A smooth projective variety has a canonical divisor K_X, an integer divisor measuring how top-dimensional differential forms vanish. For each positive m, the nonzero pluricanonical forms of degree m produce rational maps from X into projective space. The largest dimension of their images, as m grows, is the Kodaira dimension k(X). When k(X)>=0, the maps stabilize to a canonical fibration whose base field is generated by ratios of pluricanonical forms; this is the Iitaka fibration.
https://doi.org/10.2969/jmsj/02320356

The first theme is uniformity. Fix the dimension d. In characteristic zero, one degree m(d) works for every smooth projective d-fold X with k(X)>=0: the complete linear system |m(d)K_X| determines the Iitaka fibration, including the whole base field, not only an image of the right dimension. The second theme is an index bound for log canonical pairs: for any DCC set Ph of rational coefficients, one integer a(d,Ph) makes a(d,Ph)(K_X+B) an integral principal divisor whenever (X,B) is a projective lc pair of dimension d with coefficients in Ph and K_X+B rationally trivial.
https://doi.org/10.1007/s10240-016-0080-x
https://doi.org/10.4007/annals.2014.180.2.3

The general-type case of effective pluricanonical maps was settled by Hacon-McKernan, Takayama, and Tsuji. Their results show that a fixed multiple of the canonical divisor gives a birational map for all smooth projective varieties of general type in a bounded family.
https://doi.org/10.1007/s00222-006-0504-1
https://doi.org/10.1007/s00222-006-0503-2
https://doi.org/10.18910/6697

For lower Kodaira dimensions, the general fibre adds torsion and monodromy data. Fujino-Mori found a uniform degree for threefolds of Kodaira dimension one using their canonical bundle formula, bounding one finite order via an extended Hodge line and a middle Betti number. Viehweg-Zhang proved effective Iitaka statements when the base has dimension two. Birkar-Zhang proved a general effective Iitaka theorem with constants depending on dimension, the least nonvanishing canonical degree of the general fibre, and a middle Betti number of a canonical cover.
https://doi.org/10.4310/jdg/1090347529
https://doi.org/10.1090/S1056-3911-09-00515-3
https://doi.org/10.1007/s10240-016-0080-x

The canonical bundle formula is the bridge from a fibration to its base. For a contraction f:X-Z with K_X pulled back from Z, it writes K_X as f^*(K_Z+B_Z+M_Z), where B_Z is the discriminant divisor and M_Z is the moduli divisor. Ambro proved the key klt-trivial fibration theorem producing this pair and its b-divisor.
https://doi.org/10.1112/S0010437X04001071

The induction runs on dimension. The discrepancy and adjunction conventions are those of Kollar's Singularities of the Minimal Model Program and Kollar-Mori's Birational Geometry of Algebraic Varieties. Let I_n be the uniform pluricanonical-degree statement in dimension n, and let L_n be the uniform log canonical index statement in dimension n. The zero-boundary klt part is K_n. The proof assumes I_j and L_j for all j<n and then proves K_n, L_n, and I_n. A companion recent log abundance theorem supplies good minimal models in all dimensions.
https://doi.org/10.1017/CBO9781139547895
https://doi.org/10.1017/CBO9780511662560

The induction needs a uniform Cartier denominator for the moduli b-divisor. The proof passes to the geometric generic fibre, applies lower-dimensional index bounds, and then uses the singular Beauville-Bogomolov decomposition. That decomposition is built from the splitting and holonomy results of Druel and Greb-Guenancia-Kebekus, and the algebraic integrability theorem of Horing-Peternell.
https://doi.org/10.1007/s00222-017-0748-y
https://doi.org/10.2140/gt.2019.23.2051
https://doi.org/10.1007/s00222-018-00853-2

Degenerations of the individual Beauville-Bogomolov blocks supply the character bounds that clear the moduli denominator. Prokhorov's equivariant minimal model program handles the finite-group steps. Semistable reduction comes from the toroidal embeddings of Kempf-Knudsen-Mumford-Saint-Donat. The semistable comparison theorem and the canonical extension come from Deligne and Steenbrink. Structure-sheaf cohomology in flat Du Bois families is controlled by Kollar's family theory, and differential forms on log canonical spaces are controlled by Greb-Kebekus-Kovacs-Peternell.
https://doi.org/10.1070/RM9990
https://doi.org/10.1007/BFb0070318
https://doi.org/10.1007/BFb0061194
https://doi.org/10.1007/BF01403146
https://doi.org/10.1017/9781009346115
https://doi.org/10.1007/s10240-011-0036-0

For a nonabelian block, a good special-fibre model gives a log canonical component with a degree-one generator. A fixed-point argument on the possibly singular central fibre relies on the Lefschetz-Riemann-Roch theorem of Donovan. The reduced component is normal lc and has smaller dimension, so the lower-dimensional index hypothesis bounds the associated character.
https://doi.org/10.24033/bsmf.1680

Torsion over rationally connected bases is handled separately. Birkar's recent boundedness theory for generalized Calabi-Yau pairs applies in every fixed dimension. Kummer theory and the etale fundamental group comparison of SGA1 identify Weil divisor torsion with homology of the regular locus, while Delfs-Knebusch provide the semialgebraic triangulation needed to show only finitely many topological types occur.
https://arxiv.org/abs/2305.18770v2
https://arxiv.org/abs/math/0206203
https://doi.org/10.1515/crll.1982.335.122

The absolute index argument reduces an unbounded-index sequence to terminal varieties with countable birational groups. Countability is proved via the Picard group, the finite stabilizer mechanism, and Matsumura's theorem excluding positive-dimensional linear algebraic groups as birational automorphism groups of varieties with nonzero pluricanonical systems.
https://www.bdim.eu/item?id=RLINA_1963_8_34_2_151_0

The scalar estimates need curves and vanishing-order bounds. The chain and tracking method goes back to the parameter-differentiation technique of Ein-Kuchle-Lazarsfeld. Generic semipositivity is the Campana-Paun theorem, and the semistability of tensor products in characteristic zero is the theorem of Ramanan-Ramanathan, after restriction results of Mehta-Ramanathan.
https://doi.org/10.4310/jdg/1214457231
https://doi.org/10.5802/aif.2945
https://doi.org/10.1007/BF01450677
https://doi.org/10.2748/tmj/1178228852

For Iitaka bases of arbitrary dimension, weak positivity of direct images supplies a fibre-to-total volume comparison. The relevant twisted weak positivity theorem is Fujino's, and the rational connectedness of maximal rationally connected quotients is the Graber-Harris-Starr theorem.
https://doi.org/10.2969/aspm/07510073
https://doi.org/10.1090/S0894-0347-02-00402-2

When the index cover is large, the diagonal argument shows that products of the pullback polarization have Seshadri constants growing only quadratically in the number of factors while the covering degree remains independent. This uses Frobenius truncation and hard-earned slope estimates of Langer, the Frobenius diagonal filtration of Sun, the filtration stability results of Kitadai-Sumihiro, and the ordinary-to-Frobenius jet comparison of Mustata-Schwede.
https://doi.org/10.4007/annals.2004.159.251
https://doi.org/10.1007/s00222-008-0125-y
https://doi.org/10.2748/tmj/1215442876
https://doi.org/10.1007/s00208-013-0976-4

The final contradiction uses compatible chains of low-cost curves on many diagonal quotients. Large covering degree forces each leaf to project finitely onto every slot, and the projection degrees cannot grow exponentially under a polynomial bound. A degree-one forgetting map then turns holomorphic flows into uncountably many birational self-maps, contradicting the countability of the birational group.
https://doi.org/10.1090/S1056-3911-2012-00574-8

The normal lc index theorem is then assembled. BCHM supplies the minimal model program and extraction steps, while Birkar's bounded complement theorem handles the Fano-type part. Global ACC of Hacon-McKernan-Xu and generalized global ACC of Birkar-Zhang reduce DCC coefficient sets to finitely many possibilities. Xu's klt induction turns zero-boundary klt bounds into boundary bounds, and the arithmetic Stein-degree bound passes from klt pairs to normal lc pairs without an index theorem for nonnormal pairs.
https://doi.org/10.1090/S0894-0347-09-00649-3
https://doi.org/10.4007/annals.2019.190.2.1
https://doi.org/10.4007/annals.2014.180.2.3
https://doi.org/10.1007/s10240-016-0080-x
https://arxiv.org/abs/1905.00297v2

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 log canonical index just proved gives nonemptiness. If the base is positive-dimensional, the denominator bound plus polarized effective birationality of Birkar-Zhang gives a fixed degree whose pluricanonical ratios recover the full base field. A common multiple over the finitely many possible base dimensions completes the induction.
https://doi.org/10.1007/s10240-016-0080-x