Uniform indices for semi-log-canonical log Calabi–Yau pairs

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

Their equality is equivalent to \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 [36, Proposition 4.1] applies to h. Its lower-dimensional normal-index hypotheses are supplied by Theorem 2.2. 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

Here the last equivalence follows from the exact canonical bundle formula and h_*\OO_V=\OO_Y. The proof of the rationally connected generalized torsion result [36, Proposition 4.5] shows that these bases are uniformly generalized \epsilon-lc and applies Birkar [2, 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

which proves the second bound. For b=1 use the normal curve itself.□

Proof. For a birational morphism p:W\to U between smooth projective varieties, p^*:H^n(U,\Q)\to H^n(W,\Q) is an injective Hodge morphism, with left inverse p_*. Birational invariance of holomorphic top forms gives H^{n,0}(W)=p^*H^{n,0}(U). The image p^*T(U) therefore contains T(W) by minimality. Its inverse image of T(W) is a Hodge substructure containing H^{n,0}(U), and hence equals T(U). Thus T(W)=p^*T(U). Common resolutions give the assertion for birational maps.

The kernel of cup product with a rational divisor class is a rational Hodge substructure, with the usual Tate twist in the target. It contains H^{n,0}(U) because H^{n+1,1}(U)=0, so it contains T(U). Finally, f^* vanishes on holomorphic top forms when its image has dimension less than n. Its kernel is again a rational Hodge substructure, which proves the last assertion.□

Proof. Fix R and write \omega=D+RH, Q=RH. Let \gamma be the \omega-harmonic representative of z. In an \omega-unitary coframe diagonalizing Q, write its eigenvalues as b_i\in[0,1] and set b_I=\sum_{i\in I}b_i for an n-element subset I of \{1,\ldots,2n\}. Keeping \gamma fixed while differentiating its squared norm with respect to \log R gives, relative to the present volume form,

Indeed the volume contributes \Tr Q, while the covariant holomorphic and antiholomorphic factors contribute -b_I and -b_J. Define the nonnegative pointwise deficit

We will prove

Put N_n=\binom{2n}{n}. If \rank Q<n, then \Tr Q\le n-1 and e\ge|\gamma|^2. If \rank Q=n, denote its n-dimensional kernel by K, the coefficient \gamma_{K\bar K} by a, and put

Every coefficient other than a involves a positive eigenvalue in at least one index set. Subtracting that eigenvalue from \Tr Q leaves a sum of at most n-1 numbers at most one. Consequently

Since 0\le b_i\le1, we also have

Consider the top-degree forms

In normalized exterior bases, the coefficient of A is a signed sum of the N_n diagonal coefficients of \gamma. The coefficient of B is p a, with the same sign as the a term in A. Thus Cauchy–Schwarz and (3.4) give

The same inequalities hold when \rank Q<n: then B=0 and e\ge|\gamma|^2.

The form A is harmonic, because multiplication by \omega commutes with the Laplacian. The form B is exact, because Q is closed and

They are therefore L^2-orthogonal. Integrating (3.5) yields

which is (3.3).

For completeness, varying the metric does not require differentiating the harmonic representative. At the current value of R, keeping it fixed supplies an upper derivative for the minimum defining \norm{z}_{D+RH}^2. These minimum norms are locally Lipschitz in \log R, by comparison of nearby metrics. Equations (3.2) and (3.3) therefore imply almost everywhere

Integration from 1 to R proves (3.1).□

Proof. Choose a resolution \rho:U\to X, isomorphic over the smooth locus, obtained by blowups, and an effective exceptional divisor N such that -N is \rho-ample. The canonical form trivializing K_X gives

Put H=\rho^*h and l=H^n\in[1,2]. We also denote by H a smooth closed semipositive representative, obtained by pulling back a Fubini–Study form from an embedding given by a multiple of h. Let

be the blowup of the diagonal, with exceptional divisor E. Subscripts 1,2 indicate pullback from the two factors, on every subsequent model. A subvariety of any such model will be called double dominant if its maps to both copies of U are dominant.

Diagonal conditions and the number of sections

Fix rational numbers

and choose a rational a_2 depending only on n,c, sufficiently large that a_2>L/c and a_2>\tau(1+1/(10n)). Set a_1=10na_2 and A=a_1H_1+a_2H_2. These choices give the strict inequalities

They remain true after sufficiently small rational perturbations of a_1,a_2,t. We first prove that

where \mathbf B denotes the stable base locus.

For sufficiently divisible positive integers m, put

The conormal bundle of the diagonal is \Omega_U^1. Its infinitesimal layers therefore give

We need an estimate uniform in j, because j grows with m.

The reflexive cotangent sheaf of X is slope semistable of degree zero with respect to h. This follows from generic semipositivity for the cotangent bundle and K_X\equiv0; the precise singular-variety semistability and restriction statements we use are [19, Theorem 5.3 and Proposition 5.4]; see also [6, Theorem 1.3]. Choose a fixed sufficiently positive general complete-intersection flag on U, cut by pullbacks of hypersurfaces on X, terminating in a smooth curve C avoiding the exceptional locus. Its members are smooth and integral. The bundle \Omega_U^1|_C is semistable of degree zero, as is every symmetric power in characteristic zero.

Write D_0=m_0H, with m_0h very ample, and r_j=\binom{j+n-1}{n-1}. On C, a negative twist of \Sym^j\Omega_U^1|_C has no sections. For k\ge0, subtracting k\deg_C D_0+1 points gives

Negative twists also have no sections on higher flag members: general curves of this kind sweep out a dense subset, and their restrictions have negative slope. If a hypersurface cut in the flag has degree e with respect to D_0, write b_{Z,j}(k) for the section dimension on the member Z. The restriction sequences and negative-twist vanishing give

Since D_0^{i-1}\cdot Z'=e(D_0^i\cdot Z), summing the degree-(i-1) leading term on Z' divides its coefficient by ei. Induction gives, on an i-dimensional flag member Z,

with the implied constant independent of j. Since \sum_{j<q}r_j=q^n/n!+O(q^{n-1}), (4.3) becomes

All asymptotic errors in this proof concern the fixed variety and fixed divisors; their thresholds need not be uniform in X. On the other hand,

The first inequality in (4.1), together with l\ge1, makes the difference of the leading coefficients positive. Thus G is big.

We next show that, for all sufficiently large divisible m, there is no divisorial valuation v over Y whose centre is double dominant and satisfies

Here v(V_m) is the order of the base ideal of V_m\subset H^0(Y,mA), and A_Y(v) is the log discrepancy. The Seshadri assumption and a_2c>L imply that ma_2H separates jets modulo the Lm-th power of the maximal ideal on a dense open subset of U, for all sufficiently large divisible m. Indeed, at a smooth point satisfying the Seshadri bound the divisor a_2h-LE_x on its blowup is ample. Serre vanishing supplies the jet separation, and surjectivity of evaluation is open.

If (4.5) held, localize a resolution carrying v over the generic point of the first factor and extend to \overline{\C(U)}. The divisor remains horizontal; its discrepancy and its base-ideal order become the corresponding quantities on this smooth geometric generic fibre. Double dominance ensures that its centre meets the second-factor jet-separation open. At a closed point y of that intersection, the restricted base ideal I would satisfy \lct_y(I)\le1/m. For J=I+\mathfrak m_y^{Lm}, the sum-of-ideals inequality gives

[8, Lemma 2.18]. The threshold–colength inequality [7, Theorem 1.1] then gives

Jet separation allows us to choose s sections g_1,\ldots,g_s of ma_2H, defined over \C, whose images modulo J are linearly independent over \overline{\C(U)}. Namely, after scalar extension the images of any complex basis span the quotient, so one can select a basis among them. If f_1,\ldots,f_q is a complex basis of H^0(U,ma_1H), their qs products f_i g_j are linearly independent modulo V_m. To see this, restrict a complex linear relation to the geometric generic fibre and reduce modulo J. Independence of the g_j there makes each coefficient \sum_i c_{ij}f_i zero at the generic point. These are complex linear relations among the f_i, so all c_{ij} vanish.

Consequently (4.6) gives the lower leading coefficient

for m^{-2n}\dim(Q_m/V_m). By the second inequality in (4.1), this exceeds the upper coefficient in (4.4), a contradiction. The numerical bounds are uniform over the chosen jet-separation open, so the sufficiently large m can be fixed independently of the hypothetical valuation.

This exclusion controls the discrepancies of a general divisor with the prescribed diagonal vanishing along every double-dominant centre. We now turn that discrepancy control into the second assertion of (4.2), by comparing an adjoint ample model with the diagonal blowup near such centres.

Removing base components that dominate both factors

Fix such an m, also large enough that |mG| has generically finite map. First choose a log resolution p_0:W\to P of its base ideal, together with E and N_K=\pi^*K_Y. Then choose \Delta_P\in\frac1m|mG| general. Bertini makes its free part smooth and transverse to the fixed normal crossings divisor, so the resulting boundary on W is log smooth. Write

Every double-dominant divisor on W has coefficient less than one in \Delta_W. To verify this, use

For the finitely many exceptional or fixed divisors on this resolution, a general member has valuation v(V_m)/m after pushforward to Y. Its coefficient in \Delta_W is therefore

when its centre is double dominant, by the exclusion of (4.5). Components of the general free part have coefficient 1/m<1. Formula (4.7) remains valid if E occurs in the fixed part of |mG|: pulling back the pushforward restores exactly the prescribed multiplicity m\tau E.

Replace negative coefficients of \Delta_W by zero, and decrease all coefficients at least one to rational numbers strictly below one. The resulting boundary B_W is klt. Set L_W=K_W+B_W; then

where N_K\ge0, G_e\ge0 is p_0-exceptional, and every component of F_N\ge0 fails to be double dominant. The assertion about G_e follows because \Delta_P is effective: negative coefficients can occur only on p_0-exceptional divisors. The assertion about F_N is the preceding coefficient bound.

The divisor K_W+B_W is big. Indeed, K_W is effective, and B_W contains with positive coefficient the big free part of |mG|. The existence of ample models for big klt adjoints [3] gives a projective klt pair (V,B_V), the ample rational adjoint divisor D_V=K_V+B_V, and a common smooth resolution

such that

We choose divisor representatives compatibly throughout these formulas.

Suppose that \mathbf B(A-tE) has a double-dominant component. Its image in Y contains a point z=(x,y) satisfying two conditions: neither x nor y lies on the image of a nonconstant rational curve in U, and z avoids the images of N_K and F_N. Such a point exists. The effective canonical divisor makes U non-uniruled, so images of rational curves are contained in a countable union of proper closed subsets. Double dominance, and uncountability of \C, allow us to avoid their inverse images and the finitely many other indicated subsets.

Every fibre of q is rationally chain connected because (V,B_V) is klt, by [20, Corollary 1.6]. If a fibre meets (\pi p)^{-1}(z), all its rational chains have constant image in both copies of U: a nonconstant image would be a rational curve through x or y. Thus that entire fibre maps to z. This uses the all-fibres statement; chains through special points may equivalently be obtained by proper specialization.

It follows that V maps to Y near the points lying over z. Here is the precise local argument. Let \Gamma\subset V\times Y be the closure of the graph of the birational map V\dashrightarrow Y. Every fibre of \Gamma\to V meeting V\times\{z\} is a singleton, by the preceding paragraph and the map S\to\Gamma. Properness makes this projection finite near each such fibre, and finite birationality over normal V makes it an isomorphism there. The image in Y of its non-isomorphism locus is closed and avoids z. Removing it, and the images of N_K,F_N, gives an open Y^0\ni z for which the graph is an open V^0\subset V, proper over Y^0, and S_{Y^0}=q^{-1}(V^0).

Over Y^0, Equations (4.8) and (4.9) reduce to

The divisors D' on P_{Y^0} and D_V on V^0 are relatively ample over Y^0: the first has relative class -tE, and the second is the restriction of an ample divisor. Adding an effective exceptional divisor to a birational pullback does not change its pushforward section algebra, by normality. The relative Proj of the two sides of (4.10) therefore identifies P_{Y^0}\simeq V^0. On this common model, pushforward gives D'=D_V, and (4.10) gives E_V=r_0^*G_e.

For divisible k, the model formulas give the following map of complete section spaces:

The first isomorphism pulls sections to S, multiplies by the canonical section of kE_V, and descends to W. The last uses L_W+F_N=p_0^*(D'+N_K)+G_e and removes the effective exceptional divisor G_e by pushforward. Large divisible multiples of D_V are globally generated. Their images in (4.11) generate above z: there F_N=N_K=0, and E_V=r_0^*G_e shows that multiplication and removal of the exceptional factors cancel. Thus under P_{Y^0}\simeq V^0 the transported section is exactly the original section of kD_V. Finally N_K is a fixed divisor and can be removed. Indeed, under Y\to X\times X, K_Y is effective exceptional and A is pulled back from X\times X, so

for divisible k. Its pullback N_K has zero coefficient along E; dividing by its canonical section therefore preserves the vanishing condition imposed by -ktE. Hence

Generation above z contradicts the assumed base component. This proves (4.2).

An ample class with controlled corrections

The preceding argument controls where base conditions occur, rather than their number or multiplicity. This is enough: on each correction, Lemma 3.1 annihilates the pullback of T(U) from at least one factor.

Choose an ample rational divisor on P of the form

with m' large and \lambda>0 small. For sufficiently small rational \varepsilon>0, the stable base locus of A-tE-\varepsilon J still has no double-dominant component. In fact

The first three terms satisfy (4.2) by the strict margins in (4.1); adding the last effective term can introduce base components only in its non-double-dominant support.

Take a divisible multiple whose base locus is the stable base locus, and principalize its base ideal on a smooth w:\widetilde P\to P, with normal crossings support also including E. This can be done without changing the complement of the base locus. Write

where M is semiample and F_{\mathrm{fix}}\ge0 has only non-double-dominant components. Choose an effective w-exceptional T with -T relatively ample. For sufficiently small rational u>0, \varepsilon w^*J-uT is ample. Consequently

is ample and has a Kähler representative with D\ge H_1+H_2. If F is the strict transform of E, then

Its components are smooth and fail to be double dominant. All further constants C depend only on n,c, and may increase from line to line. Volume monotonicity in (4.12) gives

For general x\in U, the fibre \widetilde P_x over the first projection is smooth and birational to U. All the correction components avoid the fibre above the diagonal point (x,x): for each one, at least one projection has proper image. Thus the original exceptional space there is unchanged and

where D_x=D|_{\widetilde P_x}. The last inequality follows from D_x\ge H_2|_{\widetilde P_x}.

The diagonal class and its norm

We will construct a middle-degree class v whose intersection with D^n is fixed, but whose norm is small when r(U) is large. A test class with the same intersection and the improved growth of Lemma 3.2 will then force an upper bound for r(U).

Put r=r(U)>0; positivity follows from the canonical generator, since H^{n,0}(U) is one-dimensional. By Lemma 3.1, T(U) is primitive. Choose a Hodge-homogeneous basis \alpha_1,\ldots,\alpha_r of T(U)_\C, orthonormal for its Hodge–Riemann Hermitian form, so that

Put \beta_i=\overline{\alpha_i}/c_i and, on \widetilde P, set

using pullback from Y. The graded product rule and the dual-basis normalization give

On the diagonal, the class v has integral l; hence its restriction to F is H_1^n|_F. On every component of Z, one projection has image of dimension less than n, so Lemma 3.1 kills the corresponding factor in v. The projection formula and (4.12) now give

For the second identity, integrate over the first factor and use \int_{F_x}D_x^{n-1}=t^{n-1} from (4.14).

For R\ge1, consider \omega=D+RH_1. We claim that

By Hodge-star duality, it suffices to pair with harmonic (n-1,n-1)-forms g. On the general fibre, Lefschetz decomposition of the (1,1)-class [F_x] bounds its squared Hodge norm by a dimensional constant times

This is bounded by (4.14), since the two numerators are t^{n-2} and t^{2n-2}. Consequently

Here replacing a closed restricted form by its harmonic representative can only decrease its norm. Integrate against H^n on U and apply Cauchy–Schwarz. Pointwise restriction to the vertical tangent space does not increase the norm, while

Fibre integration outside the measure-zero exceptional sets therefore gives

Since \int_UH^n=l\le2, this proves (4.17).

Decompose v=v_0+v_1 into its primitive and nonprimitive parts for \omega. Lefschetz linear algebra has a dimension-only inverse bound on the nonprimitive middle-degree part. Equations (4.15), (4.16), and (4.17) imply

Hodge–Riemann positivity on v_0, and orthogonality of Lefschetz summands for both the Hodge norm and the middle intersection pairing, then give

Indeed, the signed self-pairing of v_0 is its squared norm, whereas the absolute self-pairing of v_1 is bounded by its squared norm; (4.15) supplies the remaining term l^2/r.

To test the fixed intersection in (4.16), set

Thus H_1^n z'=0, and |c'|\le C by H_i\le D and (4.13). The same inequalities give \norm{z'}_{D+H_1}\le C: the indicated smooth representatives have bounded pointwise norm for D+H_1, whose volume is bounded. The form H_1 has rank at most n, so Lemma 3.2 yields

Since vH_2=0, cup-product Cauchy–Schwarz and (4.18) now imply

Choose R depending only on n,c so large that the term C R^{-\delta_n/2} is at most t^n/2. The remaining inequality bounds r in terms of n,c, as required.□

Proof. We explain the two scalar-order arguments from [36] 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

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 5.2 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 uth infinitesimal neighbourhood has rank at most

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

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 [36], 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 4.1 forces \varepsilon(P_i)\to0, and hence (5.6) holds. Take small projective \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 5.3 produces members of a fixed dimension e<n on a subsequence. With W_i their smooth models and L_i=P_i|_{W_i},

Here K_{V_i}\sim0, so C' can be fixed. Input 5.4 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 \Q-factorial good model Z_i' of this big adjoint exists by [3]; write N_i' for its big semiample adjoint. Then

On a common resolution, with maps x_i to U_i and y_i to Z_i', there is also the effective comparison

To prove it, write x_i^*(K_{U_i}+\Gamma_i)\sim_{\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_{\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 (5.7).

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 \Q-factorial klt, and satisfy

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 (5.7). 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 (5.5), a contradiction. This is the dimension-decreasing part of the first scalar-estimate proof in [36], stopped before its separate fibration contradiction.□

Proof. Its leaves have positive dimension. A leaf cannot fill V, since (5.2) and (5.5) would give \gamma(P;V)\le n\eta(P^n)^{1/n}, contradicting (5.1). Its function field is relatively algebraically closed in \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,

Choose H_M general for this resolution as well. With the canonical generator used to represent K_{\widetilde T}, terminality gives

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 (5.10) 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

The map \widetilde\phi is constant on C^*, so (5.9) 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 5.2, the section-ratio field of J is contained in \C(M). The reverse inclusion follows from ratios of hyperplane pullbacks, since H_M is very ample. Thus this section-ratio field is exactly \C(M).

For a small positive rational \varepsilon, the pair (T,\varepsilon J) is klt and its adjoint is pseudo-effective. Theorem 2.3 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 \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 \Q-factoriality, gives

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

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\PP^b with f^*\OO_{\PP^b}(1)=\OO_{\widehat Y}(H). For every effective real Weil divisor L on \widehat Y,

In fact f^*f_*L-L\ge0 and f_*L\sim_{\R}\ell(L)\OO_{\PP^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 (5.14) 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)_{\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

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

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

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

We use its numerical formulation, as in [32, Chapter III] and [4]. Fix the dimension-dependent integer M=M(n) in Birkar’s Calabi–Yau effective birationality theorem [1, 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

Here the trace of the one-class positive product is the positive part of the divisorial Zariski decomposition; see [5, 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

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 2.3 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 \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

Its hypotheses are exactly that \overline X is projective klt, K_{\overline X}\sim_{\R}0, and \overline A is big and integral; no Cartier-index bound is required.

A single rational number u\ge0 gives inclusions

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 \vol(A+uE_0)\ge\vol(\overline A). For all sufficiently small t>0 we have N_0'\ge tuE_0, and hence

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 (5.15) and continuity, it takes the value (2M)^{-1} at some t>0. At this parameter,

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 \Q-factorial, the rational class has a rational Cartier divisor representative. This proves (5.13).□

Proof. If \mu<0, a nonzero section would yield a subsheaf of slope at least zero, contradicting semistability. Suppose \mu\ge0. For b\ge2, the Grauert–Mülich–Spindler theorem [17, 38], in its torsion-free form [23, Corollary 3.1.6], bounds the highest degree a of the splitting on a general line by a\le\mu+(u-1)/2\le\mu+u. For b=1, the splitting theorem on \PP^1 and semistability give the stronger equality a=\mu.

A general linear flag gives, for any torsion-free sheaf with highest general-line splitting degree a,

Indeed the successive restrictions can be chosen torsion-free: embed the sheaf in a vector bundle and choose the cuts general for the associated primes of the quotient as well. Testing general lines shows that twists by integers below -a have no sections. Starting with the bound u(a+1) on a line, the hyperplane restriction sequences and the identity \sum_{i=0}^a {i+b-1\choose b-1}={a+b\choose b} prove (5.25) inductively.

For an integer t\ge1, let

Finite separable pullback preserves semistability [41, Lemma 2.1]. Since f_t is flat, \mathcal G_t=f_t^*\mathcal G\otimes\OO_{\PP^b}(b(t-1)) is torsion-free and semistable, of rank u and slope t\mu+b(t-1). Moreover

To see this, multiply the pulled-back sections by the t^b homogeneous monomials

On x_0\ne0, these monomials form a basis of the upper function field over the field generated by the t-th powers of the affine coordinates. A linear relation among their products with pulled-back sections therefore separates into linear relations among the original sections, which proves (5.26).

Apply (5.25) to \mathcal G_t, whose highest general-line degree is at most t\mu+b(t-1)+u. Divide the resulting bound by t^b and let t\to\infty, keeping \mathcal G and u fixed. This yields

and hence the asserted bound, for example with C_b=b^b/b!.□

Completion of the proof of Theorem 5.1. Choose a finite linear projection f:\widehat Y\to\PP^b for the embedding defined by H. It has degree N=H^b and f^*\OO_{\PP^b}(1)=\OO_{\widehat Y}(H). The torsion-free sheaf f_*\mathcal E_m has rank Nr_m and

For this, let \mathcal A be its first Harder–Narasimhan term. Adjunction gives a nonzero map f^*\mathcal A\to\mathcal E_m. After removing torsion and extending across codimension two, it gives a map from the reflexive pullback f^{[*]}\mathcal A to the reflexive sheaf \mathcal E_m; denote its torsion-free image by \mathcal Q. Finite separable reflexive pullback preserves semistability and multiplies slopes for these polarizations by N [41, Lemma 2.1]. Consequently

proving (5.27). No injectivity of the adjunction map is required.

Apply Lemma 5.8 to the semistable Harder–Narasimhan factors of f_*\mathcal E_m and sum their section bounds. Their ranks sum to Nr_m, and N is bounded in terms of n. For all sufficiently large divisible m, this gives

This is the target estimate (5.17). Divide by m^n and use the volume asymptotic in divisible degrees. Since n=s+b, we obtain

The lower calibration bound v(D)\ge c\delta then implies c^b/n!\le C\eta^s. This is impossible for a sufficiently small positive \eta depending only on n. There are only finitely many pairs b,s>0 with b+s=n, so choose \eta for all of them at once, also satisfying the earlier verticality and chain requirements, and then impose the rank threshold in Lemma 5.5. The resulting contradiction bounds r(T) in terms of n alone.□

Proof. The extraction and \Q-factorialization theorem for klt pairs [3, 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,

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

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

Let \mathcal T_R(-\log\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 \C-algebra

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 6.3. 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

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 \C(P_i) in \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}_{\C(P_i)}\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 \C(P)/\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

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\C(R_i)^*. Pulling (6.2) 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 (6.1); proportionality of the forms proves that each h_i is B-birational.□

Proof. By Han and Jiang [21, 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\PP^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,

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 [26, 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 [36, 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 \Q-factorial terminalization of Y. It still has principal canonical divisor, and the two properties just stated are birational invariants. Theorem 5.1 therefore bounds \dim_{\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

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 3.1, birational invariance of the Hodge structure generated by the top form gives

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

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

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 6.1. For n=0 the pair is a point and the character is trivial, so take L(0,m)=1. Suppose n>0. Lemmas 6.2 and 6.4 express c^{n!} as in (6.1). The rationally connected factor satisfies the hypotheses of Lemma 6.5; in particular it remains rationally connected by the cover-stable choice in Input 6.3. Every other factor satisfies Lemma 6.6. Define

Each d_i^{E(n,m)}=1, and (6.1) 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 [29, Proposition 14] to the morphism Y\to\Spec\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\OO_Y. That proposition provides a birational comparison commuting exactly with the degree-m residue maps, and hence the displayed identity. The divisor identities (7.2) 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 \PP^1-links [29, Theorem 10]. Over the function field of a link’s base, the two sections are 0,\infty on \PP^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 [39, Resolution Lemma, p. 633]. Resolve the map to T and its boundary, using smooth blowup centres having normal crossings with the accumulated boundary [27, Definition 25 and Theorem 35]. This gives a common smooth model p:W\to S, q:W\to T. Write

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

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

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

Choose positive real numbers \alpha_1,\ldots,\alpha_k linearly independent over \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 \C(W). Its value group is \sum_j\Z\alpha_j, and its residue field is \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 \Q-linearly independent, then

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 (7.4), the values of the x_i are the rows of E applied to (\alpha_j); they are positive and independent. Apply (7.6) to \OO_{T,\eta_C}. Since q is birational, its fraction field and that of \OO_{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 \C(Z)=\C(C). This proves (7.5).

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 (7.3), with A'=C and the birational map induced through Z. Finally (7.2) shows that this map is B-birational.□

Proof of Theorem 1.1. First let the ground field be \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

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_\Q0. By Theorem 2.2, 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

Choose projective crepant \Q-factorial dlt modifications (Y_i,\Gamma_i)\to(X_i,\Delta_i) [29, 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\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

The ratio is therefore a nonzero constant in the conductor function field. Write

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 7.2 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 7.1. 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

The product respects the orientations of the walk. Moreover \dim A\le d-1 and m(K_A+\Gamma_A)\sim0 by (7.2). Figure 1 displays the composition for a three-edge cycle. The same argument includes loops, including the involution at a nonsplit node.

Choose

where L(q,m) is given by Theorem 6.1. Applying that theorem to (7.9) shows that the product of r_e^L around every closed walk is one. Consequently there are constants c_i\in\C^* satisfying

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\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 (7.7) 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

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_*\OO_U=\OO_X, the last equality following from S_2. Hence \mathcal F_a is invertible. Equivalently, divide \sigma by the ath 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 \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 \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 \C with the same integer (7.10).

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 \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_{\C},\OO_{X_{\C}}) =\C. Its one-dimensional space over k_0 therefore has a nonzero section whose evaluation becomes an isomorphism over \C and hence is already an isomorphism over k_0. Thus \mathcal F_a\simeq\OO over k_0, and extending to k proves the theorem.□