A smooth integral projective variety X has a canonical divisor K_X. Think of K_X as the bookkeeping device for the zeros and poles of a rational volume form on X. For each positive integer m, the nonzero m-pluricanonical forms are the nonzero sections of O_X(mK_X); whenever these sections exist they send X into a projective space. The limiting growth of those images is measured by one integer, the Kodaira dimension k(X). If k(X) is nonnegative, then for all sufficiently divisible large m the pluricanonical maps stabilize: their birational images are equivalent and they package into the Iitaka fibration of X, whose base records every ratio of pluricanonical sections in one degree. The main theorem below says that a single degree m(d), depending only on the dimension d, already realizes this stable fibration for every smooth projective d-fold. Kodaira's elliptic surfaces are the model example: the canonical pluriforms of an elliptic surface distinguish the base of the elliptic pencil and ignore the elliptic fibre direction.
Iitaka 1971, On D-dimensions of algebraic varieties: ./Iitaka1971.pdf
    > The D-dimension of an algebraic variety is defined and studied.
Kodaira 1963, On compact complex analytic surfaces I: https://www.jstor.org/stable/pdf/2373157.pdf
    > Elliptic surfaces and their canonical bundles organize the fibre structure.

The companion statement controls singular pairs. A pair (X,B) consists of a normal projective variety X and an effective rational boundary divisor B with coefficients in [0,1]. Singularities are measured by discrepancies: log canonical means every divisorial discrepancy is at least -1, while kawamata log terminal means it is strictly greater than -1. The log canonical divisor is K_X+B. The theorem says that for each dimension d and each coefficient set Phi satisfying the descending chain condition, there is one integer a(d,Phi) with the following property: every projective log canonical pair (X,B) of dimension d with coefficients in Phi and K_X+B rationally linearly trivial satisfies a(d,Phi)(K_X+B)~0, so the multiple is an integral principal divisor, not merely a numerically trivial class. The descending-chain condition form follows from the finite-set form and the global ascending chain condition for lc thresholds, so the proof may reduce every coefficient set to a finite set.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > The effective Iitaka fibration conjecture is reduced to polarized pairs.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.

For varieties of general type, k(X)=d and the pluricanonical systems already separate points eventually. Hacon-McKernan, Takayama, and Tsuji proved that a fixed multiple of K_X gives a birational map for every smooth projective variety of general type, the strongest possible statement for that case. The theorem needed here generalizes this from general type to every nonnegative Kodaira dimension. The cost is that the general fibre of the Iitaka fibration now contributes its own torsion, its own canonical index, and monodromy data. Those contributions cannot be ignored because in Kodaira dimensions between 0 and d-1 the pluricanonical sections see through to a base of smaller dimension.
Hacon-McKernan 2006, Boundedness of pluricanonical maps of varieties of general type: ./0504327v3.pdf
    > A bounded pluricanonical degree separates a bounded family.
Takayama 2006, Pluricanonical systems on algebraic varieties of general type: https://doi.org/10.1007/s00222-006-0503-2
    > Pluricanonical maps are birational in uniformly bounded degree.
Tsuji 2007, Pluricanonical systems of projective varieties of general type II: ./ojm44_03_12.pdf
    > The boundedness statement is completed for projective varieties of general type.

Earlier effective Iitaka theorems had constants depending on the general fibre. Fujino-Mori proved a uniform degree for threefolds of Kodaira dimension one from a canonical bundle formula. Viehweg-Zhang handled two-dimensional Iitaka bases. Birkar-Zhang proved a general effective Iitaka theorem whose constant depends on the dimension, the least nonvanishing canonical degree of the general fibre, and a middle Betti number of a smooth model of a canonical cover. The missing step is to show those fibre-theoretic numbers have a uniform bound controlled only by the dimension; that is exactly what the denominator theorem and the index theorem below accomplish.
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: ./0707.4287v3.pdf
    > Effective Iitaka fibrations are obtained for two-dimensional bases.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > The general effective Iitaka theorem depends on explicit invariants of the general fibre.

The canonical bundle formula is the machine that turns a fibration into data on the base. Let f:X->Z be a contraction of normal projective complex varieties with K_X rationally pulled back from Z. Then K_X is rationally equivalent to f^*(K_Z+B_Z+M_Z), where B_Z is the discriminant divisor recording singular fibres and M_Z is the moduli divisor recording the variation of the fibers. Ambro proved independence of the b-divisor and produced the klt-trivial fibration theorem that supplies B_Z and the moduli b-divisor. The uniform problem is that B_Z must have coefficients in a fixed descending-chain set, and M_Z must have a Cartier multiple whose size depends only on dim X. The section below obtains both uniform bounds without assuming a bound on middle Betti numbers of the fibres.
Ambro 2005, The moduli b-divisor of an lc-trivial fibration: ./0308143.pdf
    > The moduli b-divisor of an lc-trivial fibration is constructed.

The induction runs on dimension and interleaves three statements. Write I_n for the uniform pluricanonical-degree theorem in dimension n, L_n for the uniform normal lc index theorem in dimension n for all rational descending-chain coefficient sets, and K_n for the special zero-boundary klt case of L_n. The step assumes I_j and L_j for all j<n. It proves K_n by contradiction, then obtains L_n by adjunction and a Stein-degree descent, and finally obtains I_n from the canonical bundle formula and polarized effective birationality. The proof uses one standing input beyond the standard minimal model program: every projective log canonical pair with pseudo-effective adjoint has a good log minimal model, and hence its adjoint is semiample on a suitable model. The standard klt minimal model program and discrepancy theory of the cited references are also used throughout.
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

The denominator theorem is the main relative input. It fixes n>=1 and assumes L_j for every j<n. It says there are positive integers p0 dividing p, depending only on n, with the following property: if f:X->Z is a contraction with dim X=n, dim Z>0, X klt, and K_X rationally linearly equivalent to a pullback from Z, then there is a rational function psi on X and canonical bundle formula data satisfying
K_X+(1/p0)div(psi)=f^*D_Z, D_Z=K_Z+B_Z+M_Z,
where B_Z has coefficients in a rational descending-chain set depending only on n, the generalized pair (Z,B_Z+M_Z) is generalized klt, and pM_Z is b-Cartier for the moduli b-divisor. The proof first uses the geometric generic fibre. That fibre is klt of dimension less than n with rationally trivial canonical divisor, so the lower-dimensional index theorem gives a common integer p0 trivializing it. Cartierness and the one-dimensionality of the trivializing section space descend by flatness, so the same p0 works over the function field of Z.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > The effective Iitaka fibration conjecture is reduced to polarized pairs.

Next choose psi so that p0K_X+div(psi) has no horizontal component. Starting from a rational Cartier divisor D^0 with K_X~_Q f^*D^0, the difference p0K_X+div(psi)-p0f^*D^0 is principal and vertical after replacing it by a multiple. A rational function whose divisor is vertical is constant on the projective geometric generic fibre, so it comes from the base field; that produces D_Z. Ambro's rank-one condition holds because the relevant generic direct image has rank one when rounded discrepancies over the generic base point are allowed to be exceptional. The discriminant coefficient at a prime P is 1-t_P, where t_P is the lc threshold of f^*P over the generic point of P. Threshold ACC forces these coefficients into a fixed rational descending-chain set.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.
Ambro 2005, The moduli b-divisor of an lc-trivial fibration: ./0308143.pdf
    > Ambient phrasing for the rank-one condition is avoided; the generic direct image has rank one.

The remaining denominator control is local over one prime divisor P of a smooth determination W of the base. Let alpha be the coefficient of the pullback of D_Z at P and let t be the threshold computed with the crepant total-space data. Because canonical divisors are integral, the denominator of M_W at P is the denominator of alpha+t. One computes alpha+t on a generic transverse curve C through a point c of P. After resolving the slice and cutting W with sufficiently general very ample divisors, the sliced equality is
K_V+B^c+(1/p0)div(psi_C)=h^*D_C, coeff_cD_C=alpha,
where h:V->C is the resolved slice and B^c is its crepant boundary. The geometric generic fibre of h is a normal klt variety with the original zero-boundary data. Let z be a local parameter at c, let theta be a rational top form defining K_V, and on the generic fibre set phi=(theta/dz)^{p0} psi_C. For a relative pluricanonical form omega of degree m define the weight
w_z(omega)=inf_E [ord_E(omega wedge dz/z)+m]/[m ord_E(z)],
where E runs over vertical divisorial valuations centered over c. The crepant equation gives
w_z(phi)=alpha-1+t.
The denominator is therefore controlled if a fixed multiple of w_z(phi) is an integer.
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.

Three elementary rules govern weights. Finite extension of the fibre function field with the base fixed does not change the weight of a pulled-back form: ramification multiplies both the numerator and denominator by the same ramification index. A curve extension of ramification l at the marked point multiplies the pulled-back weight by l. Taking a tensor power leaves the weight unchanged, while multiplying a degree-m form by a base function a adds ord_z(a)/m. These rules convert geometric degenerations on a semistable model into arithmetic statements about units and roots of unity.
Deligne 1970, Equations differentielles a points singuliers reguliers: https://link.springer.com/content/pdf/10.1007/BFb0061194.pdf
    > Equations differentielles a points singuliers reguliers

The geometric generic fibre is now decomposed by the singular Beauville-Bogomolov theorem. After a finite extension of the curve function field, a finite quasi-etale cover of that fibre is a product of an abelian block, if present, and irreducible Calabi-Yau or irreducible holomorphic symplectic blocks. Taking a Galois comparison cover R and enlarging the curve field preserves geometric integrity and separates the factor fields for a subgroup of index at most d!, where d is the dimension of R. On each factor R_i choose a volume generator eta_i. After equivariant semistable reduction and normalization by a power of the uniformizer u, arrange w_u(eta_i)=0. The product form also has weight zero. The pullback of phi and the p0-th power of the product both generate the generic p0-canonical line on R, so they differ by a base function a:
phi=a(wedge_i eta_i)^{p0}, l w_z(phi)=ord_u(a)/p0.
Druel 2018, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five: ./1606.09006v1.pdf
    > 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: ./1704.01408v5.pdf
    > Holonomy, differential forms, and fundamental groups
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle

Let g be a finite-order lift of base inertia to R and put b=n!, so g^b preserves every block. Write (g^b)^*eta_i=c_i eta_i and lambda_i=c_i(c). Because equal weights force ord_u(c_i)=0, each c_i is a unit and lambda_i is a root of unity. The phi-invariance of the original form gives the inertia identity
zeta^{b ord_u(a)} prod_i lambda_i^{p0}=1,
where zeta is the action on the tangent line at the marked point. If one integer N kills every lambda_i, then the identity raised to the N-th power forces l to divide bN ord_u(a), and hence p0 b N w_z(phi) is an integer. By the weight equation this one integer clears the denominator of alpha+t at P. The difficult point is to bound the orders of the lambda_i uniformly while allowing l to grow without an a priori bound.
Prokhorov 2021, Equivariant minimal model program: https://doi.org/10.1070/RM9990
    > Equivariant minimal model program
Kempf-Knudsen-Mumford-Saint-Donat 1973, Toroidal Embeddings I: https://link.springer.com/content/pdf/10.1007/BFb0070318.pdf
    > Toroidal Embeddings I

For an abelian block of dimension s, the normalized form is a local frame of the extended Hodge line in degree s under semistable reduction. Its eigenvalue lambda is an eigenvalue of an integral matrix obtained from monodromy and finite base rotation. The rank of the relevant integral local system is binomial(2s,s), so the cyclotomic degree of lambda is bounded by that rank. There are only finitely many possible orders for each s<n. This is the same finiteness principle used in the elliptic-fibre Hodge calculations of Fujino-Mori, but it is implemented systematically in all dimensions.
Steenbrink 1976, Limits of Hodge structures: https://link.springer.com/content/pdf/10.1007/BF01403146.pdf
    > Limits of Hodge structures
Deligne 1970, Equations differentielles a points singuliers reguliers: https://link.springer.com/content/pdf/10.1007/BFb0061194.pdf
    > Equations differentielles a points singuliers reguliers

For a nonabelian block, the proof constructs a G-equivariant projective dlt model Y whose central fibre T is reduced and whose canonical divisor is semiample over the curve. The quotient by G is made klt by adding a small strong twist, and the good-model theorem together with an MMP with scaling yields the model; normalization pulls each step back upstairs. The relative log divisor of eta wedge du/u is effective and rationally trivial; this forces it to vanish identically, so eta wedge du/u is a degree-one log generator. Its residue makes K_T trivial and provides a generator of K_S+Theta_S on every preserved component S of T. Degree-one implies that Theta_S is integral, hence reduced.
Horing-Peternell 2019, Algebraic integrability of foliations with numerically trivial canonical bundle: ./1710.06183v2.pdf
    > Algebraic integrability of foliations with numerically trivial canonical bundle
Kollar 2023, Families of Varieties of General Type: ./modbook-final.pdf
    > Families of Varieties of General Type

The existence of a preserved component follows from structure-sheaf cohomology. Flat nearby fibres are klt with rational singularities, and their cohomology is computed from the algebra of differential forms of the block. For a Calabi-Yau block the alternating trace on H^*(T,O_T) is 1+(-1)^s lambda^{-1}, and for a symplectic block a power of exponent at most s+1 has nonzero trace of the induced map on even cohomology. The coherent Lefschetz theorem of Donovan then gives a fixed point on the possibly singular central fibre. Around a fixed point at most s+1 components meet, so a further power of exponent dividing (s+1)! preserves one such component. The finite-character lemma, proved from L_s, bounds the order of the corresponding character, hence of lambda. Taking p=p0 b N completes the denominator proof.
Donovan 1969, The Lefschetz-Riemann-Roch formula: ./BSMF_1969__97__257_0.pdf
    > The Lefschetz-Riemann-Roch formula
Greb-Kebekus-Kovacs-Peternell 2011, Differential forms on log canonical spaces: ./1003.2913v4.pdf
    > Differential forms on log canonical spaces

When the base of the denominator theorem is rationally connected and K_X~_Q0, a principal multiple of K_X is needed, not merely a Cartier bound. The auxiliary proposition says: for fixed d, fixed rational DCC set I, and fixed moduli denominator p, if Z is a normal rationally connected base with generalized adjoint D=K_Z+B+M_Z~_Q0, then there is an integer ell(d,I,p) such that ell D is an integral principal divisor. The proof first extracts marked valuations by the existence of a klt boundary perturbation, using BCHM extraction. Global ACC gives a finite coefficient set and a uniform positive lower bound for generalized log discrepancies, so Birkar's boundedness theorem applies to this class in every dimension.
Birkar-Cascini-Hacon-McKernan 2010, Existence of minimal models for varieties of log general type: ./0610203.pdf
    > Existence of minimal models for varieties of log general type
Birkar 2025, Singularities on Fano fibrations and beyond: ./2305.18770v2.pdf
    > Singularities on Fano fibrations and beyond

Boundedness in codimension one makes the homology groups of the regular loci take only finitely many isomorphism types. The regular locus U of Z is invariant under small codimension-one modifications, and semialgebraic triviality plus semialgebraic triangulation over real closed fields shows H_1(U(C),Z) is finitely generated with finitely many possible types. For each fixed Z the class group Cl(Z) is finitely generated because a rationally connected resolution has vanishing Pic^0. Kummer theory and SGA1 give, for every a>=1,
Cl(Z)[a] ~ H^1_et(U,mu_a) ~ Hom(H_1(U(C),Z),mu_a(C)).
It follows that H_1(U(C),Z) is finite, and a common multiple T of the finitely many exponents kills every torsion class. Since pM_Z is integral and the finite coefficient set makes qD integral for a suitable q, the divisor TqD is principal. The pullback identity then makes rK_X principal for a uniform r.
Birkar 2025, Singularities on Fano fibrations and beyond: ./2305.18770v2.pdf
    > Singularities on Fano fibrations and beyond
SGA1 2003, Revetements etales et groupe fondamental: ./0206203.pdf
    > Revetements etales et groupe fondamental
Delfs-Knebusch 1982, On the homology of algebraic varieties over real closed fields: https://www.degruyter.com/document/doi/10.1515/crll.1982.335.122/pdf
    > On the homology of algebraic varieties over real closed fields

The next step reduces unboundedness of the zero-boundary klt index to a very rigid sequence. If the canonical indices of projective klt n-folds with K_X~_Q0 were unbounded, there would be projective Q-factorial terminal n-folds V_i of canonical indices r_i tending to infinity such that each V_i has irregularity zero on a resolution and countable birational group; every positive-dimensional rational image of smaller dimension has a rationally connected resolution; every effective Weil divisor with positive Iitaka dimension is big; and the cyclic index cover Y_i->V_i is terminal with K_{Y_i}~0, while every equivariant rational fibration with positive-dimensional base and fibre has a general-type geometric generic fibre. The same fibration assertion holds for V_i without any group action.
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > Log canonical thresholds satisfy the ascending chain condition.

The reduction uses the product decomposition of klt varieties with trivial canonical class. If the Beauville-Bogomolov cover has at least two positive-dimensional blocks, the finite-character lemma bounds the canonical index of the quotient. If the cover is purely abelian or has one big abelian block, the eigenvalue of the canonical character acts on H^n of an abelian variety, whose rank is binomial(2n,n); that also gives a uniform bound. Thus an unbounded sequence can only come from a single nonabelian Calabi-Yau or symplectic block. Such a variety has no reflexive one-forms and no intermediate reflexive forms, so the BDPP criterion and the absence of non-uniruled rational images force every lower-dimensional rational image to be rationally connected.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension
Graber-Harris-Starr 2003, Families of rationally connected varieties: ./0109220v1.pdf
    > Families of rationally connected varieties

Terminality and countability arise as follows. Noncanonical members can be discarded because their index is bounded by the rationally connected torsion proposition. A canonical variety has a Q-factorial terminalization by BCHM extraction. Countability of Bir(V_i) follows because terminal varieties with torsion canonical class and irregularity zero have countable divisor class group; birational self-maps act on that group, and within each class the maps form a finite set because the connected automorphism group of a projective embedding is linear. Matsumura's theorem excludes a positive-dimensional linear group as a birational automorphism group of a smooth model with a nonzero pluricanonical system. Thus Bir(V_i) is countable.
Matsumura 1963, On algebraic groups of birational transformations: ./Matsumura1963.pdf
    > On algebraic groups of birational transformations
Birkar-Cascini-Hacon-McKernan 2010, Existence of minimal models for varieties of log general type: ./0610203.pdf
    > Existence of minimal models for varieties of log general type

The scalar estimates are quantitative preparations for the diagonal contradiction. For a big nef semiample rational Cartier divisor P on V and a smooth point x, let gamma(P;V) be the very general value of the normalized vanishing order of complete plurisections through x after placing V over a smooth projective model. Let epsilon(P;V) be the very general Seshadri constant, the infimum of P*C/mult_xC over curves C through x. Asymptotic Riemann-Roch gives gamma(P;V)>=(P^n)^{1/n}, because a section vanishing to order ell imposes at most binomial(d+ell-1,d) conditions. The scalar theorem says: if V is normal projective canonical with K_V~_Q0, a finite group G acts on V preserving the class of an ample rational Cartier divisor P, and every G-equivariant intermediate rational fibration has a general-type fibre, then gamma(P;V)<=C_n(P^n)^{1/n} and epsilon(P;V)>=c_n(P^n)^{1/n}, with constants depending only on n.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles

The upper gamma bound is proved by contradiction. If no C_n exists, rescale a sequence to have P^n tending to 0 while gamma(P;V)>1. The dimension-one case is trivial because section order is bounded by degree. In dimension at least two, choose d+1 equally spaced levels below the lower gamma bound. The parameter-differentiation chain lemma produces a covering family of smaller-dimensional members with controlled polarization volume and controlled canonical degree. Passing repeatedly to Iitaka fibres, the dimensions decrease. The new input is the small-volume lemma: if a sequence of smooth e-folds has L^e->0 and K_W L^{e-1}<=C'L^e, then after passing to Iitaka fibres their general fibres have fixed positive dimension, Kodaira dimension zero, and volume of K_U+L|_U tending to 0.
Ein-Kuchle-Lazarsfeld 1995, Local positivity of ample line bundles: ./9408003v1.pdf
    > Local positivity of ample line bundles

The small-volume lemma combines Fujino's weak positivity of direct images with birational flattening. For an Iitaka morphism f:W->T, weak positivity of the direct image of l(K_{W/T}+L) means a symmetric power can be twisted by a big multiple of the base polarization and generated by global sections over a big open set. This allows enough fibre sections to be lifted to global sections of a bounded adjoint on W. The asymptotic volume of K_U+L|_U is then comparable from below to the volume of a multiple of K_W+L, so it must tend to zero. The second inequality of the scalar theorem, the lower Seshadri bound, follows from the upper gamma estimate and a thick-fibre dimension count: if epsilon(P) were arbitrarily small, a moving family of low-cost curves would create a fake chain leaf, whose small degree conflicts with P^n>=1; a section of large degree vanishing on the thickened fibre would have normalized order exceeding the gamma bound.
Fujino 2017, Notes on the weak positivity theorems: ./weak-posi11.pdf
    > Notes on the weak positivity theorems
Campana-Paun 2015, Orbifold generic semi-positivity: ./1303.3169v4.pdf
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://link.springer.com/content/pdf/10.1007/BF01450677.pdf
    > 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

The diagonal jet bound is the central positive-characteristic estimate. Let V be a normal projective canonical complex n-fold with K_V~_Q0 and canonical index r; let pi:Y->V be its cyclic cover. Suppose L is an ample rational Cartier divisor with 1<=L^n<=2 and gamma(L;V)<=C1. Choose b_0>0 with 3(2b_0)^n<1/4 and b_0(C1+1)<3/4. For each t>=2 form the diagonal quotient Z_t=Y^t/mu_{r,diag}, with natural map theta_t:Z_t->V^t, and set P_t=theta_t^*(L pushforward over all slots). Then P_t is ample and
epsilon(P_t)<=(n+1)t^2/b_0.
The proof compares two orders of vanishing of one determinant generated by ordinary jets specialized to large characteristic.
Langer 2004, Semistable sheaves in positive characteristic: https://doi.org/10.4007/annals.2004.159.251
    > Semistable sheaves in positive characteristic

Suppose the displayed epsilon bound fails. Ordinary jets at a very general point then give a finite set of sections whose products span a thick Frobenius point. Specialize a finitely generated integral model over Z to an algebraically closed fibre of sufficiently large characteristic p. Relative Frobenius turns the spanning sections into a column map of full generic rank on the diagonal product. If the full rank is R=p^{nt}, the columns are indexed by t-tuples of nonnegative exponents a_i with sum q=2tr. The diagonal rank is at most R/4 by slope estimates for the truncated symmetric algebra of the relative cotangent bundle. A nonzero maximal minor therefore must vanish to high order along the reduced diagonal.
Sun 2008, Direct images of bundles under Frobenius morphism: ./0611360v2.pdf
    > 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: ./1203.1081v1.pdf
    > A Frobenius variant of Seshadri constants

Three one-dimensional lemmas make the slope estimate quantitative. For a vector bundle E of rank rho on a smooth projective curve, the truncated symmetric algebra A_j(E) is the symmetric algebra modulo locally p-th powers of degree-one elements. Its graded pieces have maximum slopes at most (p-1)deg E+rho(rho-1)max(0,2g-2). A flag lemma bounds global sections of a bundle on W by sections of its restriction to a complete-intersection curve C: if mu_max(E|_C)<=M and C is cut by l_iH, then h^0(W,E)<=rank(E)(M+1)prod(1+M/(l_iH*C)). The external-product lemma says a nonzero section of an external tensor product of line bundles has order at a product point bounded by the sum of the individual maximal orders. These are then refined by slope data from generic semipositivity.
Campana-Paun 2015, Orbifold generic semi-positivity: ./1303.3169v4.pdf
    > Orbifold generic semi-positivity
Mehta-Ramanathan 1982, Semistable sheaves on projective varieties and their restriction to curves: https://link.springer.com/content/pdf/10.1007/BF01450677.pdf
    > Semistable sheaves on projective varieties and their restriction to curves

The characteristic-zero movable curve test is then applied to the same maximal minor before specialization. On a blow-up of V at a very general x, the order bound defines a strongly movable curve beta with
sigma^*D^+ - d_0 J not pseudo-effective,
where D^+=bL+(q+1)K_W and d_0=b(C1+1). The determinant's slot line bundles have classes whose average coefficients are controlled, so the ordering of each nonzero slot section is less than R times b(C1+1). Applying the external-product lemma gives an order strictly less than 3R/4. The diagonal-rank loss gives an order at least 3R/4. The contradiction proves the diagonal jet bound. This is where the quadratic growth in t comes from, independent of r.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

The high-index contradiction is now assembled from low-cost curves. Suppose there is a sequence of finite cyclic covers pi_nu:Y_nu->V_nu with V_nu normal projective n-folds, G_nu of order r_nu tending to infinity, Bir(V_nu) countable, and ample rational Cartier divisors L_nu with epsilon(L_nu)>=c and epsilon(pi_nu^*L_nu)>=c r_nu^{1/n}. Suppose further that for every t>=2 the diagonal quotient polarization P_{nu,t} satisfies epsilon(P_{nu,t})<=C2 t^2. Fix a large T. For each t<=T choose a marked covering family of curves with P_t-degree over marked multiplicity bounded by C2t^2+1, and close the family under deck translates, permutations of slots, and coordinate projections.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

The chain lemma converts these curve families into a rational quotient of V^t with a geometric generic leaf H_t of dimension h_t>0 and degree bound P_t^{h_t}*H_t<=(h_t(C2t^2+1))^{h_t}. Projection compatibility makes the images of upper leaves lie in lower leaves, and the quotient is invariant under slot permutations. The key finiteness statement is that for each fixed T and sufficiently large r, the single-slot projection from H_t is generically finite onto its image. If it were not, a positive-dimensional fibre over a very general point would carry a degree at least (c r^{1/n})^a, while the total leaf degree would be at most a polynomial in t; this is impossible as r->infinity. Consequently h_t<=n.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

Now let d_t be the degree of the geometric generic leaf over its first-slot image. Comparing the lower and upper degree bounds gives d_t<=KT^{2n} with K depending only on n,c,C2. Projection compatibility implies h_t is nondecreasing as t decreases. Choosing a constant stretch of length about T/n and selecting T so large that (T-1)/n-1>log_2(KT^{2n}) forces one adjacent projection to be birational. Permutation invariance gives a birational coordinate-forgetting map for every omitted coordinate. Thus, generically, one coordinate is rationally recovered from the other k-1 coordinates lying on the same leaf.
Boucksom-Demailly-Paun-Peternell 2013, The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension: ./0405285v1.pdf
    > The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension

The tangent distribution of the leaf projects isomorphically to each slot and therefore defines a rational plane distribution A(x) on V and rational transport isomorphisms R(x,y):A(x)->A(y) satisfying the cocycle relation. Choosing a reference point gives independent rational vector fields v_1,...,v_h. For small z=(z_1,...,z_h), let E_z be the composition of the local holomorphic flows of these vector fields. The simultaneous flows preserve the leaf quotient, so the rational inverse of the forgetting map expresses
E_z(y)=last-coordinate-of-inverse(F,E_z(tuple),quotient(tuple,y)).
The right side is rational in y for fixed z, and the inverse-time composition gives a rational inverse. Hence every sufficiently small E_z is a birational self-map of V. Because the vector fields are independent, the map z->E_z(y_0) has nonzero derivative and uncountable image. This gives uncountably many birational self-maps of V, contradicting the countability of Bir(V). Therefore no such high-index sequence exists.
Matsumura 1963, On algebraic groups of birational transformations: ./Matsumura1963.pdf
    > On algebraic groups of birational transformations

The contradiction proves K_n, the zero-boundary klt index theorem, in every dimension. The normal lc index theorem L_n is then reconstructed in three moves. First, a crepant dlt modification plus a small boundary perturbation gives a klt pair whose canonical class is not pseudo-effective, so BCHM produces a Mori fibre space f:(X',B')->Z'. Second, if the base is a point, the variety is Fano type and Birkar's bounded complements theorem supplies a bounded complement and hence a bounded index. Third, if the base has positive dimension, a component S of the horizontal boundary is used for divisorial adjunction. Lower-dimensional L_{n-1}, the vertical-descent lemma, and adjunction give a common form degree q.
Birkar 2019, Anti-pluricanonical systems on Fano varieties: ./1603.05765v4.pdf
    > Anti-pluricanonical systems on Fano varieties
Birkar-Cascini-Hacon-McKernan 2010, Existence of minimal models for varieties of log general type: ./0610203.pdf
    > Existence of minimal models for varieties of log general type

The descent step is explicit. If f:X->Z is a contraction and a rational function on X has a vertical divisor, that function belongs to the base field. If (X,B) is a rational pair with K_X+B~_Q0 and q(K_{X_eta}+B_eta)~0 on the generic fibre, there are a rational q-canonical form s and a rational Cartier divisor D~_Q0 on Z such that div(s)+qB=f^*D. Restricting to a normalized horizontal lc divisor S with normalization S^nu, taking residues, and using the trivializing lower-dimensional form gives a vertical divisor and ultimately
(deg h)D=div_Z(N_{k(Y)/k(Z)}a)
after Stein factorization S^nu->Y->Z. The degree deg h is bounded by the arithmetic Stein-degree theorem applied to the generic log Calabi-Yau fibre. Thus a uniform multiple of K_X+B is principal. This proves L_n.
Xu 2019, Some results about the index conjecture for log Calabi-Yau pairs: ./1905.00297v2.pdf
    > Some results about the index conjecture for log Calabi-Yau pairs
Hacon-McKernan-Xu 2014, ACC for log canonical thresholds: ./1208.4150v1.pdf
    > ACC for log canonical thresholds

The final uniform pluricanonical degree I_n follows. Let X be a smooth projective n-fold with k(X)>=0. A good minimal model V of X exists, and the semiample contraction of K_V represents the Iitaka fibration f:V->Z with K_V~_Q f^*D for a rational ample divisor D. If Z is a point, then K_V~_Q0 and the already proved index theorem gives a nonempty pluricanonical system. If dim Z>0, the denominator theorem provides p0 and a base adjoint rationally equivalent to D, so D is big. Polarized effective birationality of Birkar-Zhang then gives a uniform degree whose rounded base system is birational. The section comparison
H^0(X,mK_X)=psi^{m/p0}f^*H^0(Z,floor(mD_Z))
transfers this field to X. Taking a common multiple over the finitely many possible base dimensions preserves all section ratios. This proves the theorem over the complex numbers.
Birkar-Zhang 2016, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs: ./s10240-016-0080-x.pdf
    > Polarized effective birationality supplies a uniform birational rounded system.

The statement then descends from the complex numbers to every algebraically closed field of characteristic zero. The variety, the finitely many divisors, and a rational principalization of a sufficiently divisible multiple descend to a finitely generated subfield; extending scalars to C preserves smoothness, singularity conditions, section-space dimensions, and linear triviality. A uniform section on C whose divisor is principal descends because the relevant line bundle becomes trivial after extension. Field generation is checked after a sufficiently large divisible degree: the comparison rational map is birational if and only if it is birational after the extension. Therefore the same integers m(d) and a(d,Phi) work over every algebraically closed characteristic-zero field.
Kollar-Mori 1998, Birational Geometry of Algebraic Varieties: https://doi.org/10.1017/CBO9780511662560
    > Birational Geometry of Algebraic Varieties