A smooth projective variety X has a canonical divisor K_X. For each positive m, the nonzero m-pluricanonical forms map X into projective space; the limiting dimension of those images is the Kodaira dimension k(X). When k(X)>=0, sufficiently divisible degrees stabilize to the Iitaka fibration, whose base field is generated by ratios of pluricanonical forms. Kodaira's elliptic surfaces are the prototype.
Iitaka 1971, On D-dimensions of algebraic varieties: https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-23/issue-2/On-D--dimensions-of-algebraic-varieties/10.2969/jmsj/02320356.pdf
    > The D-dimension of an algebraic variety is defined and studied.
Kodaira 1963, On compact complex analytic surfaces I: https://doi.org/10.2307/2373157
    > Elliptic surfaces and their canonical bundles organize the fibre structure.

The first uniform statement fixes dimension d. One integer m(d) works for every smooth projective d-fold X with k(X)>=0: |m(d)K_X| defines the Iitaka fibration and recovers the whole Iitaka base field. The second statement is an index bound: for each descending chain condition set Phi of rational coefficients, one integer a(d,Phi) makes a(d,Phi)(K_X+B) an integral principal divisor for every projective log canonical pair (X,B) of dimension d with coefficients in Phi and K_X+B rationally trivial.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: https://arxiv.org/pdf/1407.1098
    > The effective Iitaka fibration conjecture is reduced to polarized pairs.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: https://arxiv.org/pdf/1212.1354
    > Log canonical thresholds satisfy the ascending chain condition.

The general-type case was settled by Hacon-McKernan, Takayama, and Tsuji: a fixed multiple of the canonical divisor gives a birational map for every smooth projective variety of general type.
Hacon-McKernan 2006, Boundedness of pluricanonical maps of varieties of general type: https://arxiv.org/pdf/math/0509057
    > A bounded pluricanonical degree separates a bounded family.
Takayama 2006, Pluricanonical systems on algebraic varieties of general type: https://arxiv.org/pdf/math/0306410
    > Pluricanonical maps are birational in uniformly bounded degree.
Tsuji 2007, Pluricanonical systems of projective varieties of general type II: https://ir.library.osaka-u.ac.jp/repo/ouka/all/6697/ojm44_03_12.pdf
    > The boundedness statement is completed for projective varieties of general type.

For lower Kodaira dimensions the general fibre contributes torsion and monodromy. Fujino-Mori treated threefolds of Kodaira dimension one via their canonical bundle formula. Viehweg-Zhang handled two-dimensional Iitaka bases. Birkar-Zhang obtained a general effective Iitaka theorem depending on dimension, the least nonvanishing canonical degree of the general fibre, and a middle Betti number of a canonical cover.
Fujino-Mori 2000, A canonical bundle formula: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-56/issue-1/A-Canonical-Bundle-Formula/10.4310/jdg/1090347529.pdf
    > A canonical bundle formula relates the total space to the base.
Viehweg-Zhang 2009, Effective Iitaka fibrations: https://arxiv.org/pdf/0707.4287v3
    > Effective Iitaka fibrations are obtained for two-dimensional bases.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: https://arxiv.org/pdf/1407.1098
    > The general effective Iitaka theorem depends on explicit invariants of the general fibre.

The canonical bundle formula turns a fibration into data on the base. For a contraction f:X->Z with K_X pulled back from Z, K_X=f^*(K_Z+B_Z+M_Z), where B_Z is the discriminant divisor and M_Z is the moduli divisor. Ambro proved the klt-trivial fibration theorem producing B_Z and the moduli b-divisor.
Ambro 2005, The moduli b-divisor of an lc-trivial fibration: https://arxiv.org/pdf/math/0308143
    > The moduli b-divisor of an lc-trivial fibration is constructed.

The induction runs on dimension. The discrepancy and adjunction conventions follow Kollar; the minimal model program and cone machinery go back to Mori, Kawamata, and Shokurov. Write I_n for the uniform pluricanonical-degree statement, L_n for the normal log canonical index statement, and K_n for the zero-boundary klt case. The proof assumes I_j and L_j for j<n, then proves K_n, L_n, and I_n. A recent log abundance theorem supplies good minimal models in all dimensions.
Kollar 2013, Singularities of the Minimal Model Program: https://doi.org/10.1017/CBO9781139547895
    > Singularities of the Minimal Model Program
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties

A uniform Cartier denominator for the moduli b-divisor uses the geometric generic fibre, lower-dimensional index bounds, and the singular Beauville-Bogomolov decomposition. Druel and Greb-Guenancia-Kebekus supplied the splitting and holonomy parts; Horing-Peternell supplied algebraic integrability.
Druel 2018, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five: https://arxiv.org/pdf/1508.03734
    > A decomposition theorem for singular spaces with trivial canonical class
Greb-Guenancia-Kebekus 2019, Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups: https://arxiv.org/pdf/1704.01408v5
    > Holonomy, differential forms, and fundamental groups
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: https://arxiv.org/pdf/1610.03016
    > Algebraic integrability of foliations with numerically trivial canonical bundle

Degenerations of the individual Beauville-Bogomolov blocks clear the moduli denominator. Prokhorov's equivariant minimal model program handles finite groups. Semistable reduction comes from Kempf-Knudsen-Mumford-Saint-Donat. Deligne and Steenbrink supply the semistable comparison theorem and the canonical extension. Kollar's family theory controls structure-sheaf cohomology in flat Du Bois families, and Greb-Kebekus-Kovacs-Peternell control differential forms on log canonical spaces.
Prokhorov 2021, Equivariant minimal model program: https://arxiv.org/pdf/2007.15548
    > Equivariant minimal model program
Kempf-Knudsen-Mumford-Saint-Donat 1973, Toroidal Embeddings I: https://doi.org/10.1007/BFb0070318
    > Toroidal Embeddings I
Deligne 1970, Equations differentielles a points singuliers reguliers: https://doi.org/10.1007/BFb0061194
    > Equations differentielles a points singuliers reguliers
Steenbrink 1976, Limits of Hodge structures: https://doi.org/10.1007/BF01403146
    > Limits of Hodge structures
Kollar 2023, Families of Varieties of General Type: https://web.math.princeton.edu/~kollar/FromMyHomePage/modbook-final.pdf
    > Families of Varieties of General Type
Greb-Kebekus-Kovacs-Peternell 2011, Differential forms on log canonical spaces: https://arxiv.org/pdf/1007.1426
    > Differential forms on log canonical spaces

For a nonabelian block, a good special-fibre model produces a normal log canonical component with a degree-one log generator. A fixed-point argument on the possibly singular central fibre uses Donovan's Lefschetz-Riemann-Roch theorem; lower-dimensional index bounds then bound the associated character.
Donovan 1969, The Lefschetz-Riemann-Roch formula: https://www.numdam.org/article/BSMF_1969__97__257_0.pdf
    > The Lefschetz-Riemann-Roch formula

Torsion over rationally connected bases is handled separately. Birkar's boundedness theory for generalized Calabi-Yau pairs works in every fixed dimension. Kummer theory and SGA1 identify Weil divisor torsion with homology of the regular locus, while Delfs-Knebusch semialgebraic triangulation gives only finitely many topological types.
Birkar 2025, Singularities on Fano fibrations and beyond: https://arxiv.org/pdf/2305.18770v2
    > Singularities on Fano fibrations and beyond
SGA1 2003, Revetements etales et groupe fondamental: https://arxiv.org/pdf/math/0206203
    > Revetements etales et groupe fondamental
Delfs-Knebusch 1982, On the homology of algebraic varieties over real closed fields: https://doi.org/10.1515/crll.1982.335.122
    > On the homology of algebraic varieties over real closed fields

An unbounded-index sequence reduces to terminal varieties with countable birational groups. Countability uses the Picard group, finite stabilizers, and Matsumura's theorem excluding positive-dimensional linear algebraic groups as birational automorphism groups of varieties with nonzero pluricanonical systems.
Matsumura 1963, On algebraic groups of birational transformations: https://www.bdim.eu/item?id=RLINA_1963_8_34_2_151_0
    > On algebraic groups of birational transformations

The scalar estimates need curves and vanishing-order bounds. The chain-and-tracking method comes from Ein-Kuchle-Lazarsfeld parameter differentiation. Campana-Paun provide generic semipositivity. Tensor products of semistable bundles in characteristic zero are semistable by Ramanan-Ramanathan, after restriction results of Mehta-Ramanathan.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-2/Local-positivity-of-ample-line-bundles/10.4310/jdg/1214457231.pdf
    > Local positivity of ample line bundles
Campana-Paun 2015, Orbifold generic semi-positivity: https://arxiv.org/pdf/1412.0526
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://doi.org/10.1007/BF01450677
    > Semistable sheaves on projective varieties and their restriction to curves
Ramanan-Ramanathan 1984, Some remarks on the instability flag: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-36/issue-2/Some-remarks-on-the-instability-flag/10.2748/tmj/1178228852.pdf
    > Some remarks on the instability flag

For arbitrary-dimensional Iitaka bases, weak positivity of direct images gives a fibre-to-total volume comparison. Fujino's twisted weak positivity extends Viehweg's original theorem. The maximal rationally connected quotient is controlled by Graber-Harris-Starr.
Fujino 2017, Notes on the weak positivity theorems: https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf
    > Notes on the weak positivity theorems
Graber-Harris-Starr 2003, Families of rationally connected varieties: https://arxiv.org/pdf/math/0107108
    > Families of rationally connected varieties

When the index cover is large, diagonal products of the pullback polarization have Seshadri constants growing only quadratically in the number of factors, independent of covering degree. The argument uses Langer's Frobenius truncation and slope estimates, Sun's Frobenius diagonal filtration, Kitadai-Sumihiro filtration stability, and Mustata-Schwede's ordinary-to-Frobenius jet comparison.
Langer 2004, Semistable sheaves in positive characteristic: https://arxiv.org/pdf/math/0304341
    > Semistable sheaves in positive characteristic
Sun 2008, Direct images of bundles under Frobenius morphism: https://arxiv.org/pdf/math/0611360v2
    > Direct images of bundles under Frobenius morphism
Kitadai-Sumihiro 2008, Canonical filtrations and stability of direct images by Frobenius morphisms: https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-60/issue-2/Canonical-filtrations-and-stability-of-direct-images-by-Frobenius-morphisms/10.2748/tmj/1215442876.pdf
    > Canonical filtrations and stability of direct images by Frobenius morphisms
Mustata-Schwede 2014, A Frobenius variant of Seshadri constants: https://arxiv.org/pdf/1208.0486
    > A Frobenius variant of Seshadri constants

The final contradiction uses compatible chains of low-cost curves on diagonal quotients. Large covering degree forces leaves to project finitely onto every slot, and projection degrees cannot grow exponentially under a polynomial bound. A degree-one forgetting map turns holomorphic flows into uncountably many birational self-maps, contradicting countability.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: https://arxiv.org/pdf/0907.1265
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

The normal lc index theorem is assembled using BCHM for the minimal model program and extraction, Birkar's bounded complements for the Fano-type part, Hacon-McKernan-Xu global ACC and Birkar-Zhang generalized global ACC, Xu's klt induction, and an arithmetic Stein-degree bound passing from klt to normal lc pairs.
Birkar-Cascini-Hacon-McKernan 2010, Existence of minimal models for varieties of log general type: https://arxiv.org/pdf/math/0610203
    > Existence of minimal models for varieties of log general type
Birkar 2019, Anti-pluricanonical systems on Fano varieties: https://arxiv.org/pdf/1607.07165
    > Anti-pluricanonical systems on Fano varieties
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: https://arxiv.org/pdf/1212.1354
    > ACC for log canonical thresholds
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: https://arxiv.org/pdf/1407.1098
    > Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs
Xu 2019, Some results about the index conjecture for log Calabi-Yau pairs: https://arxiv.org/pdf/1905.00297v2
    > Some results about the index conjecture for log Calabi-Yau pairs

Finally, the uniform pluricanonical degree follows. On a good minimal model the semiample canonical divisor realizes the Iitaka fibration. If the base is a point, the proved index gives nonemptiness. If the base is positive-dimensional, the denominator bound and Birkar-Zhang polarized effective birationality recover the base field, and a common multiple over the finitely many base dimensions completes the induction.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: https://arxiv.org/pdf/1407.1098
    > Polarized effective birationality supplies a uniform birational rounded system.