Canonical ampleness of compact hyperbolic Kähler manifolds

1. Two uses of the same variation

Maximize $J=d+P_q-\delta A_h$ among finite-area discs with a fixed center. The area penalty gives attainment, and comparison with dilations supplies boundary regularity. Compute the Hessian along centered variations in a boundary-$h$-orthonormal holomorphic frame. The choice $h=k,q=R_k,\delta\downarrow0$ bounds the Ricci integral of every test disc; $q=-h,\delta=1$ gives a lower volume bound under $R_h\ge-h$.

Proposition 3.2; Lemma 3.3 · pp. 6–8 · Proposition 5.2 · pp. 17–18 · Proposition 6.4; Lemma 6.3 · p. 21 · Proposition 7.1 · p. 22

2. Feed the nef class into the volume estimate

Put $\alpha=2\pi c_1(K_X)$. Nefness makes $\alpha+t[k]$ a Kähler class for $t>0$. The negative-sign Monge–Ampère equation supplies $[h_t]=\alpha+t[k]$ with $R_{h_t}=-h_t+tk$. Integrate the uniform lower bound and let $t\downarrow0$ in cohomology to obtain $\int_Xc_1(K_X)^n>0$.

Monge–Ampère existence [MA] · §1, equation (1.1) · print pp. 366–367 / PDF pp. 3–4 · Proposition 8.2 · pp. 23–24

3. Prepare the Morse criterion and obtain projectivity

For a nef line bundle $L$, put $\beta=2\pi c_1(L)$ and choose $\Theta_\varepsilon\ge-\varepsilon k$. On the locus with a negative eigenvalue,

The integral on the right is $O(\varepsilon)$ by cohomology. If $\int_X\beta^n>0$, the integral on the nondegenerate locus of index at most one is positive as well. The Morse inequalities make $L$ big and $X$ Moishezon. Apply this with $L=K_X$, then use Namikawa’s projectivity theorem with the retained Kähler and smoothness assumptions. Projectivity was not an initial hypothesis.

Lemma 9.1 · pp. 24–25 · Holomorphic Morse inequalities [HM] · (0.7); Theorem 0.8(a) · print pp. 193–194 / PDF pp. 6–7 · Theorem 9.2 · p. 25 · Moishezon projectivity [MP] · Theorem 6 · p. 3

4. From absence of rational curves to ampleness and an embedding

A nonconstant map $\mathbb P^1\to X$ would contradict hyperbolicity after restriction to $\mathbb C$. Now $X$ is smooth and projective and $K_X$ is big, so the hypotheses of the criterion recorded by Diverio–Trapani hold. Proposition 9.3 proves that criterion again through the cone theorem as follows.

Choose a smooth very ample hyperplane section $H$. Compare the growth of sections from bigness with the $O(m^{n-1})$ bound on $H$ from the weak Morse inequality, using the restriction exact sequence. This yields $mK_X\sim H+E$ for some $m>0$ and effective $E$.

Holomorphic Morse inequalities [HM] · Theorem 0.1(a) · print p. 189 / PDF p. 2

Choose a sufficiently small positive rational $\eta$ so that $(X,\eta E)$ is klt. Absence of rational curves and the log cone theorem make $N=K_X+\eta E$ nef. Hence

is ample. Clearing the positive rational coefficient makes $K_X$ itself ample, and a sufficiently high positive tensor power gives the projective embedding.

Proposition 9.3 · pp. 25–26 · Canonical ampleness criterion [DT] · Lemma 2.1 · p. 3 · Log cone and ampleness criteria [LC] · Theorem 1.1(5) · print pp. 728–729 / PDF pp. 2–3 · Log cone and ampleness criteria [LC] · §4.4; Theorem 4.10 · print pp. 737–738 / PDF pp. 11–12 · §9; completion of Theorem 1.1 · pp. 24–26

1. Area penalty and the boundary tail

Lemma 3.1 carries out Brody reparametrization to bound the central derivative. On an outer annulus, $-\delta A_h$ dominates the logarithmic term; Fatou’s lemma and normal convergence in the interior give attainment. Comparing with $f(rz)$ yields $A_q(f)\ge-d(f)$ and $A_k(f)-A_{k,r}(f)\le C_f(1-r)$. Local mean-value estimates and summation over dyadic annuli give continuity up to the boundary and $W^{1,p}$ regularity for $2<p<4$. No uniformity of $C_f$ over all metrics is asserted.

Proposition 3.2; Lemma 3.3 · pp. 6–8 · Proposition 3.4 · pp. 8–9

2. Realize the directions by genuine disc families

Lemma 4.1 uses the Cauchy–Green right inverse and the Oka principle to trivialize a holomorphic bundle over a slightly larger disc. Proposition 4.2 normalizes the frame in a weighted Hardy space; rotation-difference estimates for a Toeplitz operator then supply boundary $W^{1,p_0}$ regularity and nonsingularity. Proposition 4.3 subtracts the central value from the right inverse, $Hg=Tg-(Tg)(0)$, and applies the complex Banach implicit function theorem to

This preserves both the center and the prescribed first variation. The background on spectral factorization is not treated as automatically supplying the required boundary regularity.

Lemma 4.1; Proposition 4.2 · pp. 10–12 · Proposition 4.3 · pp. 13–16 · Cauchy–Green operator [CG] · §4.1 · p. 11 · Oka principle [OK] · Theorem 1.3; §1, example (A) · pp. 1–3 · Banach implicit functions [IF] · Theorem 2.3(ii), (d) · p. 11

3. Pass through the weak boundary limit

The inequality $p_0/2>1$ justifies parameter differentiation of the area and Poisson integrals. Closedness gives the local identity $\mathcal Lq(F_z,F_z)=\partial_z\partial_{\bar z}(|z|^2\operatorname{tr}(B^*qB))$. For $N=B^*hB$, the boundary value $N=I$ allows replacement of $\partial_r\operatorname{tr}N$ by $\partial_r\log\det N$ along suitable radii. The error vanishes by a radial $W^{1,2}$ estimate. Green’s formula gives the three identities and $\log\det N(0)=P_{R_h}(f)$; maximality then gives $\mathcal LJ\le0$. No holomorphic extension across the boundary is used.

Proposition 5.2 · pp. 17–18 · Lemma 5.1 · p. 17 · Corollary 5.3 · p. 19

1. From maximizers to every test disc

Take $h=k,q=R_k$ and maximize for $0<\delta\le1$. Corollary 5.3 and $A_{R_k}(f)\ge-D$ imply $\operatorname{tr}N(0)\le n+D+M$, where $M=\sup_X|\operatorname{tr}_kR_k|$. The determinant identity and arithmetic–geometric mean bound $P_{R_k}(f)$ uniformly from above. Fix an arbitrary finite-area disc $g$, compare it with the maximum, and only then let $\delta\downarrow0$. This gives $P_{R_k}(g)\le C$ without a uniform area bound on the maximizing discs in this limit.

Proposition 6.1 · pp. 19–20 · Corollary 5.3 · p. 19

2. Turn envelope homogeneity into a canonical metric

On $Y=(\det T^{1,0}X)\setminus0_X$, put $\ell(s)=\log|s|_k^2$ and let $v$ be its Poisson envelope. The preceding estimate on projected discs gives $\ell-C\le v\le\ell$, excluding the identically $-\infty$ alternative in the Poletsky–Rosay theorem. Homogeneity $v(\lambda s)=v(s)+\log|\lambda|^2$ gives $v-\ell=u\circ\pi$ with $-C\le u\le0$. For local frames $s_i$ and $a_i=\log|s_i|_k^2$, the weights $a_i+u$ are bounded and psh. Setting $|s_i^*|^2=e^{-(a_i+u)}$ on the dual frames yields curvature $-R_k+i\partial\bar\partial u\ge0$ on $K_X$.

Proposition 6.2 · pp. 20–21 · Poletsky–Rosay envelopes [PR] · Theorem 1.1 · print p. 1407 / PDF p. 1

3. Smooth without assuming continuity

Lemma 6.3 supplies smoothing of the bounded potential. Appendix A compares local suprema of psh potentials at fixed multiples of a radius, bounding overlap errors by $O(1/|\log b|)+O(b)$. Choose a radius making this error small and glue by a regularized maximum after cutoff corrections. The curvature loss can be made arbitrarily small. Applying this to Proposition 6.2 proves the claim. The alternative citation to Demailly’s regularization theorem mentioned in the manuscript is not additionally checked here.

Proposition 6.4; Lemma 6.3 · p. 21 · Appendix A; Lemma A.1 · pp. 27–30

1. Compare with the constant disc

At each point choose a maximizer with $q=-h,\delta=1$. Comparison with the constant disc, whose value is zero, gives $P_h(f)+A_h(f)\le d(f)\le D$. The curvature hypothesis implies $P_{R_h}(f),A_{R_h}(f)\ge-D$. Insert $\operatorname{tr}_h(-h)=-n$ into the maximizing inequality to obtain $\operatorname{tr}(B(0)^*kB(0))\le2n+D$.

Proposition 7.1 · p. 22 · Corollary 5.3 · p. 19

2. Cancel the frame determinant

Lemma 5.1 gives $\det(B(0)^*hB(0))=e^{P_{R_h}(f)}\ge e^{-D}$. The preceding trace bound controls the denominator determinant by arithmetic–geometric mean. Taking their ratio cancels $|\det B(0)|^2$ and gives

The auxiliary disc, frame and size of the variation may depend on $h$ and the point. None appears in the final constant.

Lemma 5.1 · p. 17 · Proposition 7.1 · p. 22

1. Existence for an arbitrary positive volume form

Put $\alpha=[-R_k]$ and use nefness to choose $-R_k+i\partial\bar\partial a_t\ge-(t/2)k$. Apply Theorem 8.1 to $\gamma_t=-R_k+tk+i\partial\bar\partial a_t>0$ to obtain

The prescribed volume form is $e^{a_t}k^n$; no prior normalization of its total volume is needed because the additive constant of the solution enters the right-hand side. Logarithmic differentiation gives $R_{h_t}=-h_t+tk\ge-h_t$.

Theorem 8.1 · p. 23 · Monge–Ampère existence [MA] · §1, equation (1.1) · print pp. 366–367 / PDF pp. 3–4 · Proposition 8.2 · pp. 23–24

2. Take the limit of the intersection polynomial

Apply Proposition 7.1 with the same constant for every $t>0$. Since $[h_t]=\alpha+t[k]$,

The left-hand side is a polynomial in $t$. Evaluation at zero gives $(2\pi)^n\int_Xc_1(K_X)^n\ge c_{n,D}\int_Xk^n>0$. No convergence of $h_t$ or its potentials is assumed in this step.

Proposition 7.1 · p. 22 · Proposition 8.2 · pp. 23–24

1. Meet the hypotheses of the input

Theorem 1.1 makes $X$ a smooth connected projective variety with $K_X$ ample. Hence the Fujita freeness input from catalogue 038 applies with $L=K_X$. This input is not used retroactively in the proof of the main theorem.

Theorem 1.1 · p. 1 · Fujita freeness [FF] · Corollary 6.3 · p. 23

2. Preserve the quantifier over all integers

For every integer $m\ge n+2$, one has $m-1\ge n+1$. Corollary 6.3 of the input therefore makes

globally generated. The all-exponents corollary, rather than only the smallest-exponent main theorem, is the required input. The article on 038 describes that extension.

Corollary 10.1 · p. 26 · Fujita freeness [FF] · Corollary 6.3 · p. 23

1. Remove the extra factor in the classification

First use Theorem 1.1 to make $X$ projective, then apply Theorem 1.1 of 058 to its ordinary universal cover. The resulting decomposition is $\widetilde X\simeq D\times\mathbb C^m\times F$, with $F$ simply connected, normal and projective. Smoothness of the product forces $F$ to be smooth. Hyperbolicity passes to covers and factors, so $m=0$ and $F$ is Brody hyperbolic. The proof of the classification theorem itself is treated as an external input.

Corollary 10.2 · pp. 26–27 · Semialgebraic universal covers [SC] · Theorem 1.1 · p. 1

2. Use finite automorphisms to obtain a product action

If $\dim F>0$, reapply Theorem 1.1 to $F$ to make $K_F$ ample. Thus $c_1(F)<0$, and Kobayashi’s theorem makes $\operatorname{Aut}(F)$ finite. Finiteness is immediate when $F$ is a point. The $D$-component of a deck transformation is constant on compact connected $F$; applying the same observation to the inverse gives an automorphism of the base. The remaining $F$-component is a family in a finite group over connected $D$, hence independent of the base. The argument includes the case where $D$ is a point.

Corollary 10.2 · pp. 26–27 · Finite automorphisms [KA] · Theorem · print p. 184 / PDF p. 1

3. Check the action of the finite-index kernel

Let $\Gamma_0$ be the kernel of the homomorphism from the deck group $\Gamma$ to $\operatorname{Aut}(F)$. It has finite index and acts faithfully and freely on $D$. For compact $K\subset D$, the set $K\times F$ is compact, so proper discontinuity of the deck action descends to $D$, and $\Gamma_0$ is discrete. The quotient is $(D/\Gamma_0)\times F$. Its map to $X$ is a connected finite unramified cover, hence a finite étale cover of the projective variety. Compactness of this cover makes $D/\Gamma_0$ compact, proving cocompactness as well.

Corollary 10.2 · pp. 26–27