Proof. Choose a common smooth projective resolution \(W\), and use one rational top form to represent \(K_W\). Write the two pulled-back forms as \(u_X\eta^{\otimes m}\) and \(u_Y\eta^{\otimes m}\). If \(K_W+\Delta_W^X\) and \(K_W+\Delta_W^Y\) are the corresponding crepant pullbacks, then \[m(K_W+\Delta_W^X)=-\mathop{\mathrm{Div}}(u_X),\qquad m(K_W+\Delta_W^Y)=-\mathop{\mathrm{Div}}(u_Y).\] Their equality is equivalent to \(\mathop{\mathrm{Div}}(u_Y/u_X)=0\). A rational function with zero divisor on the normal projective integral variety \(W\) is a nonzero constant. ◻

Proof. The bounded moduli denominator proposition of OpenAI (2026d, Proposition 4.1) applies to \(h\). Its lower-dimensional normal-index hypotheses are supplied by Theorem 3. It gives a generalized klt pair \((Y,B_Y+M_Y)\) with effective boundary in a dimension-dependent rational DCC set, b-nef moduli data with a uniformly bounded b-Cartier denominator, and \[K_Y+B_Y+M_Y\sim_{\mathbb Q}0.\] Here the last equivalence follows from the exact canonical bundle formula and \(h_*\mathcal O_V=\mathcal O_Y\). The proof of the rationally connected generalized torsion result (OpenAI 2026d, Proposition 4.5) shows that these bases are uniformly generalized \(\epsilon\)-lc and applies Birkar (2025, Theorem 1.7) to obtain boundedness up to isomorphism in codimension one. Its normal bounded comparison models give \(\widehat Y\).

Choose very ample divisors from bounded projective embeddings of this family. Their top degrees are bounded. A general curve cut out by \(b-1\) such divisors is smooth and avoids the singular locus, and its degree and genus are bounded. Adjunction gives \[K_{\widehat Y}\cdot H^{b-1} =2g(C)-2-(b-1)H^b,\] which proves the second bound. For \(b=1\) use the normal curve itself. ◻

Proof. We explain the two scalar-order arguments from (OpenAI 2026d) used here, since the additional fibration hypothesis in its full scalar-estimate Proposition 6.1 is unnecessary for this conclusion.

From small Seshadri constants to large ambient orders. Suppose that \(1\le P_i^n\le2\) and that the very general Seshadri constants \(\varepsilon(P_i)\) tend to zero. Then \[ \gamma(P_i;V_i)\longrightarrow\infty \tag{31}\] after passing to a subsequence. Otherwise the orders are bounded by some \(D_0\). Curves realizing arbitrarily small Seshadri ratios spread into marked covering curve families. Input 10 produces leaves of a fixed dimension \(h\) with \((P_i|_{F_i})^h\to0\). Since \(P_i^n\ge1\), eventually \(0<h<n\). Resolve the leaf fibration and choose a very general smooth fibre \(F_i\); put \(q=n-h\). Its normal bundle is trivial. Therefore restriction to its \(u\)th infinitesimal neighbourhood has rank at most \[\binom{u+q-1}{q}h^0(F_i,mP_i|_{F_i}).\] Set \(u=\lceil Dm\rceil\) for a fixed \(D>D_0\). For each fixed fibration the leading coefficient of this expression in \(m^n\) is \[\frac{D^q}{q!h!}(P_i|_{F_i})^h,\] whereas the ambient section count has coefficient at least \(1/n!\). First take \(i\) large, then \(m\) sufficiently divisible and large. A nonzero ambient section vanishes on that neighbourhood, so has order at least \(u\) at a very general point of \(F_i\). This contradicts the bound \(D_0\). This argument is the proof of the second scalar inequality in (OpenAI 2026d), with its order bound as an explicit hypothesis.

From large ambient orders to small orders on moving members. If the lemma were false, choose counterexamples with \(r(T_i)\to\infty\) and rescale \(P_i\) rationally to have \(1\le P_i^n\le2\). Lemma 8 forces \(\varepsilon(P_i)\to0\), and hence (31) holds. Take small projective \(\mathbb Q\)-factorializations and pull back \(P_i\). They are klt, have principal canonical divisor, and preserve the complete section spaces. Choose \(n+1\) order levels with spacing \(\Delta_i\to\infty\) between \((P_i^n)^{1/n}\) and \(\gamma(P_i;V_i)\). Input 11 produces members of a fixed dimension \(e<n\) on a subsequence. With \(W_i\) their smooth models and \(L_i=P_i|_{W_i}\), \[\kappa(W_i)\ge0,\qquad L_i^e\longrightarrow0,\qquad K_{W_i}L_i^{e-1}\le C'L_i^e.\] Here \(K_{V_i}\sim0\), so \(C'\) can be fixed. Input 12 gives positive-dimensional Iitaka fibres \(U_i\) with \(\kappa(U_i)=0\) and adjoint volume tending to zero.

Represent \(L_i|_{U_i}\) by a divided general free member \(\Gamma_i\) for which \((U_i,\Gamma_i)\) is klt. A \(\mathbb Q\)-factorial good model \(Z_i'\) of this big adjoint exists by (Birkar et al. 2010); write \(N_i'\) for its big semiample adjoint. Then \[(N_i')^{\dim Z_i'}\longrightarrow0, \qquad K_{Z_i'}\preceq N_i', \qquad\kappa(Z_i')=0.\] On a common resolution, with maps \(x_i\) to \(U_i\) and \(y_i\) to \(Z_i'\), there is also the effective comparison \[ x_i^*(L_i|_{U_i})\preceq y_i^*N_i'. \tag{32}\] To prove it, write \(x_i^*(K_{U_i}+\Gamma_i)\sim_{\mathbb Q}y_i^*N_i'+F_i\) with \(F_i\ge0\) exceptional over \(Z_i'\). Since \(\kappa(U_i)=0\), choose an effective \(D_i\sim_{\mathbb Q}K_{U_i}\). In divisible degrees, the pullback of \(D_i\) plus a general member of the free polarization system contains the fixed part \(F_i\). The latter general member contains none of the finitely many components of \(F_i\), hence \(F_i\le x_i^*D_i\). Subtracting proves (32).

If \(\gamma(N_i';Z_i')\to0\) along a subsequence, stop. Otherwise take a subsequence on which it has a positive lower bound. More generally, the data at this stage have fixed dimension \(d\), are \(\mathbb Q\)-factorial klt, and satisfy \[N_i^d\to0,\qquad \kappa(Z_i)\ge0,\qquad K_{Z_i}\preceq a_0N_i,\qquad \gamma(N_i;Z_i)\ge a_1>0\] for fixed positive \(a_0,a_1\). Choose \(d+1\) equally spaced fixed levels below \(a_1\); they eventually exceed \((N_i^d)^{1/d}\). The tracking and Iitaka-fibre inputs apply again, since their canonical-degree bound is at most \((a_0+2d/\Delta)L_i^e\). They produce another good model of strictly smaller positive dimension, with polarization volume tending to zero, Kodaira dimension zero, canonical class bounded above by the new polarization, and comparison (32). At this new stage we make the same dichotomy: either the orders tend to zero along a subsequence and we stop, or they have a positive lower bound on a subsequence and the construction repeats. Every repetition decreases dimension. In dimension one the order is bounded by the degree, which tends to zero, so the construction must terminate with order tending to zero.

Trace these members through the successive covering families and Iitaka fibres to the original \(V_i\). Fibres sweep the preceding member; the birational comparisons are isomorphisms on opens met by the next geometric generic members. Thus the composite members map birationally to their images and have Kodaira dimension zero. Each effective comparison restricts to the next member, which is not contained in its extra divisor. Restricting successively and pulling back to a common smooth model of the final descendant compares the original restricted polarization with the final polarization by an effective difference. Multiplication by that difference injects their complete section spaces. At a generic mark outside the difference it preserves order. Thus the resulting members satisfy \(\gamma(P_i|_{C_b};C_b,u(b))\to0\). They eventually satisfy (30), a contradiction. This is the dimension-decreasing part of the first scalar-estimate proof in (OpenAI 2026d), stopped before its separate fibration contradiction. ◻

Proof. Its leaves have positive dimension. A leaf cannot fill \(V\), since (27) and (30) would give \(\gamma(P;V)\le n\eta(P^n)^{1/n}\), contradicting (26). Its function field is relatively algebraically closed in \(\mathbb C(T)\) by the geometric integrality of its leaves.

Choose a smooth projective base model \(M\) for this field. For \(\phi:T\dashrightarrow M\), let \(J\) be the closure on \(T\) of the inverse image of a general member \(H_M\) of a very ample system on \(M\). The map is defined in codimension one, and these divisors form a movable system. On a smooth resolution \(p:\widetilde T\to T\) resolving \(\phi\), \[ p^*J=\widetilde\phi^*H_M+E_M, \qquad E_M\ge0\text{ exceptional over }T. \tag{34}\] Choose \(H_M\) general for this resolution as well. With the canonical generator used to represent \(K_{\widetilde T}\), terminality gives \[ K_{\widetilde T}\ge aE_M \tag{35}\] for some positive rational \(a\).

We claim that all ratios of sections of divisible multiples of \(J\) are constant on the generic members of the covering family, transferred to \(\widetilde T\). Here is why (35) may be used on a member. Cut the parameter space by the dimension of a generic evaluation fibre, choosing hyperplanes with independent generic coefficients. The resulting incidence evaluates generically finitely and dominantly to \(\widetilde T\). Nonempty zero-dimensional cuts of the generic evaluation fibre avoid the added boundary in a projective closure. The universal choices of hyperplanes still dominate the original parameter space. After a geometric component and a resolution are chosen, the ramification formula on this total space, restricted to its smooth geometric generic parameter fibre \(C^*\), gives \[K_{C^*}\succeq aE_M|_{C^*}.\] The map \(\widetilde\phi\) is constant on \(C^*\), so (34) identifies the restricted polarization with \(E_M|_{C^*}\). Two independent restricted sections in any common degree, followed by a suitable power and multiplication by the effective canonical difference, would give two independent sections of a pluricanonical system on \(C^*\). That would contradict \(\kappa(C^*)=0\). Ambient nonzero sections remain nonzero at these generic data. The claim follows.

By the function-field characterization after Input 10, the section-ratio field of \(J\) is contained in \(\mathbb C(M)\). The reverse inclusion follows from ratios of hyperplane pullbacks, since \(H_M\) is very ample. Thus this section-ratio field is exactly \(\mathbb C(M)\).

For a small positive rational \(\varepsilon\), the pair \((T,\varepsilon J)\) is klt and its adjoint is pseudo-effective. Theorem 4 gives a terminating MMP to a good model. There are no divisorial steps: the divisor is movable, whereas the exceptional divisor of a negative divisorial contraction is contained in its fixed part. Movability persists through the flips. These steps are crepant for the zero boundary; the resulting \(T'\) is again terminal and isomorphic to \(T\) in codimension one. The semiample contraction, with Stein factorization, realizes the section-ratio field \(\mathbb C(M)\), which is already relatively algebraically closed. This proves attainment. ◻

Proof. We first construct a supporting functional for the pseudo-effective cone and then choose a point on which volume has the required scale. The hypothesis on irregularity, together with \(\mathbb Q\)-factoriality, gives \[\mathop{\mathrm{Cl}}(X)_{\mathbb R}=\mathop{\mathrm{Pic}}(X)_{\mathbb R}=N^1(X)_{\mathbb R}.\] Indeed pullback to a resolution detects numerically trivial line bundles, whose classes are torsion when the Picard variety is zero. We can therefore pass between real linear equivalence and numerical classes throughout the proof.

Choose common smooth big open subsets of \(Y\) and \(\widehat Y\). Their inverse image in \(X\) has complement of codimension at least two, so Cartier pullbacks there extend uniquely as Weil divisors on \(X\). This defines the indicated pullback on class groups. It is injective: if \(s=n-b\) and \(Q\) is an ample Cartier divisor on \(X\), the operation \[B\longmapsto h_*(B\cdot Q^s)\] on divisor classes is a left inverse to pullback, up to multiplication by the positive degree of \(Q^s\) on the generic fibre. The same operation sends effective classes to effective classes or zero, by using general representatives of sufficiently divisible multiples of \(Q\).

The degree \(\ell\) is strictly positive on \(W\cap\overline{\operatorname{Eff}}(X)\setminus\{0\}\). To see the point requiring care, choose a finite linear projection \(f:\widehat Y\to\mathbb P^b\) with \(f^*\mathcal O_{\mathbb P^b}(1)=\mathcal O_{\widehat Y}(H)\). For every effective real Weil divisor \(L\) on \(\widehat Y\), \[ \ell(L)H-L\quad\text{has an effective representative}. \tag{39}\] In fact \(f^*f_*L-L\ge0\) and \(f_*L\sim_{\mathbb R}\ell(L)\mathcal O_{\mathbb P^b}(1)\). The intersect-and-push operation just described shows that if a pullback class is pseudo-effective on \(X\), its base class lies in the closure of effective base classes. Applying (39) to approximating classes and pulling back shows that a degree-zero such class and its negative are both pseudo-effective on \(X\). The pseudo-effective cone in \(N^1(X)_{\mathbb R}\) contains no line, so the class is zero. This also proves that the formula defining \(\ell\) is well defined on \(W\).

Fix an ample class \(P_0\) on \(X\), and minimize \(\ell(L)\) on the set \[\{L\in W:P_0+L\in\overline{\operatorname{Eff}}(X)\}.\] The set is nonempty. Its subsets on which \(\ell\) is bounded above are bounded: otherwise, dividing an unbounded sequence by its norm and passing to a limit would give a nonzero pseudo-effective class in \(W\) of nonpositive degree. Closedness therefore gives a minimum \(c_0=\ell(L_0)\). Put \(d_0=P_0+L_0\). The open big cone is disjoint from \[P_0+\{L\in W:\ell(L)\le c_0\}.\] Indeed a big point in this affine half-space would remain big after subtracting a sufficiently small positive multiple of \(A\), contradicting minimality. Separating these two convex sets gives a nonzero linear functional \(\lambda\), nonnegative on the pseudo-effective cone, with \(\lambda(d_0)=0\). Its restriction to \(W\) is a nonnegative multiple of \(\ell\): it vanishes on \(\ker\ell\), and the orientation follows from the half-space being separated. That multiple is positive, since otherwise \(\lambda(P_0)=\lambda(d_0)=0\), whereas a nonzero nonnegative functional is strictly positive at an interior point of the cone. Normalize \(\lambda\) to obtain \(\ell_X\).

For \(t>0\), set \(d_t=d_0+tA\). These classes are big. For large \(t\), \(L_0+tA\) is pseudo-effective, since sufficiently large multiples of \(H\) added to any fixed real Weil class on the base are effective; convexity with the pseudo-effective class \(d_0\) then proves the assertion for every \(t>0\). On the other hand, \(A\) is not big: its restriction to the positive-dimensional generic fibre is trivial. Continuity and homogeneity of volume give \[ \frac{v(d_t)}t=v(A+t^{-1}d_0)\longrightarrow0 \qquad(t\longrightarrow\infty), \qquad \ell_X(d_t)=tH^b. \tag{40}\] It remains to bound this volume ratio from below for small \(t\). This is where the trivial canonical class enters the construction.

Let \(p:\widetilde X\to X\) be a smooth projective resolution and write the divisorial Zariski decomposition of \(\widetilde d_0=p^*d_0\) as \[\widetilde d_0=P_\sigma(\widetilde d_0) +N_\sigma(\widetilde d_0).\] We use its numerical formulation, as in (Nakayama 2004, III) and (Boucksom 2004). Fix the dimension-dependent integer \(M=M(n)\) in Birkar’s Calabi–Yau effective birationality theorem (Birkar 2023, Corollary 1.4). Suppose first that \(P_\sigma(\widetilde d_0)\) is numerically nonzero. For an ample class \(J\) on \(\widetilde X\), monotonicity of positive intersection products gives \[\begin{align*} \mathop{\mathrm{vol}}(\widetilde d_0+uJ) &=\bigl\langle(\widetilde d_0+uJ)^n\bigr\rangle\\ &\ge u^{n-1} P_\sigma(\widetilde d_0+uJ)\cdot J^{n-1}. \end{align*}\] Here the trace of the one-class positive product is the positive part of the divisorial Zariski decomposition; see (Boucksom et al. 2009, Definition 2.10, Proposition 2.13, Theorem 3.1, and Section 3.4). As \(u\downarrow0\), the intersection on the right tends to \(P_\sigma(\widetilde d_0)\cdot J^{n-1}>0\). Choose \(a>0\) such that \(p^*d_1-aJ\) is pseudo-effective. For \(0<t<1\), the identity \(d_t=(1-t)d_0+td_1\) and volume monotonicity yield \[\mathop{\mathrm{vol}}(d_t)\ge(1-t)^n \mathop{\mathrm{vol}}\left(\widetilde d_0+\frac{at}{1-t}J\right).\] Consequently \(v(d_t)/t\to\infty\) as \(t\downarrow0\). The constants in this divergence may depend on \(X\); no uniform threshold in \(t\) is needed.

Suppose instead that \(P_\sigma(\widetilde d_0)\) is numerically zero. Then \(d_0\) has a unique effective real representative \(N_0'\). For existence, push forward \(N_\sigma(\widetilde d_0)\); numerical and real linear equivalence agree here. For uniqueness, the pullback of any effective representative of \(d_0\) dominates \(N_\sigma(\widetilde d_0)\). Their difference is effective and numerically zero, hence zero. Let \(E_0\) be the reduced support of \(N_0'\). It has Kodaira dimension zero: if some multiple of \(E_0\) had a different effective representative, replacing a sufficiently small positive multiple of \(E_0\) inside \(N_0'\) would contradict uniqueness.

Choose a small positive rational \(\varepsilon\) for which \((X,\varepsilon E_0)\) is klt. Theorem 4 gives a terminating MMP for this pair and a good model \(\overline X\). Precisely the components of \(E_0\) are contracted. Any divisorial step must contract a component of its transform, since \(K\sim0\) and a curve outside the effective boundary has nonnegative intersection with it. Conversely, semiampleness on the good model and \(\kappa(E_0)=0\) force the effective pushforward of \(E_0\) to vanish. These steps are crepant for the zero boundary, so \(\overline X\) is again \(\mathbb Q\)-factorial canonical with principal canonical divisor. The pushforward \(\overline A\) of \(A\) is an integral Weil divisor and is big: the big class \(d_t\) pushes forward to \(t\overline A\). Birkar’s theorem now gives \[ \mathop{\mathrm{vol}}(\overline A)\ge M^{-n} \qquad\text{\citep[Corollary~1.4]{Birkar2023}}. \tag{41}\] Its hypotheses are exactly that \(\overline X\) is projective klt, \(K_{\overline X}\sim_{\mathbb R}0\), and \(\overline A\) is big and integral; no Cartier-index bound is required.

A single rational number \(u\ge0\) gives inclusions \[H^0(\overline X,m\overline A) \subseteq H^0(X,m(A+uE_0))\] for every sufficiently divisible \(m\). Indeed, on a common resolution pull back \(\overline A\) and push forward to \(X\). The resulting rational divisor differs from \(A\) only on \(E_0\), and its difference is bounded above by \(uE_0\). Thus \(\mathop{\mathrm{vol}}(A+uE_0)\ge\mathop{\mathrm{vol}}(\overline A)\). For all sufficiently small \(t>0\) we have \(N_0'\ge tuE_0\), and hence \[\mathop{\mathrm{vol}}(d_t)=\mathop{\mathrm{vol}}(N_0'+tA) \ge t^n\mathop{\mathrm{vol}}(A+uE_0)\ge t^n M^{-n}.\] The same conclusion includes \(E_0=0\), with \(u=0\).

In both cases the ratio \(v(d_t)/t\) is at least \(M^{-1}\) for all sufficiently small \(t>0\). By (40) and continuity, it takes the value \((2M)^{-1}\) at some \(t>0\). At this parameter, \[d_t-\frac t3A=d_{2t/3}\text{ is big}, \qquad \ell_X(d_t)=tH^b\le N_0t.\] Approximate \(t\) by a positive rational number \(\delta\) and \(d_t\) by a rational class \(D\). Openness of the big cone and continuity of volume and \(\ell_X\) preserve these assertions with, for example, lower constant \(1/(4M)\) and suitable upper constant depending only on \(n,N_0\). Since \(X\) is \(\mathbb Q\)-factorial, the rational class has a rational Cartier divisor representative. This proves (38). ◻

Proof. The extraction and \(\mathbb Q\)-factorialization theorem for klt pairs (Birkar et al. 2010, Corollary 1.4.3) extracts the exceptional divisors with discrepancy at most zero. There are only finitely many such divisors. To recall why, take a log resolution with crepant subboundary \(\sum b_jD_j\), where every \(b_j<1\). For an exceptional divisorial valuation \(v\) over this resolution, \[A_{\sum b_jD_j}(v) =A_{\sum D_j}(v)+\sum_j(1-b_j)v(D_j).\] The first term is a nonnegative integer. If it is positive and the centre is contained in the support, the displayed sum is strictly greater than one. If the generic point of the centre lies outside the support, smoothness gives log discrepancy at least two. Thus valuations of log discrepancy at most one are lc places of the reduced simple normal crossings divisor. Such places are toroidal; their positive integral weights are bounded by \(\sum_j(1-b_j)v(D_j)\leq1\). Together with the finitely many divisors on the resolution, this proves finiteness. Extracting these places gives the asserted terminal pair, and its boundary is effective because each extracted discrepancy is nonpositive.

Pull \(\theta\) back to \(V\). Its order at every prime divisor of \(V\) is nonpositive, whereas its order at every exceptional divisor over \(V\) is strictly positive by terminality. A transformation multiplying \(\theta\) by a nonzero constant cannot exchange these two kinds of divisorial valuations. Neither it nor its inverse therefore contracts a prime divisor. It is a pseudo-automorphism, and the orders of \(\theta\) show that it preserves the boundary coefficients. ◻

Proof. A cover admitting a lift. Put \(U=V_{\mathrm{sm}}\). The connected étale cover \(P_U\to U\) corresponds to a finite-index subgroup of \(\pi_1^{\mathrm{top}}(U)\). This fundamental group is finitely generated, so it has only finitely many subgroups of any fixed index. Their intersection is a finite-index characteristic subgroup contained in the subgroup defining \(P_U\). Riemann existence gives its connected algebraic étale cover. Normalize \(V\) in its function field to obtain a finite quasi-étale cover \(R\to V\) factoring through \(P\).

The pseudo-automorphism \(f\) identifies two open subsets of \(U\) with complements of codimension at least two. Removing such subsets does not change the fundamental group of a complex manifold, by transversality. Characteristicity therefore lifts this isomorphism to the chosen cover. Riemann existence makes the lift algebraic, and it induces a pseudo-automorphism \(g\) of \(R\). If \(H_R\) and \(\theta_R\) denote the pulled-back boundary and form, then \[g^*\theta_R=c\theta_R,\] and \(g\) preserves \(H_R\) in codimension one. No degree bound for \(R\to V\) is required.

Preserving the factor directions. Over the product of the smooth loci of the \(P_i\), the map \(R\to P\) is étale. The tangent directions of the factors give a splitting there, whose reflexive extension is \[\mathcal T_R=\bigoplus_{i\in I}\mathcal E_i.\] Let \(\mathcal T_R(-\log\mathop{\mathrm{Supp}}H_R)\) be the reflexive sheaf of derivations preserving the reduced ideal of every boundary component, a condition tested at codimension-one points. Consider the finite-dimensional \(\mathbb C\)-algebra \[\mathcal A=\{u\in H^0(R,\mathop{\mathrm{End}}(\mathcal T_R)): u(\mathcal T_R(-\log\mathop{\mathrm{Supp}}H_R)) \subseteq\mathcal T_R(-\log\mathop{\mathrm{Supp}}H_R)\}.\] Conjugation by \(g\) acts on \(\mathcal A\): both the tangent sheaf and the logarithmic condition are determined on a big open set.

Every member of \(\mathcal A\) has zero entries between different \(\mathcal E_i\). To see this, fix general smooth points in the complementary factors and slice in one factor direction. Each connected component over that factor’s smooth locus extends by normalization to a connected normal finite quasi-étale cover of the factor. The omitted set has codimension at least two, since a divisorial point of a finite cover lies over a divisorial, hence smooth, point of the normal factor. The ambient cover is étale on this open slice, so restricted entries are regular and extend reflexively across its complement. On the slice its tangent directions are \(\mathcal E_i\), while complementary summands are trivial. An entry in either direction involving a nonabelian boundary-free factor therefore gives reflexive one-forms or vector fields on its cover, and these vanish by Input 19. For the only remaining case, an abelian and a rationally connected factor, slice in the rationally connected direction. An entry towards the abelian summand gives one-forms; an entry in the reverse direction gives vector fields tangent to the boundary: that boundary is pulled back from the rationally connected factor, so the ideal-preservation condition restricts at every divisorial point of the slice. These vanish as well. The slices cover a dense open of \(R\), proving the assertion.

Consequently every block projection is a central idempotent of \(\mathcal A\). The primitive central idempotents give nonzero direct summands of the rank-\(n\) tangent sheaf, so there are at most \(n\) of them. Conjugation by \(g\) permutes these idempotents. Therefore \[h=g^{n!}\] fixes every primitive central idempotent. Every central idempotent is a sum of primitive central idempotents, so \(h\) fixes every block projection and preserves each factor direction.

Recovering the factor fields. Let \(K_i\) be the relative algebraic closure of \(\mathbb C(P_i)\) in \(\mathbb C(R)\), and let \(R_i\) be the normalization of \(P_i\) in \(K_i\). Equivalently, \(R\to R_i\to P_i\) is the Stein factorization of the projection. At the generic point of \(R\), the directions complementary to \(i\) span \(\operatorname{Der}_{\mathbb C(P_i)}\mathbb C(R)\). In characteristic zero their common field of constants is exactly \(K_i\). Preservation of these directions thus gives a birational self-map \(h_i\) of \(R_i\).

The map \(\psi:R\to\prod_iR_i\) is proper and has finite fibres, because its composite to \(P\) is finite. It is finite and, by dimension, surjective. The maps \(R_i\to P_i\) are quasi-étale. Indeed, since \(\mathbb C(P)/\mathbb C(P_i)\) is a regular extension, the product \(R_i\times\prod_{j\ne i}P_j\) is an intermediate normal finite cover of \(P\). Ramification over a prime of \(P_i\) would persist in this product and then in a prolongation to \(R\), contradicting quasi-étaleness of \(R\to P\). The same ramification calculation in the tower proves that \(\psi\) is quasi-étale. In particular the pulled-back pairs \((R_i,H_i)\) are klt. The relation between \(h\) and the \(h_i\) is rational equivariance of \(\psi\); no regularity assertion about the \(h_i\) is needed.

Comparing the forms. On each boundary-free \(R_i\), choose a generator \(\theta_i\) of its trivial canonical sheaf. For the rationally connected factor, restrict the degree-\(m\) trivialization pulled back from \(V\) to that factor of \(P\), using smooth complementary points and the external product of canonical lines. The resulting form has divisor \(-mH\) on that factor; pull it back to \(R_i\). This constructs the forms of degree \(p_i\) in the statement. Their product, in any fixed order, satisfies \[ \theta_R=a\,\psi^*\left(\bigwedge_i\theta_i^{\otimes m/p_i}\right) \qquad(a\in\mathbb C^*). \tag{54}\] Here the wedge denotes external product of pluricanonical lines. Indeed both sides have divisor \(-mH_R\), so their ratio is a global unit on the normal projective integral variety \(R\).

Write \(u_i=h_i^*\theta_i/\theta_i\in\mathbb C(R_i)^*\). Pulling (54) back by \(h\) shows that the pullback of \(\prod_i u_i^{m/p_i}\) is the constant \(c^{n!}\). Since \(\psi\) is dominant, the product is constant on \(\prod_iR_i\). Fixing general points in all complementary factors shows that every \(u_i^{m/p_i}\), and hence every \(u_i\), is constant. Denote it by \(d_i\). This proves (53); proportionality of the forms proves that each \(h_i\) is B-birational. ◻

Proof. By Han and Jiang (2024, Theorem 1.2), the underlying varieties are bounded modulo flops: they are isomorphic in codimension one to normal members \(Y\) of a bounded family. The theorem applies because \(-m(K_F+H)\) is Cartier and nef. In dimension one one can instead use \(F\simeq\mathbb P^1\) directly.

Let \(B_Y\) be the strict transform of \(H\). A degree-\(m\) log trivializing form has divisor \(-mB_Y\) on \(Y\), by the codimension-one identification. Thus \(m(K_Y+B_Y)\sim0\) is integral principal, and the map is B-birational by the form criterion. In particular the transformed pair is klt. Choose very ample divisors \(A\) in bounded projective embeddings of these \(Y\), with \(A^r\) uniformly bounded. Since \(mB_Y\) is integral, \[ (\mathop{\mathrm{Supp}}B_Y)A^{r-1} \leq mB_YA^{r-1}=-mK_YA^{r-1} \leq m\bigl(2+(r-1)A^r\bigr). \tag{55}\] For the last inequality, take a general complete-intersection curve of \(r-1\) members of \(|A|\). It is smooth and avoids the singular locus of the normal variety \(Y\), and adjunction gives \(-K_YA^{r-1}=2-2g+(r-1)A^r\).

Reduced subvarieties of bounded degree in bounded projective embeddings form a bounded family. Moreover, the nonzero coefficients of \(B_Y\) lie in \(\{1/m,\ldots,(m-1)/m\}\), since the pair is klt. Thus the pairs are log bounded modulo B-birational contractions. (A birational map that is an isomorphism in codimension one is a birational contraction in this terminology.)

Jiang and Liu (2021, Theorem 3.2), which applies in every dimension, now gives integers \(k,N>0\) depending only on \(r,m\) such that \(k(K_F+H)\sim0\) and the image of the degree-\(k\) B-representation has order at most \(N\). If a map acts by \(d\) on a degree-\(m\) generator and by \(e\) on a degree-\(k\) generator, then comparison in degree \(km\) gives \(d^k=e^m\). Since \(e^{N!}=1\), we obtain \(d^{kN!}=1\). We may take \(E_{\mathrm{rc}}(r,m)=kN!\). ◻

Proof. Let \(U\) be a smooth projective resolution of \(Y\). The canonical singularities and trivial canonical class give \(h^{r,0}(U)=1\). For a nonabelian factor, OpenAI (2026d, Lemma 5.2) shows that \(q(U)=0\) and that every positive-dimensional rational image of smaller dimension has rationally connected smooth resolution. Its hypotheses hold for \(Y\) itself, using the identity cover, because the defining form-algebra properties persist under all further connected normal finite quasi-étale covers. Take a crepant projective \(\mathbb Q\)-factorial terminalization of \(Y\). It still has principal canonical divisor, and the two properties just stated are birational invariants. Theorem 9 therefore bounds \(\dim_{\mathbb Q}T(U)\) in terms of \(r\). Nonabelian factors here have dimension at least two. For an abelian factor, take \(U=Y\) and use instead \[\dim_{\mathbb Q}T(U)\leq\dim_{\mathbb Q}H^r(U,\mathbb Q) =\binom{2r}{r}.\] Thus there is a common dimension bound \(b(r)\) in either case.

Let \(p,q:W\to U\) be a smooth projective common resolution of the induced birational self-map of \(U\). By Lemma 6, birational invariance of the Hodge structure generated by the top form gives \[T(W)=p^*T(U)=q^*T(U).\] Consequently \(A=p_*q^*\) preserves \(T(U)\) and the cup-product pairing there; its inverse is \(q_*p^*\). Pullback and pushforward are integral on cohomology modulo torsion, so \(A\) preserves the full lattice \[T(U)\cap\bigl(H^r(U,\mathbb Z)/\text{torsion}\bigr).\] Moreover, \(T(U)\) is primitive for every rational ample class: the kernel of cup product by that class is a rational Hodge substructure containing \(H^{r,0}(U)\). The Hodge–Riemann relations therefore turn the cup pairing and Hodge decomposition on \(T(U)\) into a positive-definite Hermitian form, preserved by \(A\).

The scalar \(d\) on the canonical volume form is an eigenvalue of this integral operator. It is an algebraic integer, and all its conjugates occur among the eigenvalues of \(A\), hence have modulus one. Kronecker’s theorem makes \(d\) a root of unity. If its order is \(e\), the degree of its cyclotomic polynomial is Euler’s totient \(\varphi(e)\), so \[\varphi(e)\leq b(r).\] There are only finitely many positive integers with this property. Their least common multiple is a permissible choice of \(E_0(r)\). ◻

Proof of Theorem 17. For \(n=0\) the pair is a point and the character is trivial, so take \(L(0,m)=1\). Suppose \(n>0\). Lemmas 18 and 20 express \(c^{n!}\) as in (53). The rationally connected factor satisfies the hypotheses of Lemma 21; in particular it remains rationally connected by the cover-stable choice in Input 19. Every other factor satisfies Lemma 22. Define \[E(n,m)=\operatorname{lcm}_{1\leq r\leq n} \{E_{\mathrm{rc}}(r,m),E_0(r)\}, \qquad L(n,m)=n!E(n,m).\] Each \(d_i^{E(n,m)}=1\), and (53) gives \(c^{L(n,m)}=1\). Every bound depends only on the factor dimensions and \(m\), and therefore only on \(n,m\). ◻

Proof. There is nothing to prove if the pair is klt. Otherwise apply Kollár (2016, Proposition 14) to the morphism \(Y\to\mathop{\mathrm{Spec}}\mathbb C\). Its hypotheses hold in degree \(m\): the pair is dlt, its log canonical divisor is rationally trivial, and \(\omega_Y^{[m]}(m\Gamma)\simeq\mathcal O_Y\). That proposition provides a birational comparison commuting exactly with the degree-\(m\) residue maps, and hence the displayed identity. The divisor identities (57) imply B-birationality.

The exact equality, rather than proportionality, can also be seen in the proof of that proposition. The minimal centres are joined by \(\mathbb P^1\)-links (Kollár 2016, Theorem 10). Over the function field of a link’s base, the two sections are \(0,\infty\) on \(\mathbb P^1\), and the logarithmic form is a base form times \((dz/z)^{\otimes m}\). Its residues at the two sections agree for even \(m\). Composition along the links preserves this equality. ◻

Proof. The form identity makes \(\tau\) B-birational. If one pair is klt, both are klt, and we take \(A=S\), \(A'=T\), and \(\psi=\tau\). Suppose therefore that there are proper lc centres. Put \(n=\dim S\) and let \(k>0\) be the largest codimension of an lc centre of \(S\). Choose such a centre \(A\).

Corresponding strata on a common resolution. Take a projective log resolution of \(S\) which is unchanged over the simple-normal-crossing neighbourhood of the generic point of \(A\) (Szabó 1994, Resolution Lemma, p. 633). Resolve the map to \(T\) and its boundary, using smooth blowup centres having normal crossings with the accumulated boundary (Kollár 2007, Definition 25 and Theorem 35). This gives a common smooth model \(p:W\to S\), \(q:W\to T\). Write \[K_W+\Delta_W=p^*(K_S+\Delta_S)=q^*(K_T+\Delta_T),\] with compatible canonical divisors. The subboundary \(\Delta_W\) need not be effective, but its total support is simple normal crossing.

There is a codimension-\(k\) component \(Z\) of an intersection of \(k\) coefficient-one divisors of \(\Delta_W\) dominating \(A\). To verify this during the chosen blowups, start with \(A\) on the initial snc open. A blowup whose centre misses the generic point of the currently chosen intersection leaves that intersection unchanged there. If its centre contains that generic point, the centre is locally defined by a subset of the \(k\) boundary parameters: any additional tangential equation would cut the intersection properly and so could not contain its generic point. For a nontrivial such blowup, the subset has size \(h\ge2\) and the exceptional divisor has coefficient \(h-(h-1)=1\); in each relevant coordinate chart, the exceptional divisor and the surviving strict transforms again have a codimension-\(k\) intersection dominating the old one. This proves the assertion inductively. In making the resolution, include the full total transforms of both boundaries among the normal-crossing divisors, including exceptional components of crepant coefficient zero.

Let \(C=q(Z)\). The intersection \(Z\) determines an lc place: for \(k=1\) it is itself a coefficient-one divisor, and for \(k>1\) blow up its generic intersection. Hence \(C\) is an lc centre of \(T\). Since \(\dim C\le\dim Z=n-k\), its codimension is at least \(k\). Repeating the same construction with \(S,T\) interchanged shows that the largest codimensions of their lc centres agree. It follows that \(\operatorname{codim}_T C=k\), and both \(A\) and \(C\) are minimal strata. In particular \(Z\to A\) and \(Z\to C\) are generically finite.

Birationality and the exact residue comparison. We claim that these two maps are birational and that pullback through either one commutes exactly with the degree-\(m\) residue. We prove this for \(q\). At the generic points of \(C\) and \(Z\), choose regular parameters \(x_1,\ldots,x_k\) and \(z_1,\ldots,z_k\) for the respective coefficient-one divisors, completed by separating coordinates \(y_1,\ldots,y_{n-k}\) and \(w_1,\ldots,w_{n-k}\) along the strata. The total transforms have normal crossings, so \[ q^*x_i=u_i\prod_{j=1}^k z_j^{e_{ij}}, \qquad u_i\in\mathcal O_{W,\eta_Z}^{*},\quad e_{ij}\in\mathbb Z_{\ge0}. \tag{59}\] In these coordinates the logarithmic Jacobian is the matrix expressing \(q^*(dx_i/x_i),q^*dy_l\) in terms of \(dz_j/z_j,dw_l\). Its determinant is a unit at \(\eta_Z\): the pullback of \(\theta_T\) has exactly the logarithmic poles prescribed by the coefficient-one components through \(Z\), and no other zero or pole there. Modulo the ideal of \(Z\), the matrix has the form \[\begin{pmatrix} E&*\\0&J_C\end{pmatrix}, \qquad E=(e_{ij}),\] where \(J_C\) is the Jacobian along \(Z\to C\). In particular \(\det E\ne0\).

We now use birationality of the ambient map to strengthen this to \[ |\det E|=1,\qquad \mathbb C(Z)=\mathbb C(C). \tag{60}\] Choose positive real numbers \(\alpha_1,\ldots,\alpha_k\) linearly independent over \(\mathbb Q\), and take the monomial valuation at \(\eta_Z\) assigning \(z_j\) the value \(\alpha_j\). It can be constructed in the completed regular local ring and restricted to \(\mathbb C(W)\). Its value group is \(\sum_j\mathbb Z\alpha_j\), and its residue field is \(\mathbb C(Z)\).

Here is the local calculation that also identifies this valuation from the target. If a real valuation \(v\) is centred on a regular local ring \(R\) with parameters \(t_1,\ldots,t_k\), and their positive values are \(\mathbb Q\)-linearly independent, then \[ v\bigl((\operatorname{Frac}R)^*\bigr)=\sum_i\mathbb Zv(t_i), \qquad \kappa(v)=\kappa(R). \tag{61}\] To check the assertions, expand any nonzero \(f\in R\) modulo \(\mathfrak m_R^N\) as a polynomial in the \(t_i\), with coefficients lifting residue classes. Nonzero coefficients have value zero, distinct monomials have distinct values, and the remainder has value at least \(N\min_i v(t_i)\). Taking \(N\) so large that this exceeds \(v(f)\) gives a unique term of least value. Thus \(v(f)\) is an integral combination of the parameter values. For a quotient of two elements of equal value, independence forces their least terms to have the same exponent vector, so its residue is the ratio of their coefficients in \(\kappa(R)\). Residue classes already have lifts in \(R\), proving the reverse inclusion as well.

By (59), the values of the \(x_i\) are the rows of \(E\) applied to \((\alpha_j)\); they are positive and independent. Apply (61) to \(\mathcal O_{T,\eta_C}\). Since \(q\) is birational, its fraction field and that of \(\mathcal O_{W,\eta_Z}\) are the same. The value groups and residue fields computed in the two rings must therefore agree. Equality of the value groups says that \(E\) is unimodular, while equality of the residue fields gives \(\mathbb C(Z)=\mathbb C(C)\). This proves (60).

Taking the logarithmic residue of \(q^*\theta_T\) along \(Z\) now gives the pullback of \(\theta_C\) multiplied by \((\det E)^m=1\). The tangential Jacobian \(J_C\) is precisely the one appearing in pullback of the stratum form. The identical argument for \(p\) compares the residue along \(Z\) with \(\theta_A\) and proves that \(Z\to A\) is birational. The equality \(q^*\theta_T=r p^*\theta_S\) therefore gives (58), with \(A'=C\) and the birational map induced through \(Z\). Finally (57) shows that this map is B-birational. ◻

Proof of Theorem 1. First let the ground field be \(\mathbb C\). Write \(D=K_X+B\) and let \(\nu:\bar X\to X\) be the normalization, with conductor divisor \(\bar C\). The normalization formula is \[\nu^*D=K_{\bar X}+\bar C+\nu_*^{-1}B.\] Denote its component pairs by \((X_i,\Delta_i)\), \(i\in I\). They are projective integral lc \(d\)-folds with effective boundary coefficients in \(\Phi\cup\{1\}\) and \(K_{X_i}+\Delta_i\sim_\mathbb Q0\). By Theorem 3, there is a common principal multiple depending only on \(d,\Phi\). Choose an even multiple \(m=m(d,\Phi)\) which also clears the coefficients of \(\Phi\). Fix rational \(m\)-canonical forms \(\theta_i\) with \[\mathop{\mathrm{Div}}(\theta_i)=-m\Delta_i.\] Choose projective crepant \(\mathbb Q\)-factorial dlt modifications \((Y_i,\Gamma_i)\to(X_i,\Delta_i)\) (Kollár 2016, Proposition 5). Use the same notation for the pulled-back forms, whose divisors are \(-m\Gamma_i\).

Form a finite graph with vertices \(I\) and one edge for each generic point of the double locus of \(X\). At a split node its two branches give conductor components on the normalization, possibly on the same \(X_i\). Their strict transforms are strata \(S\subset Y_i\), \(T\subset Y_j\), and the identification of the branch function fields gives a birational map \(\tau:S\dashrightarrow T\). At a nonsplit node there is one conductor component with a quadratic function-field extension over the node; use it twice and take \(\tau\) to be the nontrivial involution. This gives a loop at the corresponding vertex. The graph may have multiple edges and loops. We can treat its connected components separately.

The assumed rational linear triviality of \(D\) supplies a principal Cartier multiple \(MD\), with \(M\) divisible by \(m\). A generator in degree \(M\) pulls back to \(b_i\theta_i^{\otimes M/m}\) for constants \(b_i\in\mathbb C^*\): on each projective normal component the ratio has zero divisor. Its branch residues satisfy the node rule. Since \(M\) is even, for an oriented edge \(e:i\to j\) this gives \[\left(\frac{\tau^*\theta_T}{\theta_S}\right)^{M/m} =\frac{b_i}{b_j}.\] The ratio is therefore a nonzero constant in the conductor function field. Write \[ \tau^*\theta_T=r_e\theta_S, \qquad r_e\in\mathbb C^*,\qquad r_{\bar e}=r_e^{-1}. \tag{63}\] In particular the induced map of the two adjoint pairs is B-birational. The integer \(M\) is used only to establish these constant comparisons; it will not enter the uniform bound.

For each oriented edge, Lemma 24 supplies minimal strata on its two branches and a birational comparison with multiplier \(r_e\). By adjunction these strata are also minimal strata of the corresponding \(Y_i\). Along any closed walk in the graph, join successive choices within a vertex by Lemma 23. These intermediate maps have multiplier one. Composing all comparisons gives a B-birational self-map \(f\) of one projective integral klt pair \((A,\Gamma_A)\), with \[ f^*\theta_A=\left(\prod_{e\text{ in the walk}}r_e\right)\theta_A. \tag{64}\] The product respects the orientations of the walk. Moreover \(\dim A\le d-1\) and \(m(K_A+\Gamma_A)\sim0\) by (57). Figure 1 displays the composition for a three-edge cycle. The same argument includes loops, including the involution at a nonsplit node.

Choose \[ L=\operatorname{lcm}\{L(q,m):0\le q\le d-1\}, \qquad a=mL, \tag{65}\] where \(L(q,m)\) is given by Theorem 17. Applying that theorem to (64) shows that the product of \(r_e^L\) around every closed walk is one. Consequently there are constants \(c_i\in\mathbb C^*\) satisfying \[ c_i=c_jr_e^L\quad\text{for every oriented edge }e:i\to j. \tag{66}\] For completeness, choose a spanning tree in each connected component, set \(c_i=1\) at one vertex, and determine the other constants along its tree paths. Every additional edge is consistent because it closes a walk with product one. A loop imposes precisely \(r_e^L=1\), already proved.

The forms \(c_i\theta_i^{\otimes L}\) now have equal residues on every pair of conductor branches. Regard their tuple as a rational \(a\)-canonical form \(\sigma\) on \(X\): it is an element of the product of the degree-\(a\) canonical lines of the component function fields, a rank-one module over the total quotient ring \(\prod_i\mathbb C(X_i)\). At every normal codimension-one point, its divisor says that it generates the line corresponding to \(aD\). At each generic node, its branches are generators and have matching residues, so the local rule (62) makes it a generator there as well; \(a\) is even and \(B\) avoids those points.

We explain why this proves Cartierness, rather than only triviality after normalization. Choose a big open immersion \(j:U\hookrightarrow X\) containing all codimension-one points, such that \(X\) is Gorenstein on \(U\), \(aB\) is Cartier on \(U\), and the preceding local comparisons make \(\sigma\) a generator of \[\mathcal L_{a,U}=\omega_U^{\otimes a}(aB|_U).\] The divisorial sheaf associated with \(aD\) is its \(S_2\) extension \(\mathcal F_a=j_*\mathcal L_{a,U}\). The generator identifies this sheaf with \(j_*\mathcal O_U=\mathcal O_X\), the last equality following from \(S_2\). Hence \(\mathcal F_a\) is invertible. Equivalently, divide \(\sigma\) by the \(a\)th power of a rational canonical form, chosen to generate at the generic nodes. The ratio is an invertible element of the total quotient ring whose principal divisor is \(-aD\) on \(U\) and therefore on \(X\). Thus \(aD\) is principal Cartier. Isolated vertices of the conductor graph cause no difficulty: their chosen forms already satisfy the required condition.

Finally let the ground field \(k\) be any algebraically closed field of characteristic zero. All the data descend to an algebraically closed subfield \(k_0\subset k\) of finite transcendence degree over \(\mathbb Q\), including the normalization, conductor, a principal Cartier multiple of \(D\), and a log resolution of the normalized pair. The field \(k_0\) embeds in \(\mathbb C\). The hypotheses persist under these algebraically closed field extensions: \(S_2\) and the smooth or nodal codimension-one charts are preserved, normalization and conductor commute with them, and the coefficients on the chosen log resolution test log canonicity after extension. We therefore obtain the degree-\(a\) conclusion over \(\mathbb C\) with the same integer (65).

On a fixed big Gorenstein open over \(k_0\), form \(\mathcal F_a\) as above. The open immersion is quasi-compact and separated, since \(X\) is noetherian. Flat base change for quasi-coherent sheaves therefore shows that this extension by \(j_*\) commutes with field extension, so its pullback to \(\mathbb C\) is trivial. Invertibility descends faithfully flatly. For the resulting line bundle, proper cohomology and field extension identify its space of global sections after extension with \(H^0(X_{\mathbb C},\mathcal O_{X_{\mathbb C}}) =\mathbb C\). Its one-dimensional space over \(k_0\) therefore has a nonzero section whose evaluation becomes an isomorphism over \(\mathbb C\) and hence is already an isomorphism over \(k_0\). Thus \(\mathcal F_a\simeq\mathcal O\) over \(k_0\), and extending to \(k\) proves the theorem. ◻