Exercise: The main theorem. What does the abelianity conjecture say, and is it proven here?
Campana's abelianity conjecture predicts that the fundamental group of any special compact Kähler manifold has an abelian subgroup of finite index. The paper proves it in full: for every connected smooth compact Kähler manifold X, if X is special, then π₁(X) is virtually abelian. Equivalently, there is a connected finite unramified cover X' → X with π₁(X') abelian. No projectivity, no linearity, no residual finiteness, no prescribed Kodaira dimension is assumed; the finite index and the cover are allowed to depend on X; torsion in π₁(X) is allowed.
Claudon-Höring 2013, Compact Kähler manifolds with compactifiable universal cover: pdf/claudon-horing-2013-compact-kahler-manifolds-with-compactifiable-universal-cover.pdf
Campana 2004, Orbifolds, special varieties and classification theory: pdf/campana-2004-math0110051-orbifolds-special-varieties-and-classification-theory.pdf

Exercise: Kähler manifold. What is a compact Kähler manifold, and what are the basic examples?
A complex manifold is a topological space locally modelled on ℂⁿ with holomorphic transition maps. It has real dimension 2n. A Kähler structure is a Riemannian metric whose associated 2-form is closed, which forces compatibility between metric and complex structure. Compactness means every open cover has a finite subcover (in ℝⁿ, by Heine-Borel, this is equivalent to closed and bounded). Examples: complex projective space Pⁿ, complex tori ℂⁿ/Λ, smooth hypersurfaces in projective space, all compact Riemann surfaces. The Kähler condition buys strong global Hodge theory, which is used in the deep input theorems.
Ueno 1975, Classification Theory of Algebraic Varieties and Compact Complex Spaces: https://doi.org/10.1007/BFb0070570

Exercise: Fundamental group. What is π₁(X), and how does it relate to covers?
Fix a basepoint x₀. π₁(X, x₀) is the set of loops x₀ → x₀ up to continuous homotopy, with concatenation as product. For a compact manifold it is finitely presented. Up to conjugation it does not depend on x₀. Finite-index subgroups of π₁(X) are in bijection with connected finite unramified covers X' → X: the cover X' corresponds to the subgroup π₁(X') ≤ π₁(X). So "virtually abelian", "abelian after a finite cover", and "finite-index abelian subgroup" are equivalent statements.

Exercise: Virtually abelian. What does virtually abelian mean?
A group G is virtually abelian if it contains an abelian subgroup of finite index. Equivalently, some power-map trick (multiplication of generators killed in the commutator) allows passage to a torsion-free free abelian subgroup of finite index if G is finitely generated. For π₁(X) with X compact Kähler, this is the conclusion of the main theorem.

Exercise: Line bundle and Kodaira dimension. What are they?
A holomorphic line bundle L on X is a holomorphic family of 1-dimensional ℂ-vector spaces over X; sections are holomorphic functions into the fibres. The space H⁰(X, L^{⊗m}) is finite-dimensional for X compact. The Iitaka dimension κ(X, L) is: −∞ if h⁰(X, L^{⊗m}) = 0 for all m > 0; otherwise the unique integer d ≥ 0 such that h⁰(X, L^{⊗m}) grows like m^d up to a bounded factor, equivalently the dimension of the image of the m-th plurisection map to projective space for large divisible m. A point has κ = 0. Ample line bundles have κ = dim X. The canonical bundle K_X = det Ω^1_X satisfies κ(X) = κ(X, K_X), the Kodaira dimension.
Ueno 1975, Classification Theory of Algebraic Varieties and Compact Complex Spaces: https://doi.org/10.1007/BFb0070570

Exercise: Holomorphic p-forms. What is Ω^p_X?
The holomorphic cotangent bundle Ω^1_X is the complex dual of the holomorphic tangent bundle; a holomorphic section is a local holomorphic 1-form dz_i with holomorphic coefficients. Its p-th exterior power is Ω^p_X, sections of which are holomorphic p-forms. For X compact Kähler, H⁰(X, Ω^p_X) = H^{p,0}(X) has the same dimension as H^p(X, O_X) by complex conjugation. Hodge symmetry gives h^{p,q}(X) = h^{q,p}(X), and Serre duality gives h^{p,q}(X) = h^{n-p,n-q}(X).

Exercise: Sheaves. What is a sheaf, and what is a sheaf morphism?
For our purposes, a sheaf on X assigns to each open set U a vector space (or abelian group) of "local data" of a fixed type, glued by a consistency rule. A line bundle L is an example: L(U) is the space of holomorphic sections over U. A sheaf morphism L → F assigns to each open U a linear map L(U) → F(U), compatible with restrictions. Line bundles, Ω^p, and their tensor powers are all sheaves. Bogomolov sheaves are nonzero sheaf morphisms L → Ω^p with a Kodaira-dimension constraint on κ(X, L).

Exercise: Bogomolov sheaf. What is a Bogomolov sheaf?
A Bogomolov sheaf on a compact Kähler X is a nonzero sheaf morphism L → Ω^p_X from a line bundle L to the sheaf of holomorphic p-forms, with κ(X, L) = p for some 1 ≤ p ≤ dim X. Bogomolov's theorem gives κ(X, L) ≤ p always; equality means "the p-form image has an Iitaka dimension as large as possible", which is a strong positivity property. Such a sheaf exhibits X as having a positive-dimensional "general type base" hidden inside the p-form data. Specialness is exactly the absence of Bogomolov sheaves.
Bogomolov 1979, Holomorphic tensors and vector bundles on projective varieties: https://www.mathnet.ru/eng/im1965
Campana 2004, Orbifolds, special varieties and classification theory: pdf/campana-2004-math0110051-orbifolds-special-varieties-and-classification-theory.pdf

Exercise: Special. What is a special compact Kähler manifold?
Let X be compact Kähler of complex dimension n. X is special if for every 1 ≤ p ≤ n and every nonzero sheaf morphism L → Ω^p_X from a line bundle L, one has κ(X, L) ≤ p − 1 (i.e., strictly less than p). Equivalently, X is special iff there is no Bogomolov sheaf on X. A point is special. Pⁿ is special. Every Calabi-Yau and irreducible symplectic manifold is special. Conversely, all smooth compact Kähler manifolds of general type (κ = n) and all fibrations onto a general type base contain a Bogomolov sheaf, so they are not special.
Campana 2004, Orbifolds, special varieties and classification theory: pdf/campana-2004-math0110051-orbifolds-special-varieties-and-classification-theory.pdf

Exercise: Why "special". What is the intuition behind the name?
Campana's classification program partitions every compact Kähler manifold into a base of general type plus "special" fibres, in analogy with the classification of curves (base = curve of Riemann surface, fibres = genus 0 or 1) and surfaces. On a manifold of general type, holomorphic tensors with κ = p detect the "base" completely, so the fundamental group sees a piece of it. The conjecture says special manifolds should have "hidden rigidity" in the opposite direction: their π₁ has no interesting non-abelian behaviour after finite covers. The paper makes this precise.

Exercise: Albanese map. Define it precisely.
Let H⁰(X, Ω¹_X) be the ℂ-vector space of global holomorphic 1-forms (finite-dimensional, since X is compact). Integration over 1-cycles yields a natural embedding of H_1(X, ℤ) modulo torsion as a lattice in H^0(X, Ω¹_X)^*, by Hodge theory of a Kähler manifold. The Albanese torus is
Alb(X) = H⁰(X, Ω¹_X)^* / H_1(X, ℤ),
a compact complex torus of dimension q = b_1(X)/2 (equal to the dimension q = h^{1,0}(X) by Hodge symmetry for X Kähler). Fixing a basepoint gives a holomorphic map α: X → Alb(X), which sends 1-cycles to the corresponding lattice coset. It satisfies the universal property: any holomorphic map X → torus factors through α up to translation.
Ueno 1975, Classification Theory of Algebraic Varieties and Compact Complex Spaces: https://doi.org/10.1007/BFb0070570

Exercise: Albanese surjectivity. Why is the Albanese map of a special X surjective with connected fibres?
Campana's 2004 paper proved this, but there was an error in the treatment of multiple fibres; the 2023 correction restores surjectivity and connectedness for Kähler special manifolds. The nonprojective Albanese torus requires careful meridian bookkeeping, which the paper does not avoid—it uses actual meridian orders throughout Section descent.tex. The corrected result is all the paper needs from that direction.
Campana 2004, Orbifolds, special varieties and classification theory: pdf/campana-2004-math0110051-orbifolds-special-varieties-and-classification-theory.pdf
Campana 2023, Albanese map of special manifolds: a correction: pdf/campana-2023-albanese-map-of-special-manifolds-a-correction.pdf

Exercise: Maximal b₁. Why can one choose a finite cover with maximal first Betti number, and what does it give?
Specialness is preserved by finite étale covers (Campana 2004) and finite étale covers of compact Kähler manifolds are compact Kähler. Every special X has surjective Albanese, so b_1(X) ≤ 2 dim X. Choose a finite étale cover X' → X with b_1(X') maximal. Pullback along a finite unramified cover is injective on H¹(−, ℚ) by transfer, and if X'' → X' were a cover with larger b_1, then X'' → X is a cover of X with larger b_1, contradicting maximality. So all further covers have b_1 = b_1(X'). One may work on X' with this convenient property.
Campana 2004, Orbifolds, special varieties and classification theory: pdf/campana-2004-math0110051-orbifolds-special-varieties-and-classification-theory.pdf

Exercise: Albanese exactness. State the exact sequence and sketch its proof.
Let α: X → A = Alb(X), G = π₁(X), S = a smooth general Albanese fibre, N = im(π₁(S) → G). Then 1 → N → G → π₁(A) → 1 is exact (Lemma: Albanese Exactness). Proof: over the smooth Albanese fibration locus, the homotopy exact sequence of a smooth proper family gives exactness after passing to the open space. The monodromy homomorphism ψ: π₁(A°) → R = G/N has finite-order image at every meridian, since a loop around the discriminant lifts to a loop in X whose image is killed by N. Filling in the discriminant gives a well-defined map π₁(A) → R; the natural map G → R = G/N factors through π₁(A), and the two factorizations are inverse, so R ≅ π₁(A). The case A = {pt} reduces to N = G.
Juanyong Wang 2021, On the Iitaka conjecture C_{n,m} for Kähler fibre spaces: pdf/juanyong-wang-2021-on-the-iitaka-conjecture-c-n-m.pdf

Exercise: N normal. Why is N normal in G, and finitely generated?
Over a smooth Albanese fibration locus, the homotopy exact sequence makes N = ker(π₁(X) → π₁(A)) normal in G. The group N is finitely generated because the general Albanese fibre S is a compact complex manifold, and the fundamental group of any compact manifold is finitely presented. Base-point transport in the smooth proper family changes N up to conjugation, which does not affect normalcy.
Ueno 1975, Classification Theory of Algebraic Varieties and Compact Complex Spaces: https://doi.org/10.1007/BFb0070570

Exercise: FAb group. What is an FAb group?
A group N is FAb if every finite-index subgroup of N has finite abelianization (i.e., H₁(−, ℤ) finite). The property is inherited by finite-index subgroups and by quotients: a quotient of an FAb group is FAb.

Exercise: N is FAb. Sketch the proof, with the role of Botong Wang's theorem.
Proposition (Fiber FAb): N is FAb. Pass to a finite-index cover X', let N' = N ∩ π₁(X'), Λ = π₁(A), V = Hom(N', ℂ) (a finite-dim. complex vector space, since N' is finitely generated). Λ acts on V by conjugation; write T_λ for the operator on V. For a finite-order character β: Λ → ℂ*, the Koszul resolution of Λ with coefficients in ℂ_β is contractible, so H^q(Λ, ℂ_β) = 0 for all q > 0. Inflation-restriction therefore identifies H¹(G', ℂ_β) ≅ (V ⊗ ℂ_β)^Λ. Now suppose V ≠ 0: the commuting T_λ have a common eigenvector v with nonzero eigenvalue, defining a character χ. If χ ≠ 1 then H¹(G', ℂ_β) ≠ 0 for β = χ^{−1}, and Botong Wang's torsion-translate theorem says this support point is a torsion character of Λ. Kill χ by an isogeny A_0 → A and pull back to a cover X_0; the argument descends, keeping V ≠ 0. Now V^Λ ≠ 0. Untwisted inflation-restriction gives an exact sequence 0 → H¹(Λ, ℂ) → H¹(G', ℂ) → V^Λ → H²(Λ, ℂ) → H²(G', ℂ), and the pullback map H²(A') → H²(X') is injective (Kähler fibre integration plus Poincaré duality), which forces V^Λ = 0. Combined with V = 0, this proves Hom(N', ℂ) = 0 for every finite-index N'. Since N' is finitely generated, its abelianization is finite; N is FAb.
Botong Wang 2016, Torsion points on the cohomology jump loci of compact Kähler manifolds: pdf/botong-wang-2016-1312.6619-torsion-points-on-the-cohomology-jump-loci.pdf

Exercise: Finite-by-abelian. State the finite-by-abelian lemma.
If 1 → F → E → ℤ^r → 1 is exact and F is finite, then E is virtually abelian and residually finite. In particular E has a finite-index subgroup disjoint from F. Proof (Lemma: Finite-by-Abelian): the centralizer E_0 of F has finite index; its commutator is central in F, so powers of generators of E_0 commute. Pass to the abelian subgroup generated by large powers and to its core in E; intersect sufficiently many conjugates to separate F.

Exercise: Realize finite index. State the realization lemma.
If G is finitely generated, N is a finitely generated normal subgroup, and G/N ≅ ℤ^r, then for every finite-index N_0 ≤ N there is a finite-index G_0 ≤ G with G_0 ∩ N ≤ N_0 (Lemma: Realize Finite Index). Proof: take the characteristic finite-index K ≤ N inside the core of N_0; then G/K is finite-by-ℤ^r; by the finite-by-abelian lemma it has a finite-index subgroup disjoint from N/K; its inverse image is G_0.

Exercise: Reducing to N finite. Why is it enough to prove N is finite?
Suppose the Albanese exact sequence 1 → N → G → ℤ^r → 1 has finite N. By the finite-by-abelian lemma, G is virtually abelian. Conversely, if N is infinite, the rest of the proof constructs a Bogomolov sheaf on a finite cover of X, contradicting specialness. So the entire paper proves N is finite.

Exercise: Γ-reduction. What is Campana's Γ-reduction?
Given a homomorphism ρ: π₁(M) → P from a smooth compact Kähler manifold, there is a fibration γ_ρ: M ⇢ B_ρ with connected general fibre, Kähler base model B_ρ, such that a smooth general fibre has finite ρ-image, and γ_ρ contracts every compact irreducible subvariety through a very general point whose π₁ has finite ρ-image. dim B_ρ is the Γ-dimension γd(M, ρ). ρ is generically large if γd(M, ρ) = dim M. The theorem is due to Campana; the arbitrary-quotient form is used in the paper.
Campana 1994, Remarques sur le revêtement universel des variétés kählériennes compactes: pdf/campana-1994-remarques-sur-le-revetement-universel-des-varietes.pdf
CCE 2015, Linear representations of Kähler groups: pdf/cce-2015-1302.5016-linear-representations-of-kahler-groups.pdf

Exercise: Minimal ℓ. What is minimized, and how does a Barlet-type space give the factorized map?
Assume N is infinite. Consider all finite étale covers X' → X and all fibrations S ⇢ T of a very general good Albanese fibre S, where C = im(π₁(T) → N) and N/C is infinite. Minimize ℓ = dim T over all such data. The identity fibration S ⇢ {pt} is admissible (N infinite), so ℓ ≥ 1. Compactified family language (Barlet space of compact analytic cycles, Campana's meromorphic Γ-reduction) globalizes the minimizer to a factorization X ⇢ Z →^g A with connected general fibres and dim Z − dim A = ℓ (Proposition: Minimal Base).
Barlet 1975, Espace analytique réduit des cycles analytiques complexes compacts: https://doi.org/10.1007/BFb0080243
Campana 1994, Remarques sur le revêtement universel des variétés kählériennes compactes: pdf/campana-1994-remarques-sur-le-revetement-universel-des-varietes.pdf

Exercise: Inertial boundary. What is Δ_Z on a prepared model of Z?
Fix R = G/B_0 where B_0 is the actual group image of the general Z-fibre group. On a prepared smooth model Z (smooth compact Kähler, resolved source, SNC deleted divisor, maps holomorphically to A), for a prime divisor D ⊂ Z let m_D be the order in R of a meridian around a general point of D under the compatible monodromy ψ: π₁(Z°) → R. The inertial boundary is Δ_Z = ∑_D (1 − 1/m_D) D. It is a ℚ-Cartier divisor with SNC support and coefficients in [0,1); the pair (Z, Δ_Z) is klt. On a very general good Albanese fibre Y ⊂ Z, the restriction Δ_Y = Δ_Z|_Y has the same integer meridians, and the same order.
Juanyong Wang 2021, On the Iitaka conjecture C_{n,m} for Kähler fibre spaces: pdf/juanyong-wang-2021-on-the-iitaka-conjecture-c-n-m.pdf

Exercise: Two cases. What is the case distinction after minimality?
The quotient Q = N/B_0 is an infinite FAb subgroup of R. Either every finite-dimensional complex linear image of Q is finite ("Case A"), or Q has an infinite complex linear image ("Case B"). In Case A, L² spectral analysis plus Hilbert tensor positivity produces a big cotangent line on Y, hence K_Y + Δ_Y big. In Case B, FAb rules out virtually solvable images, Selberg's lemma produces a torsion-free image in a simple semisimple group, and CCE linear general-type gives K_Y big. Both cases then contradict torus descent.

Exercise: Spectral theorem. What does "zero in the Dolbeault spectrum" mean?
Theorem (Zero in the Dolbeault Spectrum): if M is a connected compact Kähler orbifold and U → M a connected normal orbifold cover with infinite deck group P, then in some (p, 0) degree zero belongs to the L² spectrum of the Dolbeault Laplacian Δ'' on scalar (p, 0)-forms. Equivalently, there are L² (p, 0)-forms on U with Δ''-eigenvalues → 0 (equivalently, the infimum of the spectrum is zero). The proof places the diagonal {(u, u)} into the compact quotient D = (U × U)/P; a positive lower bound on the spectral gap would let weighted smoothing shrink the pairing of its class with the diagonal test form to zero while preserving the class—impossible. Weighted Monge–Ampère potentials with cellwise exponential integrability are used. This is the building block for constructing holomorphic tensors with Hilbert coefficients.
Atiyah 1976, Elliptic operators, discrete groups and von Neumann algebras: pdf/iyah-1976-elliptic-operators-discrete-groups-and-von-neumann.pdf
Dodziuk 1977, De Rham–Hodge theory for L²-cohomology of infinite coverings: https://doi.org/10.1016/0040-9383(77)90013-1

Exercise: Spectral → Hilbert tensors. How does the spectral theorem produce nonzero holomorphic Hilbert-valued tensors?
Fold low-spectrum (p, 0)-forms on a normal orbifold cover U → Y down to the compact base, using the regular representation λ_Q of Q on ℓ²(Q). The result is a holomorphic section of Ω^p_Y ⊗ λ_Q in the ultrapower as the spectral parameter → 0. Taking an exterior power / rank-minimalization gives a line tensor. For the amenable case, Shalom's reduced-cohomology theorem produces the analogous Hilbert coefficient. Combine to get Proposition (Hilbert Coefficients): a separable unitary representation ρ: Q → U(H) with no nonzero finite-dim. invariant subspace and a nonzero holomorphic section of Ω^p_Y ⊗ H_ρ over the orbifold.
Shalom 2006, The algebraization of Kazhdan's property (T): pdf/shalom-2006-the-algebraization-of-kazhdan-s-property-t.pdf
Atiyah 1976, Elliptic operators, discrete groups and von Neumann algebras: pdf/iyah-1976-elliptic-operators-discrete-groups-and-von-neumann.pdf

Exercise: Line tensor. What is a line tensor?
A holomorphic section τ ∈ H⁰(Y_0, (Ω^1_{Y_0})^{⊗m} ⊗ B_ρ) whose "coefficient-line map" (dual of cotangent tensor factor → coefficient Hilbert space) has generic rank one. So the tensor is (locally) a holomorphic 1-form-line times a holomorphic coefficient vector; the coefficients vary inside a projective Hilbert line. Lemma (Plucker) produces a line tensor whose coefficient lines span an infinite-dimensional space: take an exterior power r of chained coefficients and antisymmetrize to get a decomposable rank-one entry inside a large tensor power.

Exercise: Minimal rank s. What is minimized, and how is the minimal case characterized?
Among all line tensors with infinite-dimensional coefficient span over finite orbifold covers, minimize the generic complex rank s of the coefficient-line map F: U° → PP(B_ρ). Rank zero would make F constant, giving a 1-dim. span; so 1 ≤ s ≤ ℓ. Proposition (Discrete Monodromy): after shrinking the finite-index subgroup Γ ≤ Q defining the cover, the isometry monodromy of F on its connected regular image-germ manifold Z acts properly, discretely, and effectively. Proof: build Z as a connected second-countable Hausdorff complex manifold of germs of rank-s plaques, use Myers–Steenrod/Manoussos to make the isometry group act properly, use Matveev–Troyanov and Tao no-small-subgroups to make it a Lie group, and show the identity component acts trivially on Z (otherwise minimality of s would be contradicted on a locally injective open set).
Matveev–Troyanov 2017, The Myers–Steenrod theorem for Finsler manifolds of low regularity: pdf/matveev-troyanov-2017-1605.03850-the-myers-steenrod-theorem-for-finsler-manifolds.pdf
Manoussos 2010, On the action of the group of isometries on a locally compact metric space
Tao 2014, Hilbert's Fifth Problem and Related Topics: pdf/tao-2014-hilbert-s-fifth-problem-and-related-topics.pdf

Exercise: Compact fibres from mass. How compactify the I-orbit fibres of rank-s maps if the target is not compact?
Lemma (Compact Fibres): let V be a smooth compact Kähler modification of the finite-cover source, and T a closed positive (1,1)-current on V that restricts on the smooth locus V° to a positive multiple of the transverse Kähler form f^*ω_Z from the local rank-s submersion f: V° → Z/I. The mass bound ∫_V T^s ∧ ω^{d−s} < ∞ follows from the pp-form calculus for truncated Monge–Ampère potentials (Boucksom). Coarea then shows almost every fibre has finite volume. El Mir's extension theorem: the integration current of such a finite-volume fibre extends as a positive closed current across the analytic complement; Siu's theorem: the Lelong superlevel sets of a closed positive current are analytic, so the support is compact analytic of pure dimension d − s. Those components meeting V° have fundamental group mapping into a finite stabilizer of I, so the Γ-reduction contracts them.
El Mir 1984, Sur le prolongement des courants positifs fermés: https://doi.org/10.1007/BF02392374
Siu 1974, Analyticity of sets associated to Lelong numbers: https://doi.org/10.1007/BF01389965
Boucksom 2002, On the volume of a line bundle: pdf/boucksom-2002-math0201031-on-the-volume-of-a-line-bundle.pdf

Exercise: Minimality contradicts smaller s. How?
Suppose the minimal line tensor has s < ℓ. Apply the minimality hypothesis (M) to Q_0 = Γ: get a connected finite cover S' → S with image Q' ⊆ Γ, and for every infinite quotient ψ: Q' → P an infinite image H, the Γ-reduction of ψ ∘ ν' has base dimension ≥ ℓ. Now ψ = im(Q') ⊆ I is an infinite quotient (since Γ's image in I is infinite), so the transverse Kähler current on V yields gdim(V, ρ) ≤ s (Lemma: Compact Fibres). That contradicts gdim ≥ ℓ. So s = ℓ.

Exercise: Campana–Păun. How does s = ℓ finish Case A?
Take a finite Galois orbifold cover dominating the minimal-rank cover. Tensor all deck-group translates of the descended line bundle: the result is a G-linearized line bundle M whose invariant metric has positive curvature, and which admits a nonzero map into a symmetric power of the orbifold cotangent tensor bundle. Equivariant descent gives a positive orbifold line bundle M_0 → a nonempty open subset with strictly positive curvature. A positive power of M_0 descends to an ordinary big line bundle A on Y (Boucksom volume criterion). Big line + Kähler ⇒ Moishezon ⇒ projective (Moishezon's theorem). On the Campana–Păun adapted Kawamata cover, π^*A injects into a tensor power of the adapted orbifold cotangent bundle; the Campana–Păun Theorem 7.11 (orbifold cotangent positivity) then forces K_Y + Δ big.
Boucksom 2002, On the volume of a line bundle: pdf/boucksom-2002-math0201031-on-the-volume-of-a-line-bundle.pdf
Campana–Păun 2019, Foliations with positive slopes and birational stability of orbifold cotangent bundles: https://doi.org/10.1007/s10240-019-00105-w

Exercise: Linear general-type. State the CCE linear general-type criterion.
Theorem (Linear General-Type): if a compact Kähler manifold M has a generically large complex linear representation ρ with torsion-free image and connected simple semisimple Zariski closure, then M is of general type. Proof: CCE (2015) give a Shafarevich morphism sh_ρ: M → Sh_ρ(M) with normal projective target of general type; generic largeness forces the general fibre to be a point; the morphism is bimeromorphic; bimeromorphic invariance of Kodaira dimension finishes.
CCE 2015, Linear representations of Kähler groups: pdf/cce-2015-1302.5016-linear-representations-of-kahler-groups.pdf

Exercise: Selberg's lemma. How does it give a torsion-free linear image?
In Case B, an infinite complex linear image of Q → GL(V) is possible but the Zariski closure L is reductive-or-solvable. Pass to a finite-index subgroup of Q with connected L. Since Q is FAb, L has no infinite virtually solvable (or solvable) quotient; the semisimple quotient L/Rad(L) is nontrivial. Take a simple factor H of L_adj = (L/Rad(L))_ad. The projection Q → H(ℂ) has a Zariski-dense image. In the faithful adjoint representation, that image is a finitely generated linear group; Selberg's lemma gives a finite-index torsion-free subgroup. Pull back to Q, realize via the Realize-Finite-Index Lemma. The image in H(ℂ) stays Zariski dense (finite-index subgroup of a dense subgroup of the connected group H(ℂ) is still dense).
Selberg 1960, On discontinuous groups in higher-dimensional symmetric spaces
CCE 2015, Linear representations of Kähler groups: pdf/cce-2015-1302.5016-linear-representations-of-kahler-groups.pdf

Exercise: Meridian order m_D. How does it define the inertial boundary?
Let p: X → Z be a dominant holomorphic map and ρ: π₁(X) → R a compatible homomorphism over a Zariski-open smooth fibration locus. For a prime divisor D ⊂ Z, let μ_D be a meridian loop around a general point of D (well-defined up to conjugacy). The order m_D = ord_R(ψ(μ_D)) is finite, because the preimage of D in X is a compact hypersurface whose transverse boundary maps trivially in R. Define Δ_Z = ∑_D (1 − 1/m_D) D. This is the "inertial boundary" and it is an invariant of the map and quotient, not of the model.

Exercise: Inertia divisibility. State it.
Lemma (Inertia Divisibility): let X → Z be dominant holomorphic of smooth complexes, ψ: π₁(Z°) → R compatible, p: Z → W smooth proper surjective with connected fibres. Suppose Δ_Z = p^*Δ_W globally (nonzero components of Δ_W are divisor orders m_D ≥ 2), and every other divisorial meridian is trivial under ψ. If E ⊂ X dominates D ⊂ W under p ∘ f, then m_D divides ord_E((p ∘ f)^*D). This includes the case where f(E) is contained in p^{-1}(D) as a proper subvariety. Proof: at a general point of D, the fundamental group of the local smooth preimage of D is ℤ; the winding number around D is ord_E((p ∘ f)^*D); the compatibility of ψ with ρ forces its m_D-th power to be trivial.

Exercise: Small ample boundary. State it.
Lemma (Small Ample Boundary): let u: X → B be a surjective holomorphic map from a smooth compact Kähler X to a positive-dim. torus, and Δ = ∑ (1 − 1/m_D) D an ample rational divisor on B with m_D ≥ 2 and m_D | ord_E(u^*D) for every prime divisor E dominating D. Then X is not special. Proof: choose ε > 0 such that ε v_E ≤ a_E for every prime divisor E (where v_E and a_E are the vanishing multiplicities of u^* Δ and of the top differential of u along E, respectively); finitely many divisors give a bound. The pullback of a plurisection of mεΔ, multiplied by the m-th power of the top differential of u, gives a holomorphic section of L^{⊗m} for the saturated top-differential line L. So κ(X, L) ≥ dim B; specialness forces κ(X, L) ≤ dim B; contradiction.

Exercise: Neat pullback. State it.
Lemma (Neat Pullback): suppose u: X̂ → W is a neat fibration (smooth, every u-exceptional divisor disappears under ν: X̂ → X), Δ_W = ∑ (1 − 1/m_D) D with m_D ≥ 2, and m_D | ord_E(u^*D) for every dominant component. Then K_W + Δ_W big ⇒ X is not special. Proof: pull back a big pluricanonical section, cancel the boundary pole by dividing by the differential with e − 1 ≥ e(1 − 1/m_D) vanishing, extend across codim ≥ 2, get a Bogomolov sheaf on X. Neat models exist by Juanyong Wang's Lemma 1.4 for any meromorphic fibration over a base.

Exercise: Birational push-down. State it.
Lemma (Birational Push-down): if μ: Z' → Z is a modification between smooth base models of the same fibration and quotient, then for sufficiently divisible m there is a natural injection H⁰(Z', m(K_{Z'} + Δ_{Z'})) ↪ H⁰(Z, m(K_Z + Δ_Z)). Proof: view a section upstairs as a meromorphic pluricanonical form on the common function field; divisorial poles satisfy the same bounds on strict transforms; extend across codim ≥ 2.

Exercise: Common-fibres. State it.
Lemma (Common Fibres): let g: Z → A and h: Z → W be surjective holomorphic maps with connected general fibres and Z smooth. Let L be a rational line bundle. If L restricts big to a very general smooth g-fibre, then L restricts big to every component of a general fibre of (h, g): Z → J ⊆ W × A. This is purely a properness + coherent-direct-image argument: choose the very general g-fibre where base change holds for all m-sheaves; the relative evaluation map has full generic rank; a general common fibre meets the non-degeneracy open; the section map restricts injectively.

Exercise: klt pair. What is a klt pair (Z, Δ)?
Δ is an effective ℚ-divisor on Z, typically with simple normal crossing support and coefficients in [0, 1). The pair (Z, Δ) is klt (Kawamata log terminal) if all coefficients of Δ are < 1 and Δ has SNC support, and more generally iff every divisorial "discrepancy" of the pair is strictly greater than −1. Intuitively, (Z, Δ) is mild enough that positivity and vanishing theorems still apply. Wang's subadditivity theorem and Campana–Păun's positivity theorem work for klt pairs. On a prepared model of the minimal base, the inertial boundary Δ_Z makes (Z, Δ_Z) klt.

Exercise: Wang on tori. State the two Juanyong Wang theorems used.
Theorem (Wang on Tori): (a) if Z is smooth compact Kähler, (Z, D) klt, and g: Z → A a fibration onto a complex torus, then κ(Z, K_Z + D) ≥ κ(F, K_F + D|_F) for a general smooth fibre F (subadditivity); (b) if κ(Z, K_Z + D) = 0, the Albanese morphism of Z is surjective with connected fibres. Both are Theorems A(II) and C of Wang 2021, valid over arbitrary complex tori (no projectivity).
Juanyong Wang 2021, On the Iitaka conjecture C_{n,m} for Kähler fibre spaces: pdf/juanyong-wang-2021-on-the-iitaka-conjecture-c-n-m.pdf

Exercise: Iitaka fibration. What is it?
If κ(Z, L) = d ≥ 0, the m-th plurisection spaces H⁰(Z, mL) for large divisible m define meromorphic maps to projective space whose images eventually stabilize to a d-dimensional variety. The resulting fibration h: Z ⇢ W is the Iitaka fibration of L; it has connected general fibres, and K_Z + Δ restricted to a general fibre has Iitaka dimension 0. The torus descent theorem applies to the Iitaka fibration of L = K_Z + Δ_Z.

Exercise: Torus descent. Sketch the torus descent theorem.
Theorem (Torus Descent): if M is special compact Kähler with α: M → A surjective connected fibres and M ⇢ Z →^g A is a factorization by fibrations with d = dim Z − dim A > 0, then on every prepared model of Z and for the true inertial boundary Δ_Z for R = π₁(M)/im(π₁(F_c)), K_Y + Δ_Y is not big for any very general g-fibre Y. Proof: assume big. Wang (a) gives κ(Z, K_Z + Δ_Z) ≥ d; let h: Z_1 → W be its Iitaka fibration with W projective, dim W = b ≥ d. On a very general h-fibre H, the common-fibres lemma makes K_H + Δ_1|_H big; Wang (a) + the effective-divisor-on-torus lemma then forces the Albanese map of H birational and H ≅ Ab(H) up to modification. Components of Δ_1|_H are birational-exceptional, so on the torus-bundle model Z_* = W_1 ×_{A_1/T_1} A_1, the true inertia Δ_Z has no horizontal part over W_1. So Δ_{Z_*} = p^*Δ_{W_1} and K_{Z_*} = p^*K_{W_1}; the birational push-down lemma gives κ(W_1, K_{W_1} + Δ_{W_1}) ≥ b. Choose a neat model; the inertia divisibility lemma + neat pullback then produce a Bogomolov sheaf on a finite cover of M, contradicting specialness.

Exercise: Effective divisor on a torus. State it.
Lemma (Torus Divisor): for a nonzero effective rational divisor D on a complex torus A, there is a quotient p: A → B with connected fibres to a positive-dim. abelian variety and an ample effective rational D_B with D = p^*D_B; in particular κ(A, D) > 0. Proof: average the integration current over translations; the resulting translation-invariant semipositive (1,1)-form has integral rational radical W; quotient by K = W/(W ∩ Λ). Every prime component of D is invariant under translations by K, so its image is a divisor on B and the pullback equals D with multiplicity one. The induced form on B is positive definite, so D_B is ample.

Exercise: Endgame. How do the two cases and the group lemmas close the proof?
Apply Proposition (Minimal Base): Q = N/B_0 is infinite, FAb, has minimal-base Z and prepared fibres Y with true inertia Δ_Y. Case A: s = ℓ gives K_Y + Δ_Y big; torus descent on the cofinal collection of prepared models contradicts specialness. Case B: torsion-free Selberg image → generically large representation → linear general-type gives K_Y big; same torus descent contradicts. Both exclude N infinite; so N is finite; by the finite-by-abelian lemma, G = π₁(X) has a finite-index abelian subgroup. The initial covering X' → X was finite étale, so π₁(X) is virtually abelian for the original X as well.

Exercise: Kodaira-zero. State and prove the κ = 0 corollary.
Corollary (Kodaira-Zero Abelianity): if X is smooth compact connected Kähler with κ(X) = 0, then π₁(X) is virtually abelian. Proof: Campana's Theorem 6.7 (κ = 0 ⇒ X special) reduces to the main theorem.
Campana 2011, Orbifoldes géométriques spéciales et classification bimeromorphe des variétés kählériennes compactes: pdf/campana-2011-0705.0737-orbifoldes-geometriques-speciales-et-classification-bimeromo.pdf

Exercise: Conjecture S. State the Conjecture S and Iitaka uniformization corollary.
Corollary (Conjecture S and Iitaka Uniformization): if X is connected compact Kähler of complex dim n and either (i) X is special with γd(X) = n, or (ii) the ordinary universal cover of X is biholomorphic to ℂⁿ, then there is a connected finite étale cover X' → X bimeromorphic to a complex torus. Proof: the main theorem gives virtual abelianity; take the torsion-free abelian cover X'; π₁(X') → H_1(X', ℤ) is then an isomorphism; the Albanese map of X' is surjective with connected fibres; the group of a general Albanese fibre has trivial image in π₁(X'); maximality of ordinary Γ-dimension forces the general fibre to have dimension 0; so the Albanese is bimeromorphic. Assertion (ii) is Campana's implication (Conjecture S ⇒ the stated bimeromorphic Iitaka conclusion), from Remark A.4(2) of Claudon-Höring.
Claudon-Höring 2013, Compact Kähler manifolds with compactifiable universal cover: pdf/claudon-horing-2013-compact-kahler-manifolds-with-compactifiable-universal-cover.pdf

Exercise: Holomorphic convexity. State the corollary and sketch it.
Corollary (Special Shafarevich/Convexity): if X is smooth compact Kähler special, then the ordinary universal cover of X is holomorphically convex. Proof: G = π₁(X) is finitely generated and virtually abelian; take the finite-index abelian subgroup H; write H = ℤ^r ⊕ T; T finite. Construct a faithful finite-dimensional complex representation of H via powers of 2 on the ℤ-factors and roots of unity on the T-factors; induce to G. The induced rep is faithful because in the direct-sum realization indexed by left cosets of H, the kernel lies in H and acts faithfully there. Campana–Claudon–Eyssidieux's Kähler linear Shafarevich theorem (author's Corollary 5.9) gives holomorphic convexity of the ordinary universal cover.
CCE 2015, Linear representations of Kähler groups: pdf/cce-2015-1302.5016-linear-representations-of-kahler-groups.pdf

Exercise: Compactifiable covers. State the two-part corollary.
Corollary (Compactifiable Covers): (i) if X is connected normal compact Kähler and p: V → X a connected infinite étale Galois covering with deck group Γ, and V is a Zariski-open subset of a normal compact complex space, then Γ is virtually abelian. (ii) if X is a connected compact Kähler manifold and U the ordinary universal cover, and U is a Zariski-open subset of a normal compact Kähler space, then there is a connected finite étale X' → X whose Albanese α_{X'}: X' → A = Alb(X') is a locally trivial holomorphic fibre bundle with simply connected compact Kähler fibre F; writing q = dim_C A, U ≅ F × ℂ^q. Proof: (i) minimal-counterexample reduction of Claudon–Höring to a smooth special base; main theorem. (ii) apply (i); the Kähler compactification hypothesis is the Kähler analogue of Definition 2.3 in Claudon–Höring, and Theorem 1.5 gives the Albanese bundle on a connected finite étale cover X'. The pullback of the principal Aut(F)-bundle over A to ℂ^q is holomorphically trivial (Oka–Grauert), so U ≅ F × ℂ^q.
Claudon-Höring 2013, Compact Kähler manifolds with compactifiable universal cover: pdf/claudon-horing-2013-compact-kahler-manifolds-with-compactifiable-universal-cover.pdf

Exercise: What wasn't assumed. What hypotheses does the proof not use?
The proof needs no projectivity of X (Wang's theorems hold for arbitrary complex tori), no linearity of π₁(X) (Case A assumes no infinite linear image but does not require linearity), no maximality of ordinary Γ-dimension, no residual finiteness of π₁(X), and no prescribed value of κ(X). Compact Kähler is genuinely used—in Wang's subadditivity theorems, in Campana–Păun orbifold positivity, and in the Kähler Albanese story. The special sub-case κ(X) = 0 does not need anything beyond the main theorem plus Campana's essentially free κ = 0 ⇒ specialness.

Exercise: Correctness bookkeeping. What is the role of the 2023 correction of Campana's Albanese theorem?
Campana's 2004 paper claimed the Albanese map of a special manifold is surjective with connected fibres and has no multiple fibres. The multiple-fibre part had an error; the 2023 correction fixes it and requires a projective target for the multiple-fibre exclusion. The paper avoids that by not needing the multiple-fibre exclusion, using only surjectivity plus connectedness (which the correction preserves). The rest of the argument—especially the torus descent—depends on carefully tracking meridian orders in R = G/B_0 rather than assuming they vanish. This is why the descent section spends so much on the "actual inertial boundary" rather than a "canonical" one.
Campana 2023, Albanese map of special manifolds: a correction: pdf/campana-2023-albanese-map-of-special-manifolds-a-correction.pdf
Juanyong Wang 2021, On the Iitaka conjecture C_{n,m} for Kähler fibre spaces: pdf/juanyong-wang-2021-on-the-iitaka-conjecture-c-n-m.pdf