The abelianity conjecture for special compact Kähler manifolds

1. Exactness and FAb on finite covers

Finite étale covers preserve specialness, and their Albanese maps are surjective with connected fibres, so $b_1\le2\dim X$. Choose a cover attaining the maximum; subsequent finite covers have the same first Betti number. Lemma 6.1 establishes

It does not extend the absence of multiple fibres unconditionally to nonprojective tori. If a meridian has nontrivial actual order in $G/N$, the corresponding effective boundary on the torus is pulled back from an ample divisor on a projective torus quotient. Pulling back a sufficiently small rational multiple through differentials cancels the poles and produces a forbidden Bogomolov sheaf. Thus the meridians vanish and the sequence is exact.

For a further finite cover, Proposition 6.2 considers $V=\operatorname{Hom}(N',\mathbb C)$ with $N'=N\cap G'$. Nontrivial eigencharacters of the lattice action are isolated points of a rank-one jump locus, hence torsion by [BW16]. An isogeny makes an eigenvector invariant. Inflation–restriction and injectivity of $H^2(A',\mathbb C)\to H^2(X',\mathbb C)$ would then give extra $H^1$, contradicting the Albanese dimension. Lemma 2.7 realizes every prescribed finite-index condition on $N$ by a finite cover of the total space, proving that $N$ is FAb.

[Cam04, Theorems 2.26; 5.12, pp. 27, 56] · [Cam23, Theorem 1.2(1)–(2), print p. 1904 / PDF p. 3] · Lemma 6.1 · pp. 40–41 · Proposition 6.2 · pp. 41–42 · [BW16, Theorem 1.3, p. 2]

2. A minimal relative base carrying an infinite quotient

Assume $N$ is infinite. Across all compatible finite covers and general Albanese fibres $S$, minimize $\dim T$ among fibrations $S\dashrightarrow T$ for which $N/C$ is infinite, where $C$ is the fibre-group image. Write the minimum as $\ell>0$. Proposition 6.3 globalizes a minimizer, using cycle families and $\Gamma$-reduction, to $X\dashrightarrow Z\to A$. For the actual fibre-group image $B_0$ of the first map,

and the general torus fibre carries a surjection $\pi_1^{\mathrm{orb}}(Y,\Delta_Y)\twoheadrightarrow Q$. Boundary coefficients use actual meridian orders in $R$, and $Q\hookrightarrow R$ preserves those orders on restriction. For every finite-index $Q_0$, choose one cover first, $S'\to S$, whose image $Q'\subset Q_0$ satisfies $\operatorname{gdim}(S',\psi\nu')\ge\ell$ for every infinite quotient of $Q'$. This is condition (M) of Theorem 4.1; choosing a different cover separately for each infinite quotient would be weaker.

Proposition 6.3 · pp. 44–46 · Lemmas 2.6–2.7 · pp. 10–11

3. The branch with an infinite linear image

An infinite linear image of the FAb group $Q$ cannot be virtually solvable. Pass to finite index, take a simple adjoint factor of the semisimple quotient of its Zariski closure, and use Selberg's lemma and Lemma 2.7 to realize a torsion-free image on a finite cover of the total space. Finite-order meridians die in that image, so the representation extends to the ordinary group $\pi_1(Y)$. If it were not generically large, composing its $\Gamma$-reduction with $S'\dashrightarrow Y$ would give an admissible base of dimension below $\ell$. The semisimple general-type criterion of [CCE14] makes $K_Y$, and hence $K_Y+\Delta_Y$, big.

§6.5 · pp. 46–48 · [CCE14, Theorems 1; 6.3, pp. 2, 29]

4. The branch where every finite-dimensional linear image is finite

Ordinary finite-dimensional representations do not exclude this case. The discussion of Theorem 4.1 constructs holomorphic tensors with unitary Hilbert coefficients and forces their minimal coefficient-map rank to be full by (M). This gives bigness of $K_Y+\Delta_Y$. Both branches retain the actual inertia of a fixed group quotient and choose a fresh very general fibre on each prepared model, using the same minimum. Bigness on a single model would not supply the descent hypothesis.

Theorem 4.1 · p. 23 · §6.5 · pp. 46–48

5. Descent, a finite kernel, and the covering formulation

The discussion of Theorem 5.8 pulls orbifold pluriforms from a positive-dimensional general-type base to a Bogomolov sheaf on a finite cover, contradicting specialness. Thus $N$ is finite. Lemma 2.6 passes to the finite-index centralizer of the finite kernel and kills its finite central commutator subgroup by taking sufficiently large powers of generators. The resulting abelian subgroup has finite index. This conclusion passes from the initial finite cover back to the original group. Lifting the complex structure to the connected finite topological cover corresponding to that subgroup gives the equivalent unramified holomorphic covering statement.

Theorem 5.8 · pp. 37–40 · Lemmas 2.6–2.7 · pp. 10–11 · §6.5 · pp. 46–48

1. A nonzero tensor with infinite-dimensional coefficient span

The nonamenable case uses Theorem 3.1: on a normal cover of a compact Kähler orbifold with infinite deck group, zero lies in the spectrum of the scalar $(p,0)$-Dolbeault Laplacian for some $p\in\{0,\ldots,n\}$. Under contrary spectral gaps, manuscript §3 constructs weighted inverses and finite-propagation smoothing. They preserve a nonzero pairing of the compact diagonal class, while norm and support-volume estimates force that pairing to zero. Proposition 4.3 folds low-spectrum forms down to regular coefficients and obtains a nonzero holomorphic form in a Hilbert ultrapower.

In the amenable case, an infinite finitely generated group lacks property (T), so [Sha06] supplies nonzero reduced $H^1$. Remove the contribution from finite-dimensional subrepresentations and use the Kähler type decomposition to obtain a holomorphic one-form. In either case the coefficient representation can have no nonzero finite-dimensional invariant subspace. Degree zero is excluded because a holomorphic section would be parallel and supply an invariant line, so $p>0$. Exterior powers produce coefficient rank one with infinite-dimensional span, hence a coefficient-line map and an orbifold line with positive curvature. Mere nonvanishing of a tensor would not contradict specialness.

Theorem 3.1 · p. 11 · Lemma 3.6; §3.6; (35) · pp. 21–23 · [Sha06, Theorem 4.2, print p. 1300 / PDF p. 1310] · Proposition 4.3; Lemmas 4.4–4.5 · pp. 24–27

2. Minimal rank and discrete monodromy

Minimize the rank of such coefficient-line maps on finite orbifold covers, obtaining $s\in\{1,\ldots,\ell\}$. Proposition 4.6 works with the manifold of regular image germs and a Lie-group closure of the monodromy. If its identity component moves the image, spectral decomposition of the unitary coefficients produces line tensors of smaller rank. Minimality and the finite-linear-image hypothesis force that component to act trivially, leaving an infinite discrete image.

Lemma 4.7 uses finite mass of the curvature current and coarea to give almost every fibre finite volume, with compact analytic closure. Its group image lies in a finite stabilizer of the discrete action, so the $\Gamma$-reduction contracts it and $\operatorname{gdim}\le s$. Pull back to the single cover $S'$ selected by (M); for the same infinite monodromy quotient, (M) gives $\operatorname{gdim}\ge\ell$. Thus $s=\ell$.

Proposition 4.6 · pp. 27–30 · Lemma 4.7 · pp. 30–31 · §4.5 · pp. 31–32

3. Descending an ordinary line before invoking projectivity

On a Galois orbifold cover, tensor the deck translates of the line and take a power killing chart stabilizers to descend to an ordinary line bundle. Full-rank curvature is strictly positive on a nonempty open set, so the volume criterion of [Bou02] makes this line big. Thus $Y$ is Moishezon and, being Kähler, is projective. Only then is the tensor inclusion interpreted on an adapted projective cover and Theorem 7.11 of [CP19] applied to make $K_Y+\Delta$ big. Neither projectivity nor pseudoeffectivity of $K_Y+\Delta$ was assumed at the outset.

§4.5 · pp. 31–32 · [Bou02, Theorem 1.2, p. 2] · [CP19, Theorem 7.11, p. 42]

1. Minimal log Kodaira dimension and the auxiliary boundary

Klt subadditivity over tori in [JW21] gives $\kappa(Z,K_Z+\Delta_Z)\ge d>0$. Choose its minimum $b$ in that cofinal collection and resolve the Iitaka map to $h:Z_1\to W$. At this step introduce an auxiliary klt boundary $\Delta_1$, distinct from true inertia, whose exceptional coefficients are sufficiently close to one so that

This preserves sections and the Iitaka map; a general Iitaka fibre $H$ has log Kodaira dimension zero. The argument does not assume birational invariance of the log Kodaira dimension of the true inertial adjoint on arbitrary models.

Theorem 5.8 · pp. 37–40 · [JW21, Main Theorem A(II), print p. 815 / PDF p. 4]

2. A torus-bundle model and recovery of the true boundary

Apply Theorem C and subadditivity of [JW21] to $H$. Restricting bigness to common general fibres of $h$ and $g$ shows that $H$ maps birationally by an isogeny onto a torus translate. After a finite isogeny, a birational model is the principal torus bundle

True inertia has no horizontal component, so $\Delta_{Z^*}=p^*\Delta_{W_1}$ and $K_{Z^*}=p^*K_{W_1}$. An equivariant comparison for the finite deck group returns to a cofinal prepared model over the original torus; the minimum $b$ then makes $K_{W_1}+\Delta_{W_1}$ big. A key point is to recover the true boundary rather than substitute the auxiliary exceptional coefficients for it.

Theorem 5.8 · pp. 37–40 · [JW21, Theorem C, print p. 818 / PDF p. 7]

3. Cancellation of poles on a neat model

Bigness remains obtainable after any necessary further modification, so the map $M_1\dashrightarrow W_1$ can be made neat. Lemma 5.2 ensures that every boundary order divides the multiplicity of every source divisor above it. Lemma 5.5 cancels the poles of orbifold pluriforms with differential vanishing, producing a line in $\Omega^{\dim W_1}$ with Iitaka dimension $\dim W_1>0$. This contradicts specialness of the finite étale cover $M_1$.

Definition 5.1; Lemmas 5.2–5.5 · pp. 33–35 · Theorem 5.8 · pp. 37–40

1. Input specialness with zero boundary

Apply Theorem 6.7 of [Cam11] to $(X\mid0)$ and then Theorem 1.1. The hypothesis is $\kappa=0$; it is not replaced by $c_1(X)=0$. The manuscript also mentions Orbifold Iitaka in catalogue 033 as another record of this established specialness implication. It is not an additional direct input to this proof.

[Cam11, Theorem 6.7, p. 92] · Theorem 1.1 · p. 2 · Corollary 1.2 · p. 2

1. Specialness and maximal ordinary Γ-dimension

Theorem 1.1 gives a finite étale cover $X'$ with torsion-free abelian fundamental group. Specialness and maximal ordinary $\Gamma$-dimension persist: a positive-dimensional subvariety with finite group image upstairs would, under the finite map, yield one with finite group image downstairs. The map from $\pi_1(X')$ to its Albanese lattice is an isomorphism. A smooth general Albanese fibre therefore has trivial group image, so maximal $\Gamma$-dimension forces it to be zero-dimensional. Connectedness makes it a single point, proving that $X'\to\operatorname{Alb}(X')$ is bimeromorphic.

Corollary 1.3 · p. 3 · [Cam23, Theorem 1.2(1)–(2), print p. 1904 / PDF p. 3]

2. Affine space as universal cover

The manuscript uses Remarks A.4(2)–(3) in Appendix A of [CH13]. Abelianity yields Conjecture S as above, and Conjecture S gives the bimeromorphic Iitaka conclusion for $U\simeq\mathbb C^n$. The latter is a separate geometric implication supplied by the appendix, rather than a direct assertion that the shape of the universal cover makes the Albanese map an isomorphism.

[CH13, Appendix A, Remarks A.4(2)–(3), print p. 371 / PDF p. 18] · Corollary 1.3 · p. 3

1. Construct a faithful representation of the whole group

The finite-index abelian subgroup $H\simeq\mathbb Z^r\oplus T$ of the finitely generated group $G$ has a faithful finite-dimensional complex representation $\sigma$. Its induced representation $\operatorname{Ind}_H^G\sigma$ is faithful: a kernel element preserves the summand indexed by the coset $H$, hence lies in $H$, and acts there through the faithful $\sigma$. Corollary 5.9 of [CCE14] applies to this faithful linearity of $G$, giving holomorphic convexity of the ordinary universal cover. Neither projectivity nor maximal $\Gamma$-dimension is added.

Corollary 1.4 · pp. 3–4 · [CCE14, Corollary 5.9, p. 26]

1. (i) Exclude a minimal-dimensional counterexample

Choose a counterexample with non-virtually-abelian deck group and minimal base dimension. Proposition 1.4 of [CH13] makes the base smooth and special. Theorem 1.1 makes its fundamental group virtually abelian, as is its quotient deck group, a contradiction. This step requires neither a Kähler compactification nor extension of the deck action to it.

[CH13, Proposition 1.4, print p. 357 / PDF p. 4] · Corollary 1.5 · pp. 4–5

2. (ii) The Albanese bundle and simply connected fibre

If $\pi_1(X)$ is infinite, apply (i) to the universal cover; if finite, it is already virtually abelian. Now the Kähler compactification hypothesis and Theorem 1.5 of [CH13] give a locally trivial holomorphic Albanese bundle with simply connected fibre on a finite étale cover. The fibre is compact Kähler; projectivity is not asserted.

[CH13, Theorem 1.5, print p. 358 / PDF p. 5] · Corollary 1.5 · pp. 4–5

3. The product decomposition of the universal cover

Pull the bundle back along $\mathbb C^q\to A$. With structure group the automorphism group of the compact complex manifold $F$, the bundle is topologically trivial over this contractible base. The Oka–Grauert principle over the Stein base makes it holomorphically trivial. Since $F$ is simply connected, $F\times\mathbb C^q$ is the ordinary universal cover of $X'$, hence also of $X$. This follows the connection on manuscript pp. 4–5, separately from the bundle conclusion supplied by [CH13].

Corollary 1.5 · pp. 4–5