Uniform log Iitaka fibrations and bounded moduli denominators

Proof. The divisorial sheaf \OO_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 \OO_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 \OO_V is an isomorphism after scalar extension, hence already an isomorphism. A nonzero section therefore trivializes the line bundle and principalizes D.□

Proof. A generic section extends after a sufficiently positive twist from C. Choose an ample rational divisor A on C such that

We may represent A by many general points with small positive coefficients. Then (X,\Delta+f^*A) is dlt, and is klt if (X,\Delta) is klt. Choose an effective ample rational scaling divisor G, represented by general members with small coefficients, such that (X,\Delta+f^*A+G) is lc and K_X+\Delta+f^*A+G is nef. The good-model theorem just recalled and [33, Theorem A] give a terminating MMP for K_X+\Delta+f^*A with scaling of G. The resulting log adjoint is nef and hence semiample by the nef case of [LA, Theorem 11.1].

Every step of this program is over C. Indeed, suppose inductively that the current variety X_i has a morphism f_i:X_i\to C, and let R be the negative extremal ray chosen at that step. Since f_i^*A is nef, R is also negative for K_{X_i}+\Delta_i. The lc length bound [9, Theorem 1.1(5)] supplies a rational curve \Gamma spanning R with

If \Gamma dominated C, then f_i^*A\cdot\Gamma\ge\deg A, giving

a contradiction. Thus the ray is vertical. The morphism to C descends through its contraction, and any flip is also over C. On a vertical ray the two adjoints have the same intersection numbers, so these are also steps of a (K_X+\Delta)-MMP over C.

The transform of the added divisor remains f_i^*A. Dlt preservation for the augmented pairs, followed by decreasing the boundary, keeps (X_i,\Delta_i) dlt throughout; the usual discrepancy comparison preserves klt in the klt case. Finally, subtracting a divisor pulled back from C does not change relative semiampleness. Hence the semiampleness of K_{X'}+\Delta'+f'^*A proves the assertion.□

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 (4.1), \ord_E a=(\ord_c a)\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

Divisorial valuations restrict and prolong to divisorial valuations. When the constant field is unchanged, \ord_{E'}z=a\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

Equation (4.4) 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 (4.3).□

Proof. Denote the minimum on the right by w. After shrinking the curve around c, the rational divisor

is effective. For every divisorial valuation over c, pullback of this equality gives

The last two terms are nonnegative. Thus every quotient in (4.1) 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

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 4.4 computes w and gives the effective error E_W in (4.6) 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 3.1 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_{\Q,C}0.

Pushforward of (4.6) now gives, near the marked fibre,

The restriction of E_N to the generic fibre is effective and \Q-linearly trivial, hence zero. Thus E_N is vertical, and it is \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 (4.7) to (4.9) 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 (4.8).

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, \Q-factoriality of N makes K_{N_\eta} \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(\OO_{N_\eta})=K, and has K_{N_\eta}+H_\eta\simq0. Its prime boundary component S_\eta has coefficient one. Theorem 2.3, applied with coefficient lower bound 1, bounds the relative algebraic closure of K in K(S_\eta). This field is exactly \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

Its degree-m Poincaré residue is the rational m-canonical form

This construction is independent of the local equation and commutes with tensor powers. Adjunction and (4.8) give the equality of actual divisors

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 (4.11), 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 (4.11) is exactly

It remains to compute the weight at c'.

Near h^{-1}(c'), the right side of (4.11) is ew h^*c'. For every valuation over c' the corrected order is consequently

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 (4.10).□

Proof of Theorem 4.2. When s=0, geometric integrality gives V=\Spec K, and \phi\in K^*. The definition gives w_z(\phi)=\ord_c(\phi)/m, so q(0,m)=m works.

Proceed by induction on s. Use Proposition 4.5. If its horizontal boundary has a coefficient-one component, Lemma 4.6 and the induction hypothesis give

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 \Q-Cartier canonical divisor. Proposition 6.7, 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, Theorem 1.3]. Its boundary-free factors have the stated properties by the singular Beauville–Bogomolov theorem [17, Definition 1.4 and Theorem 1.5]. 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\simq0 gives K_F+H\simq0.□

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

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 \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,\OO_F)=0. Its Lie algebra therefore vanishes. In characteristic zero this Lie algebra is precisely the space of vector fields tangent to \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 5.1, 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 5.1 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 5.2 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

Consider the finite-dimensional k-algebra

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 5.2 and 5.3. 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

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 5.1 and Lemma 5.3 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 5.4 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 2.4 descends this trivialization in degree p_i, giving (5.1). The product in (5.2) 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, Chapter IV, Section 3] 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 [U, Section 4]; the marked resolution and finite-group extension are also described in [23, Paragraphs 16 and 21]. 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 4.4 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

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

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 4.3.

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 (6.2).□

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

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 6.1. 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

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 \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 3.1 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, Complement 3 and Paragraph 21]. 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 \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 \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, Section 3, before Lemma 16].

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 (6.4). 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_{\Q,C'}0. The logarithmic divisor on the left of (6.4) 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 (6.4).□

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

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 2.1 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 6.4 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(\OO) near that fibre [21, Theorem 2.62 and Corollary 2.64]. 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

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(\OO) 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, Corollary 5.5] 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 (6.4) 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

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

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 6.5 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 6.3 and 6.6 give a common integer Q=Q(s,m)>0 with \lambda_i^Q=1 for every i. Equation (6.3) implies

Together with (6.2), this gives

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 4.2.□

Proof. Put K=\C(Z) and let F=X\times_Z\Spec K be the generic fibre. Since f is a contraction and the characteristic is zero, F is geometrically integral. It is normal, its induced pair (F,B_F) is geometrically lc, and K_F+B_F\simq0. These assertions may be checked on a log resolution: after shrinking the base, the resolution and its marked strata have the required generic smoothness, and the discrepancy formula restricts to the fibres. We choose the canonical divisor on F by dividing a rational top form on X by a rational top form on Z.

Theorem 2.1, applied to the geometric generic fibre, gives a principal multiple in a degree depending only on its dimension and \Phi. Taking a common multiple over dimensions at most d, and also clearing \Phi, gives p_0. The theorem applies over the algebraic closure of K by characteristic-zero comparison: all data descend to an algebraically closed field admitting an embedding into \C.

By Lemma 2.4, this trivialization descends to K in the same degree. Consequently we may choose \psi\in\C(X)^* such that

has no component dominating Z.

It remains to show that this particular vertical divisor is a pullback. Choose n>0 and h\in\C(X)^* with n(K_X+B-f^*L)=\Div(h). Then

The divisor on the right is vertical. Its defining function has zero divisor on the normal projective generic fibre and hence belongs to K^*, because H^0(F,\OO_F)=K. Write h^{p_0}\psi^n=f^*a for a\in K^*. We obtain

Thus D_Z=L+(np_0)^{-1}\Div(a) has all the asserted properties.□

Proof. Choose a smooth projective common resolution U mapping to X and to W, and denote the latter morphism by g. With a compatible rational top form \theta on U, write

Here B_U^c is the crepant sub-boundary. Resolve so that its support, the exceptional divisors over X, and the support of g^*P are SNC. We may include the finitely many canonical and principal supports occurring in (7.5) among the markings.

Write r=\dim Z. If r>1, take a sufficiently general smooth complete-intersection curve

of very ample members, and choose a transverse point c\in C\cap P over a general point of P. If r=1, take C=W and c=P. Here are the avoidance and smoothness conditions used in this choice. The fibre-dimension jumping locus of g has codimension at least two: a jump over a divisor would give a proper inverse image of dimension \dim U. There are also finitely many smooth marked intersections on U. We avoid their images when those images have codimension at least two. For each intersection whose image is a divisor, generic smoothness gives a dense smooth open of that divisor over which its map is smooth; we avoid the complement. All these excluded closed sets have codimension at least two in W, so a general complete intersection curve misses them. Choose c also away from the other divisor supports.

For global smoothness of the cut, apply Bertini successively to the basepoint-free systems g^*|L_j| on U and simultaneously on its smooth marked intersections. This gives a smooth variety \widetilde U=g^{-1}(C) with restricted SNC markings. This argument uses the smoothness of U, not smoothness or flatness of g along P.

The morphism g has connected fibres, as follows from Stein factorization and the fact that \C(W) is relatively algebraically closed in \C(U). Its restriction h:\widetilde U\to C has the same fibres over points of C. Thus \widetilde U is connected and, being smooth, is integral; h is a contraction. In particular its function field is regular over \C(C).

We record why the threshold is retained in this cut. On the SNC model, the threshold over the generic point of P is

where the marked prime divisors dominating P have crepant boundary coefficients b_i and multiplicities a_i in g^*P. Transversality gives the same coefficients and multiplicities in the fibre over c. Components with image a proper closed subset of P are avoided by the choice of c. The SNC criterion for sub-lc pairs then shows that this minimum is also the threshold on the cut.

Exact adjunction fixes the coefficient of the base divisor as well. For each j choose L'_j\sim L_j avoiding c, and choose a rational function on W with divisor L'_j-L_j. Multiply \theta by the pullbacks of these functions and take iterated residues along the cuts. The resulting rational top form \widetilde\theta on \widetilde U has the canonical divisor prescribed by adjunction, namely the restriction of \Div(\theta)+\sum_jg^*L'_j. Restricting (7.5) therefore gives

All restrictions are defined by generality. Since L'_j avoid c and C meets P transversely, \operatorname{coeff}_cD'_C=\alpha. When r=1, the sums and the residue operations are empty.

To obtain the effective generic presentation, return to the original model X. Its normal generic fibre over Z is geometrically normal over its perfect characteristic-zero ground field, and is geometrically integral since f is a contraction. Generic flatness and openness of geometric normality and integrality therefore give a dense open subset of W where W\to Z is an isomorphism and the original fibres have these properties. Shrink this open set further using generic smoothness of the resolution strata, so that the restricted resolution computes the fibre discrepancies also after algebraic closure. Exceptional centres that do not dominate the base disappear there; centres that dominate it retain codimension at least two in the fibres by the dimension formula. These are conditions at the generic point of C, which is chosen in this open set even when the marked point c is outside it. Hence the generic fibre V is a normal projective geometrically lc pair with the effective boundary B_V induced by B, and \widetilde B^c compares crepantly with it.

Choose a rational uniformizer z at c and define the relative form

Restriction of (7.6) to the generic fibre gives \Div(\phi)+mB_V=0. The absolute form used to compute its weight is

For a divisor E over the cut lying above c, put a_E=\ord_E(z)>0 and let b_E be its crepant boundary coefficient. Equation (7.6) gives

Taking the infimum after division by a_E proves (7.4), since the infimum of (1-b_E)/a_E is exactly the preserved threshold t_P.□

Proof of Theorem 7.2. Let P be any prime divisor of the smooth determination W, and put \alpha=\operatorname{coeff}_P D_W. From (7.3),

Lemma 7.3 realizes \alpha-1+t_P as the weight of an exact degree-p_0 form with an effective lc generic presentation of dimension s=\dim X-\dim Z. Theorem 4.2 consequently gives

because K_W is integral. Take p to be a common multiple of p_0 and q(s,p_0) for 0\leq s\leq d-1. This choice works for every P without any bound on the complexity of W. Thus pM_W is an integral Weil divisor on the smooth variety W, hence Cartier. It is nef by the lc-trivial fibration theorem.□

Proof. For a rational function u, the inequality \Div(u)+\floor{\ell D}\geq0 is equivalent to \Div(u)+\ell D\geq0, since function orders are integral. Also, effectivity of a \Q-Cartier divisor is preserved by surjective pullback. It is detected by such a pullback: over the generic point of each prime divisor downstairs, a local defining parameter pulls back with positive order along some divisor upstairs.

For (i), divide a section upstairs by \psi^{\ell/a}. The result v satisfies

It has no pole on the normal projective generic fibre, so is a base function. For v\in k(Z) the displayed divisor equals f^*(\Div(v)+\ell D_Z). The effectivity comparison just noted proves both inclusions in (8.1).

For (ii), pull an effective divisor \Div(u)+\ell D to T and push it to X'. Since q_*E=0, this gives \Div(u)+\ell D'\geq0. Conversely, pull the latter inequality to T, add \ell E, and push to X. This yields the original inequality. The argument uses \Q-Cartier pullbacks, so it does not require \ell D or \ell D' to be Cartier.□

Proof. Use Proposition 7.1 to obtain (7.1), and choose a smooth projective determination \tau:W\to Z of its moduli b-divisor. After a further resolution we may suppose that the strict transform of B_Z and the exceptional divisor of \tau have SNC support. Theorem  7.2 gives a uniform p for which pM_W is nef Cartier. Let

Then (W,A) is log smooth and lc. Its coefficients lie in the fixed DCC set for the discriminant, enlarged by 1. Moreover,

where E is exceptional: at nonexceptional primes A and B_W agree, and at exceptional primes A has coefficient one whereas B_W has coefficient at most one.

The divisor in (8.3) is big. The effective birationality theorem for polarized pairs [4, Theorem 1.3] now applies to (W,A) and M_W: the pair is projective lc, its boundary coefficients belong to a fixed DCC set, pM_W is nef Cartier, and the adjoint sum is big. Consequently

is birational for every n divisible by an integer depending only on \dim W, the DCC set, and p. The notation in that theorem uses round down, as in the displayed system.

Pushing the rational section inequalities to Z shows that this birational system is a subsystem of |\floor{nD_Z}| under the identification \C(W)=\C(Z). Thus the latter system is nonempty and its ratios generate \C(Z). Choose a common such n=N over 1\leq\dim Z\leq d, also divisible by p_0. Equation (8.1) then proves the asserted nonemptiness and ratio-field equality upstairs.

It remains to compare with all degrees. If s,s_0 are nonzero sections in degree \ell, their ratio can be represented in every multiple degree b\ell as

These products are sections in degree b\ell because b\floor{\ell D}\leq\floor{b\ell D}. Choose b so that p_0\mid b\ell and apply (8.1). Every such ratio belongs to f^*\C(Z), proving K(D)=f^*\C(Z).□

Proof. The rounded divisorial sheaves commute with these field extensions. One may check this on the smooth big open set where the prime divisors are Cartier, and then use reflexive extension across its complement. Base change for global sections gives

Choose a nonzero section s_0 over k. Dividing any section over k' by s_0 expresses it as a finite k'-linear combination of ratios over k. This proves F'_\ell=k'F_\ell; taking all degrees proves the analogous assertion for K(D).

For descent of the equality, we use the following intersection fact. If F is an intermediate field k\subset F\subset k(X), then

inside k'(X'). Indeed X is geometrically integral, so k(X)\otimes_k k' injects into k'(X'). If u\in k(X)\cap k'F, write

where the c_i are linearly independent over k and the denominator is nonzero. Such an expression is obtained by collecting a finite list of constants into a k-basis. After clearing the denominator, injectivity of the tensor product map and linear independence give ub_i=a_i for every i. Some b_i is nonzero, so u\in F. Applying (8.6) to F=F_m proves descent of F'_m=K(D'). The forward implication follows from the compositum identities, and nonemptiness follows in either direction from (8.5).□

Proof of Theorem 1.1. First let the ground field be \C, and put D=K_X+B. The hypothesis \kappa(X,D)\geq0 implies that D is pseudo-effective. Take a crepant \Q-factorial dlt modification \mu:(Y,\Delta)\to(X,B). Its adjoint is \mu^*D, and its boundary coefficients belong to \Phi\cup\{1\}. Theorem 2.2 and termination with ample scaling [33, Theorem A] give a terminating (K_Y+\Delta)-MMP to a \Q-factorial dlt pair (X',B'). The nef case of the good-model theorem makes its adjoint semiample. The MMP extracts no divisors, so the coefficients of B' still belong to \Phi\cup\{1\}. With compatible canonical divisors, crepancy of \mu and the discrepancy comparison for this MMP give, on a common resolution,

see the minimal-model discrepancy comparison and negativity lemma in [22]. Lemma 8.1(ii) identifies all rounded section spaces as subspaces of the common function field. It therefore suffices to prove the result on (X',B').

The adjoint D'=K_{X'}+B' is semiample. Its semiample contraction f:X'\to Z satisfies D'\simq f^*L with L an ample \Q-Cartier divisor on the normal projective base. If \dim Z>0, Proposition 8.2, with coefficient set \Phi\cup\{1\}, gives a uniform nonempty degree whose ratios generate the entire field K(D'). If Z is a point, then D'\simq0, and Theorem 2.1 gives a uniform principal multiple. Its system is nonempty and all ratios in that degree are constants. For any other degree, pass to a common multiple by (8.4); ratios in the latter degree are again constants. Hence K(D')=\C in this case. A common multiple of the two uniform degrees works in both cases, again by (8.4). Denote it by m(d,\Phi). The rounded comparison transfers the conclusion to X.

Now let k be any algebraically closed field of characteristic zero. Descend X, its prime boundary components and coefficients, the canonical and \Q-Cartier data, and a log resolution to a finitely generated subfield of k. Let k_0\subset k be the algebraic closure of that field inside k. Enlarging the finitely generated field if necessary, the descended variety X_0 is geometrically integral and normal, its pair (X_0,B_0) is lc, and its adjoint D_0 is \Q-Cartier. These conditions can be checked after the faithfully flat extension to k, using the chosen resolution for the discrepancy inequalities. The field k_0 embeds into \C.

Equation (8.5) shows that nonzero sections in some positive degree, and hence nonnegative Kodaira dimension, descend from k to k_0 and persist after extension to \C. The complex case gives the conclusion in the same integer m(d,\Phi) for (X_0,B_0)_\C. Lemma 8.3 first descends the nonemptiness and exact equality of ratio fields to k_0, and then extends them to k. This proves the theorem over the stated ground field.□