Proof. The generic fibre \(X_\eta\) is normal and projective, with \(H^0(X_\eta,\mathcal O_{X_\eta})=k(Z)\). A rational function with vertical divisor restricts to a function with neither zeros nor poles at any prime divisor of \(X_\eta\). Normality makes it regular, and hence a member of \(k(Z)\).

For the second assertion, extend a trivializing rational form on \(X_\eta\) to a rational \(q\)-canonical form \(s\) on \(X\), using any rational top form on \(Z\) to identify relative and absolute canonical lines. The divisor \(H=\mathop{\mathrm{div}}(s)+qB\) is vertical. Choose \(a>0\) and a rational \(aq\)-canonical form \(\tau\) with \(\mathop{\mathrm{div}}(\tau)+aqB=0\). The function \(s^{\otimes a}/\tau\) has divisor \(aH\), so the first assertion writes it as \(f^*h\) for some \(h\in k(Z)^*\). Thus \(H=f^*((1/a)\mathop{\mathrm{div}}(h))\), as required. ◻

Proof. The assumptions give the zero-boundary klt index bounds through dimension \(n\). Xu’s klt induction theorem [43] therefore supplies the assertion when \((X,B)\) is klt. We prove the non-klt case without an slc index statement.

A Mori fibre space and a horizontal lc divisor. Take a crepant \(\mathbb Q\)-factorial dlt modification, and enlarge \(\Gamma\) by \(1\). Write \(T=\lfloor B\rfloor\ne0\). For sufficiently small rational \(\epsilon>0\), the pair \((X,B-\epsilon T)\) is klt and \[K_X+B-\epsilon T\sim_{\mathbb Q}-\epsilon T\] is not pseudo-effective: its intersection with the \((n-1)\)st power of an ample divisor is negative. The klt minimal model program [2] ends in a Mori fibre space \(f:X'\to Z\).

The full log canonical divisor remains crepant through this program. Indeed, fix compatible canonical divisors and write \(r(K_X+B)=\mathop{\mathrm{div}}(h)\). Divisorial contractions push forward this same identity, and flips preserve it under their codimension-one identification. On a common resolution, both pullbacks equal \(r^{-1}\mathop{\mathrm{div}}(h)\). Thus the full pairs on all models are lc, even though they need not be dlt. The exact order of their log canonical divisors is unchanged. We henceforth write \((X,B)\) for the last model.

If \(\dim Z=0\), the klt log Fano pair \((X,B-\epsilon T)\) shows that \(X\) is of Fano type. The bounded complement theorem [3], with a fixed hyperstandard set containing \(\Gamma\), gives a bounded integer \(c\) and a complement \(B^+\ge B\) with \(c(K_X+B^+)\sim0\). The effective divisor \(B^+-B\) is numerically trivial, hence zero by intersection with an ample class. This settles that case.

Suppose \(\dim Z>0\). On the Mori fibre space the surviving transform of \(T\) is relatively ample, since \(\epsilon T\) is relatively \(\mathbb Q\)-linearly equivalent to \(-(K_X+B-\epsilon T)\). In particular, some prime component \(S\) of \(T\) dominates \(Z\). Let \(\nu:S^\nu\to S\) be its normalization. Divisorial adjunction gives \[ (K_X+B)|_{S^\nu}=K_{S^\nu}+\Theta. \tag{2}\] Here \((S^\nu,\Theta)\) is a normal lc pair and \(\Theta\) is effective; its coefficients belong to the rational DCC set of differents determined by \(\Gamma\) [27]. Its adjoint is \(\mathbb Q\)-linearly trivial. The lower-dimensional assertion \(L_{n-1}\), together with global ACC as in the conventions, therefore bounds a trivializing multiple of \(K_{S^\nu}+\Theta\).

A common form degree. The geometric generic fibre of \(f\) is a normal projective lc pair of dimension less than \(n\), with the same coefficient set. The assertions \(L_j\), \(j<n\), bound a trivializing multiple there, and linear triviality descends in that same degree to \(k(Z)\). To see the last point, the corresponding divisorial sheaf becomes an invertible trivial sheaf after algebraic closure. Invertibility descends faithfully flatly. Flat base change for global sections then gives a one-dimensional space over \(k(Z)\), and its evaluation map is an isomorphism because it becomes one after the field extension.

Choose a common integer \(q\), depending only on \(n\) and \(\Gamma\), which clears \(\Gamma\) and trivializes both this fibre adjoint and \(K_{S^\nu}+\Theta\). Lemma 5 gives (1), with \(D\sim_{\mathbb Q}0\). At the generic point of \(S\), the form \(s\) has precisely its logarithmic pole of order \(q\). Its \(q\)-fold residue is a nonzero rational \(q\)-canonical form \(v\) on \(S^\nu\), and adjunction gives the equality of actual divisors \[ \mathop{\mathrm{div}}(v)+q\Theta=(f\circ\nu)^*D. \tag{3}\] For this equality one may first take a sufficiently divisible tensor power and apply the usual pluricanonical adjunction isomorphism; dividing the resulting equality of divisorial orders gives (3). No Cartier assumption in degree \(q\) along the whole of \(S\) is needed.

Stein factorization and norm. Choose a trivializing \(q\)-log canonical form \(v_0\) on \((S^\nu,\Theta)\). Then \(a=v/v_0\in k(S)^*\) satisfies \[\mathop{\mathrm{div}}(a)=(f\circ\nu)^*D.\] Factor the map as \(S^\nu\xrightarrow{g}Y\xrightarrow{h}Z\), where \(g\) is a contraction and \(h\) is finite, with \(Y\) normal. The first part of Lemma 5 gives \(a\in k(Y)^*\). Injectivity of pullback for rational Cartier divisors under the surjective map \(g\) yields \[h^*D=\mathop{\mathrm{div}}_Y(a).\] One can check this injectivity at a prime divisor of \(Y\), where a nonzero order remains nonzero at a prime divisor above it. The norm identity for the finite morphism \(h\) now gives \[ (\deg h)D=\mathop{\mathrm{div}}_Z\bigl(N_{k(Y)/k(Z)}a\bigr). \tag{4}\]

It remains to bound \(\deg h\), not the degree of any gluing cover. Apply the arithmetic Stein-degree theorem [32] to \((X_\eta,B_\eta)\) over \(k(Z)\), with its prime coefficient-one divisor \(S_\eta\) and parameter \(t=1\). This fibre is normal, integral and projective, and its field of global functions is \(k(Z)\). The relative algebraic closure of \(k(Z)\) in \(k(S_\eta)\) is exactly \(k(Y)\). Hence \(\deg h\) is bounded in terms of the fibre dimension, and therefore in terms of \(n\).

Equations (1) and (4) show that \(q(\deg h)(K_X+B)\) is principal. Taking the least common multiple of the finitely many possible degrees, and then a common multiple with the klt and base-point cases, proves the proposition. The earlier crepant comparisons return the same multiple to the original pair. ◻

Proof. Form the quotient \(\pi:V\to W=V/\langle\gamma\rangle\). Define the boundary \(E_W\) by the finite adjunction formula \[K_V+E=\pi^*(K_W+E_W).\] At a prime divisor with ramification index \(e\), its coefficient is \(1-1/e\) if the upstairs coefficient is zero, and \(1\) if the upstairs coefficient is one. Thus the coefficients belong to the fixed rational DCC set \(\{1-1/e:e\in\mathbb N\}\cup\{1\}\). The finite discrepancy formula makes \((W,E_W)\) lc [27]. A power of \(\omega\) invariant under the finite cyclic group descends to a rational pluricanonical form downstairs; the ramification formula shows that it trivializes a multiple of \(K_W+E_W\). This also establishes its \(\mathbb Q\)-Cartierness and \(\mathbb Q\)-linear triviality.

The assertion \(L_j\) and global ACC supply a uniform \(c_j\) for this DCC set such that \(c_j(K_W+E_W)\sim0\). Pull back a trivializing log pluricanonical form. It is an invariant generator upstairs, necessarily a scalar multiple of \(\omega^{\otimes c_j}\), since \(V\) is projective and integral. Therefore \(\lambda^{c_j}=1\). For \(j=0\) the same conclusion holds with \(c_0=1\). ◻

Proof. If \(\omega^{\otimes m}\) is invariant, it descends at the function field level to a rational \(m\)-canonical form on \(X\). Quasi-étaleness means that its divisor has order zero at every prime divisor of \(X\). It therefore trivializes \(\mathcal O_X(mK_X)\) as a divisorial sheaf, so \(mK_X\) is principal. Conversely, a trivializing form downstairs pulls back to a scalar multiple of \(\omega^{\otimes m}\) and is invariant. ◻

Proof. Intrinsic preservation of the product directions. Pullback on the common smooth big open, followed by reflexive extension, gives \[ \mathcal T_R=\bigoplus_{i=1}^t\mathcal E_i. \tag{6}\] There are no global homomorphisms between two distinct summands. This can be checked after extending \(k\) to an algebraic closure. At least one of the corresponding factors is nonabelian, say \(B_i\). Fix general smooth points in all complementary factors. The resulting slices of \(R\) are normal and finite over \(B_i\): generic flatness and geometric normality in characteristic zero allow this after shrinking the complementary parameter space. The locus omitted from the quasi-étale map has codimension at least two on a general slice. Each connected component of the slice is consequently a connected normal quasi-étale cover of \(B_i\).

On the smooth big open of that component, \(\mathcal E_i\) is its tangent sheaf, while the complementary summands are trivial. A homomorphism in either direction gives a vector field or a one-form, which extends reflexively and vanishes by the block properties. These slices cover a dense open of \(R\), proving the claimed vanishing.

The block projections in (6) are therefore central idempotents of the finite-dimensional algebra \(\operatorname{End}_R(\mathcal T_R)\). Its primitive central idempotents give nonzero direct summands of positive generic rank, so there are at most \(d\) of them. The action of \(G\) permutes this finite set. Its kernel \(G_0\) has index at most \(d!\) and fixes each block projection, since a central idempotent is a sum of primitive central idempotents. This reasoning is unchanged for semilinear algebra automorphisms. In particular, the idempotent argument is carried out over \(k\) itself; it does not require extending the finite group action to an algebraic closure.

Recovering the finite factor covers. Let \(F_i\) be the relative algebraic closure of \(k(B_i)\) in \(k(R)\), and let \(R_i\) be the normalization of \(B_i\) in \(F_i\). Equivalently, \(R\to R_i\to B_i\) is the Stein factorization of the projection. The intermediate fields \(F_i\) are regular over \(k\), because \(k(R)\) is regular over \(k\); hence the \(R_i\) are geometrically integral. Stein factorization commutes here with algebraic closure: the extensions \(k(R)/F_i\) are regular in characteristic zero. At the generic point, the complementary directions in (6) span \(\operatorname{Der}_{k(B_i)}k(R)\). For a finitely generated extension in characteristic zero, the common constants of these derivations are exactly the elements algebraic over \(k(B_i)\). Consequently \(G_0\) preserves \(F_i\) and induces birational factor transformations.

The extension obtained by adjoining the other product factors to \(k(B_i)\) is regular, and hence is linearly disjoint from the finite extension \(F_i/k(B_i)\). Thus the function field of the normal product \(R_i\times\prod_{j\ne i}B_j\) is intermediate between \(k(P)\) and \(k(R)\). Normalization gives finite maps \[R\longrightarrow R_i\times\prod_{j\ne i}B_j \longrightarrow P.\] Ramification over a divisor of \(B_i\) would persist in this product and then at a prolongation to \(k(R)\), contradicting quasi-étaleness. Thus \(R_i\to B_i\) is quasi-étale. The product map \(R\to\prod_iR_i\) is proper with finite fibres, because its composite to \(P\) is finite. Its image has dimension \(d\), equal to the dimension of the integral target product, so it is surjective and finite. The same ramification argument shows it is quasi-étale. In particular every fibre of \(R\to R_i\) has dimension \(d-\dim R_i\).

Why the factor transformations are regular. Fix \(g\in G_0\), and let \(\Gamma_i\) be the closed graph of its induced birational transformation on \(R_i\). The two morphisms from \(R\) to \(R_i\) give a surjection \(R\to\Gamma_i\). All its fibres have dimension at least \(d-\dim R_i\). If the first projection \(\Gamma_i\to R_i\) had a positive-dimensional fibre, its preimage in \(R\) would have dimension strictly larger than \(d-\dim R_i\), contradicting the preceding equidimensionality. Thus this projection is finite and birational. Normality of \(R_i\) makes it an isomorphism. Applying the same argument to \(g^{-1}\) proves that the induced transformation is an automorphism. For a semilinear transformation, replace the target by its field conjugate; the argument is identical.

Finally, each geometric \(R_i\) is a quasi-étale cover of its geometric block. It is canonical and has trivial canonical divisor. These properties descend to \(k\); linear triviality descends by the same one-dimensional-section argument used in Proposition 6. The product of its volume generators pulls back under the finite quasi-étale product map to a generator of \(K_R\). Any two generators differ by a nonzero constant. This proves all three assertions. ◻

Proof. For a divisorial valuation of ramification index \(e\) in a finite function-field extension, the canonical ramification formula is \[\mathop{\mathrm{ord}}_{E'}(\omega\wedge dz/z)+m =e\bigl(\mathop{\mathrm{ord}}_E(\omega\wedge dz/z)+m\bigr).\] With the base fixed, the denominator also multiplies by \(e\). Divisorial valuations restrict and prolong to divisorial valuations, which proves the first assertion. After curve ramification, \(z=u^l\) times a unit. The logarithmic base differentials differ by a unit, whereas \(\mathop{\mathrm{ord}}_{E'}(u)=e\mathop{\mathrm{ord}}_E(z)/l\), proving the second assertion. One may take the curve extension Galois; regularity of the original function field over its constant field makes the prolongations at the different base branches conjugate, so any chosen branch gives the same infimum. The last assertions follow directly from the numerator of (9). ◻

Proof. Abelian blocks. Let the block have dimension \(s\). On its projective semistable model over a disc, the direct image of the relative log top line is the extended Hodge line \(F^s\) in degree \(s\). This is the semistable comparison theorem for logarithmic de Rham cohomology and the canonical extension with nilpotent residue [8, 38]. The normalized form is a local frame: it is a regular relative log form but cannot be divided by \(u\) while remaining regular, because its order on one component is zero. Its eigenvalue on the central fibre of \(F^s\) is \(\lambda_i\).

The integral local system in degree \(s\) has rank \(\binom{2s}{s}\). Choose a coordinate in which the finite base action is a rotation. On the universal cover of the punctured disc, the lifted transformation, combined with parallel transport, acts by an integral matrix \(A\) commuting with the unipotent monodromy. A power of \(A\) is a power of monodromy, because the original transformation has finite order. Passing from flat frames to canonical-extension frames multiplies \(A\) by the exponential of a scalar multiple of the logarithm of monodromy. This is a commuting unipotent factor, so it does not change the eigenvalues. Hence a primitive root of the order of \(\lambda_i\) occurs among the eigenvalues of an integral matrix of rank \(\binom{2s}{s}\). Its cyclotomic degree is at most that rank. There are only finitely many possible orders, uniformly for \(s<n\).

Nonabelian blocks: a good special-fibre model. Let \(s\) be the block dimension, write \(\eta\) for its normalized form, and let \(G\) be the finite cyclic group generated by the transformation being considered. We will bound the character by applying Lemma 7 to a normal component of the special fibre. This requires a degree-one log generator on that component and a bounded power of the transformation preserving it. We first construct an equivariant good model on which the normalized logarithmic form generates along the whole special fibre; we then use structure-sheaf cohomology to find the required power. We claim that, near the marked point, there is a \(G\)-equivariant projective model \(Y\) such that \[ Y\text{ is klt},\qquad T=Y_c\text{ is reduced Cartier}, \qquad (Y,T)\text{ is dlt},\qquad K_Y\text{ is semiample over the curve}. \tag{14}\] The geometric generic fibre need only remain birational to the block.

Start from an equivariant smooth projective compactification \(Q\) of the semistable model, preserving its SNC special fibre. A relative \(K_Q\)-MMP is also a \((K_Q+Q_c)\)-MMP, because \(Q_c\) is a fibre. For existence and termination, take the finite quotient \(\overline Q=Q/G\) and its branch boundary \(\Delta\), so that \(K_Q\) is the finite pullback of \(K_{\overline Q}+\Delta\). The quotient is \(\mathbb Q\)-factorial and \((\overline Q,\Delta)\) is klt. Its adjoint is pseudo-effective over the quotient curve: an effective pluricanonical form on the generic fibre upstairs gives an invariant one after multiplying its finitely many translates.

Choose a divisor \(H\) of sufficiently large degree on the quotient curve, represented by many general points with small rational coefficients. The pair obtained by adding its pullback stays klt, and its adjoint is pseudo-effective absolutely, since a generic section extends after a sufficiently positive base twist. Choose also \(\deg H>2\dim Q\). The good-model theorem of [33], followed by a terminating MMP with scaling to that good model, now applies. Every step is over the curve: otherwise the extremal-ray length bound for \((\overline Q,\Delta)\) would give a horizontal generating curve \(C_0\) with \(-(K_{\overline Q}+\Delta)\cdot C_0\leq2\dim Q\), while the pullback of \(H\) has degree greater than this on \(C_0\). Inductively the base pullback and the morphism to the curve persist.

Lift each step by normalization in the upstairs function field, using Stein factorization for the contractions. Finite pullback preserves the relative negativity and ampleness needed for flips. No divisors are extracted, and the branch-canonical equality remains valid in codimension one. This gives the finite-group equivariant MMP; invariant \(\mathbb Q\)-factoriality suffices upstairs [36]. The steps factor over the upstairs curve as well, by normalization in its function field. They preserve dlt for the pair with the special fibre added. Special-fibre multiplicities remain one because there is no extraction. The klt total space is Cohen–Macaulay, so its Cartier fibre has no embedded associated components; generic reducedness therefore makes the fibre reduced scheme-theoretically. Relative semiampleness pulls back from the downstairs good model. This proves (14).

The degree-one log generator. The logarithmic divisor of \(\eta\wedge du/u\) on \(Y\) is effective near \(T\): this held on the semistable model, and the program extracts no divisors. The generic fibre has Kodaira dimension zero. Consequently the semiample contraction of \(K_Y\) has zero-dimensional image on the generic fibre; its Stein image over the curve is the curve itself. Thus \(K_Y\), and also \(K_Y+T\), is rationally linearly trivial over a neighbourhood of \(c\). The effective log divisor has no horizontal part, since its restriction to the generic fibre is effective and rationally trivial. A principalization of a multiple uses a rational function with vertical divisor, hence a base function. The log divisor is therefore \(aT\) for some \(a\in\mathbb Q_{\geq0}\). If \(a>0\), every weight quotient is at least \(a\): it is \(a\) plus the log discrepancy of \((Y,T)\) divided by the fibre multiplicity. This contradicts \(w_u(\eta)=0\). Hence \[ \mathop{\mathrm{div}}_{\log}(\eta\wedge du/u)=0. \tag{15}\] In particular \(K_Y+T\), and therefore \(K_Y\), is Cartier near \(T\).

Adjunction makes \(T\) Gorenstein with trivial dualizing line, generated by the residue of (15). The transformation acts on that generator by \(\lambda\): the residue of its action on \(du/u\) is one. If a prime component \(S\subset T\) is preserved by a power of the transformation, the residue on \(S\) generates \(K_S+\Theta_S\), where \(\Theta_S\) is its different. Dlt adjunction makes \((S,\Theta_S)\) normal lc and effective. Moreover \(\Theta_S\) is integral, since the divisor of this degree-one rational form is integral and equals \(-\Theta_S\). Its coefficients are at most one, so it is reduced. The character on this component generator is the corresponding power of \(\lambda\).

Thus a bounded power preserving one component will reduce the character bound to Lemma 7. We obtain such a power from the structure-sheaf cohomology of \(T\).

Finding a preserved component. The family is flat over the smooth curve. Its central fibre is slc by (14), and its nearby fibres are klt after shrinking; these fibres are Du Bois. Cohomology and base change for proper flat Du Bois families makes \(h^j(\mathcal O_{Y_t})\) locally constant [26]. The nearby fibres are birational to the block and have rational singularities. Their structure-sheaf cohomology is therefore that of a smooth resolution of the block. Extension of differential forms on klt spaces, together with Hodge symmetry, identifies these dimensions with the form algebra of the block [17].

For an irreducible Calabi–Yau block, the only nonzero groups \(H^j(T,\mathcal O_T)\) occur in degrees \(0\) and \(s\), and each has dimension one. Serre duality and the log generator give alternating trace \[1+(-1)^s\lambda^{-1}.\] If \(\lambda\) has order greater than two, this is nonzero. In the symplectic case there is one dimension in each even degree and none in odd degree. For some power of the transformation of exponent at most \(s+1\) the trace is nonzero. Indeed, if the power sums of the finitely many nonzero eigenvalues all vanished through their number, Newton’s identities would force their product to vanish.

The coherent Lefschetz theorem implies that a finite-order transformation with nonzero alternating trace has a fixed point [10]. Its vanishing statement when there are no fixed points applies to the possibly singular projective scheme \(T\), for example by equivariant embedding and coherent Lefschetz localization. At a fixed point at most \(s+1\) components of \(T\) meet: intersections of components of a dlt boundary are lc strata, with the generic SNC codimension description. A further power of exponent dividing \((s+1)!\) therefore preserves a prime component through that point. Its reduced log Calabi–Yau pair has dimension \(s<n\) and the degree-one generator just constructed. Lemma 7, which follows from \(L_s\), bounds the order of its character. This bounds the order of \(\lambda\) as well. Taking a common multiple over all \(s<n\) and both types of block proves the lemma. ◻

Proof. Take a smooth determination \(\tau:W\to Z\) on which \(pM_W\) is nef Cartier, and resolve the boundary support. Let \(\Gamma\) be the strict transform of \(B_Z\) plus the reduced exceptional divisor. Generalized klt gives \[K_W+\Gamma+M_W=\tau^*D_Z+E, \qquad E\geq0\text{ exceptional}.\] This is a big adjoint for a log smooth lc pair whose coefficients belong to a fixed DCC set, with fixed Cartier denominator for the nef part. Polarized effective birationality [5] gives a uniform degree whose rounded system is birational. Such sections push forward as rational sections of the rounded system of \(D_Z\), because the two divisors agree at the nonexceptional generic divisorial points. Thus a uniform rounded system downstairs is birational. Replacing its degree by a common multiple preserves its section ratios and makes the degree divisible by \(p_0\).

For every \(m\) divisible by \(p_0\), the exact identity is \[ H^0(X,\mathcal O_X(mK_X)) =\psi^{m/p_0}f^* H^0\bigl(Z,\mathcal O_Z(\lfloor mD_Z\rfloor)\bigr), \tag{16}\] where both sides are viewed as divisorial spaces of rational functions with compatible canonical representatives. One inclusion follows by pulling back an effective rational Cartier divisor. For the converse, divide an upstairs section by the common factor \(\psi^{m/p_0}\). Its restriction to the projective generic fibre is a regular function, hence belongs to \(\mathbb C(Z)\). Checking orders over each prime of \(Z\) shows that \(\mathop{\mathrm{div}}(h)+mD_Z\geq0\), equivalently \(\mathop{\mathrm{div}}(h)+\lfloor mD_Z\rfloor\geq0\). This proves (16). The birational system on \(Z\) then gives precisely its full function field, not merely a finite subfield. ◻

Proof. We follow the generalized torsion argument of [34], keeping track of why no dimension-four restriction is needed.

We need two uniform integers: one making \(D\) integral, and one killing the resulting torsion class in the Weil divisor class group \(\mathop{\mathrm{Cl}}(Z)\). Global ACC, after the extractions below, gives a finite coefficient set and a uniform positive lower bound for generalized log discrepancies. The latter permits boundedness; topology of the regular loci and Kummer theory then give a common torsion exponent.

Extraction and global ACC. We first justify the small modifications and one-divisor extractions used with generalized global ACC. Choose a high log resolution \(g:W\to Z\) on which the nef data descend and, if a valuation \(E\) is marked, on which \(E\) appears. Write \[K_W+B_W+M_W=g^*D.\] The boundary \(B_W\) is sub-klt and need not be effective. There is an effective exceptional Cartier divisor \(F\) with \(-F\) relatively ample; construct the resolution as a sequence of blowups and take suitable positive combinations of the successive exceptional divisors. Resolve all supports. If \(E\) has generalized log discrepancy \(a\in(0,1)\), choose \(\delta>0\) small enough that \(B_W+\delta F\) remains sub-klt and \(\delta\operatorname{mult}_EF<a\). For a sufficiently ample Cartier divisor \(A\) on \(Z\), choose a general effective rational divisor \[\Lambda\sim_{\mathbb Q}M_W+g^*A-\delta F\] by dividing a general very ample member. Then \((W,B_W+\delta F+\Lambda)\) is sub-klt. Pushing down and comparing the rational principal divisors shows that \(\Xi=B+g_*\Lambda\) is an effective ordinary klt boundary on \(Z\). Its discrepancy at the marked \(E\) is \(a-\delta\operatorname{mult}_EF\in(0,1)\).

The extraction theorem [2] applied to \((Z,\Xi)\) extracts exactly \(E\) when \(E\) is exceptional, or gives a small \(\mathbb Q\)-factorialization if the marked list is empty. If the marked valuation is already a prime on \(Z\), use the empty list. Restore the original generalized data on this model \(Z^+\). The boundary is effective: it is the strict transform of \(B\), together with \((1-a)E\) in the extraction case. The generalized adjoint remains rationally trivial and the nef denominator remains \(p\).

If the nef data are numerically nontrivial, write them as the single nef Cartier term \(pM_W\) with weight \(1/p\). Generalized global ACC [5] applies. If they are numerically trivial, take a common higher model \(\pi:W'\to Z^+\) determining the nef data. Since \(Z^+\) is \(\mathbb Q\)-factorial, its nef trace is rational Cartier. The difference \(M_{W'}-\pi^*M_{Z^+}\) is exceptional and numerically trivial over \(Z^+\). Negativity applied to it and its negative shows that the difference is zero. The generalized discrepancies are then ordinary discrepancies, and ordinary global ACC [20] applies instead. Apply this discussion first with no marked valuation. It places the original boundary coefficients in a fixed finite subset \(I_0\) of \(I\).

The same argument gives a uniform \(\epsilon>0\) for generalized \(\epsilon\)-lc singularities. Otherwise choose marked valuations whose discrepancies decrease strictly to zero. Extract them as above. The new coefficients \(1-a_i\) form a strictly increasing sequence, so adjoining them to \(I\) still gives a DCC set. Global ACC would force them into a finite set, a contradiction.

Bounded topology and class-group torsion. Birkar’s boundedness theorem [4] applies in every fixed dimension: the varieties are projective and rationally connected, their generalized adjoints are real-linearly trivial, and they are uniformly generalized \(\epsilon\)-lc. Consequently they are bounded up to isomorphism in codimension one. The bounded comparison models can be normalized, still in a bounded family and still isomorphic to \(Z\) in codimension one.

Put \(U=Z_{\mathrm{reg}}\). The groups \(H_1(U(\mathbb C),\mathbb Z)\) are finitely generated and range over only finitely many isomorphism types. First, this group is invariant under isomorphism in codimension one of normal varieties. On the regular loci the common open is obtained by removing subsets of complex codimension at least two. After stratifying those subsets, transversality moves paths and their homotopies off every stratum, whose real codimension is at least four; the fundamental group is unchanged. Second, stratify a finite-type parameter space for the bounded comparison family until it is flat. The relative smooth locus then has fibres equal to the regular loci of the selected normal fibres. Cover the parameter strata by finitely many affine opens and embed the family projectively over each. Complex projective space is a compact real algebraic set via its realization as rank-one Hermitian projection matrices. Thus the entire relative smooth locus, not just individual affine fibre charts, maps semialgebraically to each parameter open. Semialgebraic triviality gives a finite partition on which these fibres are homeomorphic, and semialgebraic triangulation gives each the homotopy type of a finite complex [9]. This proves the asserted finite list of finitely generated homology groups.

For each individual \(Z\), the class group \(\mathop{\mathrm{Cl}}(Z)\) is finitely generated. Indeed a rationally connected smooth resolution \(Y\) has \(\operatorname{Pic}^0(Y)=0\), so \(\operatorname{Pic}(Y)\) is finitely generated by Neron–Severi finiteness; divisor pushforward surjects onto \(\mathop{\mathrm{Cl}}(Z)\). Since \(Z\) is normal and projective, \(\Gamma(U,\mathcal O_U^*)=\mathbb C^*\) and \(\operatorname{Pic}(U)=\mathop{\mathrm{Cl}}(Z)\). Kummer theory and comparison for finite etale covers [18] give, for every \(a\geq1\), \[ \mathop{\mathrm{Cl}}(Z)[a]\simeq H^1_{\mathrm{\acute et}}(U,\mu_a) \simeq\operatorname{Hom}\bigl(H_1(U(\mathbb C),\mathbb Z),\mu_a(\mathbb C)\bigr). \tag{17}\] The left side has bounded order as \(a\) varies for this fixed \(Z\). Thus \(H_1(U(\mathbb C),\mathbb Z)\) has no free summand and is finite. There are only finitely many such finite groups in our bounded topological list. A common multiple \(T\) of their exponents kills every torsion class in every \(\mathop{\mathrm{Cl}}(Z)\), by (17).

Clearing the actual divisor. The trace \(pM_Z\) is an integral Weil divisor, being the pushforward of Cartier data. Choose \(q\) divisible by \(p\) and by the denominators of the finite set \(I_0\). Then \(qD\) is integral. Its class in \(\mathop{\mathrm{Cl}}(Z)\) is torsion because \(D\sim_{\mathbb Q}0\). Therefore \(TqD\) is principal. Taking \(\ell=Tq\) proves the proposition. ◻

Proof. The divisor \(D_Z\) in (7) is rationally linearly trivial. Indeed a Cartier multiple whose pullback is trivial is itself trivial by the projection formula and \(f_*\mathcal O_X=\mathcal O_Z\). Proposition 14 gives a uniform \(\ell\) and a function \(v\in\mathbb C(Z)^*\) with \(\ell D_Z=\mathop{\mathrm{div}}(v)\). Choose a common multiple \(r\) of \(\ell\) and \(p_0\). Then \[rK_X=\mathop{\mathrm{div}}\left((v\circ f)^{r/\ell}\psi^{-r/p_0}\right).\] All constants depend only on \(n\), since the dimensions of \(Z\) range over a fixed finite set and Proposition 10 supplies a fixed coefficient set and nef denominator. ◻

Proof. A regular one-form on a smooth projective model of \(X\) pulls back injectively to a reflexive one-form on \(P\). Indeed, the rational map from the normal variety \(P\) to the proper smooth target is defined at every codimension-one point. The pulled-back form is regular there and extends reflexively. The block has no reflexive one-forms, so Hodge symmetry gives irregularity zero.

It remains to exclude a dominant rational map to a non-uniruled smooth projective variety \(T\) with \(0<j=\dim T<n\). By the pseudo-effectivity criterion of BDPP [6], \(K_T\) is pseudo-effective. Nonvanishing in LA gives a nonzero \(\eta\in H^0(T,mK_T)\) for some \(m>0\). Resolving the composite map from \(P\), we obtain \(u:W\to P\) and \(f:W\to T\), with \(W\) smooth. Pullback of differentials gives a nonzero section \[s=f^*\eta\in H^0\bigl(W,(\Omega_W^j)^{\otimes m}\bigr).\] At the generic point it has the form \(s=a\beta^{\otimes m}\), where \(a\in\mathbb C(P)^*\) and \(\beta\) is a nonzero decomposable rational \(j\)-form, obtained by pulling back a rational top form on \(T\).

The divisorial zero locus of \(s\) is exceptional over \(P\). To prove this, let \(H=u^*H_P\) for an ample Cartier divisor \(H_P\) on \(P\). Canonicality and \(K_P\sim0\) imply that \(K_W\) is rationally linearly equivalent to an effective \(u\)-exceptional divisor. In particular, \[K_W H^{n-1}=0.\] Generic semipositivity of the cotangent bundle [7] implies that every torsion-free quotient of \(\mathcal V=(\Omega_W^j)^{\otimes m}\) has nonnegative degree against \(H^{n-1}\). Here one first uses an ample perturbation \(H+\epsilon H_0\). On a sufficiently high complete-intersection curve, the Harder–Narasimhan factors of \(\Omega_W^1\) restrict to semistable bundles of nonnegative slope [30]. Tensor products of semistable bundles in characteristic zero are semistable [37], and exterior powers are quotients of tensor powers. This proves the assertion for \(\mathcal V\) with the perturbed polarization; passage to the limit gives the assertion for \(H\).

The saturation of the line generated by \(s\) is \(\mathcal O_W(Z)\), where \(Z\ge0\) is its divisorial zero divisor. Since \(c_1(\mathcal V)\) is a positive multiple of \(K_W\), the quotient inequality gives \(ZH^{n-1}\le0\). Every nonexceptional prime has strictly positive \(H^{n-1}\)-degree. Thus \(Z\) is exceptional.

At the generic point of a prime divisor of \(P\), write \(\beta\) in a regular frame and let \(b\) be the minimum order of its coefficients. The absence of zeros and poles of \(s\) there gives \[\mathop{\mathrm{ord}}(a)+mb=0.\] All coefficients of \(\mathop{\mathrm{div}}_P(a)\) are therefore divisible by \(m\). Normalize \(P\) in a field component obtained by adjoining an \(m\)th root \(z\) of \(a\). This is a finite quasi-étale cover: over a divisorial discrete valuation, removing the uniformizer to its \(m\)-divisible order leaves the equation for a root of a unit, which is unramified in characteristic zero. The rational form \(z\beta\) has no divisorial pole, hence is a nonzero reflexive \(j\)-form on the cover. It remains decomposable.

This contradicts the defining form algebra of the block. In the Calabi–Yau case there are no intermediate-degree reflexive forms on any connected quasi-étale cover. In the symplectic case the only possibilities are scalar multiples of \(\sigma^q\) with \(0<2q<n\), where \(\sigma\) is generically nondegenerate. Contraction into \(\sigma^q\) is injective on the nondegenerate locus, whereas a nonzero decomposable \(j\)-form with \(j<n\) has a contraction kernel of dimension \(n-j\). These forms cannot agree.

Finally, a smooth resolution of any proper rational image has a maximal rationally connected quotient whose smooth projective base is non-uniruled [15]. A positive-dimensional such base would be a forbidden rational image of \(X\). The quotient is therefore a point, proving rational connectedness. ◻

Proof. The cover is connected by minimality of the index and is quasi-étale. The finite discrepancy formula makes \(Y\) terminal: an exceptional divisorial valuation over \(Y\) restricts to an exceptional valuation over \(V\), and its log discrepancy is the positive ramification index times a number greater than one. Also \(K_Y\sim0\).

Let \(\mathbb C(T)\subset\mathbb C(Y)\) be the relatively algebraically closed base field of an equivariant fibration. Its invariant field \(\mathbb C(T)^G\) lies in \(\mathbb C(V)\), has the same transcendence degree, and is relatively algebraically closed there. Indeed, an element of \(\mathbb C(V)\) algebraic over \(\mathbb C(T)^G\) is algebraic over \(\mathbb C(T)\), hence lies in \(\mathbb C(T)\) and then in its invariant field. This gives a descended intermediate rational fibration \(V\dashrightarrow Z\).

Resolve the maps and bases, obtaining \(u:U\to Y\) with \(U\) smooth, a morphism \(f:U\to T'\) to the unquotiented base, and a morphism \(g:U\to Z'\) constant on the generic fibres of \(f\). Pull back a general very ample hyperplane \(H\) on \(Z'\) by the rational map from \(V\), and close up the divisor to obtain \(D\) on \(V\). The properness of \(Z'\) ensures that this map is defined in codimension one. The hyperplane pullbacks form a moving linear system with a nonconstant section ratio; thus \(\kappa(V,D)>0\) and \(D\) is big.

For \(q=\pi\circ u\), one has \[ q^*D=g^*H+E, \tag{18}\] with \(E\ge0\) exceptional over \(Y\). At every nonexceptional prime the two pullbacks agree, because its image on \(V\) is a prime at which the original rational map is defined. After fixing the resolution, choose \(H\) not to contain the generic image of any of the finitely many exceptional primes. Its pullback has zero coefficient at each such prime, whereas \(q^*D\) is effective. This proves the effectivity assertion in (18).

Let \(F\) be the smooth geometric generic fibre of \(f\). The class \(q^*D|_F\) is big: decompose the big class \(q^*D\) into an ample rational class and an effective rational class, and restrict to the generic fibre, which is not contained in the effective support. Since \(g^*H|_F\sim0\), the class \(E|_F\) is big. Terminality gives \[K_U\sim_{\mathbb Q}\sum_a d_a E_a, \qquad d_a>0,\] where the sum includes every \(u\)-exceptional prime. Some positive rational multiple of \(E\) is coefficientwise dominated by this sum. Restriction and adjunction therefore make \(K_F=K_U|_F\) big. The proof for \(V\) is the same with the finite cover omitted. ◻

Proof. We suppress the sequence index. First, for every fixed positive rational \(a\), \[ \mathop{\mathrm{vol}}(aK_W+L)\le(aC'+1)^eL^e. \tag{26}\] Choose a divided general member of a sufficiently divisible free multiple of \(L/a\), giving a klt effective boundary \(\Gamma\sim_{\mathbb Q}L/a\). The big klt adjoint \(K_W+\Gamma\) has a good model by BCHM. On a common resolution, let \(P_a\) be \(a\) times the pullback of its semiample adjoint. The negative-map comparison gives \[aK_W+L\sim_{\mathbb Q}P_a+F_a, \qquad F_a\ge0,\] where \(F_a\) is exceptional over that model. Thus \(P_a^e=\mathop{\mathrm{vol}}(aK_W+L)\). With \(L\) pulled back to this resolution, nef mixed-intersection inequalities give \[(aC'+1)L^e\ge(aK_W+L)L^{e-1} \ge P_aL^{e-1}\ge(P_a^e)^{1/e}(L^e)^{(e-1)/e},\] which proves (26). Replacing \(W\) by a higher smooth model and retaining the pullback of \(L\) changes neither this volume nor the intersection estimates: additional canonical terms are effective exceptional divisors.

By \(I_e\), a fixed integer \(m_0\) defines every Iitaka fibration under consideration. Pass to a subsequence where \(j=\kappa(W)\) is constant. The case \(j=e\) is impossible for large indices. The resolved system \(|m_0K_W|\) would have an integral free moving part with generically finite map and positive integral top intersection; hence \(\mathop{\mathrm{vol}}(K_W)\ge m_0^{-e}\), contrary to (26). If \(j=0\), the assertion follows directly from that inequality with \(a=1\).

Suppose henceforth that \(0<j<e\). Resolve the Iitaka system and its base, obtaining a morphism with connected fibres between smooth projective varieties \[f_0:W_0\longrightarrow T_0.\] Write \(L_0\) for the pulled-back polarization. The hyperplane class of the system’s image pulls back to a big basepoint-free Cartier divisor \(H_0\) on \(T_0\), and \[ f_0^*H_0\preceq m_0K_{W_0}, \qquad H_0^j\ge1. \tag{27}\] The smooth geometric generic fibre \(U\) has Kodaira dimension zero by the Iitaka-fibration theorem. The restriction \(L_0|_U\) is big: the moving fibres meet the generically finite locus of the semiample map of \(L_0\). Nefness and semiampleness restrict as well.

We arrange a model on which sections over a big open of the base can be used without losing volume on \(W_0\). Apply flattening by a modification of \(T_0\), and choose a smooth projective \(T\) dominating that modification. Let \(X\) be the normalization of the flat principal family after this base change, and resolve \(X\) by \(W\) without changing its smooth generic fibre, which we continue to denote by \(U\). The resulting maps fit into \[\begin{array}{ccccc} W&\longrightarrow&X&\longrightarrow&W_0\\ &&\downarrow&&\downarrow f_0\\ &&T&\longrightarrow&T_0. \end{array}\] Write \(f:W\to T\), \(\rho:W\to W_0\), \(L=\rho^*L_0\), and \(H\) for the pullback of \(H_0\) to \(T\). Both horizontal maps in the top row are proper and birational. The morphism \(X\to T\) is equidimensional: flatness gives the generic fibre dimension before normalization, and finite normalization preserves it. Hence a prime divisor \(E\) on \(W\) whose image in \(T\) has codimension at least two is exceptional over \(X\). Its image in \(X\) has dimension at most \(e-2\), so its image in \(W_0\) also has codimension at least two. Thus \(E\) is \(\rho\)-exceptional. This property will allow extension of sections on \(W_0\). Relation (27) gives \(f^*H\preceq m_0K_W\) on the higher smooth model, and \(H^j=H_0^j\ge1\).

Choose a general projection of the morphism defined by \(H\) to \(\mathbf P^j\). It is a generically finite morphism: a general \((j+1)\)-dimensional linear subsystem has no common zero on the \(j\)-dimensional image. Its ramification divisor yields \(K_T+(j+1)H\succeq0\). Therefore \[ K_{W/T}+L+2f^*H \preceq \bigl(1+(j+3)m_0\bigr)K_W+L. \tag{28}\]

Choose, separately for this fibration, a sufficiently divisible integer \(l\ge1\) so that the \(l\)-Veronese of the section ring of \(K_U+L|_U\) is generated in degree one. This follows from finite generation of the big klt adjoint ring, after representing \(L|_U\) by a divided general free member. Enlarge \(l\) to clear all rational divisors and equivalences used below. On \(W\), similarly represent \(L\) by an effective klt boundary. Twisted weak positivity [13] gives weak positivity of the torsion-free direct image \[E_l=f_*\mathcal O_W\bigl(l(K_{W/T}+L)\bigr).\] There is \(b>0\) for which \((\operatorname{Sym}^b E_l)^{**}\otimes\mathcal O_T(bH)\) is generated by global sections at the generic point. A big \(H\) suffices for this consequence of weak positivity: choose \(kH=A+F\) with \(A\) ample and \(F\ge0\), apply weak positivity with symmetric exponent \(k\beta\) and twist \(\beta A\), and multiply by the section of \(\beta F\).

For every multiple \(q\) of \(b\), multiplication of those global sections, followed by fibrewise multiplication in the adjoint ring, gives sections over a big open of \(T\) of \[ql(K_{W/T}+L)+qf^*H\] spanning all degree-\(ql\) sections on the generic fibre. To see the asserted domain of definition, remove only the codimension-two sets where the relevant torsion-free sheaves differ from their reflexive hulls or are not locally free. The multiplication maps are defined at every codimension-one point. Generation is needed only at the generic point, so its possible failure along a divisor does not require removing that divisor.

Choose a basis over \(\mathbb C(T)\) among these sections. Its size is \(h^0(U,ql(K_U+L|_U))\). Multiplying it by a basis of \(H^0(T,qlH)\) gives linearly independent products over \(\mathbb C\): a relation would first contradict the generic-fibre independence over \(\mathbb C(T)\) and then the independence of the base sections. If necessary, multiply by a fixed nonzero section of \((ql-q)H\) to place these products in degree \(ql\) of the left side of (28). Multiplying further by the effective difference in that equation, after clearing denominators, embeds them into sections of its right side over the same big open.

Let \(T^\circ\subset T\) be the big open just obtained by removing the codimension-two defects of the relevant torsion-free sheaves. The resulting rational pluriforms are regular on \(f^{-1}(T^\circ)\). Every prime divisor on \(W\) in its complement is \(\rho\)-exceptional, by the flattening construction. Thus the forms are regular at the generic point of the strict transform of every prime divisor on \(W_0\). Push them to \(W_0\) with their \(L_0\)-twist. They have no divisorial pole and therefore extend, by smoothness, to sections of \[H^0\!\left(W_0, \mathcal O_{W_0}\!\left(ql\bigl((1+(j+3)m_0)K_{W_0}+L_0\bigr)\right)\right).\] Their independence is preserved by this birational identification.

Put \(g=e-j\). With this fibration and \(l,b\) fixed, let \(q\) tend to infinity through sufficiently divisible multiples of \(b\). The asymptotic section counts on \(T\) and \(U\) give \[ \mathop{\mathrm{vol}}\bigl((1+(j+3)m_0)K_{W_0}+L_0\bigr) \ge \frac{e!}{g!j!}\,H^j\mathop{\mathrm{vol}}(K_U+L|_U). \tag{29}\] The left side tends to zero along the original sequence by (26), whereas \(H^j\ge1\). This proves (25). No uniform choice of \(l\) or \(b\) is needed: the asymptotic estimate is obtained separately for each fibration before taking the sequence limit. ◻

Proof of the first inequality in Proposition 19. The assertion in dimension one follows directly from degree, so assume \(n\ge2\). If no constant \(C_n\) exists, rational rescaling gives a sequence with \[P^n\longrightarrow0, \qquad\gamma(P;V)>1.\] Take small projective \(\mathbb Q\)-factorializations and pull back \(P\). We describe a dimension-decreasing construction. At a stage, after passing to a subsequence, there is a fixed dimension \(d\) and data \((Z,N)\) such that

1. \(Z\) is projective, \(\mathbb Q\)-factorial and klt; \(N\) is big, nef and semiample; and \(N^d\to0\);

2. a smooth model of \(Z\) has nonnegative Kodaira dimension;

3. \(K_Z\preceq a_0N\) for a fixed positive rational \(a_0\);

4. \(\gamma(N;Z)\) has a positive lower bound.

These conditions hold initially by canonicality and crepantness of the small modification. They cannot hold in dimension one, because the order of a section is bounded by its degree.

For \(d\ge2\), choose \(d+1\) fixed equally spaced positive rational levels below the order lower bound. Eventually they lie above \((N^d)^{1/d}\). Lemma 21 gives a covering family with a smooth projective model \(W\) of its geometric generic member, of dimension \(0<e<d\). Set \(L=N|_W\), using pullback notation. It is big, nef and semiample, and \(\kappa(W)\ge0\) by the covering assertion. The tracking inequalities and restriction of \(K_Z\preceq a_0N\) give \[L^e\longrightarrow0, \qquad K_WL^{e-1}\le C'L^e\] for a fixed \(C'\). Lemma 22 therefore gives a positive-dimensional Iitaka fibre \(U\), possibly all of \(W\), with \(\kappa(U)=0\) and \(\mathop{\mathrm{vol}}(K_U+L|_U)\to0\).

Choose an effective klt boundary \(\Gamma\sim_{\mathbb Q}L|_U\) from a divided general free member, and take a \(\mathbb Q\)-factorial good model \(Z'\) of \(K_U+\Gamma\). Let \(N'\) be its semiample big adjoint. Then \((N')^{\dim Z'}\to0\) and \(K_{Z'}\preceq N'\). On a common resolution with maps \(x\) to \(U\) and \(y\) to \(Z'\), one also has \[ x^*(L|_U)\preceq y^*N'. \tag{30}\] Here is the fixed-part argument for this useful comparison. Write \[x^*(K_U+\Gamma)=y^*N'+F, \qquad F\ge0,\] with \(F\) exceptional over \(Z'\). Choose an effective \(D\sim_{\mathbb Q}K_U\), which exists because \(\kappa(U)=0\). In sufficiently divisible degrees, add a general member of the free system of \(L|_U\) to this representative. Its pullback must contain the fixed part \(F\). The general free member contains none of the finitely many components of \(F\), so \(F\le x^*D\). Subtracting gives (30).

If \(\gamma(N';Z')\to0\) on a subsequence, stop. Otherwise pass to a subsequence with a positive lower bound and repeat. All stage conditions persist: the smooth-model Kodaira dimension of \(Z'\) is zero because it is birational to \(U\), and the dimension strictly decreases. The impossibility of the stage conditions in dimension one forces termination.

Trace the resulting families back to the original \(V\). This produces covering members \(A\subset V\) of positive dimension whose smooth geometric generic models satisfy \[ \kappa(A^*)=0, \qquad\gamma(P;A)\longrightarrow0. \tag{31}\] We make the section-order comparison explicit. All intermediate constructions take place on geometric generic members and spread after dominant parameter extension. Fibres sweep the preceding member, and each birational comparison is an isomorphism on an open met by the next generic member. Composite members therefore map birationally to their images. At every stage the effective polarization comparison restricts, since a sweeping member is not contained in its extra divisor. On common resolutions it gives an injection from the complete section space of the restricted preceding polarization into that of the next one, in divisible degrees. Multiplication by the effective difference preserves order at a generic point outside its support. Proper birational pushforward preserves these complete section spaces. Iteration compares the complete spaces on a resolution of \(A\) with the final spaces whose normalized order tends to zero. This proves (31), not merely an assertion about restrictions of ambient sections.

Mark these members at generic smooth points and include all their \(G\)-translates. Their order bounds are unchanged because \(g^*P\sim_{\mathbb Q}P\). Apply Lemma 20. If the chain leaf had full dimension, its order estimate would imply \(\gamma(P;V)\to0\), contradicting the chosen lower bound. The leaf has positive dimension, so eventually it is the generic fibre of an intermediate equivariant rational fibration.

That geometric generic leaf is covered by the Kodaira-dimension-zero members in (31). Indeed, every generic marked member is contained in its chain leaf; after extension to the geometric generic point of the quotient, the incidence marks still dominate the leaf. These extensions preserve the geometric generic members and their Kodaira dimension. The covering assertion of Lemma 21 therefore prevents the leaf from being of general type. This contradicts the hypothesis of Proposition 19 and proves the upper order bound. ◻

Proof of the second inequality in Proposition 19. Rationally rescale so that \(1\le P^n\le2\). The upper estimate now gives a uniform bound \(\gamma(P;V)\le D_0\). If no positive lower bound for \(\varepsilon(P)\) exists, choose covering marked curve families with \[\frac{P\cdot C}{\operatorname{mult}_x C} \longrightarrow0,\] and include their \(G\)-translates. Such families are obtained from curves through very general ambient marks by spreading their geometric generic data; after shrinking, the generic marked ratio satisfies the same bound. Lemma 20 gives chain leaves of a fixed dimension \(h\) on a subsequence, with degrees \(P^hH\to0\). Since \(P^n\ge1\), eventually \(0<h<n\).

Resolve the chain fibration and choose a very general smooth fibre \(F\) of dimension \(h\). Write \(s=n-h\) and retain pullback notation for \(P\). Its normal bundle is trivial, because \(F\) is a fibre of a smooth morphism near the chosen base point. For its ideal \(\mathcal I_F\), the filtration of \(\mathcal O/\mathcal I_F^u\) has associated graded terms \(\operatorname{Sym}^j(\mathcal I_F/\mathcal I_F^2)\) for \(0\le j<u\), of total rank \(\binom{u+s-1}{s}\). Consequently the rank of restriction of global degree-\(l\) sections to this thickened fibre is at most \[ \binom{u+s-1}{s}\,h^0(F,lP|_F). \tag{32}\] Take \(u=\lceil Dl\rceil\) with \(D>D_0\). For each fixed fibration, in large divisible degrees the leading coefficient in (32) is \[\frac{D^s}{s!h!}(P^hF)\,l^n.\] It tends to zero along the sequence, while the ambient section count has leading coefficient \(P^n/n!\ge1/n!\). Thus, first choosing a sufficiently late sequence member and then a sufficiently large divisible \(l\), there is a nonzero ambient section vanishing in \(\mathcal I_F^u\).

At any smooth point of \(F\), this section has order at least \(u\), because \(\mathcal I_F\subset\mathfrak m_x\). The fibre and a point on it can be chosen very general in the original birational isomorphism locus and outside all the countably many order-exceptional sets. Birational pullback preserves the ambient section spaces and their orders there. The resulting normalized order is at least \(D>D_0\), contradicting the upper bound. This proves the second inequality and completes Proposition 19. ◻

Proof. We give the canonical-connection argument underlying the Frobenius instability estimates of Langer [28]; in particular, we do not assume that tensor products are semistable in characteristic \(p\). For any rank-\(\rho\) bundle \(U\), absolute Frobenius \(F_C\) satisfies \[ \mu_{\min}(F_C^*U) \ge p\mu_{\min}(U)-(\rho-1)G. \tag{36}\] Indeed, consider the successive slopes of \(F_C^*U\). If a successive gap is larger than \(G\), take the last such gap and let \(S\) be the preceding saturated filtration step. The second fundamental map of the canonical connection, \[S\longrightarrow ((F_C^*U)/S)\otimes\Omega_C^1,\] vanishes by the slope inequality. Consequently \(S\) descends under Frobenius. Here this descent can be checked directly: over \(k(C)\), put a matrix for \(S\) in a chart with an identity minor. Differentiating its columns gives vectors in \(S\otimes\Omega_{k(C)/k}^1\) with zero identity rows, hence all matrix entries have derivative zero. They belong to \(k(C)^p\). Extracting their roots and saturating gives a subbundle \(S_0\subset U\) with \(S=F_C^*S_0\), since Frobenius is flat.

The quotient \(Q=(F_C^*U)/S\) is therefore the pullback of a quotient of \(U\), so \(\mu(Q)\ge p\mu_{\min}(U)\). All successive slope gaps of \(Q\) are at most \(G\); its smallest slope is at least its average minus \((\rho-1)G\). It is also the smallest slope of \(F_C^*U\). This proves (36). If there is no large gap, use \(S=0\) and the same average-slope argument.

Iteration gives \[ \mu_{\min}((F_C^e)^*E) \ge-(\rho-1)G\frac{p^e-1}{p-1}. \tag{37}\] Choose a line bundle \(T_e\) of integer degree \(d_e\) in the interval \[2g-\mu_{\min}((F_C^e)^*E) \le d_e<2g+1-\mu_{\min}((F_C^e)^*E).\] The bundle \(E_e=(F_C^e)^*E\otimes T_e\) is globally generated: after subtracting one point, its minimum slope is at least \(2g-1\); Serre duality gives vanishing of \(H^1\) and hence surjectivity of evaluation. If \(Q_i\) is a positive-rank torsion-free quotient of \(E^{\otimes i}\), then \((F_C^e)^*Q_i\otimes T_e^{\otimes i}\) is globally generated. Thus \(p^e\mu(Q_i)+i d_e\ge0\). Divide by \(p^e\) and let \(e\) tend to infinity, using (37), to obtain \[ \mu(Q_i)\ge-\frac{i(\rho-1)G}{p-1}. \tag{38}\] Since \(A_i(E)\) is a quotient of \(E^{\otimes i}\), its minimum slope is at least \(-c_2\) for \(0\le i\le u\).

Multiplication gives a perfect pairing \(A_j(E)\otimes A_{u-j}(E)\to A_u(E)\): a truncated monomial pairs with the unique complementary monomial. The top line is \(A_u(E)=(\det E)^{p-1}\), as is checked on diagonal and elementary changes of frame. Therefore \[A_j(E)^*\otimes(\det E)^{p-1}\simeq A_{u-j}(E).\] Its minimum slope is at least \(-c_2\), proving (35). Counting the \(p^\rho\) monomials with exponents between \(0\) and \(p-1\) gives the rank assertion. ◻

Proof. The divisor exact sequences along the flag bound \(h^0(W,E)\) by \[\sum_{k_1,\ldots,k_{n-1}\ge0} h^0\left(C,E|_C\otimes \mathcal O_C\left(-\sum_{i=1}^{n-1}k_i l_iH\right)\right).\] Each iteration terminates because a sufficiently negative ample twist of a fixed vector bundle has no sections. One may see this by embedding that bundle in a finite sum of sufficiently positive line bundles. A summand vanishes if \(\sum_i k_i d_i>M\), by its negative maximum slope. Otherwise evaluation at \(\lfloor M\rfloor+1\) distinct points is injective, again by the slope criterion, so the summand is at most \(e(M+1)\). There are at most \(\prod_i(1+M/d_i)\) possible tuples. The same proof for \(n=1\) uses an empty product. ◻

Proof. Choose bases adapted to the order filtrations of the section spaces. Initial forms with the same order are linearly independent. By Künneth, a section of the external product is a linear combination of tensor products of these bases. In its smallest occurring total degree, terms with different slot multidegrees cannot cancel; within a multidegree their initial forms are independent because the slot parameters are disjoint. Its order is therefore the smallest total order of a basis tensor with nonzero coefficient, at most \(\sum_i a_i\). ◻

Proof of Proposition 23. The map \(\theta_t\) is finite, so \(P_t\) is ample. Fix \(V,L,r,t\); all auxiliary data in this proof may depend on these fixed choices. Put \[ b=b_0/t,\qquad q=2tr. \tag{40}\] Suppose that \(\varepsilon(P_t)>(n+1)t/b\).

A fixed supply of ordinary jets. Choose a smooth very general \(z\in Z_t\) whose coordinates lie over \(V_{\mathrm{reg}}\), and a rational number \(c\) with \((n+1)t<c<b\varepsilon(P_t;z)\). On the blow-up of \(z\), the pullback of \(bP_t\) minus \(c\) times the exceptional divisor is ample. Indeed interpolate a nef class with slightly larger exceptional coefficient and an ample class with small positive coefficient. Serre vanishing on this blow-up, together with the usual pushforward of powers of its exceptional ideal, gives, for sufficiently divisible \(k_0\), surjectivity onto ordinary jets through degree \((n+1)t k_0\). Increase \(k_0\) so that \(Q=k_0bL\) is Cartier. Thus \[ H^0(Z_t,k_0bP_t)\longrightarrow \mathcal O_{Z_t,z}/\mathfrak m_z^{(n+1)t k_0+1} \quad\text{is surjective}, \tag{41}\] where a local frame is understood.

Fix a trivialization \(\eta\) of \(rK_V\). On \(V_{\mathrm{reg}}\), the index cover is the torsor of canonical frames \(\tau\) satisfying \(\tau^r=\eta\). The map \(Y^t\to Z_t\) is etale over this locus. At a chosen lift of \(z\), Künneth and the character decomposition express the pulled-back sections in (41) as combinations of tensors with slot factors \[ \tau^{-m_i}s_i,\qquad s_i\in H^0(V_{\mathrm{reg}},Q+m_iK_V),\qquad 0\le m_i<r,\qquad \sum_i m_i\equiv0\pmod r. \tag{42}\] Choose a smooth projective resolution \(u:W\to V\) which is an isomorphism over \(V_{\mathrm{reg}}\). Write \(K_W=u^*K_V+A\), with \(A\ge0\) exceptional. We use \(L,Q\) also for their pullbacks. Each \(s_i\) extends to \(Q+m_iK_W\): close its effective divisor on \(V\), pull back this rational Cartier divisor, and add \(m_iA\). Likewise \[\eta_W\in H^0(W,rK_W),\qquad \mathop{\mathrm{div}}(\eta_W)=rA.\] Fix finitely many tensors (42) spanning the required jets. Only this finite jet data will be specialized.

A fixed flag and a movable curve test. Since \(K_WL^{n-1}=0\), we may take an ample rational divisor \(H=L+\delta H_{\mathrm{amp}}\), for \(H_{\mathrm{amp}}\) fixed ample and \(\delta>0\) small rational, such that \[ \begin{split} H^n&\le3,\qquad LH^{n-1}\le H^n,\\ (q+t-1)K_WH^{n-1}&\le b_0H^n. \end{split} \tag{43}\] The canonical divisor of \(W\) is pseudo-effective. Generic semipositivity [7], followed by restriction of its Harder–Narasimhan factors [30], gives a smooth complete intersection flag in multiples \(l_iH\) ending in a curve \(C\) with \[ \mu_{\min}(\Omega_W^1|_C)\ge0. \tag{44}\] The flag may be chosen to avoid the codimension-two singular sets of the filtration quotients.

Independently choose a very general \(x\in W\), in the isomorphism locus of \(u\), where the all-degree order bound defining \(\gamma(L)\) holds. Let \(\sigma:\widehat W\to W\) blow up \(x\), with exceptional divisor \(J\), and define \[D^+=bL+(q+1)K_W,\qquad d_0=b(C_1+1).\] In divisible degrees the sections of \(D^+\) are those of \(bL\) multiplied by a fixed exceptional section: \(rK_V\sim0\), and \(u_*\mathcal O_W(E)=\mathcal O_V\) for effective integral exceptional \(E\). That fixed section has order zero at \(x\). Hence the normalized orders of sections of \(D^+\) at \(x\) are at most \(bC_1\).

It follows that \(\sigma^*D^+-d_0J\) is not pseudo-effective. Otherwise interpolation with the big divisor \(\sigma^*D^+\) would make \(\sigma^*D^+-dJ\) big for a rational \(d\) strictly between \(bC_1\) and \(d_0\), producing an excluded section. The projective movable-cone duality theorem [6] therefore gives a strongly movable curve class \(\beta\) on \(\widehat W\) satisfying \[ D^+\cdot\beta<d_0J\cdot\beta, \qquad J\cdot\beta>0, \qquad L\cdot\beta\ge0, \qquad K_W\cdot\beta\ge0. \tag{45}\] Divisors from \(W\) are pulled back in these pairings. More precisely, we may choose \(\beta=v_*(A_1\cdots A_{n-1})\), where \(v:T\to\widehat W\) is a smooth projective birational model and the \(A_i\) are very ample. The strict separation inequality holds on one such generator of the movable cone. Nonnegative pairing with \(L\) and \(K_W\) then forces \(J\cdot\beta>0\).

Specialization of the finite data. Spread all the preceding algebraic data over an integral finitely generated \(\mathbb Z\)-subalgebra of \(\mathbb C\). This includes the points, blow-up, flag, testing morphism, ample divisors, torsor, its marked lift, and the finitely many jet-spanning sections. After shrinking the base, preserve smoothness, geometric integrality, ampleness, dominance of \(v\), and the intersection numbers in (43) and (45). The fixed jet evaluation is a map of finite locally free modules, so its surjectivity persists. Invert \(r\) and the finitely many denominators in the chosen data.

We can also preserve (44). A fixed relative twist makes \(\Omega_W^1|_C\) globally generated; thus the degrees of its positive-rank quotients are bounded below uniformly by rank. There are only finitely many negative degrees and ranks to exclude. For each associated Hilbert polynomial, the relative Quot scheme is projective and has empty geometric generic fibre, by (44). Removing their images excludes all negative-degree quotients. Quotients with torsion cause no issue, since their torsion-free quotients have no larger degree. Consequently (44) holds on every remaining geometric fibre.

There are such fibres in arbitrarily large characteristics \(p\). Work over their algebraically closed residue fields, retaining the notation. The test \(\beta\) still pairs nonnegatively with every effective divisor: pull back by the retained dominant morphism \(v\) and intersect with the very ample \(A_i\). We have specialized a fixed test, not an assertion about the entire pseudo-effective cone.

Full rank at a general product point. Let \(F:W\to W'\) be relative Frobenius, where the prime indicates the base-field twist. Set \[ N=\lfloor p/k_0\rfloor,\quad B=NQ,\quad \mathcal E=F_*\mathcal O_W(B),\quad s=p^n,\quad R=s^t. \tag{46}\] The finite flat map \(F\) has degree \(s\), so \(\mathcal E\) has rank \(s\). Products of \(N\) of the fixed jet sections span jets through degree \((n+1)t k_0N\). To check this, factor each monomial of degree at most that number into \(N\) monomials of degree at most \((n+1)t k_0\), and choose corresponding sections with these initial forms. Their products give a triangular spanning set for the jet filtration. For large \(p\), \[(n+1)t k_0N\ge nt(p-1).\] Thus these products span the thick point defined by the \(p\)-th powers of all \(nt\) smooth parameters. This is the elementary comparison of ordinary and Frobenius jets used in [31]. Etaleness identifies the completed local rings at the torsor lift and its tuple in \(W^t\), and therefore identifies these thick points.

For an \(N\)-fold product of pure tensors, write \(m_i'\) for the new exponent in slot \(i\). Then \(0\le m_i'\le N(r-1)<pr\) and \(\sum_i m_i'\equiv0\pmod r\). Choose \(a_i\) in \([r,2r-1]\) with \(pa_i\equiv m_i'\pmod r\). Their sum is a multiple of \(r\) less than \(q=2tr\). Increase \(a_1\) by the difference to obtain \[a_i\ge0,\qquad \sum_i a_i=q,\qquad pa_i\ge m_i'.\] Multiplying the slot section by \(\eta_W^{(pa_i-m_i')/r}\) expresses its torsor factor as \(\tau^{-pa_i}\) times a section of \(B+pa_iK_W\). The frame factor is a nonzero constant on the thick Frobenius point, because it is a \(p\)-th power of a unit.

By projection formula, the resulting column map is \[ \begin{split} \bigoplus_{\substack{a_i\ge0\\\sum_i a_i=q}} \left(\bigotimes_{i=1}^t H^0(W,B+pa_iK_W)\right) \otimes\mathcal O_{(W')^t}\left(-\sum_{i=1}^t a_iK_{W',i}\right) \longrightarrow\mathcal E^{\boxtimes t}. \end{split} \tag{47}\] Here \(K_{W',i}\) is pulled back from slot \(i\). The spanning assertion proves that this map has rank \(R\) at one point, and hence at the generic point. The identity used is the line-bundle identity \(F^*\mathcal O_{W'}(K_{W'})=\mathcal O_W(pK_W)\); it is not the pullback map on differential forms.

Rank loss along the diagonal. On the diagonal \(W'\subset(W')^t\), every source line bundle in (47) restricts to \(\mathcal O_{W'}(-qK_{W'})\). Let \[\Delta_F=\underbrace{W\times_{W'}\cdots\times_{W'}W}_{t\text{ factors}}, \qquad h:\Delta_F\to W',\qquad g:\Delta_F\to W\] be the structure map and the first projection. Finite base change and projection formula identify \[h_*\mathcal O_{\Delta_F}\left(\sum_i\operatorname{pr}_i^*B +pq\operatorname{pr}_1^*K_W\right) = (\mathcal E^{\boxtimes t})|_{W'}\otimes\mathcal O_{W'}(qK_{W'}).\] After twisting, the columns restrict to global sections of this bundle, so their rank at every diagonal point is bounded by the dimension of that global section space.

Filter by the ideal of the reduced diagonal in \(\Delta_F\) and push forward by \(g\). The graded pieces are \[ \mathcal G_j=\mathcal O_W(tB+pqK_W) \otimes A_j((\Omega_W^1)^{\oplus(t-1)}), \quad 0\le j\le n(t-1)(p-1). \tag{48}\] Indeed, in etale coordinates the \(t-1\) difference slots have \(n(t-1)\) variables with \(p\)-th powers zero. Their degree-one transitions are those of the cotangent bundle. This is the many-factor form of the canonical Frobenius-diagonal filtration [39], [24]. If \(e_j\) is the rank of the truncated factor, then \(\sum_j e_j=p^{n(t-1)}\).

Apply Lemma 24 on \(C\) with \(\rho=n(t-1)\) and its constant \(c_2\), independent of \(p,j\). Put \(D=l_1\cdots l_{n-1}H^n\), so \(l_iH\cdot C=l_iD\). Using \(B\cdot C\le pbL\cdot C\), (43), and \(K_W\cdot C\ge0\), we obtain \[\begin{align*} \mu_{\max}(\mathcal G_j|_C) &\le \bigl(tB+pqK_W+(p-1)(t-1)K_W\bigr)\cdot C+c_2\\ &\le M_p:=2b_0pD+c_2. \end{align*}\] Lemma 25 now gives \[ \begin{split} h^0(W,\mathcal G_j) &\le e_j(M_p+1)\prod_{i=1}^{n-1}(1+M_p/(l_iD))\\ &=e_jp^n\bigl((2b_0)^nH^n+O(p^{-1})\bigr). \end{split} \tag{49}\] The error is uniform in \(j\): all the flag data and \(c_2\) were fixed before varying \(p\). The leading coefficient follows from \(D^n/\prod_i(l_iD)=H^n\). Summing the filtration bounds, and using (33), gives \[ \operatorname{rank}\bigl(\text{column map at a diagonal point}\bigr) \le p^{nt}\bigl((2b_0)^nH^n+O(p^{-1})\bigr) \le R/4 \tag{50}\] for sufficiently large \(p\).

We have obtained the two rank bounds. It remains to apply the characteristic-zero curve test to a nonzero maximal minor, after specialization, and compare its order with this diagonal rank loss.

The determinant contradiction. Choose \(R\) columns with nonzero determinant. If \(a_i^{(\nu)}\) is the slot-\(i\) weight in column \(\nu\), their determinant is a section of the external product of the line bundles \[\mathcal Q_i=(\det\mathcal E)^{\otimes R/s} \otimes\mathcal O_{W'}\left(\sum_{\nu=1}^R a_i^{(\nu)}K_{W'}\right).\] Set \(\lambda_i=R^{-1}\sum_\nu a_i^{(\nu)}\), so \(0\le\lambda_i\le q\). Identifying numerical classes under the base-field twist, \[ \frac{c_1(\mathcal Q_i)}R =\frac Bp+\left(\frac{p-1}{2p}+\lambda_i\right)K_W. \tag{51}\] For a direct verification, the two-factor version of (48) filters \(F^*\mathcal E\) by \(\mathcal O_W(B)\otimes A_j(\Omega_W^1)\). Among all \(p^n=s\) truncated monomials, the total exponent of each variable is \(s(p-1)/2\). Hence \[c_1(F^*\mathcal E)=sB+\frac{s(p-1)}2K_W.\] Frobenius pullback multiplies numerical divisor classes by \(p\), which proves (51) and agrees with [39].

The normalized class in (51) pairs with the specialized test \(\beta\) to at most \(D^+\cdot\beta\), because \(B/p\) is a nonnegative scalar multiple of \(L\) no larger than \(bL\), and the coefficient of \(K_W\) is at most \(q+1\). For any nonzero slot section \(s_i\) of \(\mathcal Q_i\), set \(m_i=\mathop{\mathrm{ord}}_{x'}s_i\). Its divisor has effective strict transform of class \(\sigma^*c_1(\mathcal Q_i)-m_iJ'\) on the twisted blow-up. Thus (45) gives \[m_i(J\cdot\beta) \le c_1(\mathcal Q_i)\cdot\beta \le R D^+\cdot\beta<Rd_0(J\cdot\beta),\] and \(m_i<Rd_0\). The determinant is nonzero, so all its slot section spaces are nonzero. Lemma 26 gives the strict upper bound for its order at \((x',\ldots,x')\): \[Rt d_0=Rb_0(C_1+1)<3R/4.\] On the other hand, (50) lets us make at least \(R-\lfloor R/4\rfloor\) rows of its matrix vanish at that point, by constant invertible row operations in local frames. Every determinant term then lies in the corresponding power of the maximal ideal, giving order at least \(3R/4\). The contradiction proves (34). ◻