Proof. The divisorial sheaf \(\mathcal O_V(D)\) becomes the trivial line bundle after extension to \(\overline F\). Divisorial sheaves commute with this extension, as can be seen on a regular big open and by reflexive extension. Faithfully flat descent makes \(\mathcal O_V(D)\) invertible. Proper base change shows that its space of sections is one-dimensional over \(F\). The evaluation map from this space tensored with \(\mathcal O_V\) is an isomorphism after scalar extension, hence already an isomorphism. A nonzero section therefore trivializes the line bundle and principalizes \(D\). ◻

Proof. If \(z'\) is another uniformizer, \((dz'/z')/(dz/z)\) is a base unit at \(c\). This proves independence of \(z\). For every valuation in (2), \(\mathop{\mathrm{ord}}_E a=(\mathop{\mathrm{ord}}_c a)\mathop{\mathrm{ord}}_E z\); this gives the first formula, and the tensor-power formula follows by multiplying numerator and denominator.

For a divisorial valuation \(E'\) above \(E\) in a finite extension of characteristic-zero fields, with ramification index \(a\), the canonical ramification formula gives \[ \mathop{\mathrm{ord}}_{E'}(\Omega)+m =a\bigl(\mathop{\mathrm{ord}}_E(\Omega)+m\bigr). \tag{5}\] Divisorial valuations restrict and prolong to divisorial valuations. When the constant field is unchanged, \(\mathop{\mathrm{ord}}_{E'}z=a\mathop{\mathrm{ord}}_E z\), so the individual quotients, and hence their infima, agree.

For a curve extension, the pullback of \(dz/z\) is a unit times \(du/u\) at \(c'\), and \[\mathop{\mathrm{ord}}_{E'}u=\frac{a}{l}\mathop{\mathrm{ord}}_E z.\] Equation (5) therefore multiplies each quotient by \(l\). Regularity of \(F/K\) makes \(F\) and \(K'\) linearly disjoint over \(K\). The Galois action on their compositum is thus available to move a prolongation to the chosen branch \(c'\). Every valuation downstairs has such a prolongation, proving (4). ◻

Proof. Denote the minimum on the right by \(w\). After shrinking the curve around \(c\), the rational divisor \[ E_W=\frac1m\mathop{\mathrm{div}}_W(\Omega)+T_W+H_W-wF_W \tag{7}\] is effective. For every divisorial valuation over \(c\), pullback of this equality gives \[ \frac{\mathop{\mathrm{ord}}_E(\Omega)}m+1 =w\mathop{\mathrm{ord}}_E z+A(E;W,T_W+H_W)+\mathop{\mathrm{ord}}_E E_W. \tag{8}\] The last two terms are nonnegative. Thus every quotient in (2) is at least \(w\). A component realizing the displayed minimum gives equality. ◻

Proof. Choose a smooth projective model \(W\to C\) whose generic fibre is a log resolution of \(V\). Resolve also the closures of the horizontal boundary and exceptional divisors, together with the fibre over \(c\). Let \(B^c_{W_\eta}\) be the crepant subboundary on the generic resolution, and take \(H_W\) to be the closure of its positive part. Its coefficients are at most one, and the horizontal divisor \[\frac1m\mathop{\mathrm{div}}_W(\Omega)+H_W\] is effective and exceptional over \(V\) on the generic fibre. The markings can be chosen so that \((W,T_W+H_W)\) is log smooth. Lemma 10 computes \(w\) and gives the effective error \(E_W\) in (7) near \(c\).

On the generic fibre, \(K_W+T_W+H_W\) is rationally linearly equivalent to an effective exceptional divisor over the normal projective variety \(V\). Its Kodaira dimension is zero: a rational function whose poles are supported on that exceptional divisor descends to a regular function on \(V\), hence is constant. In particular it has a section in a sufficiently divisible degree. Lemma 6 therefore gives a terminating MMP over \(C\) for \(K_W+T_W+H_W\). Write its output as \((N,T+H)\). No divisors are extracted, so the transform of \(T_W\) is precisely \((f^*c)_{\mathrm{red}}\). The generic function field is unchanged and is regular over \(K\); Stein factorization consequently makes \(f:N\to C\) a contraction.

Discrepancy comparison preserves the divisible generic-fibre section spaces through these steps. Thus the generic adjoint on \(N\) still has Kodaira dimension zero. Its relative semiample contraction has zero-dimensional image on the generic fibre. The image is therefore a normal curve finite over \(C\). Connectedness of the fibres makes its function field equal to \(K\), so this curve is \(C\). This proves \(K_N+T+H\sim_{\mathbb Q,C}0\).

Pushforward of (7) now gives, near the marked fibre, \[ \frac1m\mathop{\mathrm{div}}_N(\Omega)+T+H=w f^*c+E_N, \qquad E_N\ge0. \tag{10}\] The restriction of \(E_N\) to the generic fibre is effective and \(\mathbb Q\)-linearly trivial, hence zero. Thus \(E_N\) is vertical, and it is \(\mathbb Q\)-linearly trivial over \(C\). Such a divisor is locally a rational multiple of a fibre. Indeed, a relative principal trivialization has vertical divisor, so its defining rational function has neither zeros nor poles on the normal projective generic fibre. It belongs to \(K^*\), because that fibre has no new global functions. Consequently \(E_N=a f^*c\) near \(c\) for some rational number \(a\).

Applying the discrepancy calculation (8) to (10) shows that all valuation quotients are at least \(w+a\): the pair \((N,T+H)\) is lc. Each component of \(T\) has coefficient one, so its quotient is exactly \(w+a\). The valuation definition still gives \(w\), since the function field and form have not changed. Hence \(a=0\), proving (9).

The generic restriction of that equality gives the claimed trivialization. A dlt pair with no coefficient-one boundary is klt; generic restriction, checked on a resolution, preserves this statement after algebraic closure in characteristic zero. Finally, \(\mathbb Q\)-factoriality of \(N\) makes \(K_{N_\eta}\) \(\mathbb Q\)-Cartier. ◻

Proof. Dlt adjunction [22] makes \(S\) normal and gives an effective rational different \(\Theta\) such that \((S,\Theta)\) is lc. Its Stein curve is normal and hence smooth. The generic pair \((N_\eta,H_\eta)\) is normal, projective and lc, has \(H^0(\mathcal O_{N_\eta})=K\), and has \(K_{N_\eta}+H_\eta\sim_{\mathbb Q}0\). Its prime boundary component \(S_\eta\) has coefficient one. Theorem 4, applied with coefficient lower bound \(1\), bounds the relative algebraic closure of \(K\) in \(K(S_\eta)\). This field is exactly \(\mathbb C(C_S)\), so it gives the asserted \(D_s\).

At the generic point of \(S\), the absolute form \(\Omega\) has a logarithmic pole of order \(m\). Choose a local equation \(x\) of \(S\) and a rational \(s\)-form \(\beta\), regular at the generic point of \(S\), whose restriction generates its canonical line. Write \[\Omega=a\bigl(dx/x\wedge\beta\bigr)^{\otimes m}, \qquad a\text{ a unit along }S.\] Its degree-\(m\) Poincaré residue is the rational \(m\)-canonical form \[\Omega_S=(a|_S)(\beta|_S)^{\otimes m}.\] This construction is independent of the local equation and commutes with tensor powers. Adjunction and (9) give the equality of actual divisors \[ \frac1m\mathop{\mathrm{div}}_S(\Omega_S)+\Theta =w(f|_S)^*c \tag{12}\] near the marked fibre. To verify the identity in this exact degree, take a tensor power for which the ambient adjunction is Cartier and apply the pluricanonical adjunction isomorphism. The resulting divisor identity is that same positive multiple of (12), and can be divided by it. This uses the rational degree-\(m\) residue already defined at the generic point; it does not require a uniform Cartier index along \(S\).

The generic fibre of \(h\) is normal, projective and geometrically integral. Its pair induced by \(\Theta\) is geometrically lc, as one sees by generic restriction of a resolution in characteristic zero. Dividing \(\Omega_S\) by \((du/u)^{\otimes m}\) in the canonical-line identification gives a rational relative form \(\phi_S\). The generic restriction of (12) is exactly \[\mathop{\mathrm{div}}(\phi_S)+m\Theta_{\eta_S}=0, \qquad \eta_S=\mathop{\mathrm{Spec}}\mathbb C(C_S).\] It remains to compute the weight at \(c'\).

Near \(h^{-1}(c')\), the right side of (12) is \(ew h^*c'\). For every valuation over \(c'\) the corrected order is consequently \[\frac{\mathop{\mathrm{ord}}_E(\Omega_S)}m+1 =ew\mathop{\mathrm{ord}}_Eu+A(E;S,\Theta).\] Log canonicity gives \(w_u(\phi_S)\ge ew\). The fibre of \(h\) has a prime divisor, and any such divisor is a component of the intersection of \(S\) with \(T\). At its generic point the two coefficient-one branches of the dlt pair are simple normal crossings; adjunction gives coefficient one in \(\Theta\). The quotient for that divisor is \(ew\), which proves (11). ◻

Proof of Theorem 8. When \(s=0\), geometric integrality gives \(V=\mathop{\mathrm{Spec}}K\), and \(\phi\in K^*\). The definition gives \(w_z(\phi)=\mathop{\mathrm{ord}}_c(\phi)/m\), so \(q(0,m)=m\) works.

Proceed by induction on \(s\). Use Proposition 11. If its horizontal boundary has a coefficient-one component, Lemma 12 and the induction hypothesis give \[q(s-1,m)e w_z(\phi)\in\mathbb Z \quad\hbox{for some }1\le e\le D_s.\] Thus \(q(s-1,m)\operatorname{lcm}(1,\ldots,D_s)\) clears the weight in this case. Otherwise the prepared generic pair is geometrically klt with \(\mathbb Q\)-Cartier canonical divisor. Proposition 24, proved independently of this induction below, gives a clearing integer depending only on \(s,m\). Taking a common multiple of these two integers completes the induction. ◻

Proof. The product decomposition, including its assertion about the boundary, is [24]. Its boundary-free factors have the stated properties by the singular Beauville–Bogomolov theorem [17]. Since \(K_V\) is rational Cartier, quasi-étaleness makes \(K_P\) rational Cartier. Restricting the canonical sheaf in product charts, with the complementary points smooth, gives the same property on every factor. Restriction of \(K_P+\pi^*B\sim_{\mathbb Q}0\) gives \(K_F+H\sim_{\mathbb Q}0\). ◻

Proof. A smooth projective resolution of \(F\) is rationally connected; one may lift rational curves through general smooth points, or use the rational connectedness results for klt varieties in [14]. Its positive-degree structure-sheaf cohomology vanishes. Klt singularities are rational, so the same is true on \(F\). The extension theorem for klt differential forms [12] then gives the asserted vanishing of reflexive one-forms.

Choose \(a>0\) and a rational \(a\)-canonical form \(\theta\) with \[\mathop{\mathrm{div}}(\theta)+aH=0.\] On the smooth locus, the density \(|\theta|^{2/a}\) defines a positive measure. Pull it to a log resolution. Every divisorial order of the pulled-back form is strictly greater than \(-a\), because the pair is klt. In SNC coordinates this is exactly the local integrability condition for the density. Thus it defines a finite nonzero measure \(\mu\) on \(F\), with no mass on proper algebraic subsets.

An automorphism preserving \(H\) multiplies \(\theta\) by a nonzero constant. Its pullback therefore multiplies \(\mu\) by a positive constant; comparison of total masses makes that constant one. Consequently every automorphism of the pair preserves \(\mu\).

There can be no nontrivial additive or multiplicative algebraic one-parameter subgroup of such automorphisms. Indeed, iterate the element \(1\in\mathbb G_a\) or \(2\in\mathbb G_m\) on a nonfixed point. The orbit map extends across infinity by projectivity, so these iterates converge to a subgroup-fixed point. The nonfixed locus has full \(\mu\)-measure. Such convergence contradicts Poincaré recurrence for the resulting invertible transformation of the finite measure space \((F,\mu)\).

The group preserving \(L\) and \(H\) is a linear algebraic group: a sufficiently high power of \(L\) realizes it as a closed subgroup of a projective linear group. Every positive-dimensional linear algebraic group over \(\mathbb C\) contains a copy of \(\mathbb G_a\) or \(\mathbb G_m\), so this group is finite. Moreover, the identity component of the automorphism group preserving \(H\) preserves \(L\). Indeed its variation of \(L\) lies in \(\operatorname{Pic}^0(F)\), which is zero because \(H^1(F,\mathcal O_F)=0\). Its Lie algebra therefore vanishes. In characteristic zero this Lie algebra is precisely the space of vector fields tangent to \(\mathop{\mathrm{Supp}}H\): a connected group preserving the support fixes its components and their coefficients. This proves the last vanishing. ◻

Proof. Among all covers in Proposition 13, choose one with \(\dim F\) minimal, taking this dimension to be zero when the factor is absent. Let \(F'\to F\) be a connected normal finite quasi-étale cover. The pulled-back pair is klt and log Calabi–Yau, and \(K_{F'}\) is rational Cartier. Apply Proposition 13 to it. If \(F'\) were not rationally connected, the resulting product would have a positive-dimensional boundary-free part: otherwise rational connectedness would descend from its rationally connected factor along a finite surjection. Replacing \(F\) by this product would therefore give a product cover of \(V\) with a smaller rationally connected factor, a contradiction. Lemma 14 now applies to \((F',H')\). ◻

Proof. We first show that a bounded power preserves the factor directions on \(R\). We then recover their constant fields and show that the resulting birational actions on the finite factor covers are regular.

The factor directions. On a smooth big open where \(\rho\) is étale, the product tangent directions pull back to a splitting. Reflexive extension gives \[\mathcal T_R=\bigoplus_{i\in I}\mathcal E_i.\] Consider the finite-dimensional \(k\)-algebra \[\mathcal A= \{u\in\operatorname{End}_R(\mathcal T_R): u(\mathcal T_R(-\log\mathop{\mathrm{Supp}}H_R)) \subseteq\mathcal T_R(-\log\mathop{\mathrm{Supp}}H_R)\}.\] Every projection onto \(\mathcal E_i\) belongs to \(\mathcal A\). We claim that every member of \(\mathcal A\) has zero entries between distinct summands.

This may be checked after extending \(k\) to an algebraic closure. Fix general smooth points in all but one factor, say \(P_i\), and take the corresponding slice of \(R\). After shrinking the space of complementary points, its connected components are normal and finite quasi-étale over \(P_i\). To see this, the generic slice is normal by localization and geometrically normal in characteristic zero. The projection to the complementary factors is projective, so generic flatness and openness of geometric normality give normality of the entire fibres after shrinking that base. Every component has dimension \(\dim P_i\): the fibre-dimension inequality gives the lower bound, and finiteness over \(P_i\) gives the upper bound. Its finite image is therefore all of \(P_i\). The closed subset omitted from the étale product charts has codimension at least two on a general slice: its components not dominating the complementary factors can be avoided, and the others have fibre dimension at most \(\dim P_i-2\). On its smooth big open, \(\mathcal E_i\) is the tangent sheaf of the slice, and every complementary summand is trivial.

If either of two distinct summands is nonabelian and boundary-free, slice in that direction. A homomorphism in either direction gives reflexive one-forms or vector fields on a finite quasi-étale cover of that block, and hence vanishes. The only remaining possibility is an abelian and a rationally connected summand. Slice in the rationally connected direction. An entry from this summand to the abelian one gives reflexive one-forms. An entry in the reverse direction gives vector fields tangent to the pulled-back boundary, because the endomorphism preserves the log tangent subsheaf. Both vanish by Lemmas 14 and 15. These slices cover a dense open of \(R\), which proves the claim.

The block projections are therefore central idempotents of \(\mathcal A\). Its primitive central idempotents give nonzero direct summands of \(\mathcal T_R\), each of positive generic rank, so there are at most \(s\) of them. Conjugation by \(G\) acts on \(\mathcal A\), since \(G\) preserves the pair; it is a ring automorphism even for a semilinear action. It permutes these primitive central idempotents. Thus \(g^{s!}\) fixes every one of them and consequently every block projection. This argument takes place over \(k\) itself and requires no extension of \(G\) to an algebraic closure.

The finite factor covers. Let \(k(R_i)\) be the relative algebraic closure of \(k(P_i)\) in \(k(R)\), and let \[R\longrightarrow R_i\longrightarrow P_i\] be the Stein factorization of the projection. At the generic point of \(R\), the complementary directions span \(\operatorname{Der}_{k(P_i)}k(R)\). In characteristic zero their common constants are exactly \(k(R_i)\). Preservation of the directions therefore gives semilinear birational transformations of \(R_i\) under every \(g^{s!}\).

The fields \(k(R_i)\) are regular over \(k\), since they lie in the regular extension \(k(R)/k\); hence \(R_i\) is geometrically integral. The extension obtained by adjoining the complementary product factors to \(k(P_i)\) is regular and linearly disjoint from \(k(R_i)/k(P_i)\). Consequently the normal product \(R_i\times\prod_{j\ne i}P_j\) is an intermediate finite cover of \(P\). Ramification over a prime of \(P_i\) would persist on this product and on a prolongation to \(R\), contradicting quasi-étaleness of \(R\to P\). This proves that \(R_i\to P_i\) is quasi-étale.

The product map \(R\to\prod_iR_i\) is proper with finite fibres, because its composite to \(P\) is finite. It is therefore finite. Its image has the dimension of the integral target, so it is surjective, and the same ramification argument makes it quasi-étale. In particular every fibre of \(R\to R_i\) has dimension \(s-\dim R_i\).

Regularity of the factor actions. Fix a transformation preserving the factor directions. The two morphisms from \(R\) to \(R_i\)—the projection and its composition with the transformation—map \(R\) onto the closed graph \(\Gamma_i\) of the induced birational factor transformation. Their target is replaced by its field conjugate in the semilinear case. Every fibre of \(R\to\Gamma_i\) has dimension at least \(s-\dim R_i\). A positive-dimensional fibre of either graph projection would therefore give a fibre of \(R\to R_i\) of larger dimension. Both graph projections are thus finite and birational; normality of their targets makes them isomorphisms. This proves regularity.

Finally, equality of divisors on \(\prod_iR_i\) can be checked after pullback along the finite surjection from \(R\). Since its entire boundary is pulled back from the rationally connected factor, invariance of \(H_R\) implies invariance of that factor’s boundary. Hence the factor actions preserve their boundaries. The asserted geometric properties of the \(R_i\) follow from their being quasi-étale covers of the chosen blocks. ◻

Proof. Apply Proposition 13 and Lemma 15 to the geometric generic pair, and define the resulting product cover over a finite extension of \(K\). Take a Galois closure \(L\) of its total function field over the original field \(K(V)\), and let \(K_1\) be the relative algebraic closure of \(K\) in \(L\). The extension \(K(V)/K\) is regular. It follows that \(K_1/K\) is finite Galois and that restriction maps \(\operatorname{Gal}(L/K(V))\) onto \(\operatorname{Gal}(K_1/K)\). Let \(R\) be the normalization of \(V_{K_1}\) in \(L\). Since \(K_1\) is algebraically closed in \(L\), \(R\) is geometrically integral.

Over an algebraic closure of \(K\), the conjugate product covers are all étale over the smooth locus of the original variety. Their compositum, and hence the Galois closure, has the same property. Thus \(R\) is finite quasi-étale over the product and over \(V_{K_1}\). Its Galois transformations preserve the crepant pullback of \(B\). Proposition 16 now constructs the \(R_i\) and supplies the factor actions.

The boundary-free blocks and their covers have Cartier trivial canonical divisor over the algebraic closure, so they possess degree-one volume forms there. Still over the algebraic closure, on the rationally connected factor, restrict the pulled-back degree-\(m\) trivialization from \(V\) to the product factor, choosing the complementary points smooth. This gives a degree-\(m\) log trivialization; pull it back to its Stein block \(R_i\).

Here \(p_iH_i\) is integral: on the rationally connected block this follows from the integrality of \(mB\), quasi-étale pullback, and restriction to the product factor; on the other blocks \(H_i=0\). Thus \(p_iK_{R_i}+p_iH_i\) is an integral Weil divisor whose divisorial sheaf becomes trivial after algebraic closure. Lemma 5 descends this trivialization in degree \(p_i\), giving (13). The product in (14) and the pullback of \(\phi\) have the same divisor on the normal projective variety \(R\). Their ratio has neither zeros nor poles and is a global unit. Since \(R\) is geometrically integral, that unit belongs to \(K_1^*\). ◻

Proof. Take equivariant projective models by closing graphs of the finitely many translates, and resolve equivariantly, including the reduced marked fibre and the horizontal markings. Near the marked fibre the map to the curve is toroidal: in local coordinates a uniformizer is a monomial in the vertical coordinates, up to a unit. The horizontal markings are among the remaining coordinates. Root extraction on the base, normalization, and the projective subdivision theorem of [19] give the reduced SNC fibre. The horizontal coordinates remain transverse throughout this construction.

Here equivariance can be retained in the subdivision step. First barycentrically subdivide the finite vertical cone complex so that stabilizers fix each ray of any stabilized cone. They then act trivially on that cone’s lattice. After removing face self-identifications by further such subdivision, choose compatible projective subdivisions on orbit representatives and pull them back. The lattices used on the combinatorial quotient are the original cone lattices, not the invariant lattices of a geometric quotient. The regular height-one subdivision theorem after sufficiently divisible ramification applies to this finite complex. It applies with identical subdivisions to the copies of strata that split after normalization.

This is the equivariant marked construction used in [28]; the marked resolution and finite-group extension are also described in [23]. Taking a common further extension handles the finite list of models. One may take a Galois closure and further roots of an original local parameter, so that the extension remains Galois over any specified original curve field. Compatible automorphisms extend to the compositum; they have finite order. Equivariant resolution away from the marked fibre gives projective models over the complete curve. ◻

Proof. On a semistable model every component of the marked fibre has multiplicity one. The computation of weights on a log-smooth model in Lemma 10 shows that \(p_iw_u(\eta_i)\) is an integer. Multiplication by \(u^n\) changes the weight by \(n/p_i\), so normalization is possible. The logarithmic divisor \[\mathop{\mathrm{div}}\bigl(\eta_i\wedge(du/u)^{\otimes p_i}\bigr) +p_i(T_i+\widehat H_i)\] is then effective near the marked fibre, and its coefficient on at least one component of \(T_i\) is zero.

We check the product assertion before passing to the comparison cover. On the fibre product of the semistable models, the local vertical equations have the form \[\prod_j x_{ij}=u \quad\text{for each factor }i.\] Horizontal coordinates are independent. Include their SNC markings in the toroidal boundary. Relative logarithmic top forms multiply over the logarithmic curve with exactly one base differential \(du/u\) in the absolute form. The product therefore acquires no additional power of \(u\). In logarithmic coordinates the resulting form has at most full logarithmic poles. This remains true on toroidal resolutions, since their logarithmic canonical generators pull back to logarithmic generators. Thus its weight is nonnegative.

For equality, choose on each factor a fibre component of zero logarithmic order and a general point away from all other markings. The morphism to the curve is smooth there. The product of these open subsets supplies a fibre component of zero logarithmic order, so the weight is zero. Finite pullback to the comparison cover preserves this weight by Lemma 9.

On its normal proper generic fibre, the pulled-back product and \(\phi\) have the same divisor: both trivialize the same degree-\(m\) log canonical divisor. Their ratio has zero divisor and hence lies in the constant field \(K'\). The scalar rule and the ramification rule for weights now give (16). ◻

Proof. Here \(p_i=1\). On the semistable model the normalized form is a regular relative logarithmic top form. It is not divisible by \(u\) as such a section, since its logarithmic order is zero on some fibre component. Consequently it frames the extended top Hodge line. This identification is the semistable comparison between logarithmic de Rham cohomology and Deligne’s canonical extension, with nilpotent residue [7, 29]. The action on the central fibre of this line is \(\lambda_i\).

The integral local system in degree \(h\), modulo torsion, has rank \(\binom{2h}{h}\). Indeed its smooth fibres are abelian varieties; our models were unchanged on the smooth generic fibre. In a small disc choose an analytic coordinate linearizing the finite base action. On the universal cover of the punctured disc the action, combined with parallel transport, is an integral matrix \(A\) commuting with the unipotent monodromy \(T\). Since the geometric action has finite order, a power of \(A\) is a power of \(T\).

Passing from flat frames to canonical-extension frames changes \(A\) by a commuting factor of the form \(\exp(c\log T)\), which is unipotent. The eigenvalues are therefore unchanged. Since the Hodge line is invariant, \(\lambda_i\) is an eigenvalue of the integral matrix \(A\). If its order is \(n\), its cyclotomic polynomial divides the characteristic polynomial of \(A\), and \[\varphi(n)\leq\binom{2h}{h}.\] Only finitely many positive integers satisfy this inequality. Their least common multiple bounds and annihilates the character orders in the stated dimension. ◻

Proof. Start with the equivariant smooth semistable model \(Q\) from Lemma 18. The pair \((Q,\widehat H_i)\) is klt; adding the reduced fibre \(Q_{c'}\) gives a dlt pair. Since this fibre is a base pullback, a \((K_Q+\widehat H_i)\)-negative program over the curve is also negative for the adjoint with this fibre added.

We construct the program equivariantly. Let \(G\) be the finite cyclic group generated by \(\sigma\) and take the quotient \(\pi:Q\to\overline Q=Q/G\). Define the branch-corrected boundary \(\overline H\) by \[K_Q+\widehat H_i =\pi^*(K_{\overline Q}+\overline H).\] Its coefficients are \(1-(1-\delta)/e\), where \(\delta<1\) is an upstairs coefficient and \(e\) a ramification index. They lie in \([0,1)\), and finite-map discrepancy comparison makes \((\overline Q,\overline H)\) klt. A finite quotient of a smooth variety is \(\mathbb Q\)-factorial.

On the generic fibre upstairs the adjoint has a nonzero section in a sufficiently divisible degree: the log-resolution error is effective and exceptional over the log Calabi–Yau factor. Multiplying the finitely many translates of this section gives an invariant section in a divisible degree downstairs. The horizontal section inequalities descend by the finite-map formula. Thus Lemma 6 applies to the quotient pair over \(C'/G\) and gives a terminating program with relatively semiample output.

Lift each step by normalization in the upstairs function field, using Stein factorization for contractions. These are the usual finite-group equivariant MMP diagrams; see also [23]. They remain over \(C'\), since functions integral over the downstairs curve extend on the normalizations. Finite pullback preserves relative negativity and relative ampleness. The canonical pullback equality continues in codimension one because no step extracts divisors. In particular the lifted pairs remain klt. The invariant divisor \(J\) is rational Cartier: it descends rationally to the \(\mathbb Q\)-factorial quotient, and its finite pullback is \(J\). The displayed canonical pullback equality makes \(K_Y+J\) rational Cartier as well, and their difference makes \(K_Y\) rational Cartier.

For completeness, ordinary \(\mathbb Q\)-factoriality upstairs is unnecessary for dlt preservation here. The lifted negative contraction or flip has the same discrepancy comparison as an ordinary MMP step. Adding the marked fibre changes the adjoint by a pullback and leaves this comparison unchanged. Discrepancies strictly improve for centres in the affected locus; hence every lc centre on the output meets the unchanged locus of an input lc centre. It therefore meets its SNC locus. This is the generic-SNC characterization of dlt, and gives dlt on each output; compare [6].

Since no divisors are extracted, every surviving marked fibre component still has multiplicity one. The fibre is Cartier, and the klt total space is Cohen–Macaulay. It has no embedded fibre components, so the fibre is reduced. Relative semiampleness pulls back from the quotient output.

It remains to prove the exact equality (18). The generic adjoint has Kodaira dimension zero, and the steps preserve its divisible section spaces. The relative semiample contraction therefore has zero-dimensional generic image. Connected fibres identify its Stein image with \(C'\), so \(K_Y+J\sim_{\mathbb Q,C'}0\). The logarithmic divisor on the left of (18) is effective near \(c'\) by normalization, and stays effective by pushforward along the program. Its horizontal part is effective and rationally trivial on the proper generic fibre, hence zero. Its remaining vertical part is relatively rationally principal. A function with vertical divisor is constant on the normal proper generic fibre, and so comes from \(K'\). Thus this divisor is a rational multiple of \(T\) near \(c'\).

If the multiple were positive, the dlt discrepancy calculation would make every weight quotient positive, contradicting \(w_u(\eta_i)=0\). It is therefore zero, proving (18). ◻

Proof. The divisor of \(\omega\) is integral, so the coefficients of \(\Delta\) belong to \(\{0,1/p,\ldots,1\}\). On the normal quotient \(A/\langle\gamma\rangle\) the crepant branch boundary has coefficients in \[\Psi_p= \left\{1-\frac{1-\delta}{e}: \delta\in\{0,1/p,\ldots,1\},\ e\in\mathbb Z_{>0}\right\}.\] This is a rational DCC subset of \([0,1]\). An invariant tensor power of \(\omega\) descends and shows that the quotient log canonical divisor is rational Cartier and rationally linearly trivial. Finite-map discrepancy comparison makes the quotient pair lc.

Apply Theorem 2 in dimension \(h\) with coefficient set \(\Psi_p\). Choose its principalizing degree \(a\) divisible also by \(p\). The pullback of a degree-\(a\) trivialization downstairs is invariant and has the same divisor as \(\omega^{\otimes a/p}\). Their ratio is constant on the normal proper variety \(A\). Consequently \(\lambda^{a/p}=1\). This gives the asserted uniform integer. ◻

Proof. Use the model of Lemma 21 and write \(h=\dim R_i\). It is flat over the smooth curve, since its integral total space is torsion-free over the local discrete valuation rings. The central fibre \(T\) is a union of lc centres of the dlt pair \((Y,T+J)\) and is therefore Du Bois. Cohomology and base change for a proper flat family with Du Bois special fibre imply local constancy of \(h^j(\mathcal O)\) near that fibre [21]. After shrinking, the other fibres are klt, as can be checked on a resolution. Rational singularities and birational invariance identify their structure-sheaf cohomology with that of a resolution of the generic factor.

For the rationally connected factor this gives \[H^j(T,\mathcal O_T)=0\quad(j>0),\qquad h^0=1.\] For a Calabi–Yau factor the only nonzero groups are in degrees \(0\) and \(h\), each of dimension one. For a symplectic factor there is one dimension in every even degree and none in odd degree. These identifications follow from extension of reflexive forms, the defining form algebras of the factors, and Hodge symmetry on resolutions [12, 17].

We use the following consequence of coherent Lefschetz: a finite-order automorphism of a projective scheme with nonzero alternating trace on \(H^\bullet(\mathcal O)\) has a fixed point. It applies to this possibly singular fibre. One can either use [3], or embed the fibre equivariantly in a smooth projective space and apply [8] to its pushed-forward structure sheaf. If the fibre has no fixed point, the sheaf restricts to zero along each ambient fixed component, and its Lefschetz contribution vanishes.

In the rationally connected case the alternating trace of \(\sigma\) is \(1\), so \(\sigma\) has a fixed point. In the symplectic case write \(\mu_1,\ldots,\mu_N\) for the eigenvalues on its nonzero cohomology groups, where \(N=h/2+1\). They are nonzero. Their first \(N\) power sums cannot all vanish: Newton’s identities would then give \(\prod_j\mu_j=0\). Thus \(\sigma^a\) has a fixed point for some \(1\leq a\leq N\).

In the Calabi–Yau case \(p_i=1\) and \(J=0\). Equation (18) makes \(K_Y\) Cartier near \(T\). The klt Cohen–Macaulay total space, and hence its Cartier fibre, is Gorenstein there. Adjunction trivializes \(\omega_T\) by the residue of \(\Omega_i\). The induced scalar on this generator is \(\lambda_i\): the action on \(du/u\) has residue one. Serre duality therefore gives alternating trace \[1+(-1)^h\lambda_i^{-1}.\] If \(\lambda_i\) has order at most two it is already bounded. Otherwise the trace is nonzero, and \(\sigma\) has a fixed point.

In every remaining case a power \(\sigma^a\), with \(a\leq h+1\), fixes a point of \(T\). At most \(h+1\) components of the coefficient-one part of a dlt boundary on an \((h+1)\)-fold meet at one point. Indeed their common intersection is a union of lc centres, and at the generic point of such a centre the pair is SNC. A further power of exponent at most \((h+1)!\) therefore preserves a prime component \(A\) through that fixed point.

By dlt adjunction \(A\) is normal and carries an effective lc different \(\Delta_A\). The rational \(p_i\)-pluriresidue \(\omega_A\) of \(\Omega_i\) satisfies \[\mathop{\mathrm{div}}(\omega_A)+p_i\Delta_A=0.\] This is an equality of divisors: at the generic smooth point it is the ordinary residue, and at every prime it follows by taking a Cartier tensor power in adjunction and dividing the resulting identity. No change of the residue’s degree is required. Naturality of residue shows that the character of the chosen power on \(\omega_A\) is the corresponding power of \(\lambda_i\).

Apply Lemma 22 with \(p_i=1\) or \(m\) and \(\dim A=h\leq s\). The power used to stabilize \(A\) was bounded in terms of \(s\), so the order of \(\lambda_i\) is bounded in terms of \(s\) and \(m\). Taking least common multiples supplies a single annihilating integer for all factors. ◻

Proof. Use the product and comparison constructions of Section 5, and then the normalized models above. Lemmas 20 and 23 give a common integer \(Q=Q(s,m)>0\) with \(\lambda_i^Q=1\) for every \(i\). Equation (17) implies \[l\mid bQ\,\mathop{\mathrm{ord}}_{c'}a .\] Together with (16), this gives \[mbQ\,w_z(\phi)=\frac{bQ\,\mathop{\mathrm{ord}}_{c'}a}{l}\in\mathbb Z.\] Thus \(q_{\mathrm{klt}}(s,m)=m\,s!\,Q(s,m)\) works. The reduced generic presentation in Section 4 is rationally Gorenstein, so this is exactly the klt assertion needed there. The residue induction therefore proves Theorem 8. ◻