Proof. We may shrink \(C\) about \(c\) throughout. In particular, we take a uniformizer \(z\in\mathbb C(C)\) with \(\mathop{\mathrm{Div}}_C z=[c]\) on this neighbourhood. The fibre \(T\) is connected and projective, since \(f\) is a projective contraction. Its irreducible components have dimension \(\dim N-1\): the divisor \(F\) is an effective Cartier divisor on the integral variety \(N\), and its support is the whole fibre. When \(\dim N=1\), the contraction \(f\) is an isomorphism of normal curves. Comparing the coefficient at \(c\) in (10) gives \(p_0\beta\in\mathbb Z\), so this case is immediate. Henceforth \(1\leq\dim T\leq3\).

Adjunction on the entire reduced fibre. Write \(T=\bigcup_i T_i\). By dlt adjunction and the structure theorem for dlt strata, each \(T_i\) is normal and \[ (K_N+T+H)|_{T_i}\sim_{\mathbb Q}K_{T_i}+\Theta_i, \tag{11}\] where \((T_i,\Theta_i)\) is lc and \(\Theta_i\) is the effective different; see [17]. We use the residue form of this adjunction [18]. If \(n\) is sufficiently divisible, the \(n\)th tensor power of the ordinary residue at the generic point of \(T_i\) extends to an isomorphism \[\mathcal O_N\bigl(n(K_N+T+H)\bigr)|_{T_i} \simeq \mathcal O_{T_i}\bigl(n(K_{T_i}+\Theta_i)\bigr).\] The different is defined so that the generic residue extends uniquely to this canonical isomorphism once the ambient Cartier index is cleared. Here \(T_i\) is normal, so its normalization does not change the target. The isomorphism therefore identifies actual rational pluricanonical forms. The same form of adjunction will be used for normal dlt components upstairs. Intersections of distinct components of \(T\) are unions of lc strata. At the generic point of an intersection of codimension two in \(N\), the pair is simple normal crossing, precisely two components of \(T\) pass through the point, and \(H\) is absent. The corresponding conductor prime on either branch occurs in \(\Theta_i\) with coefficient one.

The remaining coefficients of the differents belong to the set \[ \mathcal D(\Phi)= \left\{ \frac{r-1+b}{r}\ \middle|\ \begin{gathered} r\in\mathbb N_{>0},\quad n_\phi\in\mathbb Z_{\geq0},\\ b=\sum_{\phi\in\Phi\cup\{1\}}n_\phi\phi\leq1 \end{gathered} \right\}\cap[0,1]. \tag{12}\] This is the usual coefficient rule for the different; see [15]. The positive elements of \(\Phi\cup\{1\}\) have a positive minimum, so the possible sums \(b\leq1\) form a finite set. For each such sum, the displayed coefficients are nondecreasing in \(r\) and converge to one. Consequently \(\mathcal D(\Phi)\) is a fixed rational DCC set. We include the conductor coefficient one in this set.

To apply the index theorem to \(T\), we must glue this component adjunction on the reduced scheme itself. We will prove that \(T\) is demi-normal and that divisible residues identify its log pluricanonical sheaf with an ambient restriction. The degree needed for this adjunction may depend on the pair. Conductor-compatible descent is also the issue addressed by the admissible-section methods of Fujino [10] and Gongyo [13]; here we give the local comparison needed for the later uniform degree.

Apply the depth theorem [18] to the dlt pair \((N,T+H)\) with the effective integral divisor \(D=T\leq\lfloor T+H\rfloor\) and \(L=0\). It gives that \(\mathcal O_N\), \(\mathcal O_N(-T)\), and \(\mathcal O_T\) are Cohen–Macaulay. Here \(\mathcal O_N(-T)\) is the ideal of the reduced union of the prime divisors \(T_i\): on the normal variety \(N\), it is the intersection of their height-one prime ideals, equivalently the regular functions vanishing to order at least one at every \(T_i\). Thus the subscheme in the depth theorem is the reduced fibre used here. In particular, \(T\) satisfies \(S_2\).

The normality of the \(T_i\) and the generic normal crossing description show that \(T\) has only smooth points and ordinary double crossings in codimension one. It is also seminormal. By the square–cube criterion, an element of the total quotient ring whose square and cube are regular lies in the local ring at each codimension-one point, where seminormality is explicit. The \(S_2\) extension property then puts it in \(\mathcal O_T\). Thus \(T\) is demi-normal. Its normalization is the disjoint union of the \(T_i\), and its conductor divisor is the union of the double-crossing divisors described above.

Remove the coefficient-one conductor divisors from the \(\Theta_i\) and push the remaining boundary cycles to \(T\); denote the result by \(B_T\). Off the conductor the normalization is an isomorphism in codimension one, so this does not double count any prime. The boundary \(B_T\) is effective, has coefficients in \(\mathcal D(\Phi)\), and has no component contained in the conductor.

Choose an open immersion \(j:T^\circ\hookrightarrow T\) whose complement has codimension at least two, such that \(T^\circ\) has only smooth points and ordinary double crossings. We may also arrange that at each crossing the ambient pair is simple normal crossing with just the two vertical branches, and that the support of \(B_T|_{T^\circ}\) is contained in the smooth locus. The dualizing sheaf of \(T^\circ\) is invertible, and the relevant multiples of \(B_T|_{T^\circ}\) are Cartier. For such a multiple \(n\), interpret \(\mathcal O_T(n(K_T+B_T))\) initially as \[j_*\bigl(\omega_{T^\circ}^{\otimes n} \otimes\mathcal O_{T^\circ}(nB_T|_{T^\circ})\bigr).\] We claim that for every sufficiently divisible positive integer \(n\), the residue maps give a natural isomorphism \[ \mathcal O_N\bigl(n(K_N+T+H)\bigr)|_T \simeq \mathcal O_T\bigl(n(K_T+B_T)\bigr). \tag{13}\] Choose \(n\) even and divisible enough that the ambient sheaf is invertible and the component residue isomorphisms hold. Away from the conductor, these isomorphisms give (13) on \(T^\circ\). At a node, take local normal crossing coordinates \(x,y\) for the two branches. Hypersurface adjunction for \(xy=0\) identifies the restriction of the ambient log canonical line with the dualizing line of the node. The two iterated residues of \(dx/x\wedge dy/y\) differ by a sign; their \(n\)th tensor powers give the dualizing-sheaf gluing and have equal next residues when \(n\) is even. This proves the claimed residue identification on \(T^\circ\).

An invertible sheaf on the \(S_2\) scheme \(T\) equals the extension of its restriction from \(T^\circ\). Applying this to the left side of (13) and using the definition of the right side proves the isomorphism on \(T\). In particular, the defined extension is invertible in these degrees, and the isomorphism retains the actual rational residue forms on every normalized branch.

It follows that \(K_T+B_T\) is \(\mathbb Q\)-Cartier. Its pullback to the normalization is \(K_{T_i}+\Theta_i\), with the conductor included, so \((T,B_T)\) is slc. Equation (10), together with \(F=\mathop{\mathrm{Div}}(f^*z)\), makes \(K_N+T+H\) rationally linearly trivial near \(T\). Restricting a sufficiently divisible trivialization in (13) gives \[ K_T+B_T\sim_{\mathbb Q}0. \tag{14}\]

A uniform degree on the slc fibre. Apply global ACC to the projective normalized component pairs \((T_i,\Theta_i)\), whose adjoints are numerically trivial by (14). Their dimensions are at most three and their coefficients lie in the fixed DCC set \(\mathcal D(\Phi)\); hence their nonzero coefficients lie in a fixed finite rational set [15]. The same is true of \(B_T\). The bounded index theorem for projective slc log Calabi–Yau pairs in dimensions at most three now gives a common integer annihilating \(K_T+B_T\) [16]. Enlarging this integer, choose once and for all an even multiple \(p\) of \(p_0\), depending only on \(\Phi\) and \(p_0\), such that \[ pB_T\text{ is integral},\qquad p(K_T+B_T)\sim0. \tag{15}\] Choose a nowhere-vanishing generator \(u\) of this trivial log pluricanonical line. On the normalization it is a tuple of rational \(p\)-pluricanonical forms \(u_i\) on the \(T_i\), with the prescribed boundary poles and conductor gluing.

The comparison uses two degrees for different purposes. A sufficiently divisible degree \(n=bp\) makes \(u^b\) a generating ambient restriction and will control divisors of componentwise ratios. The conductor comparison will then take place in the uniform degree \(p\), using only the ordinary node description there.

The root cover. Write \[ p_0\beta=\frac{a}{m},\qquad a\in\mathbb Z,\quad m\in\mathbb N_{>0},\quad\gcd(a,m)=1. \tag{16}\] Thus the goal is \(m\mid p/p_0\). The case \(m=1\) is immediate; assume \(m>1\). Let \(C'\to C\) be the normalization in \(\mathbb C(C)(w)\), where \(w^m=z\), and let \(\pi\colon Y\to N\) be the normalization of the dominating component of \(N\times_C C'\). The polynomial \(w^m-z\) is Eisenstein at \(c\). Thus \(C'\) has a unique point \(c'\) over \(c\), with total ramification of degree \(m\) there. Since \(f\) is a contraction of normal varieties, \(\mathbb C(C)\) is algebraically closed in \(\mathbb C(N)\); in characteristic zero this is a regular function-field extension. It is consequently linearly disjoint from \(\mathbb C(C')/\mathbb C(C)\). The map \(\pi\) has degree \(m\) and cyclic Galois group \(\mu_m\), acting by \(w\mapsto\zeta w\). Write \(f_Y:Y\to C'\) for the induced morphism and put \(T_Y=(\pi^{-1}T)_{\mathrm{red}}\). Figure 1 records these maps and their reduced special fibres; the total space \(Y\) is the normalization of the base change.

The morphism \(f_Y\) is projective. Its function field is regular over \(\mathbb C(C')\), because regularity is preserved by this base change. Stein factorization therefore gives \((f_Y)_*\mathcal O_Y=\mathcal O_{C'}\). In particular, \(T_Y\) is connected and projective. The divisor \(p_0(K_N+T+H)\) in (10) is integral. If \(F=\sum_i e_iT_i\), comparison of coefficients in that equation and (16) gives \(m\mid e_i\) for every \(i\). At the generic point of \(T_i\), write \(f^*z=\tau^{e_i}v\) with \(v\) a unit. After dividing \(w\) by \(\tau^{e_i/m}\), the normalized extension is obtained by extracting an \(m\)th root of \(v\) and is étale. Away from \(T\) the function \(f^*z\) is a unit, and the same conclusion holds. Hence \(\pi\) is étale in codimension one.

Put \(H_Y=\pi^*H\), and choose canonical divisors compatibly. Since \(\pi\) is quasi-étale and the coefficient-one divisor pulls back without divisorial ramification, we have \[ K_Y+T_Y+H_Y=\pi^*(K_N+T+H). \tag{17}\] The pair on the left is dlt. Indeed, finite crepant discrepancy comparison multiplies a downstairs log discrepancy by the ramification index of a divisorial prolongation [19]. Thus the upstairs pair is lc, and every upstairs lc centre maps onto a downstairs lc centre. The pair downstairs is simple normal crossing near the generic point of each such centre. A finite quasi-étale map is étale over the smooth locus of its target, by purity of the branch locus [20]. Consequently the upstairs pair is simple normal crossing at the generic points of all its lc centres. A log resolution chosen to be an isomorphism over this log smooth open has no exceptional divisor of log discrepancy zero, which is the dlt condition. This argument does not require \(Y\) to be \(\mathbb Q\)-factorial. In particular, each irreducible component \(T_{Y,j}\) of \(T_Y\) is normal.

Let \(\theta\) be the rational top differential form defining \(K_N\). Define the rational \(p_0\)-pluricanonical form \[ s=(\pi^*\theta)^{\otimes p_0}\,\pi^*\psi\,w^{-a}. \tag{18}\] Since \(\pi^*F=m\mathop{\mathrm{Div}}_Y w\), equations (10)–(17) give \[\mathop{\mathrm{Div}}(s)+p_0(T_Y+H_Y)=0.\] Thus \(s\) generates the degree-\(p_0\) divisorial log pluricanonical sheaf on \(Y\). In particular its sufficiently divisible powers generate the corresponding invertible log pluricanonical sheaves.

Residue comparison on each component. Let \(T_i\) be the component of \(T\) dominated by \(T_{Y,j}\), and write \(\rho_j\colon T_{Y,j}\to T_i\) for the induced finite morphism. At their generic points the map is étale. The ordinary generic residue \[v_j=\mathop{\mathrm{res}}_{T_{Y,j}}\bigl(s^{p/p_0}\bigr)\] is therefore a nonzero rational \(p\)-pluricanonical form. The form \(\rho_j^*u_i\) is also nonzero, and the quotient \[ r_j=\frac{v_j}{\rho_j^*u_i}\in\mathbb C(T_{Y,j})^* \tag{19}\] is a rational function. We prove that \(r_j\) is constant by comparing sufficiently divisible powers.

Choose a multiple \(n=bp\) sufficiently divisible for (13) and for component adjunction upstairs. Then \(v_j^b\) is a nowhere-vanishing generator of the degree-\(n\) adjunction sheaf on \(T_{Y,j}\). The same is true of \(\rho_j^*u_i^b\). For the latter assertion, work near any point of \(T\). By (13), the section \(u^b\) is a generator of the restriction of the invertible sheaf \(\mathcal O_N(n(K_N+T+H))\). Relative to a local ambient generator it is a unit on \(T\). Such a unit lifts, after restricting the neighbourhood, to a unit on \(N\). Hence \(u^b\) is the residue restriction of a local ambient log generator. Pull that generator back to \(Y\). It remains a log generator by (17); its residue on \(T_{Y,j}\) generates the upstairs adjunction sheaf. At the generic point residue commutes with the étale pullback, so this residue is exactly the rational form \(\rho_j^*u_i^b\). This identifies the rational forms everywhere they are defined, and proves the assertion at every point of \(T_{Y,j}\).

It follows that \(r_j^b\) has zero divisor on \(T_{Y,j}\), and hence so does \(r_j\). A rational function with zero divisor on a normal projective integral variety is an invertible global function, and therefore a nonzero complex constant. We have proved \[ v_j=r_j\rho_j^*u_i,\qquad r_j\in\mathbb C^*. \tag{20}\]

Gluing the scalars and killing the character. Suppose two distinct components \(T_{Y,j}\) and \(T_{Y,j'}\) meet. Their intersection contains a stratum of codimension two in \(Y\) along which the pair is generically simple normal crossing. Its image is a codimension-two lc centre downstairs. At the generic point of this image the map \(\pi\) is étale and the downstairs pair is simple normal crossing. The two downstairs vertical branches are distinct: étale inverse images of one smooth divisor cannot give two intersecting branches. Moreover \(H\) is absent at this generic crossing.

At the generic point \(\xi\) of the upstairs crossing, choose the two normal coordinates \(x,y\) and a regular ambient \((\dim Y-2)\)-form \(\eta\) whose restriction is a nonzero top form on the smooth stratum. Such choices are available in normal crossing coordinates. Since \(s^{p/p_0}\) is a log generator, it has the local form \[s^{p/p_0} =\varepsilon \left(\frac{dx}{x}\wedge\frac{dy}{y}\wedge\eta\right)^{\otimes p}, \qquad \varepsilon\in\mathcal O_{Y,\xi}^*.\] The two next residues differ by \((-1)^p\) and are nonzero because \(\varepsilon\) is a unit. They therefore agree. The components of the tuple \(u\) have the same matching, nonzero next residues downstairs: \(u\) is a generator of the slc log pluricanonical line, whose local description at the node is the even tensor power of the dualizing line. Their étale pullbacks retain this property. Taking next residues in (20) now gives \(r_j=r_{j'}\).

Since \(T_Y\) is connected, its component-incidence graph is connected. All the constants \(r_j\) are consequently equal to one scalar \(r\in\mathbb C^*\). In tuple notation on the normalized components of \(T_Y\), we have \[ \bigl(\mathop{\mathrm{res}}_{T_{Y,j}}(s^{p/p_0})\bigr)_j =r\,\pi^*u. \tag{21}\] The tuple \(\pi^*u\) is invariant under the covering group, with the natural action also permuting the components. On the other hand, (18) shows that for \(\zeta\in\mu_m\), \[\zeta^*(s^{p/p_0})=\zeta^{-ap/p_0}s^{p/p_0}.\] Residue is natural under this action, including its permutation of the components. The nonzero scalar comparison (21) forces this character to be trivial. Thus \(m\mid ap/p_0\). Since \(\gcd(a,m)=1\), we obtain \(m\mid p/p_0\). Finally \[p\beta=\frac{p}{p_0}\frac{a}{m}\in\mathbb Z,\] which proves the lemma. ◻

Proof. Put \(L=K_X+B\) and \(K=\mathbb C(Z)\), and use the exact presentation (1). For every positive integer \(l\) divisible by \(p_0\), we first prove the equality of subspaces of \(\mathbb C(X)\) \[ H^0\!\left(X,\mathcal O_X(\lfloor lL\rfloor)\right) =\psi^{l/p_0}f^* H^0\!\left(Z,\mathcal O_Z(\lfloor lD_Z\rfloor)\right). \tag{22}\] Here \(f^*\) on the right means pullback of rational functions. For \(v\in K^*\), the exact divisor identity gives \[ lL+\mathop{\mathrm{Div}}_X\!\left(\psi^{l/p_0}f^*v\right) =f^*\!\left(lD_Z+\mathop{\mathrm{Div}}_Z(v)\right). \tag{23}\] If \(v\) is a section of the rounded system on \(Z\), then \(lD_Z+\mathop{\mathrm{Div}}_Z(v)\geq0\) by the section convention of Section 2. This divisor is rational Cartier, so local effective Cartier multiples show that its pullback is effective. Equation (23) gives the inclusion from right to left in (22).

Conversely, let \(u\) be a nonzero section on the left, and let \(X_\eta\) be the generic fibre over \(K\). The exact presentation rewrites the section inequality as \(\mathop{\mathrm{Div}}_X(u/\psi^{l/p_0})+lf^*D_Z\geq0\). Its horizontal valuations give \[\mathop{\mathrm{Div}}_{X_\eta}\!\left(u/\psi^{l/p_0}\right)\geq0.\] The generic fibre is normal, so this quotient is regular there. The contraction condition \(f_*\mathcal O_X=\mathcal O_Z\) gives \(H^0(X_\eta,\mathcal O_{X_\eta})=K\). Hence \(u=\psi^{l/p_0}f^*v\) for some \(v\in K^*\). By (23), the source section inequality is \(f^*G\geq0\), where \(G=lD_Z+\mathop{\mathrm{Div}}_Z(v)\) is rational Cartier. Effectivity of \(G\) descends to \(Z\): near the generic point of any prime \(P\subset Z\), write a Cartier multiple of \(G\) as \(aP\), with \(P\) locally principal. The local Cartier pullback of \(P\) has a prime component dominating \(P\), with positive multiplicity \(e\), because \(f\) is surjective. Its coefficient in the pullback of the chosen multiple of \(G\) is \(ae\). Thus \(f^*G\geq0\) forces \(a\geq0\). Doing this for every \(P\) proves \(G\geq0\), so \(v\) is a section on \(Z\). This proves (22). Since \(p_0\) clears \(\Phi\), the divisor \(lL\) is integral Weil. The argument applies to its reflexive sheaf even when \(lL\) is not Cartier.

We now produce birational sections on the base. Choose a smooth projective determination \(q:W\to Z\) of the nef data which also resolves the boundary, and write \[K_W+B_W^{\mathrm{cr}}+M_W=q^*D_Z.\] Let \(A\) be the strict transform of \(B_Z\) plus the reduced exceptional divisor. On nonexceptional primes \(A\) agrees with \(B_W^{\mathrm{cr}}\); on exceptional primes its coefficient is one, at least the coefficient of \(B_W^{\mathrm{cr}}\) by generalized log canonicity. Thus \[ K_W+A+M_W=q^*D_Z+E_W, \qquad E_W=A-B_W^{\mathrm{cr}}\geq0 \text{ exceptional over }Z. \tag{24}\] Put \(\Lambda=\mathcal B\cup\{1\}\). The pair \((W,A)\) is an ordinary log smooth lc pair with coefficients in the fixed DCC set \(\Lambda\), and \(pM_W\) is nef Cartier. Since \(D\) is big and \(D_Z\sim_\mathbb QD\), the adjoint on the left of (24) is big.

For each \(1\leq k\leq d\), let \(b_k=b(\Lambda,k,p)\) be the integer in Birkar–Zhang’s polarized effective birationality theorem [6]. Its conclusion is that \[\left|\left\lfloor l(K_W+A+M_W)\right\rfloor\right|\] is birational for every positive \(l\) divisible by \(b_k\) when \(\dim W=k\); the floor is taken on the whole adjoint. Set \[m=\operatorname{lcm}(p_0,b_1,\ldots,b_d).\] This integer depends only on \(d\) and \(\Phi\). Fix any positive multiple \(l\) of \(m\) and take \(k=\dim Z\). For a rational section \(v\in\mathbb C(W)=K\) of the displayed birational system, pushforward of its divisor inequality using (24) gives \[\mathop{\mathrm{Div}}_Z(v)+lD_Z\geq0.\] Thus these sections belong to the complete rounded system on \(Z\), and their ratios generate \(K\) because the system on \(W\) is birational. The complete system on \(Z\) is therefore nonempty and generates \(K\). Equation (22) transfers precisely these ratios to \(X\), since the common factor \(\psi^{l/p_0}\) cancels. The complete source system is nonempty and generates the embedded field \(K\subset\mathbb C(X)\) for every such \(l\).

The function field of the image of a system’s rational map is generated by its section ratios, so each of these maps is birationally equivalent to \(f\). To compare with the Iitaka fibration, choose for the individual pair an integer \(r>0\) such that \(rL\) is Cartier. Degrees divisible by \(\operatorname{lcm}(m,r)\) form a cofinal divisible subsequence of the Cartier section series of \(L\). Such a subsequence has the same field of section ratios: given \(u/v\) in one degree, choose \(a\geq1\) so that the multiplied degree lies in the subsequence and write \(u/v=(uv^{a-1})/v^a\). The field defining the Iitaka fibration is therefore \(K\), and \(\kappa(L)=\operatorname{trdeg}_\mathbb CK=\dim Z\). This identifies \(f\) with the Iitaka fibration up to birational equivalence. The individual index \(r\) is used only for this comparison and does not enter \(m\). ◻

Proof. There are three steps. Global ACC gives a finite coefficient set and a uniform positive lower bound for generalized log discrepancies. Boundedness up to isomorphism in codimension one then bounds torsion in the divisor class group. Finally we clear the coefficients of the actual adjoint before applying that torsion bound.

Finite coefficients and uniform discrepancies. We first construct the small modifications and the one-divisor extractions that will allow us to use global ACC. If an exceptional prime divisor \(E\) over \(Z\) is marked, assume that its generalized log discrepancy is \(a\in(0,1)\). Choose a sufficiently high projective log resolution \(g:W\to Z\) on which the nef data descend and on which \(E\) appears, when marked. Write \[K_W+B_W+M_W=g^*D.\] The support of \(B_W\) is SNC, all its coefficients are less than one, and \(B_W\) need not be effective.

We can also arrange that there is an effective exceptional Cartier divisor \(F\) on \(W\) such that \(-F\) is \(g\)-ample and \(\mathop{\mathrm{Supp}}(B_W)\cup\mathop{\mathrm{Supp}}(F)\) is SNC. To see this, construct a resolution by blowups with centres over the singular locus of the normal variety \(Z\), and then resolve the nef model, the marked valuation, and the boundary by further blowups. The latter can be taken with centres of codimension at least two on smooth models. All the resulting exceptional divisors therefore have images of codimension at least two on \(Z\). At each blowup the negative of its effective Cartier exceptional divisor is relatively ample. Adding sufficiently large multiples of the pullbacks of the divisors chosen at the preceding stages gives an effective exceptional Cartier divisor whose negative is ample for the composite morphism. Additional blowups resolving the union of the two supports preserve this construction.

Choose a positive rational number \(\delta\) sufficiently small that all coefficients of \(B_W+\delta F\) are still less than one and, when \(E\) is marked, \(\delta\mathop{\mathrm{mult}}_E F<a\). For a sufficiently positive ample Cartier divisor \(A\) on \(Z\), the divisor \[M_W+g^*A-\delta F\] is ample. Choose a general effective rational divisor \(\Lambda\sim_{\mathbb Q}M_W+g^*A-\delta F\) by dividing a general member of a sufficiently divisible very ample multiple. Bertini’s theorem and the SNC condition on the union of the supports show that \((W,B_W+\delta F+\Lambda)\) is sub-klt. There is a rational principal divisor \(P\) on \(W\) such that \[\delta F+\Lambda=M_W+g^*A+P.\] Since \(g\) is birational, \(P=g^*(g_*P)\). Pushing forward gives \[K_Z+B+g_*\Lambda=D+A+g_*P,\] and pulling this equality back gives \[g^*(K_Z+B+g_*\Lambda) =K_W+B_W+\delta F+\Lambda.\] Thus, setting \(\Xi=B+g_*\Lambda\), the pair \((Z,\Xi)\) is an ordinary effective rational klt pair. If \(E\) is marked, the general divisor \(\Lambda\) does not contain \(E\), so its ordinary log discrepancy is \[a(E,Z,\Xi)=a-\delta\mathop{\mathrm{mult}}_E F.\] This lies in \((0,1)\) by the choice of \(\delta\) and the original bound \(a<1\).

Take a log resolution of \((Z,\Xi)\) dominating \(W\), so that it exhibits \(E\) when marked. Apply the extraction corollary [4] with the prescribed set \(\{E\}\), or with the empty set when no valuation is marked. The klt pair \((Z,\Xi)\) satisfies the lc hypothesis, and \(\Xi\) itself can be used as the auxiliary klt boundary. When a divisor is marked, its discrepancy is less than one, so the extra condition for selected discrepancy-one valuations is vacuous. We obtain a birational morphism \[h:Z^+\longrightarrow Z\] with \(Z^+\) \(\mathbb Q\)-factorial and with exactly the prescribed exceptional prime divisors. The construction in the cited proof is a log terminal model over \(Z\), hence \(h\) is projective. In the empty case it is a small \(\mathbb Q\)-factorial modification.

Restore the original generalized data on \(Z^+\) by setting \[K_{Z^+}+B^++M_{Z^+}=h^*D,\] with the same nef b-divisor \(\mathbf M\). The resulting pair is generalized klt and its adjoint is rationally linearly trivial. Moreover, \(B^+\) is effective: it is the strict transform of \(B\), together with \((1-a)E\) in the extraction case. If a marked divisor is already a prime on \(Z\), the small modification given by the empty-list construction suffices.

We next apply global ACC to these effective generalized pairs, whenever their boundary coefficients belong to a fixed DCC set. There are two cases. If the nef trace is not numerically trivial on a model determining \(\mathbf M\), then on a common higher model the nef part has the form \[M_W=\frac1p(pM_W),\] where \(pM_W\) is a nef Cartier divisor that is not numerically trivial. The generalized global ACC theorem [6] applies: the dimension is fixed, the pair is projective generalized lc, its total adjoint is numerically trivial, the boundary coefficients belong to the chosen DCC set, and the single nef weight is \(1/p\). We adjoin this fixed weight to that DCC set. It follows that the possible boundary coefficients belong to a finite set.

If instead \(M_W\equiv0\), take a common higher model \(\pi:\widetilde W\to Z^+\) on which the nef data descend. Since \(Z^+\) is \(\mathbb Q\)-factorial, the trace \(M_{Z^+}\) is \(\mathbb Q\)-Cartier. The divisor \[M_{\widetilde W}-\pi^*M_{Z^+}\] is exceptional and numerically trivial over \(Z^+\). Applying the negativity lemma to it and to its negative shows that it is zero. Consequently \(M_{Z^+}\equiv0\): every curve on \(Z^+\) is dominated by a curve on \(\widetilde W\), and we use the projection formula. Since the nef data are pulled back from \(Z^+\), the generalized discrepancies are the ordinary discrepancies of \((Z^+,B^+)\). This is an ordinary klt pair with \(K_{Z^+}+B^+\equiv0\), so ordinary global ACC [15] again gives a finite set of boundary coefficients.

Applying this discussion to the small modification with no marked divisor first shows that the coefficients of \(B\) lie in a fixed finite rational set \(I_0\subset I\). It also shows that the original generalized pairs are uniformly generalized \(\varepsilon\)-lc for some \(\varepsilon=\varepsilon(d,I,p)>0\). Indeed, otherwise we could choose a sequence of such pairs and marked valuations whose generalized log discrepancies satisfy \[1>a_1>a_2>\cdots>0,\qquad a_i\longrightarrow0.\] Perform the extraction just constructed when the marked valuation is exceptional, and otherwise take the small modification. The boundary coefficients of the resulting pairs belong to \[I\ \cup\ \{1-a_i:i\geq1\},\] which is a DCC set because the added sequence is strictly increasing. Global ACC would force the coefficients \(1-a_i\) to belong to a finite set, a contradiction.

Bounded topology and class-group torsion. We have obtained the uniform singularity bound without choosing any bound for the class group. We now turn this geometric information into a finite topological list.

Birkar’s boundedness theorem [5] now applies to the original varieties \(Z\). Its hypotheses are projectivity, fixed dimension, generalized \(\varepsilon\)-lc singularities, effective boundary, a real-linearly trivial generalized adjoint, and rational connectedness. The singularity bound was just proved, the adjoint is even rationally linearly trivial, and rational connectedness follows from that of the given smooth projective resolution. Thus, for every \(Z\), the theorem gives a projective variety \(V_Z\), chosen from one bounded set, that is isomorphic to \(Z\) in codimension one. Replacing each \(V_Z\) by its finite normalization preserves this property: normalization is an isomorphism over the common normal open subset, and the complement still has codimension at least two. These normalizations still form a bounded set, as in [5]. Hence there is a projective morphism \(\mathcal V\to S\), with \(S\) of finite type over \(\mathbb C\), such that for every \(Z\) some fibre \(V_s:=\mathcal V_s\), \(s\in S(\mathbb C)\), is normal and isomorphic to \(Z\) in codimension one.

We explain explicitly how this boundedness controls torsion. Put \(U=Z_{\mathrm{reg}}\). The groups \[H_1\bigl(U(\mathbb C),\mathbb Z\bigr)\] are finitely generated and take only finitely many isomorphism types. First, this group is invariant under isomorphism in codimension one of normal varieties. The common open subset, intersected with the regular loci, is obtained from each regular locus by removing a closed subset of complex codimension at least two. Removing such a subset from a smooth complex variety does not change its fundamental group: choose a smooth stratification of the removed subset and perturb paths and homotopies, relative to their prescribed boundaries, to be transverse to its strata. All these strata have real codimension at least four, so the perturbed paths and two-dimensional homotopies avoid them. The assertion for \(H_1\) follows by abelianization.

Second, we prove the assertion for the selected fibres \(V_s\). Replace \(S\) by its reduction and partition it into finitely many locally closed strata over which the family is flat. Cover these strata by finitely many affine opens. On any one of these opens, still denoted \(S\), let \[\mathcal U=\operatorname{Sm}(\mathcal V/S)\] be the full relative smooth locus. Flatness and finite presentation give \[\mathcal U_s=\operatorname{Sm}(\mathcal V_s/\mathbb C)=(V_s)_{\mathrm{reg}}\] for each selected normal fibre, since regularity and smoothness agree over \(\mathbb C\). Embed \(\mathcal V\) in \(\mathbb P^N_{\mathbb C}\times S\). Identify \(\mathbb P^N(\mathbb C)\) with the real algebraic set of Hermitian matrices \(H\) satisfying \(H^2=H\) and \(\mathop{\mathrm{tr}}H=1\), by \([z]\mapsto zz^*/(z^*z)\), where \(z^*\) is conjugate transpose. With real and imaginary coordinates on the affine \(S\), the entire map \[\mathcal U(\mathbb C)\longrightarrow S(\mathbb C)\] is then a continuous semialgebraic map between affine semialgebraic spaces. The semialgebraic triviality theorem of Delfs and Knebusch [8] gives a finite partition of \(S(\mathbb C)\) into semialgebraic pieces over each of which the whole inverse image is a product. Thus the selected regular loci have only finitely many homeomorphism types. Each semialgebraic fibre has the homotopy type of a finite simplicial complex by [8]. The asserted finite list of finitely generated groups \(H_1\) follows.

For each individual \(Z\), the group \(\mathop{\mathrm{Cl}}(Z)\) is also finitely generated. Indeed, let \(r:Y\to Z\) be a smooth projective rationally connected resolution. Weil divisor pushforward gives a surjection \[\mathop{\mathrm{Pic}}(Y)\longrightarrow\mathop{\mathrm{Cl}}(Z):\] strict transforms give surjectivity on divisors, and principal divisors push forward to principal divisors. The Albanese map of \(Y\) is constant on every rational curve and therefore constant on \(Y\), since rational curves connect general points. Consequently \(\mathop{\mathrm{Pic}}^0(Y)=0\), and Néron–Severi finiteness makes \(\mathop{\mathrm{Pic}}(Y)\) finitely generated.

Normality and projectivity give \(\Gamma(U,\mathcal O_U^*)=\mathbb C^*\), and restriction gives \(\mathop{\mathrm{Pic}}(U)=\mathop{\mathrm{Cl}}(Z)\) because \(U\) is regular and contains every codimension-one point. The \(n\)th-power map on \(\mathbb C^*\) is surjective. For every \(n\geq1\), the étale Kummer sequence [20] gives the first isomorphism below. Riemann existence for finite étale covers [14] gives the second by classification of finite abelian torsors and abelianization: \[ \mathop{\mathrm{Cl}}(Z)[n] \simeq H^1_{\mathrm{\acute et}}(U,\mu_n) \simeq \mathop{\mathrm{Hom}}\bigl(H_1(U(\mathbb C),\mathbb Z),\mu_n(\mathbb C)\bigr). \tag{25}\] For a fixed \(Z\), the orders of the groups on the left are bounded as \(n\) varies, because \(\mathop{\mathrm{Cl}}(Z)\) is finitely generated. If \(H_1(U(\mathbb C),\mathbb Z)\) had a free summand of positive rank, the orders of the groups on the right would grow without bound. Hence \(H_1(U(\mathbb C),\mathbb Z)\) is finite. Only finitely many such finite groups occur by the preceding paragraph, so there is a uniform integer \(T\) divisible by all their exponents. Every torsion class \(c\in\mathop{\mathrm{Cl}}(Z)\) is killed by \(T\): take \(n\) equal to its order and apply (25), whose right-hand side has exponent dividing \(T\).

An integral principal multiple. Finally, \(pM_Z\) is an integral Weil divisor, being the pushforward of a Cartier divisor on a model determining the nef data. Choose a uniform integer \(q\) divisible by \(p\) and by all denominators in the finite set \(I_0\). Then \(qD\) is an integral Weil divisor. The relation \(D\sim_{\mathbb Q}0\) makes the class \([qD]\in\mathop{\mathrm{Cl}}(Z)\) torsion: if \(nD=\mathop{\mathrm{Div}}(v)\) for some \(n\geq1\) and \(v\in\mathbb C(Z)^*\), then \(n(qD)=\mathop{\mathrm{Div}}(v^q)\). Hence \(TqD\) is principal. Taking \(\ell=Tq\) proves the proposition for dimension \(d\). Taking a common multiple of the finitely many resulting integers handles the stated range of dimensions. All positive multiples of \(\ell D\) are principal, and projectivity and integrality of \(Z\) give the last assertion about sections. ◻

Proof. Apply Theorem 1 to \(f\) with the zero divisor as the rational pullback class. It gives uniformly bounded integers \(p_0\mid p\), a rational function \(\psi\in\mathbb C(X)^*\), and an exact divisor identity \[ K_X+B+\frac1{p_0}\mathop{\mathrm{Div}}(\psi)=f^*D_Z, \qquad D_Z=K_Z+B_Z+M_Z, \tag{26}\] where \((Z,B_Z+M_Z)\) is generalized klt, \(B_Z\) is effective with coefficients in a fixed rational DCC set, and \(p\mathbf M\) is b-Cartier with nef data. The divisor \(D_Z\) is rationally linearly trivial. This also follows directly from \(f^*D_Z\sim_{\mathbb Q}0\): choose a positive multiple making \(D_Z\) Cartier and its pullback linearly trivial, and use \(f_*\mathcal O_X=\mathcal O_Z\) and the projection formula to trivialize that multiple on \(Z\).

Proposition 4 gives a uniform integer \(\ell\) and \(v\in\mathbb C(Z)^*\) such that \(\ell D_Z=\mathop{\mathrm{Div}}(v)\). Choose \(r\) divisible by both \(\ell\) and \(p\), so that \(p_0\mid r\). Multiplying (26) by \(r\) gives \[r(K_X+B) =\mathop{\mathrm{Div}}\!\left((v\circ f)^{r/\ell}\psi^{-r/p_0}\right).\] Both exponents are integers, so this is an equality to the divisor of an actual rational function and includes the integrality of the adjoint multiple. All constants depend only on \(\Phi\), since the possible dimensions of \(Z\) form a finite set and the denominator theorem supplies uniform \(p_0,p\) and a fixed DCC set. ◻