Proof. Preserving the product directions. First suppose that the product has at least two positive dimensional factors. Denote these factors by \(B_1,\ldots,B_t\), counting the abelian factor, when present, as a single factor. Thus \(t\le4\), \(1\le\dim B_i\le3\), and at most one \(B_i\) is abelian. Take the normal Galois closure \(R\to X\) of the cover, with group \(G\). This is still quasi-étale: over the smooth locus of \(X\) the original cover is finite étale by purity, and its Galois closure is finite étale there as well. The map \(R\to P\) is also quasi-étale. In particular, \(R\) is canonical and \(K_R\sim0\). Here and below purity is used in its usual form for a finite generically étale map to a regular variety [38].

The product directions pull back, and extend reflexively, to a splitting \[ \mathcal T_R=\bigoplus_{i=1}^t\mathcal E_i. \tag{6}\] There are no homomorphisms between two distinct summands. To check this, choose a nonabelian factor among the corresponding two factors, say \(B_i\), and fix general smooth points on all the other factors. After shrinking the complementary parameter space, the morphism \(R\to\prod_{j\ne i}B_j\) has normal, pure-dimensional fibres. This follows from normality of the geometric generic fibre in characteristic zero, generic flatness, and openness of geometric normality. Each of its connected components is therefore normal and finite surjective over \(B_i\). The codimension-two locus excluded from the quasi-étale map \(R\to P\) meets a general slice in codimension at least two, so these component covers are quasi-étale.

On the smooth big open of such a component, \(\mathcal E_i\) restricts to its tangent sheaf and every complementary summand restricts to a trivial bundle. The component has no reflexive one-forms by the defining property of \(B_i\). It has no vector fields either: contraction with its generating volume form identifies its reflexive tangent sheaf with its sheaf of reflexive \((\dim B_i-1)\)-forms. A homomorphism in either direction between the two summands thus restricts to zero: the resulting forms or vector fields on the big open extend reflexively across its codimension-two complement. These slices sweep a dense open of \(R\), proving the assertion.

Consequently the projections in (6) are central idempotents in the finite-dimensional algebra \(\operatorname{End}_R(\mathcal T_R)\). Its center is a finite-dimensional commutative algebra. Its primitive idempotents determine nonzero direct summands of positive generic rank, whose ranks sum to four; there are at most four of them. The group \(G\) permutes these primitive idempotents. The kernel \(G_0\) of that permutation action has index at most \(4!\) and fixes every block projection, since each block projection is a sum of primitive central idempotents. It therefore preserves every \(\mathcal E_i\).

Recovering the factor fields. Let \(F_i\) be the relative algebraic closure of \(\mathbb C(B_i)\) in \(\mathbb C(R)\), and let \(R_i\to B_i\) be the normalization in this finite field extension. At the generic point, the complementary directions in (6) span \(\operatorname{Der}_{\mathbb C(B_i)}\mathbb C(R)\): this follows first on the product and then under the finite separable extension \(\mathbb C(P)\subset\mathbb C(R)\). In characteristic zero the elements annihilated by all these derivations are exactly those algebraic over \(\mathbb C(B_i)\). Thus \(G_0\) preserves \(F_i\), and acts by birational maps on \(R_i\).

The finite map \(R_i\to B_i\) is quasi-étale. For completeness, the regular extension obtained by adjoining the geometrically integral product of the other factors is linearly disjoint from the finite extension \(F_i/\mathbb C(B_i)\). Hence the function field of \(R_i\times\prod_{j\ne i}B_j\) is an intermediate field between \(\mathbb C(P)\) and \(\mathbb C(R)\). This product is normal, so the normalization property gives finite maps \[R\longrightarrow R_i\times\prod_{j\ne i}B_j \longrightarrow P.\] Ramification over a prime divisor of \(B_i\) would persist after taking this product. Its ramification index would then divide the ramification index at any prolongation to \(\mathbb C(R)\), contradicting quasi-étaleness of \(R\to P\). Thus \(R_i\) is projective and canonical with \(K_{R_i}\sim0\). Choose its generating volume \(\omega_i\) to be the pullback of a generating volume on \(B_i\).

Bounding the characters in lower dimension. The varieties \(R_i\) have dimension at most three. We now use their finite birational actions to bound the characters on their volume lines. Let \(J_i\) be the image of \(G_0\) in \(\mathop{\mathrm{Bir}}(R_i)\). Resolve the graph of its finite birational action to obtain a smooth projective \(J_i\)-variety \(W_i\) with a birational morphism to \(R_i\). One can first take the closure of the graph in the product indexed by the group, where the action permutes the factors, and then use functorial resolution [1]. Canonicality implies that the pullback of \(\omega_i\) is a regular top form on \(W_i\). For every positive integer \(m\), \[ H^0(W_i,mK_{W_i})=\mathbb C\,\omega_i^{\otimes m}. \tag{7}\] Indeed, any section on \(W_i\) gives a rational pluricanonical form on \(R_i\) with no pole at any prime divisor there. It extends reflexively, and \(\mathcal O_{R_i}(mK_{R_i})\simeq\mathcal O_{R_i}\) has only constant global sections. This also proves that the character in degree \(m\) is the \(m\)-th power of the character \(\chi_i\) on the volume line.

Form the finite quotient \(q_i:W_i\to Q_i=W_i/J_i\). Its branch boundary is \[B_{Q_i}=\sum_D(1-1/e_D)D,\] where \(e_D\) is the ramification order along \(D\); unramified primes have coefficient zero. The quotient ramification formula gives \[K_{W_i}=q_i^*(K_{Q_i}+B_{Q_i}).\] The total divisor on the right is \(\mathbb Q\)-Cartier and the pair \((Q_i,B_{Q_i})\) is klt. These are the finite quotient and discrepancy properties of a smooth variety with zero boundary [28]; \(\mathbb Q\)-Cartierness can also be seen by descending a sufficiently divisible local canonical power, or by taking a norm after trivializing near the finite fibre. Since \(J_i\) is finite, some power of \(\omega_i\) is invariant. Galois descent of rational forms, followed by the displayed ramification formula, gives \[\kappa(Q_i,K_{Q_i}+B_{Q_i})\ge0.\]

The coefficients of these boundaries lie in the fixed DCC set \(\{1-1/e:e\in\mathbb Z_{>0}\}\), and \(1\le\dim Q_i\le3\). Effective log Iitaka fibrations in these dimensions [9] therefore give a uniform integer \(m_*\), made common to the three possible dimensions, for which \[H^0\bigl(Q_i,\mathcal O_{Q_i}(\lfloor m_*(K_{Q_i}+B_{Q_i})\rfloor)\bigr) \ne0.\] A section here pulls back as a rational \(m_*\)-canonical form on \(W_i\). The ramification formula and coefficientwise effectiveness show that it has no divisorial poles, so it is a nonzero regular form. It is invariant because it was pulled back from \(Q_i\). Equation (7) now implies \(\chi_i^{m_*}=1\).

Combining the factor characters. The wedge of the rational pullbacks of the \(\omega_i\) to \(R\) is the pullback of the product volume on \(P\). It therefore generates \(K_R\), even though \(R\) need not itself split as a product. Its character on \(G_0\) is the product of the block characters, and is killed by \(m_*\). A further factor depending only on \([G:G_0]\le4!\) kills the character on all of \(G\). For example, one may multiply by the least common multiple of the integers from one to \(4!\). Equation (4) proves the desired uniform canonical multiple in this case.

The purely abelian case. If \(P=A\) is an abelian fourfold, take the same normal Galois closure \(R\to X\). The quasi-étale map \(R\to A\) is étale by purity. A connected finite étale cover of a complex abelian variety is again abelian: analytically it is obtained from a finite-index sublattice of the lattice in its universal cover, and it is projective here. Each element of \(G\) acts integrally on \(H^4(R,\mathbb Z)\), a free group of rank \(\binom84=70\). Its eigenvalue on the one-dimensional space \(H^{4,0}(R)\) is the canonical character. If that eigenvalue has order \(d\), its cyclotomic polynomial divides an integral characteristic polynomial of degree 70, and hence \(\varphi(d)\le70\). There are only finitely many such \(d\). Their least common multiple kills every value of the character, and (4) again gives a uniform trivial multiple of \(K_X\). Taking a common multiple for the two cases proves the proposition. ◻

Proof. Suppose that \(X\dashrightarrow T\) contradicts (i), and set \(j=\dim T\in\{1,2,3\}\). Non-uniruledness implies that \(K_T\) is pseudo-effective by [6]. The minimal-model and abundance theorems in dimension at most three imply \(\kappa(T,K_T)\ge0\). In dimension three, one takes a projective \(\mathbb Q\)-factorial terminal minimal model and applies abundance for minimal threefolds [26, 24]; canonical sections are unchanged on a resolution. The lower dimensional cases are the corresponding curve and surface statements, or follow by first taking a product with an abelian variety of dimension \(3-j\) and applying the same three-dimensional results.

Choose a nonzero section \(\eta\in H^0(T,mK_T)\). Resolve the composite rational map from \(P\) to \(T\), obtaining a smooth projective \(u:W\to P\) and a morphism \(f:W\to T\). Pullback by differentials gives a nonzero tensor \[ s=f^*\eta\in H^0\bigl(W,(\Omega_W^j)^{\otimes m}\bigr). \tag{8}\] It is nonzero because the map is dominant in characteristic zero. At the function-field level 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 \(T\).

We claim that the divisorial zero locus of \(s\) is \(u\)-exceptional. Choose an ample Cartier divisor \(H_P\) on \(P\), and put \(H=u^*H_P\). Since \(P\) is canonical with trivial canonical class, \(K_W\) is \(\mathbb Q\)-linearly equivalent to an effective exceptional divisor. Thus it is pseudo-effective and \(K_W\cdot H^3=0\). Generic semipositivity gives nonnegative degrees for torsion-free quotients of \(\Omega_W^1\) with respect to any ample polarization [7]. It gives the same conclusion for torsion-free quotients of \(\mathcal V=(\Omega_W^j)^{\otimes m}\).

Here is the tensor step explicitly. Use an ample rational perturbation \(H+\epsilon H_0\), where \(H_0\) is ample and \(\epsilon>0\). The Harder–Narasimhan factors of \(\Omega_W^1\) have nonnegative slopes. On a sufficiently high general complete-intersection curve the filtration restricts to a filtration by semistable bundles [31]; the curve can avoid all the codimension-two loci where the sheaves or quotients fail to be locally free. Tensor products of semistable bundles in characteristic zero are semistable, with the sum of the slopes [36]. Tensor powers consequently have filtrations whose slopes are nonnegative. Exterior powers are quotients of tensor powers, so the same lower slope bound applies to \(\mathcal V\) and to its torsion-free quotients. Letting \(\epsilon\) decrease to zero proves the required nonnegative degree against \(H^3\).

Saturate the rank-one subsheaf generated by \(s\) in \(\mathcal V\). It is a line bundle \(\mathcal O_W(Z)\), with \(Z\ge0\) the divisorial zero divisor of \(s\), and the quotient is torsion-free. The first Chern class of \(\mathcal V\) is a positive integral multiple of \(K_W\). It has degree zero against \(H^3\); the preceding quotient inequality therefore gives \[Z\cdot H^3\le0.\] Every nonexceptional prime divisor has strictly positive intersection with \(H^3\), by the projection formula and ampleness on \(P\). Since \(Z\) is effective, it follows that \(Z\) is exceptional, as claimed.

At the generic point of a prime divisor on \(P\), write the rational form \(\beta\) in a regular frame of \(\Omega_P^j\), and let \(b\in\mathbb Z\) be the minimum order of its coefficients. The absence of divisorial zeros of \(s\) at that point says \[\mathop{\mathrm{ord}}(a)+mb=0.\] Hence every coefficient of \(\mathop{\mathrm{Div}}_P(a)\) is divisible by \(m\). Normalize \(P\) in a field component of the extension obtained by adjoining \(z\) with \(z^m=a\). This cover is finite and quasi-étale: at any divisorial discrete valuation, removing the uniformizer to its \(m\)-divisible order reduces the equation to a root of a unit, which is unramified in characteristic zero. The rational form \(z\beta\) has no divisorial poles. It is therefore a nonzero reflexive \(j\)-form on the cover, and remains decomposable at its generic point.

This contradicts the defining form algebra of \(P\). For a four-dimensional irreducible Calabi–Yau factor there are no such forms in degrees one, two or three on any quasi-étale cover. For a four-dimensional irreducible symplectic factor the only possible degree is two, and every such form is a scalar multiple of the generically nondegenerate symplectic form. Its square is nonzero, whereas the square of a decomposable two-form is zero. This proves (i).

For (ii), resolve the rational image and take its maximal rationally connected fibration. Its smooth projective base is non-uniruled [17]. A positive dimensional such base would be a rational image of \(X\) of dimension less than four, contrary to (i). The base must be a point, which is exactly rational connectedness of the resolution.

Finally let \(W_X\) be a smooth projective model of \(X\). A regular one-form on \(W_X\) pulls back under the dominant generically finite rational map \(P\dashrightarrow W_X\) to a reflexive one-form on \(P\). Indeed, the rational map to the proper target is defined at all codimension-one points, and the pulled-back form is regular there; reflexive extension completes it. Pullback is injective at the function-field level in characteristic zero. The absence of one-forms on \(P\) forces the original form to vanish. Hodge decomposition and symmetry on the smooth projective \(W_X\) then give \(h^1(W_X,\mathcal O_{W_X})=h^0(W_X,\Omega^1_{W_X})=0\), proving (iii). ◻

Proof. Fix a very ample Cartier divisor \(L\) on \(X\), put \(A=u^*L\), and fix an ample Cartier divisor \(H\) on \(W\). Because \(A\) is big, there are an integer \(b>0\) and an effective Cartier divisor \(F\) such that \(bA\sim H+F\). If \(S\) is a nonexceptional prime divisor not contained in \(\mathop{\mathrm{Supp}}F\), then \(F\cdot S\) is an effective codimension-two cycle. Both \(bA\) and \(H\) are nef, and factoring their cubes gives \[\begin{split} (bA)^3\cdot S-H^3\cdot S &=F\cdot S\cdot\bigl((bA)^2+bA\,H+H^2\bigr)\ge0. \end{split}\] There are only finitely many nonexceptional primes in \(\mathop{\mathrm{Supp}}F\), and each has positive \(A^3\)-degree. Increasing a constant to handle these primes gives \[ H^3\cdot S\le C A^3\cdot S =C L^3\cdot u_*S \tag{9}\] for every nonexceptional prime \(S\).

Let \(E=\sum_i a_iE_i\) be exceptional and pseudo-effective. Choose effective real divisors \(D_n\) whose numerical classes converge to \([E]\). Write \(D_n=S_n+T_n\), where \(S_n\) has no exceptional components and \(T_n\) is supported on the finitely many exceptional prime divisors of \(u\). Projection gives \[A^3\cdot S_n=A^3\cdot D_n\longrightarrow A^3\cdot E=0.\] Equation (9) implies \(H^3\cdot S_n\to0\). Choose divisor classes \(Q_1,\ldots,Q_\rho\) forming a basis of \(N^1(W)_{\mathbb R}\), and choose \(c_j>0\) so that both \(c_jH+Q_j\) and \(c_jH-Q_j\) are ample. Effectivity gives \[|Q_j\cdot H^2\cdot S_n|\le c_j H^3\cdot S_n\longrightarrow0.\] The Hodge index theorem makes the form \((D,Q)\mapsto D\cdot Q\cdot H^2\) nondegenerate on \(N^1(W)_{\mathbb R}\). Thus \([S_n]\to0\), and consequently \([T_n]\to[E]\).

The numerical classes of the exceptional prime divisors are linearly independent. In fact, an exceptional numerically trivial linear combination and its negative are both relatively nef. The negativity lemma forces both to be nonpositive, hence forces the combination to vanish as a divisor [28]. Their positive cone is therefore closed and has unique coordinates in these classes. Since all \(T_n\) belong to that cone, so does \([E]\); independence then gives \(a_i\ge0\) for every \(i\). ◻

Proof. Choose a smooth projective resolution \(u:W\to X\) exhibiting a negative ordinary discrepancy. Its canonical divisor satisfies \[K_W=u^*K_X+E\sim_{\mathbb Q}E,\] where \(E\) is exceptional with a negative coefficient. By Lemma 11, \(K_W\) is not pseudo-effective. The uniruledness criterion of [6] implies that \(W\) is uniruled. Its maximal rationally connected quotient consequently has dimension less than four, and its base is non-uniruled [17]. If that base had positive dimension, the induced rational map from \(X\) would contradict Lemma 10(i). Thus \(W\) is rationally connected. Apply Proposition 8 directly to \((X,0)\), with zero nef data. Its hypotheses are precisely klt singularities, \(K_X\sim_{\mathbb Q}0\), and a rationally connected smooth resolution, all of which have been checked. Its uniform integer bounds the canonical order. ◻

Proof. On a fixed smooth resolution, the discrepancy divisor of \(X\) is effective. Any divisor exceptional over that resolution has positive ordinary discrepancy over the smooth resolution, and adding the pullback of an effective discrepancy divisor preserves positivity. Consequently only the finitely many exceptional divisors already on the fixed resolution can have ordinary discrepancy zero over \(X\).

The projective extraction theorem [2] extracts this finite list and gives a \(\mathbb Q\)-factorial variety. One may arrange its klt application as follows. Fix an integer \(r\) with \(rK_X\) Cartier, and choose an effective ample Cartier divisor \(H\) containing all the centers of the selected valuations. Since \(X\) is klt, choose \(c>0\) such that \((X,cH)\) is lc. Then, for every divisorial valuation \(E\), \[\mathop{\mathrm{ord}}_E(H)\le A_X(E)/c.\] Every exceptional log discrepancy greater than one is at least \(1+1/r\). Choosing a positive rational \(\delta\) with \(\delta/c<1/(r+1)\) keeps all those discrepancies greater than one, while the selected log discrepancies equal to one decrease strictly below one. Take \(\delta\) still smaller if necessary so that \((X,\delta H)\) is klt, and apply the extraction theorem to this pair and exactly the selected list. The list can be empty, in which case the theorem gives a small \(\mathbb Q\)-factorialization.

Restore boundary zero. The extracted divisors have ordinary discrepancy zero for \(X\), so the resulting morphism is crepant. All divisors still exceptional over the extracted variety have positive ordinary discrepancy; hence the variety is terminal. Finally, compatible canonical divisors satisfy \(K_V=u^*K_X\). A principal multiple downstairs pulls back to the same principal multiple upstairs, and a principal multiple upstairs pushes forward to the same one downstairs. This proves equality of their exact orders. ◻

Proof. By \(\mathbb Q\)-factoriality, \(D\) is \(\mathbb Q\)-Cartier. Choose a small positive rational number \(\delta\) for which \((V,\delta D)\) is klt. Its adjoint has Iitaka dimension \(\kappa(V,D)\), since \(K_V\sim_{\mathbb Q}0\). Apply Proposition 4, starting its birational program at \(V\), to obtain a \(\mathbb Q\)-factorial klt model \(V'\) and its semiample contraction \[f:V'\longrightarrow Z, \qquad 1\le\dim Z=\kappa(V,D)\le3.\]

We check the canonical divisor separately from the perturbed adjoint. Choose compatible canonical representatives and a rational function \(h\) with \(rK_V=\mathop{\mathrm{Div}}(h)\). Each birational contraction in the program pushes this same relation to the target, and each flip carries it to the new model by its codimension-one identification. All these models are \(\mathbb Q\)-factorial. On a common resolution their canonical pullbacks are therefore the same divisor \(r^{-1}\mathop{\mathrm{Div}}(h)\). In particular, \(K_{V'}\sim_{\mathbb Q}0\), the zero-boundary varieties are crepant, and their canonical orders are equal. They remain klt; alternatively, this follows from crepant equality starting at the terminal \(V\).

Now use Theorem 6 on the contraction \(f:(V',0)\to Z\), with zero boundary and base class zero. It gives uniform integers \(p_0\mid p\), a rational function \(\psi\), and generalized klt data satisfying \[ K_{V'}+\frac1{p_0}\mathop{\mathrm{Div}}(\psi) =f^*D_Z,\qquad D_Z=K_Z+B_Z+M_Z\sim_{\mathbb Q}0, \tag{10}\] where \(B_Z\ge0\) has coefficients in a fixed DCC set and the nef b-divisor has Cartier denominator dividing \(p\). The base \(Z\) is a rational image of \(V\), so its smooth projective resolution is rationally connected. Proposition 8 gives a uniform \(l\), which we enlarge to a multiple of \(p_0\), such that \(lD_Z\sim0\). Multiplying (10) by \(l\) then gives \(lK_{V'}\sim0\), hence \(lK_V\sim0\).

The coefficient \(\delta\) and the divisor \(D\) were used only to produce a semiample model. The two uniform statements were applied afterward with boundary exactly zero, in one of the three fixed base dimensions. Their integers thus depend on neither \(D\) nor \(\delta\). A common multiple for these dimensions is \(N_{\rm mov}\). ◻

Proof. The variety \(Y\) is terminal. In fact, an exceptional divisorial valuation over \(Y\) restricts to an exceptional valuation over \(V\), and the finite discrepancy formula multiplies its log discrepancy by its positive ramification index. The log discrepancy remains strictly greater than one. Also \(K_Y\sim0\).

Consider a \(G\)-equivariant rational fibration \(Y\dashrightarrow T\). Its base field \(\mathbb C(T)\) is relatively algebraically closed in \(\mathbb C(Y)\). The invariant field \(\mathbb C(T)^G\) lies in \(\mathbb C(Y)^G=\mathbb C(V)\), has the same positive transcendence degree as \(\mathbb C(T)\), and is relatively algebraically closed in \(\mathbb C(V)\). For the last assertion, an element of \(\mathbb C(V)\) algebraic over \(\mathbb C(T)^G\) is algebraic over \(\mathbb C(T)\), hence belongs to \(\mathbb C(T)\), and then is invariant. A projective model of the invariant field therefore gives a descended dominant rational map \(V\dashrightarrow Z\), where \(0<\dim Z<4\).

Fix smooth projective resolutions of the maps and the bases. Thus we may use a smooth projective birational morphism \(u:U\to Y\), a morphism \(f:U\to T'\) to a smooth model of the unquotiented base, and a morphism \(g:U\to Z'\) to a projective model of the quotient base. The map \(g\) is constant on the generic fibre of \(f\). Take a general hyperplane divisor \(H\) for a very ample embedding of \(Z'\). On the domain of \(V\dashrightarrow Z'\), pull it back and take its closure to obtain an effective Weil divisor \(D\) on \(V\). The domain contains every codimension-one point, because \(V\) is normal and \(Z'\) is proper. The hyperplane pulls form a moving system of linearly equivalent divisors: the rational-function ratios on the domain extend as the same principal-divisor relations on \(V\). Some ratio is nonconstant, since \(\dim Z'>0\). Thus \(\kappa(V,D)\ge1\), and \(D\) is big by hypothesis.

Write \(q=\pi\circ u\). We have \[ q^*D=g^*H+E, \tag{11}\] where \(E\) is an effective \(u\)-exceptional rational divisor. The pullback exists because \(V\) is \(\mathbb Q\)-factorial. At every nonexceptional generic divisorial point of \(U\), the two terms \(q^*D\) and \(g^*H\) agree: the finite cover takes that point to a codimension-one point of \(V\), where the original rational map is defined. Thus their difference is exceptional. After the resolution has been fixed, choose \(H\) general enough that it contains none of the generic images under \(g\) of its finitely many exceptional prime divisors. The coefficient of \(g^*H\) at each such prime is then zero, while \(q^*D\) is effective. This proves the claimed effectivity of \(E\).

Let \(F\) be the smooth geometric generic fibre of \(f\). The divisor \(q^*D\) is big on \(U\), and its restriction to \(F\) is big as well. Indeed, write a big rational class as the sum of an ample rational class and an effective rational class. The integral geometric generic fibre is not contained in the support of the effective term, and the ample term restricts amply. The class \(g^*H\) restricts trivially to \(F\), so (11) implies that \(E|_F\) is big.

Terminality and \(K_Y\sim0\) give \[K_U\sim_{\mathbb Q}\sum_i a_iE_i, \qquad a_i>0,\] where the sum runs over all \(u\)-exceptional prime divisors. Since \(E\) is effective and supported on this finite set, there is a positive rational number \(c\) such that \(\sum_i a_iE_i-cE\ge0\). Restriction to the geometric generic fibre is effective. Smooth-fibre adjunction identifies \(K_U|_F\) with \(K_F\), and therefore \(K_F\) is the sum, up to \(\mathbb Q\)-linear equivalence, of the big divisor \(cE|_F\) and an effective divisor. It is big, which proves that \(F\) is of general type.

For a rational fibration of \(V\), use the identical argument with \(Y=V\), \(\pi=\mathop{\mathrm{id}}\), and the trivial group. Terminality of \(V\) provides the same positive discrepancy comparison. ◻

Proof. Choose an algebraically closed field of definition \(k_*\) and a point \(p\) generic in \(V\) over \(k_*\). For a tuple \(a\) of points write \(K_a=\overline{k_*(a)}\) inside one sufficiently large algebraically closed extension. Fields for successive choices include the entire preceding path.

Every point encountered in a generic path is individually generic in its ambient space over \(k_*\). Indeed, over a generic mark of \(V\), all components of the geometric generic fibre of a dominant map \(S\to V\) have dimension \(\dim S-\dim V\). A parameter generic in such a component is therefore generic in \(S\). The endpoint is then generic in the incidence with that parameter, and hence in \(V\).

We will compare paths after enlarging \(K_p\). The closure of a finite path tuple over the algebraically closed field \(K_p\) is geometrically integral. A generic point of its base change realizes the same path after an algebraically closed field extension, with the same projected loci. For a fixed sequence of families, a prefix of length \(j\) has total transcendence degree \[\sum_{i=1}^j \bigl(\dim S_i-\dim V+\dim C_{b_i}\bigr)\] over its initial-point field: these are precisely the increments from the parameter and endpoint choices. Any tuple satisfying those incidences has at most this transcendence degree. A path generic after an algebraically closed field extension has the displayed degree there, and cannot have a smaller degree before the extension. Thus all prefix degrees agree with their upper bounds, proving the converse comparison as well. This observation also applies when a path is begun at a later endpoint and earlier path data are subsequently adjoined to its ground field.

Let \(d_j\) be the largest dimension of the closure over \(K_p\) of an endpoint after \(j\) generic movements; set \(d_0=0\). These loci can be obtained from components of the finite-step path spaces and their projections. Appending a step makes the new endpoint locus contain the old one. To see this, the new locus contains, after extension to the field of the preceding path and the new parameter, the whole integral member of that last step. It therefore contains the previous endpoint, and hence its closure over \(K_p\). Consequently \(d_j\) is nondecreasing, \(d_1>0\), and the first equality has the form \[d_h=d_{h+1}=e,\qquad 1\le h\le e\le\dim V.\] Choose an endpoint locus \(H_p\) of dimension \(e\) at length \(h\) and a corresponding generic endpoint \(y\).

Every possible generic next movement, made over the preceding path, has its member and endpoint in \(H_p\): its endpoint locus contains \(H_p\), and the equality of dimensions forces equality. By the field-extension observation this is also true for all generic choices over \(K_{p,y}\), without retaining the rest of the path. The new endpoint is again generic in \(H_p\) over \(K_p\). For each family and each geometric component of its parameter fibre over the generic point of \(H_p\), choose the new parameter independently of the previous path. The stabilized dimension calculation says that the endpoint image of this generic incidence is contained in \(H_p\). This is an identity on the incidence over \(H_p\), so it persists after each subsequent independent generic base change. Induction therefore keeps all later paths in \(H_p\), with endpoints generic there over \(K_p\). We use fresh generic realizations in this argument; a previously chosen endpoint need not remain generic after arbitrary old data are adjoined.

Now form endpoint loci of paths intrinsically over \(K_y\), and realize their generic choices also over \(K_{p,y}\). They are contained in \(H_p\). Because \(y\) is generic in \(V\) over \(k_*\), their maximum dimension at length \(h\) is the same number \(e\). Every maximizing locus therefore becomes \(H_p\) after extension. This proves uniqueness of the maximizing locus at \(y\). Transporting the statement between generic points of \(V\) proves uniqueness at a generic starting point and shows \[ H_y=H_p \quad\text{when }y\text{ is generic in }H_p\text{ over }K_p. \tag{12}\]

Uniqueness gives descent to \(k_*(p)\). More explicitly, choose a projective embedding over \(k_*\) and take the equation subspaces of the saturated homogeneous ideal of \(H_p\) through a degree generating that ideal. Every algebraic conjugate is again the unique maximizing locus. Its Grassmann coordinates are therefore invariant, and belong to \(k_*(p)\) in characteristic zero. They define a rational map \(\xi\) to the closure of their image, naming \(H_p\). We may replace the image by its normalization. Equality of names means equality of subvarieties.

For a next step from the point \(y\) just considered, both endpoints are generic in \(H_p\) over \(K_p\), so (12) gives equality of their names. Realizing each choice generically after field extension shows that this equality holds at a generic parameter and endpoint of each family. It is therefore a rational identity on its incidence. It follows that every generic movement from a generic point preserves \(\xi\), including movements that are not part of the original maximizing path.

For completeness, fix a generic name \(m=\xi(p)\) and work over \(K_m\). A generic point of any component of the geometric fibre of \(\xi\) is generic in \(V\) over \(k_*\). It belongs to the subvariety bearing its own name, so each such component is contained in the integral subvariety named \(m\). Conversely, a generic point of that subvariety, chosen also generically over \(K_p\), has name \(m\) by (12). Thus the closures of these fibres equal that integral subvariety. Characteristic zero gives geometric reducedness as well. The graph closure supplies a rational family with geometrically integral generic fibre; geometric integrality and constant dimension spread to an open of the base, by generic flatness and the constructibility of geometric fibre properties [19]. This proves the fibre assertions. Finally, a finite automorphism preserving the family list takes a maximizing locus to a maximizing locus. Uniqueness proves equivariance. The induced rational action on the name space can, if desired, be made regular by taking the closure of its graph in the product of the finitely many translates. This leaves the generic leaves unchanged. ◻

Proof. The lower rational name is unchanged along every projected step. Following a path that fills the upper leaf proves the first assertion. It is enough here that equality of the lower names holds at the generic point of each projected incidence: that rational identity persists when the step is realized over the additional upper path data.

For the second assertion, let \(p'\) be generic in \(V'\) and lift it to \(p\in V\). All lifts are conjugate, and equivariance gives the same image \[W_{p'}=\theta(H_p)\] for every lift. Because \(p\) is algebraic over \(p'\), their algebraic closures in the fixed overfield agree: \(K_p=K_{p'}\). The image equation subspaces descend to \(k_*(p')\) by conjugation invariance, just as in the proof of Lemma 16. They give a rational name for \(W_{p'}\). A point generic in \(W_{p'}\) over \(K_{p'}\) lifts to a point generic in \(H_p\) over \(K_p\), and hence has the same name as \(p'\). Conversely every ambient generic point belongs to the subvariety with its own name. The preceding fibre-of-the-name argument proves geometric integrality and identifies its fibre closure with \(W_{p'}\). ◻

Proof. Choose an algebraically closed field of definition \(k_*\) for the data. Let \(m\) be a generic point of the quotient base and put \(K_m=\overline{k_*(m)}\) inside the fixed algebraically closed overfield. The restriction of \(P\) to the geometric generic leaf over \(K_m\) is big: the leaf sweeps \(V\), so it meets the generically finite locus of the morphism defined by a basepoint-free multiple of \(P\). Take a smooth projective resolution \(J\) of this leaf over \(K_m\). The generic marked families and paths lift to \(J\) wherever the resolution is an isomorphism. A point generic in the leaf over \(K_m\) is generic in \(V\) over \(k_*\). For a generic mark \(p\) on this leaf, put \(K_p=\overline{k_*(p)}\). Its name \(m\) belongs to \(k_*(p)\). Lemma 16 supplies a path of length \(h\le e\) reaching a point generic in the whole leaf over \(K_p\). Every intermediate mark and endpoint is generic in \(J\) over \(K_m\): it is generic in \(V\) and has name \(m\), and consequently has transcendence degree \(e\) over \(K_m\). In particular all their evaluation maps dominate \(J\).

Suppose that, in a divisible degree \(k\), a nonzero section of \(kP\) has order at least an integer \(m_0>hkB\) at the variable generic mark \(p\) of \(J\). In a fixed \(K_m\)-basis of \(H^0(J,kP)\), the kernel of the generic jet-evaluation matrix lets us choose such a section \(\sigma(p)\) with coefficients rational in \(p\). Follow a path of length \(h\) that fills the leaf generically over \(K_p\).

Assume inductively that the same section has multiplicity at least \(m_j\) at the current mark. Let \(T\) be the closure over \(K_m\) of the path tuple through the parameter \(b\) chosen for the next step; the original point \(p\) varies in this closure. Let \(I\subset T\times J\) be the incidence obtained by adjoining the new endpoint. Both the graph of the current mark and \(I\) dominate \(J\). Over the generic point of \(T\), the generic fibre of \(I\) is the chosen member, lifted to \(J\).

Differentiate in parameter directions, keeping the variable on \(J\) fixed. Concretely, use derivations of \(K_m(T)\) over \(K_m\) on the coefficients of \(\sigma\) in the fixed section basis. A line-bundle frame from \(J\) is regular and nonvanishing at the generic point of \(J\), so it is unaffected by these derivations. At the graph of the moving mark, the multiplicity assumption is membership in the \(m_j\)th power of the graph ideal, after localizing over the generic parameter. Parameter differentiation drops its powers by at most one. This can also be tested after field extension to make the mark rational, using faithful flatness.

Put \[r_j=m_j-\lfloor kB\rfloor.\] Every iterated derivative of length less than \(r_j\) still has order strictly greater than \(kB\) at the moving mark. Each such derivative vanishes identically on the chosen member. In case (i), a nonzero restriction, pulled back to its smooth resolution, would violate the assumed order bound at the smooth mark. The resolutions used in this comparison agree near the mark and can otherwise be compared on a common resolution.

In case (ii), a section with order at least \(l\) at the ambient mark restricts to an element of the \(l\)th power of the maximal ideal of the local curve ring \(A\). On its normalized branches its order is at least \(l\,v_i(\mathfrak m_A)\). The identity \[\mathop{\mathrm{mult}}A=\sum_i v_i(\mathfrak m_A)\] then shows that a nonzero restriction has degree at least \(l\mathop{\mathrm{mult}}A\). One way to verify this identity is to use the finite semilocal normalization \(\bar A\). Its quotient by \(\mathfrak m_A^t\bar A\) has length \(t\sum_i v_i(\mathfrak m_A)\) over the algebraically closed residue field. The conductor gives a fixed \(c\) such that \[\mathfrak m_A^{t+c}\bar A \subset \mathfrak m_A^t A \subset \mathfrak m_A^t\bar A.\] Comparing leading terms in the Hilbert–Samuel lengths gives the identity. Here \(l>kB\), whereas the degree of a nonzero restricted section is \(kP\cdot C_b\le kB\mathop{\mathrm{mult}}A\), a contradiction.

Thus all parameter derivatives of length less than \(r_j\) vanish at the generic point of \(I\). We explain why this forces order at least \(r_j\) along that incidence. Let \[R=\mathcal O_{T\times J,\eta_I},\qquad F=K_m(J).\] Since \(I\) dominates both factors, this ring localizes over their generic points. It is regular, using a smooth open of \(T\). The conormal map \[\mathfrak m_R/\mathfrak m_R^2 \longrightarrow \Omega_{R/F}\otimes\kappa(I)\] is injective. Indeed the residue extension is separable in characteristic zero; in the cotangent sequence the dimensions of the ambient differentials and residue differentials differ by \(\dim R\), which is the conormal dimension. Equivalently, this is the generic smoothness computation for the incidence over \(J\). Parameter derivations span the dual of this conormal after reduction, since \(R\) is a localization of \(K_m(T)\otimes_{K_m}F\). Regularity identifies the associated graded ring with the symmetric algebra of the conormal. A nonzero initial form of degree \(a<r_j\) is detected by a length-\(a\) iterated parameter derivative, by the symmetric-power pairing and the invertibility of \(a!\). The derivative vanishings therefore imply that our section, in the chosen frame, lies in \(\mathfrak m_R^{r_j}\).

Pass to the generic parameter field and then extend it so that the independently generic endpoint is rational. The map from the ambient product to this fibre induces a local homomorphism from \(R\) to the local ring of \(J\) at that endpoint. A denominator outside the incidence prime has nonzero value at its generic point and stays a unit after this extension. The incidence ideal maps into the endpoint maximal ideal; its \(r_j\)th power therefore maps into that maximal ideal’s \(r_j\)th power. The same section consequently has multiplicity at least \[m_{j+1}=r_j=m_j-\lfloor kB\rfloor\] at the next endpoint. After \(h\) steps it still has positive multiplicity, since \(m_0>hkB\). But the endpoint is generic in \(J\) over \(K_p\), while \(\sigma(p)\) is a nonzero section defined over \(K_p\). This is impossible. Hence \(\gamma(P;H_p)\le hB\le eB\). Applying (13) on the leaf gives the second inequality of (14). ◻

Proof. The arbitrary-boundary dlt blow-up theorem [15] gives a projective birational morphism \(p:Q\to Z\), with \(Q\) \(\mathbb Q\)-factorial, such that \[ K_Q+B_Q+E=p^*(K_Z+\Theta), \qquad (Q,B_Q)\text{ dlt},\qquad E\ge0, \tag{17}\] and \(\mathop{\mathrm{Supp}}E\) lies above the non-lc locus of \((Z,\Theta)\). The theorem allows coefficients of \(\Theta\) greater than one: \(B_Q\) is the truncated strict transform together with the reduced exceptional divisor, and the remaining crepant boundary is the effective excess \(E\).

Choose a dlt stratum \(R\) minimal by inclusion among the strata mapping onto \(S\). Such a stratum exists. Indeed, over the lc neighbourhood of the generic point of \(S\), the morphism in (17) is crepant, and the centre on \(Q\) of an lc place with centre \(S\) is a dlt stratum mapping onto \(S\). Dlt adjunction, in the form of the lc-centre and different theorem [27], gives a normal \(R\) and an effective different. Adding the restriction of \(E\) gives an effective rational divisor \(\Delta_R\) with \[K_R+\Delta_R\sim_{\mathbb Q}(K_Z+\Theta)|_R.\] The restriction of \(E\) is defined and effective: \(E\) is rational Cartier and misses the generic fibre of \(R\to S\). Moreover \((R,\Delta_R)\) is klt over the generic point of \(S\). Any lc centre of the dlt different dominating \(S\) would give a smaller dlt stratum of \(Q\) dominating \(S\), contrary to the choice of \(R\); the added excess is zero over that generic point.

Let \(S^\nu\) be the normalization of \(S\) and factor the map through its normal Stein base: \[R\xrightarrow{f}S'\xrightarrow{h}S^\nu\longrightarrow S, \qquad f_*\mathcal O_R=\mathcal O_{S'},\qquad h\text{ finite}.\] Write \(D\) for the pullback of \((K_Z+\Theta)|_S\) to \(S^\nu\). The morphism \(f:(R,\Delta_R)\to S'\) satisfies the definition of a klt-trivial fibration in [14]. It is projective, generically subklt, and its adjoint is rationally linearly equivalent to \(f^*h^*D\). The rank-one condition also holds: on a resolution over the generic base point, the rounded discrepancy divisor is effective and exceptional over \(R\). Allowing poles along such a divisor does not enlarge regular functions on the normal variety \(R\), and connectedness of the fibres then gives rank one.

Ambro’s canonical bundle formula, in the form recalled in [14], therefore gives \[ h^*D\sim_{\mathbb Q}K_{S'}+B_{S'}+M_{S'}, \tag{18}\] where \(M_{S'}\) is the trace of b-nef rational data. The discriminant \(B_{S'}\) is effective. To see this even where the total pair is not lc, let \(P\) be a prime of \(S'\). Over its generic point, \(f^*P\) has a divisor of multiplicity at least one and \(\Delta_R\) has nonnegative coefficient on that divisor. The corresponding threshold is therefore at most one. It can be negative, but its discriminant coefficient, one minus the threshold, remains nonnegative.

Let \(A'\) be the pullback to \(S'\) of \(A|_{S^\nu}\). Intersect (18) with \((A')^{d-1}\). Both the effective discriminant and the b-nef moduli trace contribute nonnegatively. For the latter assertion, compute on a model where the moduli divisor is nef and apply the projection formula. If \(e=\deg h\), codimension-one ramification for the finite map of normal varieties gives \[K_{S'}\cdot(A')^{d-1} \ge e\,K_{S^\nu}\cdot(A|_{S^\nu})^{d-1}.\] These are intersections of Weil divisors with Cartier powers, so the calculation can be made on the smooth codimension-one loci. Finally, the pushforward of \(K_{S^*}\) is \(K_{S^\nu}\), and its exceptional divisors pair to zero with the pulled-back power of \(A\). Dividing by \(e\) proves (16). ◻

Proof. The moving-parameter argument follows the jet-base-locus method of [12]. We keep the subadjunction and degree estimates explicit for the rational polarizations used here. Choose a sufficiently divisible positive integer \(l\), clearing the denominators of \(N\) and of the levels. It can be chosen arbitrarily large so that every jet space under consideration is nonzero: first choose a section witnessing an order above the highest level, and then take powers. At a variable very general smooth point \(x\), write \[V_\tau(x)= H^0\bigl(Z,\mathcal O_Z(lN)\otimes\mathfrak m_x^{l\tau}\bigr).\] These spaces are the fibres of generic jet kernels, and thus have rationally varying bases after restricting the parameter space.

At each level the base locus through \(x\) is proper and has no isolated component at \(x\). For the second assertion, if its local base ideal were primary to \(\mathfrak m_x\), then \(n\) general members would have local intersection multiplicity at least \((l\tau)^n\). Successive proper cuts, discarding other base components before each cut, would give \[l^nN^n\ge(l\tau)^n,\] contrary to \(\tau>(N^n)^{1/n}\). This degree argument is also detailed below for the positive-dimensional component that we use.

The base loci are nested as the level increases. Choose successively an irreducible component through \(x\), each contained in the next. Their dimensions are in \(\{1,\ldots,n-1\}\), so two consecutive components agree. Denote their common reduced component by \(S\), and spread it over a parameter space containing the varying mark \(x\). A finite extension of the parameter field may be used to name a geometric component. The resulting incidence dominates \(Z\) because it contains the graph of the mark.

Let \(\tau\) and \(\tau+\delta\) be the two levels. We claim that the base ideal of \(V_{\tau+\delta}(x)\) has order at least \(l\delta/2\) at the generic point of \(S\). Differentiate a rationally varying basis of this jet space in parameter directions, keeping the spatial variable and a fixed basis of \(H^0(Z,lN)\) constant. A derivative of length \(r<l\delta\) has order at least \(l(\tau+\delta)-r>l\tau\) at the varying mark, so it belongs to the lower jet space and vanishes along the incidence of \(S\).

Here these derivative vanishings detect the order of the original section along the incidence. At its generic point, the ambient product is regular and both projections are dominant. Since the incidence dominates the spatial factor in characteristic zero, parameter derivations span the dual of its conormal space. The associated graded ring of the regular local ring is the symmetric algebra of that conormal space. A nonzero initial form of degree \(r\) is detected by some iterated parameter derivative of length \(r\), by the characteristic-zero symmetric-power pairing. Thus all the vanishings just obtained force order at least \(l\delta\), and in particular the asserted weaker bound. This is the same local Taylor argument used in Lemma 19. Passing to the geometric generic parameter fibre preserves it.

Put \(a=n-d\). The local ring of \(Z\) at the generic point of \(S\) is regular of dimension \(a\): a sweeping member meets the smooth open of \(Z\). Since \(S\) is a component of the higher base locus, its base ideal there is primary to the maximal ideal \(\mathfrak m\), and it is contained in \(\mathfrak m^{\lceil l\delta/2\rceil}\). General choices of \(a\) members form a parameter ideal. Their intersection length, equal to its multiplicity in this regular local ring, is at least \((l\delta/2)^a\). Globally, take the cuts successively and discard components already contained in the base locus before the next cut. No discarded component contains the generic point of \(S\), since \(S\) is itself an irreducible component of that base locus. The cuts are therefore proper at the generic point being tracked. Nefness of \(N\) makes all discarded degree contributions nonnegative, and gives \[(l\delta/2)^a(N^d\cdot S)\le l^aN^n.\] This proves (19), as well as the isolated base-point calculation used above.

Let \(t\) be the local lc threshold of this base ideal over the generic point of \(S\). A log resolution shows that \(t\) is positive and rational. Blowing up the smooth transverse centre, whose log discrepancy is \(a\), gives \[t\le\frac{2a}{l\delta}.\] For \(a=1\) the same inequality follows by testing \(S\) itself. Average sufficiently many general members of the higher jet space and multiply this average by \(t\). On a log resolution of the ideal the free parts are general and their coefficients can be made arbitrarily small. This constructs an effective rational divisor \[\Theta\sim_{\mathbb Q}bN, \qquad b=lt\le\frac{2a}{\delta},\] such that \((Z,\Theta)\) is lc near the generic point of \(S\) and \(S\) is an lc centre there. Indeed a place computing the threshold has positive base-ideal order; the primary condition makes its centre equal to \(S\) after localization. Lemma 21 with \(A=N\) now gives (20), since \(a\le n\). ◻

Proof. Pass to a smooth projective model of \(X\) and to strict transforms of the moving members. Shrink the parameter space so that the incidence has fibres of pure constant dimension. If the incidence has dimension larger than \(\dim X\), intersect the parameter space in a projective closure with general hyperplanes, in number equal to that excess. Over the generic point of \(X\) the incidence fibre embeds into the parameter space, so a general such section meets its top-dimensional part and makes the evaluation generically finite. A component of the cut incidence dominating \(X\) dominates a component of the parameter slice, by dimension.

Choose the cutting hyperplanes with independent generic coefficients over the original field of definition. A generic point of the selected parameter slice remains generic in the original parameter space over that field: every original proper closed subset has smaller dimension after the same general cuts. Thus the argument tests the original geometric generic member, rather than a special subfamily.

Resolve and compactify the cut incidence and its evaluation to \(X\). The resulting generically finite morphism between smooth projective varieties satisfies \[K_I=q^*K_X+R,\qquad R\ge0.\] Restrict to a smooth geometric generic fibre of the parameter map. It is birational to the corresponding member, and adjunction identifies the restriction of \(K_I\) with its canonical divisor. A nonzero pluricanonical section of \(X\) pulls back to a section not identically zero on this generic fibre, by dominance. This proves nonnegative Kodaira dimension. If \(K_X\) is big, restrict a rational decomposition of a multiple into an ample class and an effective divisor. The ample pullback is big on the generic parameter fibre, since evaluation is generically finite on its image. Its canonical class is then big. ◻

Proof. Suppress the sequence index. We first bound the volume of every fixed positive adjoint combination on \(W\), then compare it with the volume on an Iitaka fibre. For every fixed positive rational number \(a\), we claim that \[ \mathop{\mathrm{vol}}(aK_W+L)\longrightarrow0. \tag{22}\] Choose a general divided member of a sufficiently divisible free multiple of \(L/a\). It gives an effective klt boundary \(\Gamma\sim_{\mathbb Q}L/a\). This boundary is big, and so is \(K_W+\Gamma\), since \(\kappa(W)\ge0\). The klt log-general-type model and finite-generation theorem [2] gives a \(\mathbb Q\)-factorial klt log terminal model. Its nef big adjoint is semiample by the klt basepoint-free theorem [28]: apply it to a Cartier multiple of the adjoint, whose sufficiently large multiple minus the adjoint is again nef and big.

On a common smooth resolution, let \(P_a\) be \(a\) times the pullback of this semiample adjoint. The negative-map comparison, using the negativity lemma [28], gives \[aK_W+L\sim_{\mathbb Q}P_a+F_a, \qquad F_a\ge0\] after pullback, with \(F_a\) exceptional over the model. Sections in divisible degrees are unchanged by this exceptional part, so \(P_a^d=\mathop{\mathrm{vol}}(aK_W+L)\). All occurrences of \(L\) on the resolution mean its nef pullback. The Khovanskii–Teissier inequality gives \[(aK_W+L)\cdot L^{d-1} \ge P_a\cdot L^{d-1} \ge (P_a^d)^{1/d}(L^d)^{(d-1)/d}.\] For completeness, the nef mixed-intersection inequalities used here follow by ample perturbation and passage to the limit from the surface Hodge index theorem, applied on general complete-intersection surfaces. Combining with (21), we obtain the explicit bound \[ \mathop{\mathrm{vol}}(aK_W+L)\le (aC'+1)^dL^d, \tag{23}\] which proves the claim.

Pass to a subsequence on which \(j=\kappa(W)\) is fixed. Since \(d\le3\), effective Iitaka fibrations in the zero-boundary case give an integer \(m_0=m_0(d)>0\) such that \(|m_0K_W|\) gives the Iitaka fibration; see [9]. The case \(j=d\) is impossible for large sequence index. Resolving that linear system would produce an integral free moving part with generically finite map and top self-intersection at least one. Thus \(\mathop{\mathrm{vol}}(K_W)\ge m_0^{-d}\), whereas \(\mathop{\mathrm{vol}}(K_W+L)\to0\) by (22) and \(L\) has effective multiples.

If \(0<j<d\), resolve the Iitaka system, its Stein base, and the induced map to obtain a morphism of smooth projective varieties \[f:W\longrightarrow T\] with connected fibres. There is an integral big basepoint-free divisor \(H\) on \(T\), pulled back from the hyperplane class of the system’s image, such that \[ f^*H\preceq m_0K_W, \qquad H^j\ge1. \tag{24}\] We continue to write \(W\) for the resulting smooth model and \(L\) for its pullback. The previous intersection estimates persist: canonical divisors under further smooth resolutions acquire effective exceptional terms, and these pair to zero against \(L^{d-1}\).

Let \(U\) be a smooth integral geometric generic fibre of \(f\) and put \(g=d-j>0\). Then \(K_W|_U\sim K_U\), and \(L|_U\) is big because these fibres move. We recall explicitly that \(\kappa(U)=0\). A nonzero canonical multiple on \(W\) restricts nontrivially to the generic fibre, so \(\kappa(U)\ge0\). If a system \(|kK_U|\) had positive-dimensional image, its sections would extend over the generic base after twisting \(kK_W\) by \(vf^*H\) for a sufficiently large \(v\). Indeed the coherent direct image of \(kK_W\) has the required generic fibre of sections, and a sufficiently large multiple of the big divisor \(H\) makes it generically generated, using a decomposition into ample and effective parts. By (24), the twist is absorbed into a further canonical multiple. The resulting section ratio is nonconstant on the geometric generic fibre and hence transcendental over \(\mathbb C(T)\). Products with sections of \(|m_0K_W|\) also preserve the \(j\) independent base ratios. This would force \(\kappa(W)>j\), a contradiction. When \(j=0\), simply put \(U=W\) and \(g=d\).

We next show that \[ \mathop{\mathrm{vol}}(K_U+L|_U)\longrightarrow0. \tag{25}\] For \(j=0\) this is (22). If \(j>0\) and \(g=1\), then \(\kappa(U)=0\) implies \(\deg K_U=0\), and the volume in question equals \(\deg(L|_U)\). Set \(A=f^*H\). Nef mixed-intersection log-concavity and (24) give \[A^jL^{d-j} \le\left(\frac{AL^{d-1}}{L^d}\right)^jL^d \le (m_0C')^jL^d\longrightarrow0.\] One may add a positive multiple of \(L\) to \(A\) and pass to the limit if a mixed intersection vanishes. The projection formula and \(H^j\ge1\) show that this bounds \(\deg(L|_U)\) from above.

The only remaining case is \(d=3\), \(j=1\), and \(g=2\). Now \(T\) is a smooth curve and \(H\) is ample. A basepoint-free pencil in \(|H|\) and its ramification divisor give \(K_T+2H\succeq0\). Consequently \[ K_{W/T}+L+2f^*H\preceq(1+4m_0)K_W+L. \tag{26}\] Choose an effective klt rational boundary linearly equivalent to \(L\) from its semiample system. The twisted weak positivity theorem [16] applies to \(f\) with this boundary. Thus for sufficiently divisible \(l\) the locally free sheaf \[E_l=f_*\mathcal O_W\bigl(l(K_{W/T}+L)\bigr)\] is weakly positive. Here smoothness, projectivity, connected fibres, and effectivity of the klt boundary are all satisfied; on the smooth curve a torsion-free direct image is locally free.

Choose \(l\ge1\), separately for each member of the sequence, so that the \(l\)-th Veronese of the section algebra of \(K_U+L|_U\) is generated in degree one. Such an \(l\) exists by the same big klt adjoint model and finite-generation theorem used above. Enlarge it to clear the chosen rational linear equivalence. Weak positivity gives \(b>0\) for which \[\mathop{\mathrm{Sym}}^bE_l\otimes\mathcal O_T(bH)\] is generated by global sections at the generic point. For every positive multiple \(q\) of \(b\), symmetric multiplication and the degree-one generation on the generic fibre therefore give global sections of \[ql(K_{W/T}+L)+qf^*H\] spanning the degree-\(ql\) fibre sections. Select a basis over \(\mathbb C(T)\) among their restrictions, of cardinality \(h^0(U,ql(K_U+L|_U))\). Multiplying by a basis of \(H^0(T,qlH)\) gives linearly independent products over \(\mathbb C\): any relation can first be tested using independence over \(\mathbb C(T)\) and then independence of the base sections. Multiplication by a nonzero section of \((ql-q)H\) places these products in degree \(ql\) of \(K_{W/T}+L+2f^*H\).

Curve Riemann–Roch and the asymptotic fibre section formula, with \(l\) fixed and \(q\to\infty\) through these multiples, yield \[\begin{align*} \mathop{\mathrm{vol}}\bigl((1+4m_0)K_W+L\bigr) &\ge\mathop{\mathrm{vol}}(K_{W/T}+L+2f^*H)\\ &\ge\frac{d!}{g!}\deg(H)\mathop{\mathrm{vol}}(K_U+L|_U). \tag{27}\end{align*}\] The first expression tends to zero by (22), and \(\deg H\ge1\). This proves (25) in every case. ◻

Proof of the upper bound in Theorem 20. If no uniform \(C\) exists, rational rescaling gives a sequence of the data in the theorem, with polarizations denoted \(M\), such that \[ M^4\longrightarrow0, \qquad \gamma(M;V)>1. \tag{28}\] The proof decreases the dimension while retaining small volume and an effective canonical upper bound. At each stage we first extract a moving base component, then pass to a Kodaira-dimension-zero fibre of its Iitaka fibration, and finally replace a big adjoint by its semiample model. The process stops when its section-order invariant tends to zero. Tracing the effective divisor comparisons back will construct covering families of positive-dimensional proper subvarieties \(A\subset V\) whose geometric generic members satisfy \[ \kappa(A^*)=0, \qquad \gamma(M;A)\longrightarrow0. \tag{29}\]

All constructions can be made successively on geometric generic members. To be precise, choose countable fields of definition for the sequence. A geometric generic datum over such a field can be realized over \(\mathbb C\) to apply the model and positivity theorems used here. The resulting finite collection of varieties, morphisms, and divisors descends again to a field finitely generated over the preceding generic parameter field. Spreading it out therefore gives a dominant extension of the preceding parameter space: the old generic parameter is never specialized. Formation of section spaces and of their jet kernels commutes with field extension, so the generic all-degree order comparisons are preserved. There are only finitely many stages for each variety, and countably many varieties in the sequence.

Begin with a small projective \(\mathbb Q\)-factorialization \(Z\to V\), which exists for the klt variety \(V\) by [2], and let \(N\) be the pullback of \(M\) to \(Z\). We describe a stage of the construction. After passing to a subsequence, its dimension \(n\) is fixed and the following properties hold:

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

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

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

4. \(\gamma(N;Z)\) is bounded below by a positive constant.

Initially these assertions follow from (28), canonicality, and crepantness of the small morphism. If \(n=1\), they cannot all hold, since the order of a section on a smooth curve is bounded by its degree. We may therefore assume \(n\ge2\).

Choose \(n+1\) fixed rational jet levels, equally spaced by \(\delta>0\), below the lower bound in (iv). For all sufficiently large sequence indices they are above \((N^n)^{1/n}\). Apply Lemma 22. Write \(W=S^*\) for a resolution of its geometric generic moving member and \(L=N|_W\), and pass to a subsequence with fixed \(d=\dim W<n\). The divisor \(L\) is nef and semiample, and is big because a sweeping member meets the open set on which the semiample map of \(N\) is generically finite. Lemma 23 gives \(\kappa(W)\ge0\). Restricting the effective comparison in (iii) to the moving member and using the tracking bounds yields \[ L^d\longrightarrow0, \qquad K_W\cdot L^{d-1}\le C' L^d \tag{30}\] with the fixed constant \(C'=a_0+2n/\delta\) at this stage.

Apply Lemma 24 to these moving members. After passing to a subsequence and resolving their Iitaka systems, let \(U\) be the smooth geometric generic Iitaka fibre, with \(U=W\) when \(\kappa(W)=0\), and put \(g=\dim U\). Then \[0<g\le d<n,\qquad \kappa(U)=0,\qquad \mathop{\mathrm{vol}}(K_U+L|_U)\longrightarrow0.\] The restricted divisor \(L|_U\) is big, nef, and semiample. The generic field and spreading convention above keeps these fibres in covering families over the preceding parameter space.

To continue the dimension descent, we need a semiample polarization that still bounds the original restricted sections from above. Take an effective klt boundary \(\Gamma\sim_{\mathbb Q}L|_U\) from the free system, and take a big klt good model of \(K_U+\Gamma\), as in the proof of Lemma 24. Write \(Z'\) for its \(\mathbb Q\)-factorial klt underlying variety and \(N'\) for its big semiample adjoint. Then \[(N')^g=\mathop{\mathrm{vol}}(K_U+L|_U)\longrightarrow0.\] On a smooth common resolution with maps \(a\) to \(U\) and \(b\) to \(Z'\), the negative-map comparison reads \[ a^*(K_U+\Gamma)=b^*N'+F, \qquad F\ge0\text{ exceptional over }Z'. \tag{31}\] For any effective \(\Delta_U\sim_{\mathbb Q}K_U\), we have \[ F\le a^*\Delta_U. \tag{32}\] Indeed, for sufficiently divisible \(m\), every effective divisor in \(|mb^*N'+mF|\) contains \(mF\). Its pushforward is an effective divisor in \(|mN'|\) on the normal variety \(Z'\), and pulling that Cartier divisor back leaves exactly the exceptional summand \(mF\). But \(ma^*\Delta_U+Y\) belongs to this system when \(Y\) is a general member of the free system \(|ma^*(L|_U)|\). Choose \(Y\) avoiding every component of \(F\). Comparing coefficients proves (32). In particular, \[ a^*(L|_U)\preceq b^*N'. \tag{33}\]

If \(\gamma(N';Z')\to0\) on a subsequence, we stop. Otherwise, on a subsequence it is bounded below by a positive constant, and we repeat with \((Z',N')\). The four stage conditions hold: \(K_{Z'}\preceq N'\) because the pushed-forward boundary is effective, and a smooth model of \(Z'\) is birational to \(U\), which has Kodaira dimension zero. The dimension has decreased strictly, from \(n\) to \(g\le d<n\). The construction must stop, since on a curve maximum normalized section order is at most the degree.

We verify that the stopping condition proves (29) for the original polarization. Every moving member and fibre used above spreads to a covering family over the preceding generic parameter space. Its map to its image in that space, and ultimately in \(V\), is birational: a moving member meets the open set of isomorphism of each relevant birational model map. For a subsequent member in \(Z'\), take its strict transform on the common resolution, so it has an actual morphism back to \(U\). The effective comparisons (33) restrict to these members, since they are not contained in the extra divisors. Their successive restrictions compare the pullback of the original \(M\) with the final \(N'\) on a common resolution. Multiplication by the effective differences injects section spaces in divisible degrees, and at a generic marked point those differences do not vanish. Birational pullback between normal projective models preserves the section spaces. Thus the final image \(A\subset V\) satisfies \[\gamma(M;A)\le\gamma(N';Z')\longrightarrow0, \qquad \kappa(A^*)=\kappa(U)=0.\] Composing the dominant incidence evaluations proves that these images form a covering family. They have positive dimension and are proper because the first tracking step already decreased dimension.

Mark a generic smooth point on these members and add every \(G\)-translate of the marked family. All data can be defined over one countable field; invariance of \(M\) preserves the order bound. Lemma 16 gives a \(G\)-equivariant rational quotient, and Lemma 19 bounds the section order on its geometric generic leaf by its dimension times \(B\), where \(B\to0\). If the leaf has dimension four, this gives \(\gamma(M;V)\le4B\), contradicting (28). Otherwise it gives a \(G\)-equivariant rational fibration with positive-dimensional base and fibre. The geometric generic leaf is of general type by hypothesis.

That leaf is nevertheless covered by members of Kodaira dimension zero. To see that the assertion is about generic members, restrict the marked incidence to its generic leaf name. A component still dominates the leaf by marked-point evaluation. Its generic parameter is generic in the original parameter space over the original field, because it is obtained by geometric extension over the generic name. The quotient is generically constant on each such member, and its strict transforms give a covering family on a resolution of the leaf. This contradicts Lemma 23. The upper bound follows. ◻

Proof of the lower bound in Theorem 20. Rescale rationally so that \(1\le P^4\le2\). If no uniform positive lower bound exists, there is a sequence of marked covering families of integral curves with \[\frac{P\cdot D}{\mathop{\mathrm{mult}}_xD}\le B, \qquad B\longrightarrow0.\] Here the marks can be chosen generic over countable fields of definition. Indeed, failure of a very general bound still gives a bad mark after excluding the countably many proper closed subsets over such a field. Choose a curve witnessing a strict upper bound there and spread the curve and its mark over their field of definition. The mark evaluation is dominant, and degree and multiplicity agree with those of the geometric generic data. Include all \(G\)-translates if necessary; they obey the same ratio bound.

Take the rational quotient from Lemma 16. After passing to a subsequence with fixed leaf dimension \(h>0\), Lemma 19 gives, on its geometric generic leaf \(H\), \[ P^h\cdot H\le(hB)^h. \tag{34}\] For large sequence index it cannot have dimension four, since \(P^4\ge1\). Put \(s=4-h>0\). Resolve the leaf fibration to a morphism \(f:X\to T\) between smooth projective varieties with geometrically integral generic fibre, and let \(\pi:X\to V\) be the resulting birational morphism. Continue to write \(P\) for \(\pi^*P\).

Choose a smooth integral very general fibre \(F\) of \(f\). It can be chosen not contained in any of the countably many proper closed sets excluded by the all-degree upper order bound, nor in the exceptional locus of \(\pi\). For clarity, over a smooth open of the base, each proper closed set of \(X\) contains the whole fibre only over a proper closed subset of \(T\), by properness and upper semicontinuity of fibre dimension. Exclude these countably many subsets first, then choose a point \(x\in F\) outside their intersections with \(F\). Thus \(\pi\) is an isomorphism near \(x\) and \[ \gamma(P;V)\le C2^{1/4} \tag{35}\] holds at \(\pi(x)\) in every divisible degree. The degree \(P^h\cdot F\) equals the geometric generic degree in (34).

Let \(I=I_F\subset\mathcal O_X\). Since \(F\) is a smooth fibre over a smooth point of the \(s\)-dimensional base, it is regularly embedded and has trivial conormal bundle of rank \(s\). Consequently \[I^k/I^{k+1}\simeq \mathop{\mathrm{Sym}}^k(\mathcal O_F^{\oplus s}) \simeq\mathcal O_F^{\oplus\binom{k+s-1}{s-1}}.\] The filtration by powers of \(I\) therefore bounds the rank of restriction to the \(u\)-th thickening, for every \(u\ge1\) and every divisible \(l\), by \[ h^0\bigl(X,\mathcal O_X(lP)\otimes(\mathcal O_X/I^u)\bigr) \le\binom{u+s-1}{s}\,h^0(F,lP|_F). \tag{36}\] No vanishing theorem uniform in \(u\) is required for this inequality.

Fix \(D>C2^{1/4}\) and set \(u=\lceil Dl\rceil\). The pullback of \(P\) is big and semiample on \(X\), and its restriction to \(F\) is big and semiample because \(F\) meets the birational isomorphism locus. The semiample Hilbert-polynomial asymptotics show that the leading coefficient, after division by \(l^4\), on the right-hand side of (36) is \[\frac{D^s(P^h\cdot F)}{s!\,h!} \le\frac{D^s(hB)^h}{s!\,h!}.\] Choose the sequence index first so that this is strictly less than \(1/4!\), hence less than \(P^4/4!\). Then choose a sufficiently large divisible \(l\) for this fixed example. Since \[h^0(X,lP)=\frac{P^4}{4!}l^4+O(l^3),\] restriction to the thickening has a nonzero kernel. A kernel section lies in \(I^u\), and \(I_x^u\subset\mathfrak m_x^u\). It therefore has order at least \(u\ge Dl\) at \(x\). Normality gives \(\pi_*\mathcal O_X=\mathcal O_V\), so it is a section downstairs as well, with the same order at the point where \(\pi\) is an isomorphism. This contradicts (35). Rescaling back proves the lower bound, and completes the theorem. ◻