The abelianity conjecture for special compact Kähler manifolds

Proof. By Campana’s established specialness implication, X is special. Theorem 1.1 therefore applies.□

Proof. Theorem 1.1 supplies the Abelianity Conjecture. Campana proves that the Abelianity Conjecture implies Conjecture S in [18, Remark A.4(3)]. We spell out the finite-cover step. A finitely generated virtually abelian group has a torsion-free abelian subgroup of finite index, so take the corresponding connected finite étale cover X'. Specialness persists on X'. Maximal ordinary \Gamma-dimension persists as well. A positive-dimensional compact subvariety through a very general point with finite group image upstairs would have an image of the same dimension downstairs. The induced map between the normalizations is finite étale, so their fundamental-group images differ by finite index. The image downstairs would therefore have finite group image, contrary to the \Gamma-reduction criterion.

The Albanese map of X' is surjective with connected fibres by Theorem 2.1. Since \pi_1(X') is torsion-free abelian, its map to the Albanese lattice H_1(X',\Z)/\mathrm{torsion} is an isomorphism. The group of a smooth general Albanese fibre therefore has trivial image in \pi_1(X'). Maximal ordinary \Gamma-dimension forces that fibre to have dimension zero. Connectedness makes the general fibre one point, so the Albanese map is bimeromorphic to its torus target. This proves (i). Campana’s second implication, Conjecture S to the stated bimeromorphic Iitaka conclusion, is [18, Remark A.4(2)]; it gives (ii).□

Proof. The group G=\pi_1(X) is finitely generated because X is a compact smooth manifold. By Theorem 1.1, it has an abelian subgroup H of finite index, which is also finitely generated. Write H\cong\Z^r\oplus T with T finite. Decomposing T into cyclic factors gives a faithful finite-dimensional complex representation \sigma:H\to\operatorname{GL}(V): represent each infinite cyclic factor by powers of 2 on its own coordinate and each finite cyclic factor by a primitive root of the corresponding order on its own coordinate. For the trivial group, use the one-dimensional trivial representation.

The induced representation \operatorname{Ind}_H^G\sigma is finite dimensional because [G:H]<\infty, and it is faithful. Indeed, in its direct-sum realization indexed by the left cosets G/H, an element of its kernel must preserve the summand indexed by H, so it belongs to H; its action on that summand is given by the faithful representation \sigma. Thus G admits a faithful finite-dimensional complex linear representation. The compact Kähler linear Shafarevich theorem of Campana–Claudon–Eyssidieux [15, author’s version, Corollary 5.9] now gives the claimed holomorphic convexity of the ordinary universal cover.□

Proof. Suppose (i) has a counterexample and choose one of minimal base dimension. The cover is normal because it is étale over normal X; normalizing an analytic compactification leaves this open set unchanged. Claudon–Höring’s minimal-counterexample reduction [18, Proposition 1.4] says that this minimal base is smooth and special. Theorem 1.1 makes its fundamental group virtually abelian. Its quotient \Gamma is then virtually abelian, a contradiction.

For (ii), if \pi_1(X) is infinite, apply (i) to the ordinary universal cover; if it is finite, it is already virtually abelian. The Kähler compactification hypothesis is the one in [18, Definition 2.3], so [18, Theorem 1.5] gives the asserted Albanese bundle on a connected finite étale cover X'. Pull this bundle back along the universal cover \C^q\to A. The automorphism group of the compact Kähler fibre F is a complex Lie group, as recalled in [18, Section 2.C]. The associated principal \operatorname{Aut}(F)-bundle is topologically trivial over the contractible base \C^q, and the Oka–Grauert principle makes it holomorphically trivial over this Stein base. Thus the pullback is F\times\C^q, the product consequence also recorded in [18, Conjecture 1.1]. Since F is simply connected, the pullback is the ordinary universal cover of X', and hence also of X.□

Proof. We give the standard torus argument; see also [19]. After multiplying D by a positive integer, assume it is integral. Average its positive integration current [D] over translations of A. The result is a smooth translation-invariant semipositive (1,1)-form \theta representing c_1(\mathcal O_A(D)). Write A=V/\Lambda. The alternating form of \theta is integral on \Lambda, so its radical W is defined over \mathbb Q. It is a complex subspace because \theta has type (1,1). Thus K=W/(W\cap\Lambda) is a subtorus. Let p:A\to B=A/K be the quotient.

On every K-coset, \mathcal O_A(D) has zero real first Chern class. If its defining section is not identically zero on that coset, its zero divisor must be empty: a nonzero effective divisor on the positive-dimensional Kähler torus K has strictly positive pairing with the (\dim K-1)-st power of a Kähler class. For K=0 the same conclusion is immediate. Each coset is therefore either contained in \operatorname{Supp}D or disjoint from it. If B were a point, this argument with K=A would make the defining section of D nowhere zero, contrary to D\ne0. Hence \dim B>0.

Translations by the connected group K consequently preserve each prime component D_i of D, since a connected group cannot permute their finite set. The image \overline D_i=p(D_i) is a prime divisor on B: properness gives an analytic image, and invariance gives \dim\overline D_i=\dim D_i-\dim K=\dim B-1. Moreover p^{-1}(\overline D_i)=D_i, and smoothness of p gives p^*\overline D_i=D_i with multiplicity one. Keeping the coefficients of the D_i therefore defines an effective integral divisor D_B with D=p^*D_B.

Since p^*D_B=D, uniqueness of translation-invariant cohomology representatives identifies c_1(\mathcal O_B(D_B)) with the positive definite form induced by \theta on B. The positivity criterion for line bundles on a complex torus makes D_B ample and B an abelian variety. Dividing D_B by the initial integer proves the rational statement. Ratios of sections of its ample multiples remain independent after pullback, so \kappa(A,D)\ge\dim B>0.□

Proof. Let M be the manifold, and regard \rho as a Zariski-dense representation into its semisimple closure. By [15, author’s version, Theorem 1 and Theorem 6.3], it has a Shafarevich morphism

with normal projective target of general type. This morphism is a model of the connected \Gamma-reduction [15, author’s version, Definition 2.13]. Generic largeness gives \dim\operatorname{Sh}_\rho(M)=\dim M. The connected general fibre is then a point, so \operatorname{sh}_\rho is bimeromorphic. Bimeromorphic invariance of Kodaira dimension proves that M is of general type.□

Proof. The centralizer E_0 of F has finite index. Its commutator subgroup is contained in the finite central group F\cap E_0. Choose generators x_1,\ldots,x_t of E_0 and an integer e>0 annihilating F. Commutators are central, so [x_i^e,x_j^e]=[x_i,x_j]^{e^2}=1. The subgroup generated by the x_i^e is abelian and has finite index: its image has finite index in the finitely generated abelianization, and the commutator subgroup is finite. This abelian subgroup has a torsion-free subgroup of finite index. Its core in E is a normal free abelian subgroup of finite index. Multiples of this core have finite index in E and separate its nonzero elements; the finite quotient by the core separates elements outside it. Hence E is residually finite. Since F is finite, intersecting finitely many finite-index subgroups separates all its nonidentity elements at once.□

Proof. A finitely generated group has only finitely many subgroups of any given finite index. Intersect all automorphic images of the core of N_0 in N. The result is a characteristic finite-index subgroup K\le N contained in N_0, and hence is normal in G. The group G/K is finite-by-\Z^r. By Lemma 2.6, it has a finite-index subgroup disjoint from N/K. Its inverse image is a suitable G_0.□

Proof. Fix a nonnegative f\in C_c^\infty(U) which is strictly positive on a chart ball, and put

We solve

We give the complete continuity argument, including the global estimates which distinguish this problem from a bounded-solution existence result.

Fix an integer k>n+6 and consider

This is open in H^{k+2}. The map u\mapsto\log(\omega_u^n/\omega^n)-\lambda u is smooth from \mathcal B to H^k, by Sobolev multiplication and composition in the uniform charts. Its linearization at u is

At a smooth solution with bounded derivatives this is an isomorphism H^{k+2}\to H^k. Indeed its negative is coercive on H^1 when integrated with \omega_u^n: its quadratic form is a positive multiple of the gradient norm plus \lambda\|v\|_2^2. Lax–Milgram gives an H^1 inverse, and (1) gives the asserted higher regularity and bounded inverse. The metric \omega_u is complete and uniformly equivalent to \omega at this individual solution. A solution in \mathcal B is smooth with bounded derivatives by local elliptic bootstrapping, since initially k>n+6. The implicit function theorem therefore gives openness for

At s=0 the solution is u=0.

We next obtain estimates uniform in s, with N fixed. Every solution in \mathcal B tends to zero at infinity with its first several derivatives. At an interior maximum of u, the determinant ratio is at most one; at an interior minimum it is at least one. The same bounds hold when an extremum is approached at infinity, where u tends to zero. Consequently

In particular the determinant ratio in (7) has positive upper and lower bounds depending only on N.

Here are the second-order and higher local estimates in the order in which they are needed. Write \Delta_u=\tr_{\omega_u}\dd\dc. Since \tr_\omega\dd\dc u\ge-n,

The differential trace estimate [16, Lemma 2.2] is

It gives

In this estimate B depends on a lower bound for the background holomorphic bisectional curvature. The possible negative term involving \Delta_\omega(sF_N+\lambda u) is divided by \tr_\omega\omega_u, which is bounded below by the determinant lower bound. Apply (9) to Q=\log\tr_\omega\omega_u-Au, where A>B+1 is fixed. Since \Delta_u u=n-\tr_{\omega_u}\omega, at a maximum of Q one obtains

If the supremum is not attained, it is bounded by the value \log n at infinity. At an attained maximum, the determinant upper bound and the inverse-trace bound control \tr_\omega\omega_u there. The oscillation bound in (8) then controls it everywhere. It follows that

uniformly along the path.

This also bounds the background real Laplacian of u. Local Poisson W^{2,p} estimates, for p larger than the real dimension, give a uniform C^{1,\alpha} bound on smaller charts. The equation now has a uniformly elliptic complex Hessian and a controlled Hölder right side. Fix 0<\beta<\alpha. The local complex Monge–Ampère C^{2,\beta} estimate [39, arXiv:1111.0902v1, Theorem 1.1] applies after adding a local potential for \omega: the resulting function is strictly plurisubharmonic with local sup norm controlled by (8), its Euclidean Laplacian is bounded by (10), and the positive density is bounded below with its n-th root uniformly C^\alpha. Differentiation of the equation and local Schauder estimates then give bounds for every derivative. These are local bounds uniform over all lifted charts and all s, with constants depending on N and the derivative order; no global higher Sobolev bound has been assumed in obtaining them.

There is a global estimate with a better dependence on N. Integration by parts gives

The first mixed term is the fixed background metric, so the lower constant is independent of N. For the upper bound use u\le0 and

The integrand in (11) is absolutely integrable: away from \supp f, its absolute value is at most \lambda u^2, and on \supp f it is integrable. To justify the integration by parts, insert the complete-metric cutoffs. The mixed metrics are bounded by (10), and the cutoff errors tend to zero by u,\dd u\in L^2. Combining (12) with (2) gives

uniformly in s and N.

It remains to establish closedness in the global Banach space. In local coordinates, (7) can be written

The superscript i\bar j denotes the corresponding entry of the inverse Hermitian matrix. The coefficients have uniform ellipticity and all required derivative bounds by the preceding local estimates. Thus (1) and (13) give global bounds in every fixed Sobolev order. If s_\nu\to s, a subsequence of the corresponding solutions converges smoothly on compact sets. The limit solves (7), satisfies the uniform positive lower bound in (10), and belongs to H^{k+2} by lower semicontinuity of the global norms. It is therefore still in \mathcal B. This proves closedness and hence solvability at s=1. Denote that solution by u_N. The first three assertions of (3) and the stated regularity have been proved.

Suppose (4) fails for a fixed nonempty compact set K. Along a subsequence, \sup_K(u_N/N)\to0. Local plurisubharmonic compactness, after adding a local potential for \omega/N, gives an L^1_{\mathrm{loc}} limit v which is plurisubharmonic and nonpositive. The alternative of uniform collapse to -\infty is excluded by the supremum on K. Hartogs’ lemma implies that v attains zero on K. Since U is connected, the maximum principle gives v=0 everywhere.

The trace–determinant inequality applied to (5) gives

Pass to a further almost-everywhere convergent subsequence of u_N/N. As n\ge1, \lambda u_N=N^{1-2n}(u_N/N)\to0 almost everywhere. The right side of (14) converges in L^1 on a ball where f>0 to nf^{1/n}>0, by dominated convergence and u_N\le0. The left side converges to zero as a distribution because u_N/N\to0 in local L^1. Testing against a nonzero nonnegative function supported in that ball is a contradiction. This proves (4).□

Proof. Every cell counted in (16) contributes at least \vol(L) to the integral of |u_N(a^{-1}x)|^2. Bounded overlap and (3) prove the count.

Translate gL to L and normalize the potential by A_a(g)+1. The resulting functions are nonpositive, have \dd\dc bounded below by -\omega, and have supremum in [-1,0] on L. Plurisubharmonic compactness on a connected neighborhood of any fixed enlargement of L makes this a relatively compact family in local L^1. In particular its supremum cannot tend to -\infty on another fixed cell. There are only finitely many possible relative neighbors; this gives the upper comparison in (18), and exchanging the two cells gives the lower comparison.

On finitely many smaller charts covering L, add fixed local potentials to make the normalized functions plurisubharmonic. Each limit has a positive local exponential integrability exponent. Compactness and the semicontinuity and integrability statement of [21, Main Theorem 0.2] give a common sufficiently small exponent and a uniform integral bound. The added local potentials are bounded, so they can be removed at the cost of a constant. Summing the chart estimates and translating back gives (17).□

Proof. First work on the (m,*) complex with coefficients E^*. Its unweighted gap gives exactness and closed ranges: the bounded Green operator yields the decomposition into the images of \dbar and \dbar^*. Weighted and unweighted differential graph norms are equivalent for fixed b. Thus the domains, kernels and ranges of the closed differential d=\dbar are the same, and weighted exactness and closed ranges follow. This argument does not assert equality of the weighted and unweighted adjoint domains.

Let D'_h be the (1,0) part of the Chern connection and let *_S be the complex-linear Hodge star on form indices. On (m,q)-forms it takes values in (m-q,0), and

This formula acts trivially on the coefficient index. The derivative of the bundle metric in the adjoint cancels the corresponding Chern connection term. The Bochner–Kodaira identity and (20) imply

Here \mathcal E_{S,h}(\eta)= \|\dbar\eta\|_{S,h}^2+\|\dbar_h^*\eta\|_{S,h}^2 is the weighted Dolbeault energy. Indeed D'_h\eta=0 in top holomorphic degree, and the curvature commutator there is a sum of q curvature eigenvalues. For the last inequality drop h\ge1 from the differential norm and apply (19) to *_S\eta. The star is an isometry. Compact support approximation from Section 3.1 proves the same inequality on the form domain.

Fix a degree q and put A=dd_h^* on the closed subspace \im d. For a nonzero vector \eta in a spectral band of A up to \mu>0, its minimal weighted lift has squared norm \langle A^{-1}\eta,\eta\rangle_{S,h}, hence at least \mu^{-1}\|\eta\|_{S,h}^2. The minimal unweighted lift is also an admissible weighted lift, with norm at most e^{b/2}\delta^{-1/2}\|\eta\|_S. Consequently

On this spectral band d\eta=0 and \mathcal E_{S,h}(\eta)\le\mu\|\eta\|_{S,h}^2. Substitution in (22) gives

This holds for every nonzero lower spectral band. Solving the quadratic inequality shows that the minimal inverses of d have norm at most C(e^{b/4}+\sqrt t\,e^{b/2}) in every degree.

Finally use the conjugate-linear bundle-metric star for Serre duality, which is distinct from the linear star used above. It identifies the (m,q) complex for E^* with multiplier h with the reversed (0,m-q) complex for E with multiplier h^{-1}, interchanging the differential and its adjoint. It is an isometry for these norms. The complete-domain identities therefore transfer exactness, closed ranges, and the inverse bounds to the latter complex. This proves (21).□

Proof. Choose an even Schwartz function f_0 whose Fourier transform is smooth and compactly supported, with f_0(0)=1, and set K=f_0(\sqrt{\Delta''_1}), using the standard unweighted product metric and the fixed metric on E. The product splitting of \Delta''_1 proves the degree assertions. Commutation with \dbar follows first on its domain from the Dolbeault identities and then on currents by duality.

The wave equation has finite propagation with a speed bounded by the principal symbol. In the present setting this can be seen from the symmetric first-order operator D_1=\sqrt2(\dbar+\dbar^*): the energy identity for a moving cutoff contains only its commutator, whose norm is bounded by a fixed multiple of the cutoff gradient. It gives the domain-of-dependence estimate for e^{\ii tD_1}. Since f_0 is even, the Fourier expression for f_0(\sqrt{\Delta''_1}) uses only these finite-time wave operators. Thus K has fixed propagation. The coefficient curvature contributes only lower-order terms and does not alter this argument. The identical energy calculation in invariant charts proves it on the orbifold.

To see smoothing with uniform kernel bounds, use (1) and local Sobolev embedding to bound point evaluation, and derivative evaluation, after (1+\Delta''_1)^{-a} for a sufficiently large. The factorization

has bounded middle factor by the Schwartz decay of f_0. Evaluating the outside factors and their adjoints bounds the kernel at any two points, uniformly in the charts. Increasing a gives every derivative bound. This argument also defines K on distributions.

For a compactly supported E-valued current A, define its orbifold trace by

for smooth E_0^*-valued test forms \beta of complementary bidegree. On a covering chart [V/H]\to[V/G], this is the sum over cosets G/H, with the natural action on form and coefficient indices, summed also over the local sheets met by the support. The |G:H| terms convert the downstairs integration factor 1/|G| into the upstairs factor 1/|H|. This description applies even when H is not normal; it includes the stabilizer multiplicities at singular strata.

For a compactly supported current, the wave-evolved support stays in a fixed compact neighborhood for bounded time, by completeness and finite propagation. Its local trace is therefore a finite sum. Since the operator and coefficient metric are pulled back from the base, the local transfer commutes with the differential operator. The traced wave is thus the downstairs wave with traced initial data, by uniqueness. Integrating in time proves that trace commutes with K. On the compact base, a spectral function with value one at zero is the identity on harmonic representatives and therefore on Dolbeault cohomology, also computed by currents. This proves (iii).

It remains to check the dependence on S. For an E-valued (0,n) block of y antiholomorphic degree q, the unscaled metric on E gives

Put a_C^S(B)=\int_C|B|_S\dd V_1. The kernel bound and fixed propagation imply \|KB\|_1\le C\|(a_C^1(B))_C\|_{\ell^2}: each output cell sees only boundedly many input neighbors, and the neighbor matrix has uniformly bounded row and column sums. Since K preserves q,

The finitely many orthogonal blocks give (24). Product cells may overlap and have bounded finite chart multiplicities; these change only the fixed constant.□

Proof. By Lemma 3.4 and (30), solve \dbar v=F with

The last inequality uses \sqrt{t_j}\le C(\log N)^{1/2}N^{-1/4}. The compactly supported closed current

has the same traced cohomology class as F. The equality uses \chi_j=1 near \supp F. On any cell meeting \supp B, which is unmarked, Cauchy–Schwarz and (32) give

Here wedging with \dbar\chi_j has bounded operator norm for \Omega_S, because S\ge1. After summing squares and using bounded overlap and \dd V_S=S^n\dd V_1, we obtain

Set F_{\mathrm{new}}=KB with the fixed operator of Lemma 3.5. Its norm estimate is (34); its closedness and traced class follow from the same lemma. Fixed propagation, the marked-cell count and (33) give all support assertions. Only the compactly supported current \chi_jv is traced; no sum of the global solution v is used.□

Proof of Theorem 3.1. Continue under the assumption of positive lower bounds in all (p,0) degrees. Let \iota:M\to D be the lifted diagonal and let [\iota(M)] be its integration current of type (n,n). Orbifold integration defines this current directly by pullback of test forms to M, regardless of singularities of the coarse spaces. Project it to the component whose holomorphic degree lies entirely in the y factor. Under the natural identification with E-valued (0,n) currents, call the result T_0. This projection commutes with \dbar, so T_0 is closed and compactly supported.

Its trace to M\times M has a nonzero cohomology class. More explicitly, the canonical isomorphism K_{M\times M}\otimes E_0^*\simeq p_x^*K_M identifies p_x^*\omega^n with a closed E_0^*-valued (m,n) test form \beta. The holomorphic-degree projection used to define T_0 does not change the pairing with this form, since all other components vanish on wedging with p_x^*\omega^n. Hence

Let F_0=KT_0. It is smooth, compactly supported, and closed; its traced class has the same pairing (35). It is independent of N, and its support lies in a fixed neighborhood of the lifted diagonal. Thus it lies in \mathcal D_1 for all sufficiently large N. Apply Lemma 3.6 successively for the fixed number r of steps to obtain F_r. Its standard support volume is at most C_rN^{2n}, while

For degree (0,n), (25) and 0\le q\le n show that \|F_r\|_1\le\|F_r\|_S and \|F_0\|_S\le S^{n/2}\|F_0\|_1. The constant norm conversions are used only at these endpoints. Since S=N^2,

The pullback of the fixed test form \beta is uniformly bounded for the standard metric. Therefore Cauchy–Schwarz gives

Our fixed choice r>30n+6 more than suffices for this decay. Every step preserves the traced cohomology class, so its pairing is the strictly positive constant in (35). This is a contradiction. At least one scalar (p,0) Dolbeault Laplacian has no positive lower bound, which proves the theorem.□

Proof. Inducing a finite-dimensional representation of Q_0 to Q produces a finite-dimensional representation of Q. Its restriction to Q_0 contains the original representation, whose image is therefore finite. The group Q, and hence Q_0, is finitely generated. A nonzero homomorphism Q_0\to(\C,+) would give an infinite unipotent two-dimensional representation, proving the last assertion.□

Proof. We treat the nonamenable and amenable cases separately.

The nonamenable case. Let U\to\cY be the normal orbifold covering associated with \ker q. By Theorem 3.1, in some fixed degree (p,0) there are unit vectors with spectral support in [0,j^{-1}], for j\to\infty. Folding forms from U to the compact base identifies them with sections s_j having coefficients in the regular representation \lambda_Q on \ell^2(Q). This is the regular-coefficient bundle interpretation of covering-space analysis [1, Section 6.1]. To include the orbifold normalization, work on a chart [V/G] and put H=\ker(G\to Q). The cover has components [V/H] indexed by the cosets of \im(G\to Q) in Q. A G-invariant form with regular coefficients is exactly the corresponding family of scalar forms on these components. In the norm, the |G/H| regular coordinates cancel the downstairs factor 1/|G|, leaving the upstairs factor 1/|H|. The differential operators agree componentwise. Thus the identification preserves the L^2 norms and operators even when U has nontrivial isotropy.

Here is the compactness statement needed for these coefficients. For every integer k\ge0, elliptic estimates give

The constants do not depend on the coefficient dimension. Indeed, in a flat chart the coefficients of the operator are finite matrices tensored with the identity on the Hilbert space. Summing the scalar elliptic estimates over an orthonormal expansion gives the Hilbert estimate. The same conclusion holds on invariant orbifold charts, whose local group orders are uniformly bounded. A finite atlas then gives the displayed global estimate. Hilbert-valued Sobolev embedding, obtained from Fourier inversion and Cauchy–Schwarz, bounds every derivative uniformly on smaller charts.

Fix a nonprincipal ultrafilter. Form the Hilbert ultrapower \mathcal H^\omega and define the values and derivatives of s by the corresponding pointwise classes of those of s_j. Uniform bounds for the next derivative give uniform Taylor remainders. Thus these classes are the derivatives of a smooth section, and the exact transition identities make it a section of the ultrapower flat bundle.

This section retains its norm. The functions x\mapsto\|s_j(x)\|^2 are uniformly bounded and uniformly Lipschitz on the finite atlas. A finite \varepsilon-net shows that their pointwise ultralimit is approached uniformly along the ultrafilter: first control the finitely many net values, then use the common Lipschitz bound. Consequently

Applying the same elliptic estimates to \Delta''s_j shows convergence to zero in every fixed C^r norm, since \|(\Delta'')^{k+1}s_j\|_2\le j^{-k-1}. Hence \Delta''s=0. Integration by parts on the compact base, in antiholomorphic degree zero, gives \dbar s=0.

The ultrapower representation has no nonzero finite-dimensional invariant subspace. Otherwise such a subspace would have finite image, by the hypothesis on Q, and would be fixed by a finite-index subgroup Q_0. This subgroup is nonamenable. For a finite generating set F\subset Q_0 and some \varepsilon>0,

on its regular representation. Indeed, failure would give unit almost invariant vectors; their squared absolute values would be almost invariant probability measures in \ell^1(Q_0), yielding an invariant mean. Summing over copies gives the same inequality on the restriction of \lambda_Q to Q_0. A nonzero Q_0-fixed ultrapower vector, represented by a bounded sequence of limiting norm one, contradicts this inequality after taking the ultralimit. There are only finitely many terms on its left side.

Finally, restrict to the closed span of all coefficient values of the section and their Q-translates. Countable charts, continuity, and countability of Q make this subspace separable. It still contains the nonzero section and has no finite-dimensional invariant subspace.

The amenable case. An infinite amenable group does not have property (T): its regular representation has almost invariant vectors and no invariant vector. Shalom’s reduced-cohomology theorem therefore gives a unitary representation with nonzero reduced first cohomology [34, Theorem 4.2]. We may restrict to a separable invariant subspace generated by the values of a cocycle on the countable group.

The closed span of the finite-dimensional invariant subspaces contributes no reduced first cohomology. To see this, decompose it orthogonally into finite-dimensional irreducible summands. On any finite sum the representation has finite image. Its kernel Q_0 has H^1(Q_0,\C)=0, by Lemma 4.2; restriction of a cocycle to that kernel is zero, and averaging over the finite quotient makes the cocycle a coboundary. Finite partial sums approximate a cocycle on a finite generating set, so the same vanishing holds for reduced cohomology of the closed sum. The orthogonal complement therefore has a cocycle b that is not a limit of coboundaries and has no finite-dimensional subrepresentation.

Pullback by the surjection q preserves this nonvanishing: approximating the pullback cocycle on lifts of finitely many generators of Q would approximate b itself. The associated flat affine Hilbert bundle on \cY has a smooth section, constructed by local sections and a partition of unity, averaged in the finite orbifold charts. Its covariant differential is a smooth closed Hilbert-valued one-form \alpha representing the pulled-back cocycle.

The harmonic projection of \alpha is nonzero. Otherwise, the orthogonal decomposition of the closed de Rham Hilbert complex would put the closed form \alpha in the L^2 closure of exact forms. For fixed t>0, the heat operator e^{-t\Delta} commutes with the differential and maps L^2 convergence to C^0 convergence, by the dimension-independent elliptic estimates just used. Thus e^{-t\Delta}\alpha would be a uniform limit of smooth exact forms. Periods along finitely many fixed generator loops, with flat parallel transport of coefficients, would approximate its cocycle by coboundaries. Letting t\downarrow0 gives the same conclusion for \alpha, contrary to the choice of b. The orthogonal decomposition of the closed de Rham Hilbert complex now gives a nonzero harmonic one-form; local elliptic regularity makes it smooth.

The Kähler identities for flat unitary coefficients split this harmonic form into types. A nonzero (1,0)-part is holomorphic. If only the (0,1)-part is nonzero, complex conjugation gives a holomorphic (1,0)-form with conjugate coefficients. The conjugate representation again has no finite-dimensional invariant subspace.

In either case, degree zero is impossible: a holomorphic section of a flat unitary Hilbert bundle on a compact Kähler orbifold is parallel, by integration of the Laplacian of its squared norm. A nonzero parallel section would give an invariant line.□

Proof. Let \eta be given by Proposition 4.3, and let r be the generic rank of (\Omega_{\cY}^p)^*\to\mathcal H. Taking its r-th exterior power gives a nonzero rank-one tensor with coefficient space \bigwedge^r\mathcal H. Antisymmetrization embeds its finite-dimensional tensor factor into (\Omega_{\cY}^1)^{\otimes pr}.

If the resulting decomposable lines spanned a finite-dimensional space, Q would act on that span through a finite group. The orbit of one of these lines would be finite. A decomposable exterior line determines its r-dimensional original coefficient plane, so the corresponding orbit of planes would also be finite. Their sum would be a nonzero finite-dimensional invariant subspace of \mathcal H, a contradiction.□

Proof. Work in a uniformizing chart with a flat coefficient trivialization. Any nonzero scalar coefficient of \tau determines its meromorphic direction in the finite-rank tensor bundle. All scalar coefficients have this direction, since the coefficient rank is one. These local meromorphic directions agree on overlaps. Their saturations are coherent rank-one subsheaves of the tensor bundle. A saturated subsheaf of a locally free sheaf is reflexive in rank one, and on a smooth chart a rank-one reflexive sheaf is a line bundle. The invariant constructions give the asserted orbifold line.

Write a local line generator as e. Off its codimension-two degeneracy set the tensor has the form e\otimes u, with u Hilbert-valued and holomorphic. There are no divisorial poles, by saturation. The Hilbert-valued Hartogs extension, obtained from the Cauchy integral formula on transverse polydiscs, extends u across the remaining set.

The weights \log\|u\|^2 are plurisubharmonic. If e_\beta=g_{\alpha\beta}e_\alpha, then u_\beta=g_{\alpha\beta}^{-1}u_\alpha, so

They therefore define the metric \|e_\alpha\|^2=\|u_\alpha\|^{-2} on \cL, with positive curvature. Where u\ne0, its curvature is \dd\dc\log\|u\|^2, the pulled-back Fubini–Study form. A positive power of \cL kills the finitely many chart stabilizer actions and descends to an ordinary line bundle on the coarse space; its weights descend as psh weights.□

Proof. The manifold of image germs. At each point of U^\circ, the holomorphic constant-rank theorem writes F as a submersion onto an embedded s-dimensional plaque. This theorem also holds for the present Hilbert target: choose a finite-dimensional projection of rank s, use it for source coordinates, and observe that all derivatives in the remaining source directions vanish.

Let Z consist of the image points together with the germs of these embedded plaques at those points. A plaque supplies its own coordinate chart on Z. The map U^\circ\to Z is an open holomorphic submersion, so Z is connected and second countable. It is Hausdorff. Indeed, two germs over the same image point can both be written as graphs over one finite-dimensional transverse projection and one small connected ball. If their plaque neighborhoods shared a germ, the graph functions would agree on an open set and hence everywhere on that ball. The original germs would be equal. Points lying over different image points are separated by the Hilbert projective space.

The natural immersion \iota:Z\to\PP(\mathcal B) gives Z a Kähler metric. We use its intrinsic Riemannian length distance, which induces the manifold topology and is locally compact even when the metric is incomplete. Its isometry group acts properly [28, Corollary 4] and is a Lie group [29, author’s version, Theorem 6.3]. Let I be the closure of the \Gamma-image in this isometry group with its compact-open topology. Its elements are holomorphic. Indeed, if holomorphic isometries g_j converge to g, choose a relatively compact neighborhood whose image under g lies compactly inside one target complex chart. Eventually the g_j also take its closure into that chart, and their coordinate maps converge locally uniformly. The Weierstrass theorem makes g holomorphic there. Inversion is continuous in the isometry group, so the same argument applies to g^{-1}.

A locally compact unitary extension. Let J be the closure of the simultaneous image of \Gamma in I\times U(\mathcal B), with the strong group topology on the unitary factor. Every (g,A)\in J satisfies A\iota(z)=\iota(gz) projectively, by continuity. The projection J\to I is proper and surjective. For properness, suppose that the isometry components of a sequence converge. Choose countably many unit vectors in coefficient lines whose span is dense in \mathcal B. The images of each such vector lie in convergent projective lines. Compactness of the scalar circle gives a convergent subsequence of the unit vectors. Diagonal extraction, performed also for inverse operators, gives strong convergence to a unitary operator. This proves properness, since the spaces involved are metrizable. The image is consequently closed and contains the dense \Gamma-image, proving surjectivity.

The kernel is a closed subgroup of the scalar circle. A kernel element preserves every coefficient line; a unitary operator has the same eigenvalue on two nonorthogonal eigenlines. Nearby lines are nonorthogonal, so connectedness of Z makes this scalar the same everywhere. Their span is all of \mathcal B. It follows that J is locally compact. It has no small subgroups: use such a neighborhood in the Lie quotient I, then one in the scalar kernel. The locally compact no-small-subgroups theorem therefore makes J a Lie group [36, author’s version, Corollary 1.5.8].

Its adjoint representation, complexified, has finite image on \Gamma, by Lemma 4.2. Pass to its kernel in \Gamma, and replace both I and J by the corresponding closures. They are open finite-index subgroups of the old closures, with unchanged identity components; J\to I remains surjective and proper. The Q-cover and the map F are unchanged. Every element of the new J centralizes J^\circ, since its conjugation has identity differential on the connected group. In particular, J^\circ is abelian.

Spectral projection of holomorphic data. The spectral theorem for the continuous unitary representation of the connected abelian Lie group J^\circ, in its commutative C^*-algebra formulation [40], gives a direct integral

We take \mu finite, as is possible because \mathcal B is separable. Because \Gamma centralizes J^\circ, its operators are decomposable in this integral [40]. Write j_\gamma for the simultaneous image of \gamma in J, and put

Countability of \Gamma allows the group identities to hold on one common full-measure set, giving unitary representations \rho_\chi of \Gamma. On that set we also impose the countably many identities

We explain carefully how the holomorphic tensors pass to these fibres. These are almost-everywhere decompositions of a countable family of Taylor coefficients; no bounded point-evaluation projection from the direct integral is asserted. Let \varpi:U^\circ\to Z be the submersion. Choose a countable uniformizing atlas for \cY_\Gamma and all its lifts to the Q-cover, still a countable family. Choose also local nonvanishing holomorphic vector representatives v_\alpha of \iota on a countable cover of Z. On a source polydisc mapping into the domain of v_\alpha, the finitely many tensor coefficients satisfy vector identities

with holomorphic scalar functions c_a, obtained after shrinking by applying a bounded functional nonzero on v_\alpha. On suitable overlap domains the equivariance identities have the form

where c_{\gamma,\alpha\beta} is a holomorphic unit. Here \rho is the selected unitary representation of \Gamma. These identities are kept as vector identities before projectivizing.

Expand the tensor coefficients and the v_\alpha in Hilbert-valued power series. On every strictly smaller polydisc the sums of coefficient norms times the monomial radii are finite. Cauchy–Schwarz in the spectral variable, followed by Tonelli, gives the corresponding absolute sums in \mathcal B_\chi for almost every \chi. The projected series are therefore holomorphic there. After restricting to countably many smaller overlap polydiscs, normal convergence permits composition with the fixed coordinate maps and multiplication by the scalar functions above term by term. For each resulting coefficient sum, its partial sums converge in the direct-integral norm, while its fibre series converges absolutely almost everywhere. A subsequence converging almost everywhere identifies the two limits. There are only countably many such sums and Taylor-coefficient identities, including the tensor transition and orbifold isotropy identities. They therefore hold, together with the group identities above, on one common full-measure set and give global holomorphic tensors on \cY_\Gamma of coefficient rank at most one, and holomorphic projective maps

of rank at most s, where E_\chi is the common zero locus of the projected representatives.

The countable family of Taylor coefficients of the tensor on the lifted charts has closed span \mathcal B: the series give all coefficient values, while Cauchy integrals express their coefficients as limits of linear combinations of those values. The orthogonal projection onto the fibrewise complement of the projected coefficient spans is a measurable field, obtained by countable Gram–Schmidt orthogonalization. Its direct integral annihilates this dense family and is therefore zero. Thus the projected Taylor coefficients span \mathcal B_\chi almost everywhere. By the same series and Cauchy formulas, so do the projected coefficient values. On every nonzero such spectral fibre the projected tensor is therefore nonzero, has generic coefficient rank one, and is a line tensor whose coefficient lines span \mathcal B_\chi. We henceforth restrict to these nonzero fibres.

The identity component cannot move an image germ. The subgroup D is dense in J^\circ, since that component is open. It acts by the character \chi on each spectral fibre, so F_\chi is invariant under its action on Z. The closed analytic set E_\chi is invariant as well. Continuity extends both invariances to J^\circ.

Suppose that J^\circ acts nontrivially on Z. If some F_\chi had rank s at z\notin E_\chi, a finite-dimensional target projection of full rank and the inverse function theorem would make F_\chi locally injective at z. The connected J^\circ-orbit of z stays in the domain and in its fibre by invariance. The point z is isolated in that fibre, and its singleton is open there by isolation and closed because Z is Hausdorff. The orbit is therefore \{z\}. Applying this at every point of the nonempty open full-rank locus, each element of J^\circ fixes that open set. It is holomorphic, so the identity theorem on connected Z makes it the identity everywhere, a contradiction. Thus every F_\chi has rank strictly less than s. Minimality of s then forces \mathcal B_\chi to be finite-dimensional almost everywhere. By Lemma 4.2, \rho_\chi(\Gamma) is finite. The character \chi consequently has finite image on the dense subgroup of J^\circ just used. Continuity and connectedness force \chi to be trivial on J^\circ. Hence J^\circ acts trivially on \mathcal B. It then acts trivially on Z: each orbit maps to a point under the immersion \iota, and is connected. This contradicts the assumed nontrivial action.

We have proved that J^\circ acts trivially on Z. The surjective proper Lie-group homomorphism J\to I maps identity component onto identity component; since I is an effective isometry group, I^\circ=\{1\}. Thus I is discrete and acts properly.

Its \Gamma-image is infinite. Otherwise a finite-index subgroup would fix every image germ and would act on \mathcal B by scalars. Its scalar character has finite image; on a further finite orbifold cover the tensor would be untwisted. An untwisted Hilbert-valued tensor on a compact orbifold has finite-dimensional coefficient span: its scalar tensor space is finite-dimensional, and expansion in a basis writes the tensor as a finite sum of scalar tensors times fixed Hilbert vectors. This contradicts our choice of the line tensor.□

Proof. Let T be the current and let \omega be a Kähler form on V. The mass of T^s\wedge\omega^{d-s} on V^\circ is finite; this is the positive-current mass bound [5]. One can see the bound directly by writing T=\alpha+\dd\dc\varphi, choosing C with \beta=\alpha+C\omega>0, and truncating \varphi_j=\max(\varphi,-j). The bounded-potential products satisfy

On \{\varphi>-j\}\cap V^\circ they equal (T+C\omega)^s\wedge\omega^{d-s}, which dominates T^s\wedge\omega^{d-s}. Exhaustion gives the bound.

Use the free locus of the orbifold Z/I for the coarea formula. The discarded isotropy values, and their inverse images under the local submersions, have measure zero. With normalized volumes,

where J_f is the real normal Jacobian. Its value is the product of the squared nonzero complex singular values of the differential. The coarea formula and the mass bound show that almost every fibre has finite (d-s)-dimensional complex volume in V^\circ. The formula is applied to a countable disjoint measurable decomposition subordinate to target charts, so it counts whole fibres and requires neither compactness nor bounded geometry of the target. Moreover, this yields a full-measure set of source points: the inverse image of the exceptional null set has integral of J_f equal to zero, and J_f>0 on the submersion locus.

For a good free value, its fibre is closed analytic and pure of dimension d-s in V^\circ. Its integration current has finite mass. Extension by zero across V\setminus V^\circ is positive and closed, by the extension theorem of El Mir [23]. Siu’s analyticity theorem for positive Lelong superlevel sets [35] then makes its closure analytic of pure dimension d-s: the level set with Lelong number at least one contains the fibre, has dimension at most d-s, and agrees with the smooth fibre on V^\circ. Take its components meeting V^\circ. Equivalently, one may apply the finite-volume closure theorem directly [23].

Let \widetilde C\to C resolve one such closure component. The preimage of C\cap V^\circ has proper analytic complement in the smooth manifold \widetilde C, and its inclusion induces a surjection on fundamental groups: every loop may be perturbed off an analytic set of real codimension at least two. Along the open fibre the continued lift to Z is constant. Thus its monodromy fixes one point of Z, and lies in a finite stabilizer. The same is true of the image of \pi_1(\widetilde C), by that surjection.

The \Gamma-reduction contracts these compact subvarieties through very general points, by Theorem 2.3. The full-measure set just obtained meets the complement of its countable union of proper analytic exceptional sets. Its general fibres consequently have dimension at least d-s, proving (36).□

Proof of Theorem 4.1. Choose the minimal line tensor and the finite-index subgroup \Gamma of Proposition 4.6. Let \cL and its positive curvature current be as in Lemma 4.5. The current has rank s on the regular open set.

Suppose that s<\ell. Apply (M) with Q_0=\Gamma, obtaining S' with image Q'\subset\Gamma. The map S'\dashrightarrow Y lifts meromorphically to the coarse space of \cY_\Gamma. Indeed, the monodromy condition gives a lift on the complementary open set; the selected component of the normalized finite pullback extends its graph. Resolve this map on a smooth compact Kähler modification V of S'.

The descended power of \cL pulls back to an ordinary line bundle on V. Its psh weights pull back, since the map is dominant, giving a positive current. On a regular analytic Zariski-open subset the local coefficient-line maps give

of rank s, with monodromy the composite \pi_1(V)\to Q'\to I. The pulled-back current is a positive constant multiple of its transverse Kähler form. Lemma 4.7 gives a \Gamma-dimension at most s. The image of Q' in I is infinite: it has finite index in the infinite image of \Gamma. This image is an infinite quotient to which (M) applies. Bimeromorphic invariance of the \Gamma-reduction therefore gives a dimension at least \ell, a contradiction. Hence s=\ell.

Take a finite Galois orbifold cover \widehat{\cY}\to\cY dominating \cY_\Gamma, and pull \cL to it. If G is its deck group, form

The permutation action gives a canonical G-linearization. All its factors inject into the pullback of the same cotangent tensor bundle. Multiplication in the symmetric algebra gives a nonzero morphism from \mathcal M to the |G|-th symmetric power of that tensor bundle, and hence into a cotangent tensor power. It is nonzero because a product of nonzero vectors in a symmetric algebra is nonzero. Equivariant descent gives an orbifold line \mathcal M_0 and a nonzero cotangent-tensor morphism on \cY.

The tensor product of the translated metrics is invariant. Its curvature is positive and strictly positive on a nonempty open set, since s=\ell. A power of \mathcal M_0 kills the chart stabilizers and descends to an ordinary line bundle A on Y with the same positivity property. The volume criterion [5, author’s preprint, Theorem 1.2] makes A big. The Iitaka map of a big line bundle makes Y Moishezon. The projectivity theorem for Kähler Moishezon manifolds [30] therefore makes Y projective.

It remains to interpret the tensor inclusion on an adapted projective cover. Choose the smooth Galois Kawamata cover \pi:Y'\to Y supplied by [17, Lemma 5.2]. It is projective, being finite over Y, and its ramification order over D_i is exactly the denominator m_i. Near that component the root chart and the cover are z_i=u_i^{m_i}=w_i^{m_i}. Choosing the local root lift u_i=w_i, the cotangent generator pulls back as

Along an auxiliary ramification divisor z_j=w_j^{a_j}, the orbifold coefficient is zero and the generator is \dd z_j=a_jw_j^{a_j-1}\dd w_j. These are precisely the local generators of the adapted orbifold cotangent bundle in [17, Definition 5.3]. Thus a suitable tensor power of the descended inclusion gives a nonzero map

on this projective adapted cover. The big-line criterion for orbifold cotangent tensors [17, Theorem 7.11] now implies that K_Y+\Delta is big. Its hypotheses are satisfied because projectivity and the ordinary big line A have already been established. No pseudoeffectivity assumption is needed in that criterion.□

Proof. Take a general point w\in D, avoiding its singularities and the other components of \Delta_W. The divisor p^{-1}(D) is smooth near every point over w. Fix such a point z, and a small coordinate ball U around it. In this ball the only nontrivial inertial support is the smooth hypersurface p^{-1}(D)\cap U.

The map from the fundamental group of the original deleted-locus complement in U to \pi_1(U\setminus p^{-1}(D)) is surjective. Its kernel is normally generated by meridians around the other deleted divisors: a transverse perturbation of a null homotopy gives precisely these meridians. Their R-images are trivial. Removing the remaining analytic subsets of complex codimension at least two does not change this fundamental group, by the same transverse perturbation applied to loops and homotopies. Consequently \psi factors locally through

At a general point of E over w, choose a transverse disc whose punctured part avoids all deleted inverse images. Its image winds e=\operatorname{ord}_E((p\circ f)^*D) times around the one remaining hypersurface. Its boundary contracts in X, so compatibility with \rho gives \psi(\mu)^e=1. The order of \psi(\mu) is m_D, proving (37).□

Proof. Put r=\dim B, and let \cL\subset\Omega_X^r be the saturated differential line of u. For a local nonvanishing generator \eta of K_B, write a_E for the common divisorial vanishing order of u^*\eta along E, measured in \cL, and put v_E=\operatorname{ord}_E(u^*\Delta). If E dominates a component D, with multiplicity e, the local differential of a defining function of D gives

If u(E) has codimension at least two, the tangent rank along E is at most r-2, and the normal direction increases it by at most one. Thus every maximal minor of \dd u vanishes along E, and a_E\ge1.

The pullback of the fixed divisor \Delta has finitely many prime components on X. We may therefore choose a rational \epsilon>0 such that \epsilon\le1 and

Only components with v_E>0 impose a condition. This choice also handles components whose images lie in intersections of boundary divisors, since their v_E includes the sum of all pullback coefficients.

The torus has trivial canonical bundle. For divisible m, pullback of a section of m\epsilon\Delta, multiplied by the m-th power of its top differential, is thus a holomorphic section of \cL^{\otimes m}. The inequality above checks every divisorial pole; extension across codimension two finishes the check. The ratios of these sections retain the r parameters of the ample linear systems on B. Hence \kappa(X,\cL)\ge r, and Theorem 2.1 gives equality. This contradicts specialness.□

Proof. Let \cL\subset\Omega_X^r be the saturated differential line of the meromorphic map induced by u. A section of a divisible multiple of K_W+\Delta_W pulls back as a meromorphic section of \cL^{\otimes m}. Along a component of multiplicity e over D, its allowed pole is cancelled because

Divisors dominating W produce no boundary pole. The remaining source divisors are u-exceptional and hence disappear in codimension at least two on X. Every prime divisor of X therefore satisfies the holomorphicity condition. Extension across codimension two gives

Pullback preserves the ratios of sections because the map is dominant. Bigness gives \kappa(X,\cL)\ge r, and Theorem 2.1 again gives the forbidden equality.□

Proof. View a section upstairs as a meromorphic pluricanonical form in the common function field. Every prime divisor of Z has a strict transform with the same meridian order. Its divisorial pole bound is therefore the required bound on Z. Extension across codimension two gives the section downstairs, and the function-field identification makes this map injective.□

Proof. For each divisible m, properness gives the coherent direct image g_*\cO_Z(mL). Choose a very general smooth fibre where base change holds for all these countably many sheaves. Bigness on that fibre supplies one m for which its section map has full generic rank. On the open set where the direct image is locally free and base change holds, evaluation defines the relative meromorphic map to its projective bundle. The locus where its vertical differential has full rank contains a dense analytic Zariski open U\subset Z. This condition is intrinsic to the relative evaluation and jet maps, so it is defined over the base open, independently of a local frame of the direct image.

A general fibre component of Z\to J meets U. Otherwise the proper analytic subset Z\setminus U would contain a general component of a family of fibres covering Z, contrary to the fibre-dimension theorem. On such a component the relative section map still has injective differential at a general point: its tangent space is contained in the vertical tangent space of g. The restrictions of these sections therefore define a generically finite meromorphic map, proving bigness. General choices in J, followed by general choices in its projection to W, give the last formulation. All assertions can be restricted to the smooth loci supplied by generic smoothness.□

Proof. Assume the stated bigness on the cofinal collection. We may further prepare a model whenever necessary. By Theorem 2.5, every model in this collection satisfies

Choose a model Z_0 for which this integer is minimal, and put b=\kappa(Z_0,K_{Z_0}+\Delta_{Z_0}). Then b\ge d>0.

The Iitaka fibres. Resolve the Iitaka map on a modification \mu:Z_1\to Z_0, obtaining a fibration h:Z_1\to W with W smooth projective of dimension b. For this step introduce an auxiliary klt boundary \Delta_1. It has the strict-transform coefficients of \Delta_{Z_0} and exceptional coefficients sufficiently close to one to dominate the true inertial coefficients on Z_1. They may also be chosen so that

Indeed all discrepancies of the log-smooth klt pair are greater than -1; an exceptional coefficient less than one can be chosen above the negatives of these discrepancies and above the corresponding inertial coefficient. A simultaneous log resolution makes all the supports simple normal crossing.

Equation (41) preserves sections in divisible degrees, hence the Iitaka dimension and map. A very general smooth h-fibre H consequently satisfies

Here we use the ordinary Iitaka-fibre property, as in [38, Section 6.3]. It can also be seen directly in this setting. Write L=K_{Z_1}+\Delta_1. By Kodaira’s lemma on the projective Iitaka base, after taking a further multiple and absorbing the pullback of the effective remainder, the resolved system gives aL\sim h^*B+E for some divisible a>0, with B ample on W and E\ge0. If a multiple jL|_H had a positive-dimensional section map on a very general fibre, coherent direct image and base change, followed by a sufficiently large ample twist h^*(kB), would extend sections whose ratios vary on that fibre. Multiplication by the canonical section of kE turns them into sections of (j+ka)L. This fibrewise variation, together with the b base parameters supplied by B, contradicts \kappa(Z_1,L)=b. The restriction has a nonzero section because a general fibre is not contained in E, so its Iitaka dimension is zero. Adjunction identifies L|_H with the divisor appearing in (42).

We can arrange that a very general fibre of g:Z_1\to A has big restriction of K_{Z_1}+\Delta_1. For example, dominate Z_1 by a prepared member of the cofinal collection, perform the same auxiliary-boundary construction relative to Z_0, and replace Z_1 by this model. Its map to W is already holomorphic. The auxiliary boundary dominates its true boundary, whose fibre adjoint is big by assumption. Any further log resolution preserves this fibre bigness by pullback and an effective exceptional remainder, and (41) still holds. Lemma 5.7 now makes the log canonical divisor big on general components of common fibres of h and g. Generic smoothness, also for the finitely many boundary strata, justifies the adjunction restrictions to these common fibres.

Apply Theorem 2.5 to the klt pair in (42). Its Albanese map is surjective with connected fibres. The map g|_H factors through a homomorphism from \Alb(H) to A, followed by a translation. Quotienting by the connected kernel shows that its Stein base is a torus finite étale over its image subtorus translate. Torus subadditivity on H, using the bigness of the common fibres, forces their dimension to be zero. The Albanese map of H is therefore birational, and g|_H is, birationally, an isogeny onto a subtorus translate.

Every component of \Delta_1|_H is exceptional for this birational Albanese map. To prove this, suppose that a component maps onto a divisor D of the torus. The relative canonical divisor of the birational map to the smooth torus is effective, with positive coefficients on all exceptional primes. For sufficiently small rational \epsilon>0, it absorbs the exceptional components of the pullback of \epsilon D, while the chosen boundary component supplies its strict transform. Thus K_H+\Delta_1|_H contains that pullback up to rational linear equivalence. By Lemma 2.2, a nonzero effective divisor on a complex torus has positive Iitaka dimension. This contradicts (42).

A torus-bundle model. On the connected smooth parameter open of h, the image of H_1(H,\Z) in H_1(A,\Z) is locally constant. Transport in the proper smooth family and the total map to A identify these images. Their real span determines a fixed subtorus T\subset A, so the image of a general H is a translate of T. Write \Lambda_T for its lattice and L_H for the finite-index image lattice. Choose k>0 with k\Lambda_T\subseteq L_H, and use the isogeny A_1\to A given by multiplication by k.

Let T_1 be the identity component of the inverse image of T. The lattice of a subtorus is primitive in the ambient lattice, so the image of the lattice of T_1 is k\Lambda_T. The choice of k makes each connected lifted general H-fibre map with degree one onto a translate of T_1. This is a lattice construction; no splitting of the torus extension is required.

Pull back both the source and Z_1 to A_1. These covers are connected, since the original maps to A are surjective with connected fibres and therefore surject on \pi_1(A). Write M_1 for the source cover. The Stein factorization of the pulled-back map to W has a finite base over W; take a smooth projective model W_1 of that base. The map to A_1/T_1 is constant on its general fibres and hence descends meromorphically to W_1. Resolve this map as well. The pair of projection maps then gives a birational model

Indeed it has degree one on the general fibres just described. The space Z_* is smooth, and it is a closed submanifold of the compact Kähler product W_1\times A_1. The map p is a principal holomorphic bundle with connected torus fibre T_1. Its transition functions are translations, so an invariant top form on T_1 trivializes the relative canonical bundle. In particular,

We are free to replace W_1 by any further smooth projective modification and use its corresponding fibre product in (43).

The true inertia on Z_* has no horizontal divisor over W_1. To check this on a morphism of models, resolve the maps from the pulled-back Z_1 to W_1 and Z_*, and carry the auxiliary boundary by the rule (41). Its restriction to a general fibre still has only Albanese-exceptional components. Indeed finite étale covers of a birational torus are birational to torus covers. An exceptional prime on a further model either misses the general fibre or restricts to a divisor exceptional over the old general fibre, so it remains Albanese-exceptional. The strict transform of a horizontal prime of Z_* maps onto a divisor of its general torus fibre. It cannot occur in this auxiliary boundary, which dominates true inertia. Its meridian order is therefore one.

Every vertical prime divisor of the smooth bundle p is the full inverse image of a prime divisor of W_1. A divisor cannot map into codimension at least two by the fibre-dimension formula, and connected smooth fibres are irreducible. We have consequently obtained a rational boundary \Delta_{W_1} with

with the same finite integral meridian orders on corresponding divisors.

The true boundary is big downstairs. Let F denote the finite deck group of A_1\to A. Write \widetilde Z_1=Z_1\times_A A_1 and let \phi:\widetilde Z_1\dashrightarrow Z_* be the birational comparison. Take the closure of the simultaneous graph of \phi\circ f, for f\in F, in \widetilde Z_1\times (Z_*)^F. The group acts on this graph by its action on \widetilde Z_1 and permutation of the other factors. Resolve equivariantly and take a further equivariant Kähler modification using [26, Theorem 1.1 in the author’s version]; the finite group preserves the average of a big class. This gives a smooth Kähler V mapping to \widetilde Z_1, to Z_*, and to A_1. The action is free because its map to A_1 is equivariant for the free translation action. The quotient V/F is thus smooth Kähler and is a modification of Z_1, hence of Z_0, over the original torus A. Moreover, V\to V/F is the pullback of A_1\to A: each deck orbit maps bijectively onto the corresponding isogeny fibre.

Further prepare this quotient, and, if necessary, dominate it by a member of the cofinal collection. Pull this modification back along A_1\to A; all comparison morphisms persist. We continue to call the resulting cover V. Its true inertial adjoint is the unramified pullback of that on its downstairs quotient, by (39). The minimum in (40) therefore gives

All boundaries in this inequality are true inertial boundaries; the auxiliary exceptional coefficients of \Delta_1 are not used here. Lemma 5.6 applied to V\to Z_* gives the same lower bound on Z_*. Equations (44) and (45), together with p_*\cO_{Z_*}= \cO_{W_1}, now yield

Thus K_{W_1}+\Delta_{W_1} is big.

This argument applies after any chosen further smooth projective modification of W_1: form its fibre product, repeat the equivariant comparison, and use the same minimum on the prepared quotient over A. The newly appearing true coefficients are read on the new model. No comparison of exceptional coefficients on unrelated models is needed.

The differential contradiction. Choose W_1 sufficiently modified that the induced fibration from a resolution of M_1 to W_1 is neat, as described before Lemma 5.5. The preceding argument still gives bigness on this choice. The source resolution maps holomorphically to Z_*, using its maps to W_1 and A_1. Its fundamental-group quotient is the one used to define the true inertia on Z_*. Lemma 5.2, with the smooth map p, shows that every boundary order on W_1 divides the multiplicity of every source component dominating that divisor. Lemma 5.5 therefore gives a Bogomolov sheaf on a smooth source model of M_1. This contradicts Theorem 2.1, since M_1 is a connected finite étale cover of the special manifold M.□

Proof. Set R=G/N. Over the smooth fibration locus in A, fibrewise exactness gives a monodromy homomorphism to R, compatible with the quotient map on the total space. A meridian around a base divisor has finite order in R, since a component of its inverse image kills a positive power of that meridian. Form the actual inertial divisor D_A using these orders, as in Section 5.

Suppose D_A\ne0. Lemma 2.2 gives a quotient torus p:A\to B with connected fibres, where B is an abelian variety of dimension b>0, and an ample effective rational divisor D_B such that D_A=p^*D_B. The smooth quotient map pulls back each prime divisor with multiplicity one, so the components of D_B retain the meridian orders of their inverse images.

Lemma 5.2, applied to the smooth torus bundle A\to B, shows that each such order divides the multiplicity of every prime divisor of X dominating the corresponding divisor of B under p\alpha. This includes components whose images in A have larger codimension. For prime divisors whose images in B have codimension at least two, the top differential of p\alpha vanishes: at their general points its restriction to the divisor has rank at most b-2, so the total rank is at most b-1. There are only finitely many relevant components over the fixed support of D_B. Thus a sufficiently small rational \varepsilon>0 makes their differential vanishing orders compensate the poles allowed by \varepsilon D_B. This is the fixed-map argument of Lemma 5.3.

Let L\subset\Omega_X^b be the saturated line generated by the top differentials from B. For sufficiently divisible m, pullback of pluriforms gives

The pullback is injective, and ratios of sections retain their independence because X\to B is surjective. Since K_B is trivial and \varepsilon D_B is ample, \kappa(X,L)\ge b. The opposite inequality follows from Theorem 2.1. This contradicts specialness.

All meridians therefore have trivial image in R. The monodromy homomorphism factors through \pi_1(A), by normal meridian generation for the complement of the discriminant. Its composition with G\to\pi_1(A) is the quotient G\to R. Conversely, \alpha_* kills N and factors through R. These two factorizations are inverse because the maps from G are surjective. This proves R\cong\pi_1(A).□

Proof. We first show that every group N'=N\cap G' arising from a finite cover has \Hom(N',\C)=0. Set \Lambda=\pi_1(A') and V=\Hom(N',\C). The vector space V is finite dimensional because N' is finitely generated. Conjugation defines an action of the abelian group \Lambda on it. Write T_\lambda for the operator

where \widetilde\lambda is any lift to G'. Inner conjugation acts trivially on additive homomorphisms, so the operator is independent of this lift.

For a nontrivial character \beta:\Lambda\to\C^*, the cohomology of \Lambda with coefficients in \C_\beta vanishes in every degree. This follows, for example, from the Koszul resolution: one generator acts by a scalar different from one, making that factor contractible. Inflation–restriction gives

Choose a basis \lambda_1,\ldots,\lambda_r of \Lambda and put T_i=T_{\lambda_i}. A nonzero invariant on the right of (47) corresponds to a vector w\ne0 satisfying

Each T_i has finitely many eigenvalues, and the values on this basis determine a character. Hence only finitely many nontrivial \beta have H^1(G',\C_\beta)\ne0.

Inflation identifies the character group of \Lambda with the open and closed identity component of \operatorname{Hom}(G',\C^*): the Albanese lattice is H_1(X',\mathbb Z) modulo torsion, and the character group has finitely many components. Group and manifold cohomology agree in degree one for these local systems. The nontrivial support points just found are therefore isolated in the full degree-one rank-one cohomology jump locus. B. Wang’s torsion-translate theorem for compact Kähler manifolds [37, author’s version, Theorem 1.3] makes them torsion.

Suppose V\ne0. The commuting complex operators T_i have a common eigenvector v\ne0: successively take an eigenspace of the next operator inside the nonzero common eigenspace already chosen, which remains stable by commutativity. Their invertibility makes the eigenvalues nonzero, and they define a character \chi:\Lambda\to\C^*. If \chi\ne1, the vector v makes \beta=\chi^{-1} one of the nontrivial support points above, so \chi is torsion. If \chi=1, it is already torsion. For r=0, take any nonzero v and the unique trivial character.

Let A_0\to A' be the isogeny with lattice \Lambda_0=\ker\chi, and pull it back to X'. The total-space cover X_0 is connected because G'\to\Lambda is surjective; its group is G_0=(\alpha'_*)^{-1}(\Lambda_0), with kernel still exactly N'. Thus the vector space V=\Hom(N',\C) has not changed, and the same v is fixed by the restricted lattice. The pulled-back fibration has connected fibres. By the Albanese universal property it factors through a surjective homomorphism \Alb(X_0)\to A_0, which maximality makes an isogeny. Connectedness of the pulled-back fibres forces its degree to be one. We may therefore relabel X_0,A_0,G_0,\Lambda_0 as X',A',G',\Lambda, with the same V and V^\Lambda\ne0.

The untwisted inflation–restriction sequence contains

The last arrow is injective. To see this, its composition with the natural map to H^2(X',\C) is the pullback H^2(A',\C)\to H^2(X',\C), since a torus is a K(\Lambda,1). That pullback is injective for a surjective map from a compact Kähler manifold. In fact, if a=\dim A' and d=\dim X'-a, fibre integration of a Kähler class [\omega]^d is a positive constant. For \eta\in H^2(A',\C) and \theta\in H^{2a-2}(A',\C),

Poincaré duality proves the claim. For a=0 there is no degree-two group to consider.

The transgression in (48) is zero. Hence a nonzero invariant vector would give b_1(X')>2\dim A', contrary to the defining equality for the Albanese torus. The isogeny kept N' and V unchanged, so this contradiction proves V=0 for the original cover as well. Maximality ensured that all intervening Albanese tori were isogenous and that the pulled-back torus was itself the new Albanese torus.

Finally, let N_0\le N have finite index. By Lemma 2.7, it contains some N\cap G'. The latter has no nonzero homomorphism to \C. A homomorphism from N_0 to \C restricting to zero on a finite-index subgroup is zero. Thus \Hom(N_0,\C)=0. Since N_0 is finitely generated, its abelianization is finite.□

Proof. Choose a cover, a good fibre S, and a fibration attaining \ell. Write C\triangleleft N for its fibre-group image. The general fibration fibres give compact irreducible cycles of dimension \dim S-\ell, after taking their images in X. Choose one through a point of the fibration’s general locus in S outside all the nondominant evaluation images prepared above. Choose a family from the list, with p:W\to T and e:W\to X, containing this member. Its compact evaluation must dominate X, and hence is surjective.

The homotopy sequence of the smooth proper family W^\circ=p^{-1}(T^\circ)\to T^\circ makes its fibre-group image normal in \pi_1(W^\circ). The selected fibre is bimeromorphic to the selected cycle, so its image under e_* is C. Consequently H=e_*\pi_1(W^\circ) normalizes C. The complement of W^\circ in the smooth W is analytic, so \pi_1(W^\circ)\to\pi_1(W) is surjective and H=e_*\pi_1(W).

Put \Lambda=\pi_1(A) and \Lambda_0=(\alpha e)_*\pi_1(W). Lift the surjective holomorphic map \alpha e:W\to A to the connected complex covering A_{\Lambda_0}\to A. Its lifted image is compact analytic by proper mapping. Surjectivity downstairs forces this image to have dimension \dim A, so it is all of the connected manifold A_{\Lambda_0}. The covering is therefore compact and has finite degree. Thus \alpha_*(H)=\Lambda_0 has finite index in \Lambda.

Let L=N_G(C). It contains N and H, so Albanese exactness gives

Pass to the cover corresponding to L. Its fibre-group image is still N, by (46), and C is now normal in the total group. Continue to write X,G,A,\Lambda for this cover, its group, its Albanese torus and its lattice. The whole compact evaluation e:W\to X lifts, since e_*\pi_1(W)=H\subseteq L. Its lift has full rank somewhere, and its compact analytic image is therefore the entire connected cover. The proper fibre-dimension theorem, applied to W\setminus W^\circ\subsetneq W, shows that a general evaluation fibre meets W^\circ. Hence members with parameters in T^\circ cover general total points. Smooth proper transport gives their fibre-group image C after compatible base-point transport.

Apply Theorem 2.3 to G\to G/C, giving c:X\dashrightarrow Z. The covering cycles just constructed have dimension \dim S-\ell and are contracted through very general points. Thus

This map is over A: a compact subvariety with finite image in the quotient lattice G/N maps constantly to A. To verify the last assertion, resolve the subvariety. A finite image in the torsion-free lattice is trivial, so its map lifts to the universal cover of A, a complex vector space; compactness makes that lift constant. Consequently the Albanese map factors through c, after resolving. The induced map g:Z\to A has connected general fibres: a nontrivial finite Stein factor of g would split a general fibre of the connected Albanese map.

Let B_0 be the actual group image of a general c-fibre. It is normal in G, lies in N, and has finite image B_0C/C in G/C. If N/B_0 were finite, the image of N in G/C would be a finite union of translates of this finite image, contradicting N/C infinite. Thus N/B_0 is infinite. For the dimension comparison, restrict c to a very general good Albanese fibre. On simultaneous smooth loci, its fibres are the same general c-fibres, so they are connected and have image B_0 in N. This restricted fibration is admissible in the minimum, giving \dim Z-\dim A\ge\ell. Equality follows. No equality or inclusion between B_0 and C has been used.

For inertia use R=G/B_0, so

Fix any prepared smooth model of Z, with actual inertial boundary D_Z for R, and write \mu:\widetilde X\to X for its resolved source, with c:\widetilde X\to Z holomorphic. Choose afresh a very general good point a\in A so that

are smooth, \widetilde S\to S is a modification, and Y is transverse to every dominating stratum of the deleted divisor and avoids the images of the other strata. This choice may depend on the prepared model. The induced S\dashrightarrow Y is meromorphic, and \dim Y=\ell.

Let Z^\circ be the prepared smooth fibration locus for c, put Y^\circ=Y\cap Z^\circ, and put \widetilde S^\circ=c^{-1}(Y^\circ). The monodromy \psi_Z:\pi_1(Z^\circ)\twoheadrightarrow R has composite g_* in \Lambda, by compatibility after precomposing with the surjection from the open source group. Its restriction to \pi_1(Y^\circ) therefore lands in Q, giving the commutative diagram

The left map is surjective, by removal of an analytic subset and bimeromorphic invariance of the smooth fundamental group. The map \nu is surjective by the definition of N. Thus \psi_Y is surjective. The upper map is the map of a smooth proper fibration with connected fibres, so this is the monodromy compatible with S\dashrightarrow Y.

Transversality gives \Delta_Y=D_Z|_Y with simple normal crossing support. For a deleted prime divisor D, if E is a component of D\cap Y, its meridian \mu_E maps to a conjugate of the meridian \mu_D in Z^\circ. The injection Q\subset R therefore gives

The proper fibre-dimension theorem ensures that the deleted subsets of codimension at least two remain of codimension at least two on this very general restriction, and filling them does not change the fundamental group. Filling divisors of order one kills already trivial meridians. The m_D-th powers of the remaining meridians are also killed, so \psi_Y factors as a surjection

Finally fix a finite-index subgroup Q_0\le Q, and let N_0 be its inverse image in N. Lemma 2.7 gives a further global finite cover with group G' such that

The image Q' has finite index in Q. Choose a point of the new Albanese torus above the current a. Its fibre S' is a connected component of the pullback of S, because the new Albanese fibres are connected and the map of tori is finite. Thus S'\to S is a connected finite étale cover, and its induced homomorphism \nu' has image Q'. The good-point convention makes S' eligible for the same minimum.

Choose this cover before choosing a further quotient. For every surjection \psi:Q'\twoheadrightarrow P with P infinite, the \Gamma-reduction of \psi\circ\nu' on S' has resolved general fibre image C'\triangleleft N' with finite image in P. If N'/C' were finite, the whole image in P would be a finite union of translates of that finite image, which is impossible. The reduction is therefore an admissible fibration in the minimum, and its dimension is at least \ell. This one cover works for every infinite quotient of Q', as required by (M). The prepared model was arbitrary; repeating the choice of a very general good S on each model proves the assertion on every prepared model.□