity on the closed disc subordinate to these sets. In the ith chart put

For z\in U_i, the matrix A_i is holomorphic in the interior, continuous and invertible up to the boundary, and of class W^{1,p_0}. The map C_i is holomorphic in y and, for fixed admissible y, holomorphic in the interior source variable z. On a neighborhood of the graph of f define

Each term is used only where its target chart is defined; the compact support of \rho_i inside the corresponding source patch makes extension by zero harmless. These neighborhoods can be chosen uniformly about the graph. More precisely, on sufficiently small source patches all indices whose supports meet the patch have their charts defined on one common target neighborhood of its image under f. This follows from finiteness of the cover and subordination of the supports. By construction,

Uniform inverses and their Sobolev dependence. Work on one of these smaller source patches, with fixed target chart \psi, and put c(z)=\psi(f(z)) and e_\psi(z)=D\psi(f(z))e(z). Normalize the forward map by

Then K_z(0)=0 and D_vK_z(0)=I. Continuity of all fiber derivatives and compactness allow a common R>0 such that the maps are defined on |v|\le R and

For |\xi|<R/2, the equation

is a contraction of the closed ball of radius R into itself. Its solution V(z,\xi) is continuous in (z,\xi) and holomorphic in \xi, as follows from its uniformly convergent contraction iterations. Define E by \psi(E(z,\xi))=c(z)+e_\psi(z)V(z,\xi). The local inverses agree near \xi=0 by uniqueness and then throughout their common fiber domain by holomorphy. Taking the minimum of the finitely many radii gives a common fiber neighborhood, with

and

We next obtain a Sobolev estimate uniform in the fiber parameter. The chart formula for K_z depends smoothly on the finite coefficient list

All these coefficients are W^{1,p_0} with compact range. Fix a smaller closed fiber polydisc inside |\xi|<R/2. The strict bounds in the contraction argument persist when the coefficient list varies in a sufficiently small neighborhood of its compact range. The finite-dimensional implicit function theorem, with the coefficients regarded as real parameters, therefore makes V a smooth function of that list and \xi, holomorphic in \xi. After shrinking this coefficient neighborhood, its first coefficient derivatives are uniformly bounded on the fixed fiber polydisc. The same is true for the coordinate inverse E_\psi(z,\xi):=\psi(E(z,\xi))=c(z)+e_\psi(z)V(z,\xi). The Sobolev chain rule now shows that E_\psi(\cdot,\xi) is W^{1,p_0}, with its weak source derivatives bounded by a single L^{p_0} function independent of \xi. In particular, for some b\in L^{p_0} on the source patch,

uniformly for \xi in that smaller polydisc. One may take b to be a constant plus the sum of the absolute values of the first weak derivatives of the finite coefficient list. Derivatives with respect to \xi satisfy the same type of bound after another fixed shrink of the polydisc, by Cauchy’s formula.

Holomorphic substitution. These estimates also justify holomorphy of the substitution operator

from a neighborhood of zero in W^{1,p_0}(\D,\C^n) into the local W^{1,p_0} space. Indeed, expand in the fiber variable:

Cauchy’s formula on a fixed fiber polydisc, with weak z differentiation under its contour integral, gives

for some r_1>0, by Equation (4.16). The algebra norm \|a\|_\infty+\|\nabla a\|_{p_0} is submultiplicative. Thus \sum_\alpha E_\alpha u^\alpha converges normally in the local W^{1,p_0} space when the global W^{1,p_0} norm of u is small; the number of multi-indices of a fixed degree grows polynomially. Each term is a continuous homogeneous polynomial in u, proving the assertion. This argument only concerns fixed local target coordinates; no linear structure on X is being presumed.

The nonlinear holomorphic-map equation. The fiber coordinates now have the necessary Sobolev regularity. It remains to correct their antiholomorphic derivative in the source while keeping the center and the prescribed first-order directions. For u\in W^{1,p_0}(\D,\C^n) close to zero, the Sobolev chain rule in a target chart gives

There is no \bar\partial\overline u term, since E is holomorphic in \xi. Thus the map is weakly holomorphic precisely when

The formula is independent of the chosen target chart, because the derivative of a holomorphic chart change multiplies both terms.

Differentiating \Phi(z,E(z,\xi))=\xi with \xi fixed and using E_\xi=(D_y\Phi)^{-1} gives

For the second equality we used holomorphy in z of C_i with y fixed. In particular Q is holomorphic in \xi and jointly continuous and bounded on the closed disc times a smaller closed fiber polydisc. Cauchy’s formula gives coefficients satisfying \|Q_\alpha\|_\infty\le Cr_2^{-|\alpha|} for some r_2>0. The normally convergent series \sum_\alpha Q_\alpha u^\alpha therefore shows that

is holomorphic near zero, using the embedding of W^{1,p_0} into the sup norm.

The constant and linear terms vanish. Indeed, C_i(z,f(z))=0 and D_yC_i(z,f(z))e(z)=I. Equations (4.15) and (4.19), together with \sum_i\bar\partial\rho_i=0, therefore give

Equivalently, \mathcal Q(0)=0 and D\mathcal Q(0)=0 as maps between the indicated Banach spaces.

The implicit function theorem with the center fixed. On the full disc, let T\colon L^{p_0}\to W^{1,p_0} be the solid Cauchy right inverse from Equation (4.2). Since point evaluation is bounded on W^{1,p_0}, the operator

is bounded and satisfies

Seek a solution of Equation (4.18) in the form

Since S is holomorphic and W^{1,p_0}, the equation becomes

This is a holomorphic equation with values in L^{p_0}, and Equation (4.21) gives

The complex Banach implicit function theorem [14, Theorem 2.3(ii), (d) of the author’s preprint] supplies a holomorphic solution g(t) for t in a neighborhood of zero. Differentiating the equation at zero gives

Consequently u(t) is holomorphic with values in W^{1,p_0}, is small in the sup norm for t sufficiently small, and satisfies

where e_j here denotes the jth standard vector of \C^n.

Set F(t,z)=E(z,u(t,z)). The identities in Equation (4.12) follow from Equation (4.15) and eS=B. Equation (4.17) gives its holomorphic parameter dependence in local W^{1,p_0} coordinates. In particular it is jointly continuous up to the source boundary. For each fixed t, Equation (4.18) makes its coordinate functions weakly holomorphic on the open disc. They are therefore ordinary holomorphic functions. For each fixed z, bounded point evaluation on W^{1,p_0} and holomorphy of E in its fiber variable give holomorphy in t. Local separate holomorphy now gives joint holomorphy on \{|t|<\eta\}\times\D.

Finally F(t,\cdot) is W^{1,p_0} with p_0>2 and has compact image, so its area with respect to any smooth Hermitian metric on X is finite. Thus all parameters in a sufficiently small complex neighborhood give admissible centered discs. The construction imposes no condition on their boundary maps and requires no extension across \partial\D.□

Proof. In a holomorphic target chart, the matrix of B is holomorphic and nonsingular. Thus

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 2.1 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 (5.2). 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

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.

The local identity and its Poisson integral. Locally write q=\ddc\phi. Joint holomorphy of F on the open product of the parameter domain and the disc gives

Commuting these derivatives and using F_{t_j}(0,z)=zb_j(z) yields the intrinsic identity

In particular, the mixed source derivative on the right belongs to L^{p_0/2} by the preceding differentiability argument.

The scalar function in parentheses is continuous on \overline\D and vanishes at zero. Lemma 2.1 therefore computes its Poisson integral from its boundary values. On the boundary, the columns of B are an h-orthonormal basis, so \tr(B^*qB)=\tr_h q. This proves Equation (5.3).

The area integral and its weak boundary limit. At the identity matrix, trace and log determinant have the same differential. The boundary normalization will therefore let us replace one radial derivative by the other, with an error controlled by the radial Sobolev estimate along suitable radii. For the area identity put T=\tr N. Integrating Equation (5.5) with q=h on |z|<r gives

The left-hand side tends to \cL A_h(F_t) by integrability. The first term on the right tends to n. To handle the second term without assuming boundary differentiability, observe that

All matrix norms here are equivalent, and uniform positive definiteness gives

The radial W^{1,2} estimate and the trace N=I give

This follows first on almost every radial line by the fundamental theorem for Sobolev functions, and then by Cauchy–Schwarz and integration over the angle. Smoothness inside the disc and continuity at the boundary give the same estimate for the interior radii used below.

Since \int_{1/2}^1\|N_r\|_2^2\,\dd r<\infty, there are radii r_\nu\uparrow1 for which (1-r_\nu)\|N_r(r_\nu,\cdot)\|_2^2 is bounded. Otherwise its eventual lower bound would contradict integrability. Along these radii, Equations (5.7) and (5.8) show that the two circular derivative means differ by o(1).

On the other hand, the curvature calculation in Lemma 5.1 and Equation (2.4) give, for every interior radius,

The right-hand side tends to -A_{R_h}(f). Substitute into Equation (5.6) along the chosen radii. Its left-hand side has a limit independent of the sequence, proving Equation (5.4).□

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 \cL J(F_t)\le0. Proposition 5.2 gives the displayed inequality. The remaining assertions are Lemma 5.1 and the dilation inequality of Lemma 3.3.□

Proof. For 0<\delta\leq 1 and a prescribed center x, let f maximize

Such a maximizer exists by Proposition 3.2. Use the boundary-normalized holomorphic frame B associated to this maximizer and put N(z)=B(z)^*k(f(z))B(z). Set

The dilation inequality, Equation (3.5), gives A_{R_k}(f)\geq-d(f)\geq-D. Consequently, Equation (5.9), with h=k and q=R_k, implies

The determinant identity, Equation (5.1), and the arithmetic-geometric mean inequality yield

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

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. Write L=\det T^{1,0}X and let \pi:Y=L\setminus 0_X\to X be the complement of its zero section. Define

This is a smooth finite-valued function on the connected complex manifold Y. In a local trivialization s=\lambda s_i(x) it has the expression

Since \lambda\ne0, the first summand is pluriharmonic. Therefore

Let v=\mathcal P\ell, using test discs holomorphic past the closed unit disc. For such a disc G with G(0)=s, its projection f=\pi\circ G has finite area. Green’s formula and Equation (6.2) give

Proposition 6.1 bounds every such mean below by \ell(s)-C, while the constant disc has mean \ell(s). Thus

The Poletsky–Rosay Theorem applies even though \ell is globally unbounded, and Equation (6.3) excludes its identically -\infty alternative. Hence v is plurisubharmonic.

For every \lambda\in\C^*, multiplication by \lambda is a biholomorphism of Y and gives a bijection between the test discs centered at s and at \lambda s. Since \ell(\lambda s)=\ell(s)+\log|\lambda|^2, it follows that

Thus v-\ell is constant on every fiber. It has the form u\circ\pi, where -C\leq u\leq0. Pulling back v by the local holomorphic map s_i proves that a_i+u is plurisubharmonic. It is locally bounded because a_i is smooth and u is bounded. The local currents \ddc(a_i+u) agree and equal -R_k+\ddc u.

To check the line-bundle sign explicitly, if s_i=g_{ij}s_j then

Let s_i^* be the dual frame of K_X=L^*. The prescription

is compatible with s_i^*=g_{ij}^{-1}s_j^*. Its curvature is \ddc(a_i+u)=T\geq0. In particular T represents 2\pi c_1(K_X), rather than 2\pi c_1(L).□

Proof. Apply Lemma 6.3 with M=X, \alpha=-R_k, and the bounded function u of Proposition 6.2. The resulting functions u_\eps give smooth canonical metrics with local squared norms \exp(-(a_i+u_\eps)) in the dual frames s_i^*. Their curvatures are -R_k+\ddc u_\eps\geq-\eps k.□

Proof. Fix x\in X. Use the maximizer from Proposition 3.2 with q=-h and \delta=1, so that its functional is

The constant disc centered at x has value zero. Consequently a maximizing disc f satisfies

Both terms on the left are nonnegative. The assumed Ricci inequality therefore gives

Let B be the holomorphic frame along f normalized to be h-orthonormal on the boundary. Since \tr_h(-h)=-n, the maximum inequality, Equation (5.9), yields

Meanwhile, the determinant identity, Equation (5.1), gives

The arithmetic–geometric mean inequality for the eigenvalues of the positive Hermitian matrix B(0)^*k(x)B(0) gives

Taking the quotient cancels the common factor |\det B(0)|^2:

The point x was arbitrary, proving the assertion.□

Proof. Write

The nefness proved in Proposition 6.4 means that, for each t>0, there is a smooth real function a_t for which

It follows that

is a Kähler form. Apply Theorem 8.1 with this form and with the smooth positive volume form \Omega_t=e^{a_t}k^n. The resulting Kähler form

satisfies

Taking -\ddc of the logarithm of the ratio of the volume forms in Equation (8.1) shows, with our Ricci convention, that

Proposition 7.1 therefore applies, with the same constant for every t>0:

On the other hand [h_t]=\alpha+t[k], so closedness and Stokes’ theorem give

The expression on the right is a polynomial in t. Passing to its value at zero yields

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<\eps\le1, choose a smooth metric on L with real curvature \Theta_\eps\ge-\eps k. Thus

is positive definite. At a point where \Theta_\eps has a negative eigenvalue, write its eigenvalues relative to k as \lambda_1,\ldots,\lambda_n and choose j with \lambda_j<0. Then

Consequently

In terms of volume forms this is

on the entire locus with a negative eigenvalue. The right-hand side has integral

The implied constant is independent of the chosen metric: the remaining cohomological integral is a fixed polynomial in \eps, bounded for 0\le\eps\le1.

The degenerate locus contributes zero to \Theta_\eps^n. Removing the loci of index at least two from its total integral therefore changes that integral by at most O(\eps) in absolute value. Since \int_X\Theta_\eps^n=V, we obtain

for a sufficiently small fixed \eps. The criterion in Equation (9.1) 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_\eps^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

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}) [7, 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 [20, Theorem 35], and write

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 [13, §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 \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 [13, 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

is nef, in the numerical sense that its degree on every curve is nonnegative. Combining this with Equation (9.3) gives

The sum of a nef rational divisor and an ample rational divisor on a projective variety is ample by Kleiman’s ampleness criterion [13, 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 1.1. A rational curve in X would give a nonconstant holomorphic map \mathbb P^1\to X through its normalization. Its restriction to \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 9.3 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 1.1 makes X smooth projective with K_X ample. Apply Fujita freeness in its all-powers form [23, 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 1.1 makes X projective. The semialgebraic-cover classification [24, Theorem 1.1] gives \widetilde X\simeq D\times\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 1.1 makes K_F ample, and Kobayashi’s finite-automorphism theorem [18, 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

which implies Equation (A.1).

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 6.3. Local potentials and regularization. Choose finitely many coordinate patches U_i, smooth real functions a_i with \alpha=\ddc a_i on neighborhoods of \overline U_i, and open sets

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+\ddc 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,

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

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

near \overline W_i. Lemma A.1 and convolution show that \varphi_{i,b} is smooth and plurisubharmonic. If |y-x|\leq b/4, then

Taking suprema and then averaging proves

Comparison on overlaps. We next prove the uniform overlap comparison

All coordinate changes and their inverses have bounded derivatives on neighborhoods of the compact overlap sets \overline W_i\cap\overline W_j. After decreasing the allowed radius, coordinate balls centered on these overlaps are therefore comparable by fixed factors. Thus there are constants c,C>0, independent of the center and of b, such that an i-coordinate ball of radius b contains the j-coordinate ball of radius cb and is contained in the j-coordinate ball of radius Cb. The balls in question lie in U_i\cap U_j.

The identity \psi_i-\psi_j=a_i-a_j holds on this intersection, and the smooth function a_i-a_j varies by O(b) across any such ball, uniformly in the center. These facts, together with Equation (A.2), give constants c_1,c_2,C_2>0 for which

The analogous bounds for \varphi_{j,b}-a_j have radii b/2 and 2b. Lemma A.1, using a fixed radius available near each compact overlap, compares all four suprema. It bounds their differences by O(1/|\log b|). This proves Equation (A.3), in fact with an error bounded by O(1/|\log b|)+O(b). All constants can be chosen uniformly because there are only finitely many pairs of patches.

Gluing the local functions. Choose smooth functions \chi_i on neighborhoods of \overline W_i such that

There is a fixed constant C_0\geq0 satisfying \ddc\chi_i\geq-C_0k for every i. Given \sigma>0, choose b so small that every discrepancy in Equation (A.3) is less than \sigma, and set on W_i

Here is a precise regularized maximum with the compatibility needed when the set of available patches changes. Fix a smooth even probability density \eta_\sigma on \R, supported in (-\sigma/4,\sigma/4). For each nonempty finite index set I define

This is smooth, convex, nondecreasing in every argument, and satisfies

If t_i is more than \sigma/2 below the maximum of the other arguments, then it never attains the shifted maximum in the integrand. Its removal leaves exactly \mathcal M_{I\setminus\{i\}}, because the unused probability density integrates to one.

For x\in M, put I(x)=\{i:x\in W_i\} and define

This is globally smooth. To verify the only issue, fix x_0\in\partial W_i and choose j with x_0\in V_j. On a neighborhood of x_0 within W_i, we have \chi_i=-3 and \chi_j=0, so the overlap comparison implies

Thus branch i is inactive throughout this neighborhood, including under all the allowed shifts. The exact deletion property just proved removes it from the formula before crossing \partial W_i. There are finitely many indices, so this argument removes all indices whose patch boundaries pass through x_0, leaving a smooth formula on a neighborhood of x_0.

The curvature estimate. Work near any point in a fixed coordinate patch with local potential a_j for \alpha. Each active branch has

The difference a_j-a_i is pluriharmonic, since both functions are local potentials for \alpha. Hence

By Equation (A.4), a_j+u_\sigma is the regularized maximum of these branches. The first derivatives of \mathcal M_I are nonnegative and sum to one; its Hessian is positive semidefinite. The complex Hessian chain rule therefore expresses \ddc(a_j+u_\sigma) as a convex combination of the branch Hessians plus a nonnegative (1,1)-form. Equation (A.5) consequently gives

Choose \sigma>0 with C_0\sigma\leq\eps, and rename this function u_\eps. The resulting smooth closed form represents [\alpha] and has the required lower bound.□