Logarithmic Kodaira dimension and whole-fiber variation

Proof. Apply Theorem 1.1 with empty compactification boundaries.□

Proof. Fix a closed fiber F_v. By [11, Corollary 0.3], its non-uniruledness makes K_{F_v} pseudo-effective. The companion [40, Corollary 11.2] gives a normal projective model M_v with semiample \Q-Cartier K_{M_v} and, on a common smooth resolution p:W\to F_v, q:W\to M_v,

For any prime divisor P exceptional over M_v, pass to a higher common resolution containing P and denote the effective pullback of E again by E. Since F_v is smooth, the discrepancy identity

shows that M_v has canonical singularities. Thus every F_v admits a good minimal model in the sense used by Taji. His variation is the same whole-fiber field-of-definition invariant as (1.1), so [45, Theorems 1.1–1.2] gives the dichotomy and the special-base assertion. The case \kbar(V)=0 also follows from the second inequality and \Var(f)\geq0.□

Proof. Apply [37, Corollary 6.2]. The zero-divisor and -\infty conventions here agree with that statement. Only its projective form is used in the present paper.□

Proof. The discrepancy is exceptional because a birational morphism between smooth varieties is an isomorphism at every generic target divisor. For an exceptional prime E_0, choose boundary coordinates z_i on X, let a_i=\ord_{E_0}(\mu^*z_i), and let k_{E_0} be its Jacobian order. Pulling back a logarithmic top wedge gives k_{E_0}+1\geq\sum_i a_i: a nonzero wedge contains at most one factor du/u at E_0=(u=0). If the boundary coefficients are \delta_i\leq1, the discrepancy coefficient is therefore

Thus E is effective. A rational function with poles only on an effective exceptional divisor passes every codimension-one regularity test on the normal target, so \mu_*\cO_{X'}(mE)=\cO_X. The projection formula proves the sheaf and section identities, including their twisted, relative, and multiplicative forms.□

Proof. Stein factorization reduces to finite \pi, since a proper birational morphism onto a normal variety has direct image \mathcal O_V. Clear denominators. Each homogeneous section upstairs is integral over the downstairs section ring: its monic characteristic polynomial in the finite function-field extension has coefficients in the appropriate powers of D. Trivializing D at each prime of V shows that these coefficients are regular there, hence everywhere by normality. The two graded rings therefore have the same transcendence degree. If the downstairs ring has no positive-degree section, the norm excludes one upstairs. These observations prove the assertion in all cases.□

Proof. Proper cohomology and flat base change identify the section spaces, compatibly with multiplication, and hence preserve the dimensions of the section maps. The finite-morphism assertion follows from Lemma 2.3. Morphisms, rational maps, and the identities making two rational maps inverse use finitely many coefficients and spread after finitely many denominators are inverted. On the smooth proper locus, in each fixed degree we shrink to an open where the section dimension is constant; cohomology and base change then identify the complete fiber space. Spreading the section maps and excluding the countably many proper closed complements gives the very-general-fiber assertion.

For the last assertion, work on an affine open of I. A regular function after extension to b' is a finite sum \sum_j c_jf_j, where the c_j\in b' are linearly independent over b and f_j\in\cO(I). If it vanishes on every b-point, then every f_j does too and is zero. Thus I(b) remains Zariski dense after field extension, proving the assertion.□

Proof. For a sufficiently divisible system, compare its image with the base using the graph of the two rational maps. The fiber image has dimension bounded by the restricted complete system; the fiber dimension theorem gives the assertion, and taking the maximum over divisible systems proves the first inequality. This is the usual upper-addition estimate; see [27, Theorem 4, p. 357].

For the second claim use

After clearing denominators, multiplication by a fixed effective section of the second summand embeds the corresponding section systems of the first. Their ratios are unchanged. Taking their image dimensions proves the assertion.□

Proof. Take a smooth projective compactification of V and resolve its boundary. Since f is projective, embed U as a closed subscheme of a projective space over V and take its closure over the resulting projective base Y. Equivalently, one can take the closure of the graph of f in projective compactifications. Over V this closure is already U: the original family is proper over V, so no additional limit point is needed above that open set. Resolve the closure and its boundary, with all centers outside the smooth open U. Resolution and elimination of indeterminacy in characteristic zero give the required morphism between smooth projective varieties and the SNC boundary; see [7].

The resulting morphism has a geometrically integral generic fiber. Indeed, the original proper morphism has connected fibers and normal base, so its Stein factorization gives f_*\cO_U=\cO_V. Thus \C(V) is relatively algebraically closed in \C(U); in characteristic zero this extension is regular. The finite part of the Stein factorization of the compactified morphism is consequently birational over Y, and hence is an isomorphism because Y is normal. This proves connectedness of its fibers.

As f^{-1}(V)=U, every point of D_X lies over D_Y, and \Supp(f^*D_Y)\subseteq\Supp(D_X). A very general fiber lies over V, has no boundary, and has ordinary Kodaira dimension d. Theorem 2.1 now applies to these smooth projective varieties, connected fibers, and reduced SNC boundaries. Its conclusion is (2.1), by the compactification definition of logarithmic Kodaira dimension.□

Proof. By (2.1), K_X+D_X has nonnegative section dimension. The divisor Iitaka theorem gives a resolved map with base dimension \kbar(U) and with restriction of the divisor having section dimension zero on a very general fiber. The needed statement is [27, Theorem 3, p. 357, and its proof in §4, pp. 361–363]. The stabilized section field is relatively algebraically closed in \C(X) by [27, §3, Proposition 1, p. 359]. In the case \kbar(U)=0, the map is the map to a point.

For clarity, the fiber conclusion concerns the full restricted section system. On a resolution, a sufficiently divisible system has moving part pulled back from a big system on the Iitaka base. In a divisible degree there is consequently a section after subtracting a pulled-back ample divisor. Any extra independent plurisections on the generic fiber extend after an ample pullback twist. Multiplying by the preceding section would then enlarge the stabilized section field, which is impossible. Doing this in each degree gives the assertion on very general fibers. Formula (2.3) allows the theorem to be applied to the log divisor on each higher source model itself. Adjunction on a smooth fiber identifies its restriction with that fiber’s log canonical divisor. This proves (2.5), including after further compatible resolutions and birational changes of the Iitaka base.

Let K=\C(X), L_Y=\C(Y), and L_Z=\C(Z), viewed as subfields of K. The combined image in Y\times Z has function field L_YL_Z. Let L_W be the relative algebraic closure of this compositum in K, a finite extension of L_YL_Z. Normalize the combined image in L_W, resolve it, and resolve the resulting rational map from a higher source model. These operations give the displayed morphisms.

The field L_W is relatively algebraically closed in K by construction. Moreover, L_Y and L_Z are relatively algebraically closed in K, so each is relatively algebraically closed in L_W. In characteristic zero the three corresponding generic field extensions are regular. This proves geometric integrality of the generic fibers of x,g,h; properness and the normality of their targets give connectedness by Stein factorization. Finiteness of L_W/(L_YL_Z) gives generic finiteness to the combined image and hence generic finiteness of the indicated fiber maps.

Finally resolve D_W^0 and the source boundary and lift all the maps. Outside a proper closed subset of Z, the morphisms and every dominating boundary stratum are smooth, and the nondominating strata are absent. This follows from generic smoothness applied to the finitely many strata. It gives the asserted very general SNC fiber data. The section-ring and field arguments just given are unchanged by these birational operations.□

Proof. Work over an open subset of Z on which h and all the boundary strata are smooth. On its inverse image the logarithmic cotangent sequence is exact:

The last bundle has rank r. Since n=r+p_1, the exterior-power filtration has no term of base degree below p_1, and its first quotient is the canonical map

Pullback of logarithmic differentials under g is regular here: locally, the pullback of an equation for D_Y is a monomial in equations for D_W^0 times a unit. Thus g^*\xi is a section of the mth tensor power of the left side of (2.6). Apply the quotient tensorwise and, on W_z, evaluate on any p_1 fixed vectors in T_zZ. The result is a global section of

This construction is independent of local lifts of the chosen tangent vectors: replacing one lift by a relative tangent vector inserts r+1 relative vectors into an alternating form, so contributes zero. It is therefore valid along the entire boundary of the fiber, rather than only on its interior.

At a generic point of W_z, the differential of g has rank n, while its restriction to the tangent space of W_z has rank r. The induced normal map from T_zZ consequently has rank p_1. Some choice of the p_1 vectors gives a nonzero determinant, and hence a nonzero contracted section. When p_1=0, the same construction simply uses the relative top-degree quotient, with no tangent vectors to choose.□

Proof. First, \kappa(J)\geq0. To see this using the ordinary canonical divisor, take a very general compact Y-fiber of X'\to Y. It is a smooth birational model of the original fiber F and therefore carries a nonzero ordinary pluricanonical section because d\geq0. The restriction of this section to a general fiber of its map to W_y is nonzero: its zero divisor cannot contain all those fibers. Adjunction identifies this restriction, up to the one-dimensional determinant of the base cotangent space, with an ordinary pluricanonical section on J. Since D'|_J is effective, its log canonical divisor also has nonnegative section dimension.

Now restrict x to a very general fiber over Z:

The source and base are smooth connected projective varieties, the fibers are connected, and both displayed boundaries are reduced SNC. Every component of the inverse base boundary is in the source boundary, because both lie over D_Y. Thus Theorem 2.1 applies. Its conclusion, together with (2.5) and Lemma 2.8, is

Both terms vanish. The already proved ordinary nonnegativity, and monotonicity under addition of an effective divisor, give \kappa(J)=0 as well. The simultaneous geometric-generic and very general interpretations follow by base change in the countably many section degrees.□

Proof. The relative dimension over Y. The dimension of W_y is the image dimension of the absolute Iitaka coordinates restricted to a very general Y-fiber; the finite extension in the definition of W does not change this dimension. Those coordinates form a subsystem of a log pluricanonical system on that fiber. By the exceptional section comparison just explained, their image dimension is at most d. Conversely, upper addition for the ordinary canonical divisor on that compact fiber, whose general fiber over W_y is J, gives

Here we use the upper addition inequality for section dimensions with irreducible general fibers, as in [27, Theorem 4, p. 357]; no nonnegativity or subadditivity hypothesis on W_y is involved. Consequently \dim(W/Y)=d.

The differential of the image family. Let C_z=g(W_z)\subseteq Y with its reduced structure. The scheme image of W\to Y\times Z has geometrically integral generic fiber over Z: its fiber function field lies between \C(Z) and the regular extension \C(W). Generic flatness, followed by restriction to a smaller open, therefore yields a flat family of reduced integral subschemes C_z of Y. Hilbert representability [21, Theorems 3.1–3.2 and the discussion following Proposition 3.8] therefore gives a parameter map

We claim that the dimension of its image is exactly p_1.

Fix a very general z and a basis of T_zZ. Perform the contraction in Lemma 2.8 for every p_1-tuple of basis vectors. By Lemma 2.9, all nonzero resulting sections in degree m are proportional. Indeed, two independent sections in one degree would give a nonconstant ratio and positive Iitaka dimension.

At a general smooth point y\in C_z, the normal deformation map is

It is computed by lifting a base direction to W, applying dg, and taking its class modulo T_yC_z. This is independent of the lift and has rank p_1, since g is dominant and W_z\to C_z is generically finite. Locally trivialize the tangential determinant and the form \xi. Each contracted section is the mth power of the corresponding p_1-minor of (2.11), multiplied by a common nonzero factor. Proportionality of the sections says that the mth powers of all ratios of nonzero minors are constant as the point of W_z varies. The ratios themselves are constant: a rational function whose positive power is constant is algebraic over \C, and hence is constant on an integral complex variety. The Plücker coordinates of (2.11) thus have constant ratios. Its kernel, a subspace of the fixed vector space T_zZ, is consequently independent of the general point y.

To pass from this pointwise calculation to the parameter map, write \mathcal I_z for the ideal of C_z in Y. A Hilbert tangent vector is a homomorphism

Such a homomorphism is zero if it is zero at the generic point: the target is the structure sheaf of an integral scheme and has no torsion. On the general smooth locus of C_z, its normal value is precisely (2.11). It follows that d\tau_z has the same kernel as that normal map and therefore has rank p_1. In characteristic zero, generic differential rank equals the dimension of the reduced image. This proves the claim.

This argument includes the extremes. If p_1=0, every C_z has dimension \dim Y and is therefore Y, so the family is constant. If r=0, the smooth connected W_z is a point and its parameter map has rank \dim Y=p_1.

Construction of S and T_0. Let the function field of T_0 be the relative algebraic closure of the image parameter field in \C(Z), and take a smooth projective model. Then \dim T_0=p_1, and Z\to T_0 has geometrically integral generic fiber. Pull the universal image family back to T_0 and take the component dominated by W, with a smooth projective resolution denoted by S. After compatible resolutions all the maps in (2.8) are morphisms. Its right square commutes, and the induced map (2.9) is dominant and generically finite: over a general z it is the generically finite map to the corresponding image member.

The generic fiber of S\to T_0 has dimension r. Thus \dim S=r+p_1=\dim Y, and its dominant map \rho:S\to Y is generically finite. In fact it is birational. The maps give the field inclusions

The first extension is algebraic, whereas \C(Y) is relatively algebraically closed in \C(X). Therefore \C(S)=\C(Y). In particular, the construction embeds \C(T_0) in the original base field \C(Y).

For completeness, geometric integrality of the generic fiber of S\to T_0 can be checked in the same field tower. The field \C(T_0) is relatively algebraically closed in \C(Z) by construction, and \C(Z) is relatively algebraically closed in \C(W). It follows that \C(T_0) is relatively algebraically closed in \C(W), and hence in its subfield \C(S). The extension \C(S)/\C(T_0) is therefore regular. This also verifies the geometric connectedness independently of the universal-family description.

Dimensions and the logarithmic base fiber. The equality already proved over Y gives

Together with \dim T_0=\dim Y-r, this yields \dim Z=d+\dim T_0.

Resolve the support of D_S=(\rho^*D_Y)_{\mathrm{red}}, lifting the other maps once more. Logarithmic pullback gives a nonzero section \rho^*\xi of m(K_S+D_S). Restrict it to a very general smooth fiber S_t; this restriction is nonzero, since its zero divisor cannot contain all fibers. Adjunction, with the determinant of T_t^*T_0 a constant one-dimensional factor, gives a nonzero section of m(K_{S_t}+D_S|_{S_t}). This proves the last assertion of (2.10). All these final resolutions preserve the section and fiber properties in Proposition 2.7. We now fix an earlier resolved Iitaka source X_* and take the common source X' above it; later source modifications will continue to dominate this fixed model.□

Proof. The spaces of pluricanonical sections commute with extension to an algebraic closure of the ground field. We first make that extension and work with the smooth projective variety J over an algebraically closed field of characteristic zero. Since \kappa(J)=0, every nonzero pluricanonical system has dimension one.

Choose a rational canonical frame \tau and write \omega_J=f\tau^p. Normalize J in the field obtained by adjoining a pth root of f. This field extension has degree p. Indeed, since the ground field contains the pth roots of unity, reducibility of the Kummer polynomial would make f a qth power for some prime q\mid p. If f=a^q, then the rational (p/q)-canonical form a\tau^{p/q} has regular qth power. Its order at every prime is therefore nonnegative, so it is regular, contrary to the minimality of p. This also proves geometric connectedness before the extension of the ground field.

Denote the finite normalization by \pi_0:J^{\mathrm{nor}}\to J and a smooth resolution by \mu:\widetilde J\to J^{\mathrm{nor}}. The rational root form \eta on \widetilde J satisfies

as pluricanonical differentials. Pullback of a regular pluricanonical differential under a morphism of smooth varieties is regular. Thus (3.2) proves that \eta is regular at every prime of \widetilde J, including the resolution-exceptional primes.

Set E=\operatorname{div}(\omega_J). At a prime of J^{\mathrm{nor}} over a prime of J at which E has coefficient a\ge0, let e be the ramification index. The tame differential formula gives

If a=0, adjoining the root of a unit is unramified at that prime and e=1. If a\ge1, the right side of (3.3) is at most 2ea. This bound is a statement at the primes of the finite normalization; we do not assert it at every prime of its resolution.

For s\in H^0(\widetilde J,mK_{\widetilde J}), the rational function s/\eta^m consequently has poles at the primes of J^{\mathrm{nor}} bounded by 2m\pi_0^*E. Normality therefore gives an injection

Only codimension-one tests on J^{\mathrm{nor}} enter this injection.

Lemma 2.3 gives \kappa(J^{\mathrm{nor}},\pi_0^*E)=\kappa(J,E)=0. Since \pi_0^*E is effective, each space on the right of (3.4) has dimension one: two independent sections would give a nonconstant section ratio. The nonzero section \eta^m gives the reverse inequality on the left, proving the assertion.

The integer p and the one-dimensional generator space are unchanged geometrically. A rational generator can consequently be chosen as a section of pK_{X'/W}. The Kummer construction is algebraic over a nonempty open of W. An equivariant resolution of its projective normalization, followed by shrinking the base and generic smoothness, gives the asserted smooth projective family, with its fiberwise cyclic deck action.□

Proof. For unipotent local monodromy, the extension is the top filtration subbundle of the Deligne–Schmid extension. Nefness and compatibility with powers and log-pair pullbacks are [4, Lemmas 2.16 and 5.6]. The pullback statement follows from functoriality of the unipotent extension and the extended Hodge filtration.

Integral polarized monodromy is quasi-unipotent at the boundary. In a crossing chart, take sufficiently divisible roots of the boundary coordinates to make it unipotent. The resulting line is equivariant for the finite deck group. A common positive power kills the stabilizer characters on its fibers and descends to a line bundle in the original chart. These descents agree on overlaps: compare them on a common root cover and use uniqueness of the unipotent extension. They define the rational parabolic line, including its finite-monodromy weights. The same comparison proves compatibility with powers and further log-pair pullbacks. This proves the quasi-unipotent assertions by descent from the unipotent ones.

These comparisons can be made on a global smooth adapted alteration. For example, a finite neat level of the integral monodromy, followed by compactification and resolution, makes the boundary monodromies unipotent. Extra boundary components on which the original variation already extends have weight zero. The pullback of L is the unipotent extension there and is nef. Nefness descends under a proper surjective map: every curve downstairs is dominated by a curve upstairs, and the projection formula compares their degrees. Thus L is nef.□

Proof. At the generic point of P, correct the relative form by the factor u^{-pt_P}, where u is a local equation for P; take a tensor power to make the exponent integral. At every actual component over P its normalized order satisfies

Orders at divisors horizontal over W are nonnegative, since \omega_J is an ordinary regular pluricanonical form on the generic fiber. Near the generic point of P, the corrected form is therefore a logarithmic pluriform for the actual SNC pair (X',D').

The discrepancy inequality for a reduced SNC pair says that pullback to a higher smooth model is still logarithmic for strict transform plus reduced exceptional boundary. In particular, the normalized ordinary order of the corrected form at any higher prime is at least -1. It follows that

If Q attains the actual minimum defining t_P, then

This proves 0\le B_P\le1. If P is contained in g^{-1}D_Y, every actual component over P lies in D' and has d_Q=1. The middle term of (3.8) is then zero, so B_P=1.

Resolve, over the generic point of P, the support of the form’s zero and pole divisor together with x^*P. On this resolution the relevant conditions are monomial inequalities. The SNC discrepancy inequality shows that inequalities at its components imply the inequalities on every higher model. Thus a finite log resolution computes \lambda_P; in particular it is rational. Outside a finite divisorial subset of W, the fibers are smooth and integral, the relative form has no vertical zero or pole, and there is no vertical source boundary. There the actual component has m_Q=1, r_Q=d_Q=0, and the smooth-fiber calculation gives t_P=0 and \lambda_P=1. This proves finite support.

On a further source resolution, the old actual divisors remain. They still attain the old minimum. Equation (3.7) holds at every new actual divisor by logarithmic pullback, so the minimum cannot decrease. The all-valuation infimum is unchanged. An étale base-DVR extension preserves canonical frames up to units, multiplicities, and all the displayed orders; the same assertion holds on a fixed log resolution, including when a divisor splits. Finally resolve the finite support on the base, resolve the source map, and recompute (3.5). All previously unchanged codimension-one points keep their coefficients, and new support lies on the exceptional base divisor. The inequalities just proved apply to the recomputed data.□

Proof. Take Q attaining the first minimum in (3.5). The equality B_P=0 makes \lambda_P-t_P in (3.8) equal to one. Thus

Since m_Q is a positive integer and d_Q\in\{0,1\}, this forces m_Q=1 and d_Q=0. Equality throughout also gives the final assertion in (3.9).□

Proof. On the smooth family locus, finite change of variables gives

where c_p>0 is constant and the expression on the right is the canonical density associated with a pluricanonical form. In a local canonical frame, this density is the 2/p power of its coefficient times the usual canonical volume density.

Take a general point of P, with base coordinates (u,s_2,\ldots,s_l) and P=(u=0). Integrate (3.11) against base Lebesgue measure and the factor |u|^{-2\gamma}. Fubini and change of variables reduce local integrability to an integral on a log resolution of the total space. The factors from the absolute canonical density and from the base canonical frame cancel precisely as prescribed by K_{X'/W}=K_{X'}-x^*K_W. Thus, at a vertical prime Q, the monomial exponent in the resulting density is

The transverse integral is finite exactly when r_Q-\gamma m_Q>-1. Horizontal primes have nonnegative ordinary order and cause no further condition. A finite SNC resolution computes all these conditions, so the supremal integrability exponent is

Here local integrability is measured against base Lebesgue measure. The value at the endpoint itself is irrelevant to this supremum.

We compare this with the parabolic extension. After a transverse root coordinate killing the finite part of the monodromy, a nonvanishing canonical extension frame has squared Hodge norm bounded above and below by powers of -\log|u|. This is the one-variable nilpotent-orbit growth estimate for a polarized pure variation [44, §§4 and 6]. Accordingly, a rational section of parabolic order a has squared norm

where the notation means two-sided bounds by fixed powers of the logarithm. Its supremal exponent in (3.12) is 1+a.

There is a meromorphic coefficient to which this statement applies. The form \eta is an algebraic section of the top line on the family locus. One may use the regular-singular extension of the geometric Gauss–Manin system. Equivalently, the finiteness of the threshold in (3.12) gives a weighted L^2 bound for its holomorphic coefficient in a nonvanishing extension frame. Absorb the logarithmic factors into an arbitrarily small power of the radius. The Laurent-series integral test then bounds the negative exponents of that coefficient, excluding an essential singularity and proving meromorphic extension. The finite root coordinate gives a rational parabolic order downstairs. Clearing its denominator by a tensor power, and applying the codimension-one extension test, gives the divisor of the rational line M.

Comparing with (3.12) proves a=\lambda_P-1. There is no additional factor of p: the normalized order r_Q already contains 1/p, and the density in (3.11) uses the same normalization. Finally t_P=B_P+(\lambda_P-1) at every prime, which proves (3.10). The transverse root coordinate has only been used to compute the metric order; no section system has been descended from a ramified alteration.□

Proof. Resolve the rational map, obtaining a morphism T_1\to R. Put e=\dim T_1-\dim R. A prime divisor of T_1 with image of codimension at least two lies in the closed locus where the fiber dimension is at least e+1. There are only finitely many divisorial components in that locus. Consider those not exceptional over T_*.

For each such prime, its divisorial valuation v on K=\C(T_1) has nontrivial restriction w to L=\C(R). The latter is a discrete rank-one valuation. We justify that it is divisorial. Residue elements algebraically independent over \kappa(w) lift to elements algebraically independent over L: in a proposed relation, divide by a coefficient of least valuation and reduce to the residue field. Hence

Since v is divisorial, this gives

Conversely, lifts of algebraically independent residue elements for w, together with an element of positive valuation, are algebraically independent over \C. Thus \trdeg_{\C}\kappa(w)\le\dim R-1. Equality follows.

Choose a residue transcendence basis and lift it to units of L, together with an element generating the value group of w. They give a purely transcendental subfield of L over which L is finite. Normalize the corresponding model in L and take the center of w above the DVR cut out by that positive-valuation parameter. This realizes w as a prime divisor on a birational model of R. Extract all the finitely many restrictions in this way and dominate the result by a smooth projective model. On a compatible smooth source model, each of the original nonexceptional divisors now has divisorial image. Any new source prime is exceptional over T_*. A further birational base modification preserves the strict transform of each extracted valuation divisor, proving persistence.

For the last assertion, shrink a nonempty open in Z so that the finitely many relevant loci, their images, and their boundary strata have the expected fiber dimensions. A prime divisor contracted to codimension at least two on a fiber lies in the restriction of the fixed excess-fiber-dimension locus. A global component of codimension at least two dominating Z still has codimension at least two on a general fiber. Thus a divisorial component on a general source fiber comes from a global divisorial component of that locus, possibly after splitting. Its image over T_* has codimension at least two, and the same dimension calculation makes its image exceptional on the general fiber of T_*. This proves the fiberwise assertion.□

Proof. Apply Lemma 3.6 with fixed source X_* and base W. Retain the maps to Y and Z, resolve the finite boundary supports, and recompute (3.5). By the lemma, every divisor of a very general fiber X'_z whose image under x_z has codimension at least two is exceptional over X_{*,z}.

We also shrink the good open in Z for the finitely many supports and resolutions used in the divisorial calculation. The tests defining T then restrict to W_z. To see this explicitly, over the smooth good locus of W those tests are zero. Outside it, only restrictions of the finitely many divisors dominating Z can meet a general W_z in a divisor. Generic smoothness and adjunction preserve the multiplicities and relative-canonical orders there, with splitting of components allowed. Loci of higher codimension stay of higher codimension after the shrinking just made. The same reasoning on a fixed log resolution computes all the higher-valuation inequalities.

Write T_z=T|_{W_z}. For a sufficiently divisible integer m, take

Using the relative canonical identification, multiply its pullback by \omega_J^{m/p}. This is a rational m-canonical form on X'_z. If Q is an actual source divisor with divisor image P, its normalized order, after allowing the source boundary, is at least

by (3.7). A horizontal divisor passes the ordinary test since the generic fiber generator is regular. All other untested divisors are exceptional over X_{*,z}.

Push the rational form to X_{*,z}. At every prime of that smooth fixed fiber, its strict transform has just passed the required logarithmic test; hence the pushdown belongs to

Possible poles on a divisor exceptional over X_{*,z} have no bearing on this codimension-one test downstairs. The construction is injective. More precisely, the common factor \omega_J^{m/p} cancels in ratios, so ratios of two input sections pull back by the dominant map x_z and their rational-map dimension is preserved. The target system has Iitaka dimension zero by (3.1). The identity T_z\sim_{\Q}B|_{W_z}+M|_{W_z} from Proposition 3.5 proves (3.13).□

Proof. This is [37, Theorem 3.1], whose proof uses the full period image and boundary numerical-rank loss. The statement permits a complex summand of an integral ambient variation; it is stronger in this respect than a theorem requiring that summand itself to carry an integral lattice.□

Proof. The hypotheses meet Theorem 3.8 directly. A projective manifold belongs to class \mathcal C, a rational polarization is a real polarization, and the entire complexification of the given variation is a complex direct summand of itself. Its highest nonzero filtration step is the specified rank-one line. Thus the companion supplies \tau:B'\to P, p:B'\to R, A and c>0 with

It remains to obtain a projective source model without changing this identity or the connected-fiber property. The map P\dashrightarrow R induced by p is meromorphic between projective varieties and hence rational: its graph closure is algebraic. Resolve that graph projectively to obtain a smooth projective P' and morphisms

On a common smooth modification of B' and P', the two maps to R agree. Pulling back (3.15) there therefore identifies the pullbacks of \mu^*L and a^*A.

Pullback by a proper birational morphism v:T\to P' is injective on rational line bundles. Indeed, if a line bundle N pulls back to a torsion line bundle, some positive power v^*N^m is trivial. Normality of P' gives v_*\mathcal O_T=\mathcal O_{P'}, and the projection formula then gives N^m\simeq\mathcal O_{P'}. Applying this observation to the common modification proves

The generic fibers of a are connected as well. The original map p has connected fibers and normal target, so \mathbf C(R) is relatively algebraically closed in \mathbf C(B')=\mathbf C(P). The same field extension occurs for a. Its Stein factor is consequently finite birational over the normal variety R, and hence is R itself. Thus all fibers of a are connected.

Finally A is nef. For every rational b\geq c, K_R+bA=(K_R+cA)+(b-c)A is big. If R is a point, the line identity already gives rational triviality; the zero-dimensional bigness statement uses the usual point convention. This proves the proposition.□

Proof. Use Lemma 3.6 with fixed source T_1 and base R. On the resulting smooth source T_2, let \Delta_2 be the strict transform of \Delta plus the reduced exceptional divisor, and pull back N and A; write N_2=\pi^*N for the former pullback. The SNC discrepancy inequality gives

The section systems in the assertion therefore have the usual pullback and pushdown comparison. For a smooth birational base map \rho:R'\to R,

is still big, because K_{R'/R} is effective. Relabel the prepared base as R, its divisor as A, and its source morphism as a_2.

Choose a sufficiently divisible positive integer m for which the relative direct image

has positive rank. Such a choice is possible by restriction of a nonzero section of a multiple of K_{T_1}+\Delta to the generic fiber, followed by pullback. The pair (T_2,\Delta_2) is log canonical, T_2 is projective, and the base is smooth projective. Thus [19, Theorem 1.1] makes \mathcal E_m weakly positive.

Put L=K_R+cA and fix an ample integral divisor H on R. Increase m to clear the denominators needed below. Since L is big, choose a positive integer \alpha such that

The definition of weak positivity [19, Definition 7.2] supplies a positive integer \beta for which

At the generic point of R, multiplication of relative sections is not the zero map: the power of any nonzero section on the integral generic fiber is nonzero. Generic generation in (3.17) therefore yields a global section whose evaluation is a nonzero rational section u of

At every codimension-one point of R, the direct image is locally free and the reflexive symmetric power agrees with its ordinary symmetric power. Evaluation is regular above those points. Any remaining pole divisor of u thus has image of codimension at least two and is exceptional over the fixed source T_1.

For a sufficiently divisible positive integer q, multiply u^q by the pullbacks of all sections of

The resulting rational sections lie in the class

They are regular at every prime not exceptional over T_1, and so push down to the required absolute section system on T_1. Their ratios are the pullbacks of the ratios of sections of the big divisor \alpha mL-H. These ratios give a rational map of dimension \dim R, proving (3.16).□

Proof. Set T_1=W_z, \Delta=B|_{W_z}, and N=M|_{W_z}. For the chosen z, the fiber is smooth and all relevant boundary strata meet it transversely, including those of the enlarged divisor defining the domain of the variation. The log-pair pullback property in Lemma 3.2 therefore identifies N with the parabolic extension of the restricted variation. Moreover

by (3.1) and B\ge D_W^0. In particular, D=K_{T_1}+\Delta is rationally equivalent to an effective rational divisor.

Apply Proposition 3.9 to the restricted integral polarized variation whose entire highest piece is this line. The comparison is applied directly to this restricted variation; its entire complexification is an allowed summand in the companion theorem. On a higher smooth model it gives

On the higher source model use strict transform plus reduced exceptional boundary for \Delta. The logarithmic section comparison preserves (3.13) and the rational effectivity of D; the Hodge line is pulled back. If \dim R>0, choose a sufficiently large rational c\ge1. Lemma 3.11 gives

But

After clearing denominators, multiplication by the section of the effective last summand preserves rational-map dimensions. Thus the same positive lower bound holds for D+N, contradicting (3.13). We conclude that R is a point and N\sim_{\Q}0. Rational triviality descends through the intervening birational morphisms, since their direct image of the structure sheaf is the structure sheaf. Now (3.13) and \kappa(D)\ge0 give \kappa(D)=0.

Finally, these algebraic assertions pass to the geometric generic fiber. For the Kodaira dimension assertion, use generic base change in the countably many divisible degrees. For rational triviality, choose an integer clearing the denominator of M. For each positive integer j, simultaneous nonvanishing of sections of the jth power of that line and of its inverse is a closed condition on a smooth proper family, and is equivalent to triviality on a connected projective fiber. The countable union of these loci contains every very general z by what we proved. Hence some one of them contains a dense open of Z; otherwise a very general point would avoid them all. Generic base change gives the required rational triviality on the geometric generic fiber.□

Proof. Pass to a finite cover making the boundary monodromies unipotent. The extension pulls back, and its degree remains zero. Let \alpha be the highest-line Hodge curvature. Griffiths’ curvature formula makes it semipositive and identifies its kernel with the kernel of the projective line differential.

We check that its integral is the degree of the extension. For a smooth reference curvature \beta, write \alpha-\beta=\ddc u. Near a puncture, with \rho=-\log|z|^2, the Hodge norm and horizontal Schwarz estimates give

see [14, Theorem 5.1] and [10, Proposition 2.4 and Lemma 2.5]. Use products of cutoffs \chi((\log\rho)/N), equal to one away from a shrinking puncture neighborhood. Their second derivatives are uniformly bounded in the cusp metric and supported on escaping tails. Integration by parts bounds the error by the tail integral of (1+\log\rho)\rho^{-2}\,d\rho, which tends to zero. Thus \int_{P\setminus D}\alpha=\deg L=0. A continuous nonnegative curvature form with zero integral vanishes. Its kernel identity proves constancy on the cover and therefore on the original universal cover.□

Proof. Deligne’s determinant theorem [15, Corollary 4.2.8(iii)(b)] states that a positive tensor power of the determinant of any complex local subsystem is trivial. For a rank-one subsystem, the determinant is the subsystem itself. Its character therefore takes values in a fixed finite group of roots of unity. The theorem applies anew to the pulled-back variation on any connected smooth algebraic locus.□

Proof. The pulled-back variation is again polarized, integral and pure. Constancy of its projective highest line makes that line a rank-one complex local subsystem. Apply Corollary 3.14 to this restricted variation. No common exponent for different algebraic loci is needed.□

Proof. Put T_1=W_z and N=M|_{W_z}. Proposition 3.12 gives N\sim_{\Q}0, and the pullback property in Lemma 3.2 identifies N with the parabolic extension of the restricted entire top line. The degree of N on every complete smooth test curve is zero. Corollary 3.13 makes the projective top line constant along such curves. Complete-intersection curves through general points and tangent directions give projective constancy on the connected good locus of T_1. Corollary 3.15, applied to this restricted integral polarized variation, gives finite monodromy for its original top line.□

Proof. Kill the finite character by a finite cover of C^\circ and complete the resulting curve. Semistable reduction over a curve, followed by the usual toroidal resolution, gives, after a further finite change, a smooth projective model P_0\to C with reduced fibers; see [30, Chapter II, semistable reduction over a curve]. Take a smooth projective common dominating model of this family and the original changed family. When \Gamma is present, include the finitely many translates in the graph construction and use an equivariant resolution. This also regularizes a finite birational action: take the closure of the graph of all its maps in the product of their projective models, on which the group permutes the factors, and resolve equivariantly. Over a dense smooth open, birational pullback preserves ordinary top forms and is compatible with the cohomology local systems. The chosen top class is therefore still invariant.

The flat generator already gives a relative top form on this smooth open. Choose a log resolution of its geometric generic zero divisor. That resolution and its finitely many geometric components are defined after a finite extension of the curve’s function field. Make this change now and spread the resolution. Repeat semistable preparation at the boundary and take a smooth common dominating model, preserving the chosen resolution over a dense open. Thus the geometric divisor components used below are defined before the final P, H, and \Theta are fixed.

The global invariant cycle theorem gives a surjection from the cohomology of the smooth compactification onto the invariant part of the fiber cohomology. The restriction map is a morphism of Hodge structures, so its strictness gives a preimage of the invariant (n,0)-class in H^{n,0}(P); these are precisely holomorphic n-forms. Here we use [15, Theorem 4.1.1(ii) and Corollary 4.1.2], with smooth proper family over the good curve and smooth projective total compactification. Holomorphic forms on a smooth projective variety are closed.

To describe the relative section, choose a local coordinate t on C. The absolute form dt\wedge\Theta, divided by the base frame dt, defines s in K_{P/C}. These expressions agree under a change of coordinate on the curve. The section is holomorphic globally and is nonzero on every good fiber. Resolve its horizontal zero divisor and shrink the good open so that the resulting horizontal components and their intersections are smooth over the curve. This gives the stated relative normal crossings. Any two lifts of the fixed flat top class have the same restriction to every good fiber, and therefore give the same relative section. Thus the following change of lift does not alter s or this preparation of its zero divisor.

Put V=H^0(P,\Omega_P^n) and let N be the kernel of restriction to a good fiber. This kernel is independent of the chosen good fiber: restriction of a global closed form gives a flat cohomology section, and a holomorphic top form represents the zero cohomology class only when it is zero. Choose an ample H and a positive metric, and set

This is a positive definite Hermitian pairing on V. Indeed its integrand is a positive multiple of the pointwise norm squared when \beta=\gamma, and a nonzero holomorphic form is nonzero on an open set. For a positive smooth area form \ell from the curve, the pairing B_\ell is semidefinite. Fiber integration identifies its kernel with N. Consequently the image of the map

is the annihilator of N.

Choose the lift \Theta orthogonal to N for B_h. The functional B_h(\Theta,-) annihilates N, so it equals B_\ell(\alpha,-) for some \alpha\in V. The duality pairing of H^{n+1,1}(P) with H^{0,n}(P) is nondegenerate. Applying it to the difference proves (4.1).

In the equivariant case a tensor product of the translates of an ample bundle admits a natural \Gamma-linearization; its metric can be made invariant while keeping positive curvature. Both N and the pairing are then invariant. The unique orthogonal lift of the chosen top class transforms by its top-line character. The same character can be imposed on \alpha by projection to its character space.

Finally, a holomorphic form on a smooth projective birational model is the pullback of a holomorphic form on every smooth projective model it dominates. Applying this to P\to P_0 gives the asserted descent of \Theta. A later positive tensor power of H multiplies h by a positive integer and preserves all these conclusions, with \alpha scaled accordingly.□

Proof. In a local relative canonical frame, write \omega for the relative n-form represented by s. The normalization in (4.2) gives

Thus sv is a holomorphic section of T_P\otimes K_{P/C}: contraction with an absolute top form identifies this bundle, after choosing dt, with \Omega_P^n. Dividing by s proves the pole bound.

Use the absolute scalar-symbol sequence

Its middle sheaf consists of first-order differential operators on H with scalar symbol. Its extension class is c_1(H), up to the fixed normalizing constant; this is the connection obstruction of [3]. For the specific assertion needed here, take frames e_i=g_{ij}e_j. Local lifts of the same vector symbol differ by its contraction with d\log g_{ij}. This is the cocycle representing the stated extension class.

Tensor (4.4) by K_{P/C}. The obstruction to lifting sv is its contraction with c_1(H) in H^1(K_{P/C}) on the neighborhood in question. In Dolbeault notation and using (4.3), its restriction to D_t is, up to a nonzero constant and sign, the class obtained from h\wedge\Theta by dividing its top holomorphic degree by dt and restricting to D_t. This operation respects Dolbeault coboundaries. In fact the total space has dimension n+1, so a primitive of type (n+1,0) necessarily contains dt in submersion coordinates; division by dt commutes with \bar\partial before restriction. Equivalently this is the canonical restriction K_P|_{D_t}\simeq K_{D_t}\otimes T_t^*C with a base frame chosen.

The right side of (4.1) gives zero under this operation: a smooth area representative of \ell contains d\bar t. Hence the fiber obstruction is zero for every good t. Shrink to a Stein disk on which cohomology and base change hold for K_{P/C}. Leray and the vanishing of higher coherent cohomology on the disk identify the obstruction with a section of R^1\pi_*K_{P/C}. It is zero because its fiber values are zero. Therefore sv lifts to a holomorphic section of \operatorname{At}(H)\otimes K_{P/C} on the whole neighborhood. Dividing this lift by s produces \nabla, with the claimed symbol and pole bound. Replacing H by a tensor power uses the induced operator and leaves its symbol and its pole bound unchanged.□

Proof. We first treat a horizontal component. Choose a simultaneous very general good fiber D, so that all its positive pluricanonical section spaces have dimension one. Write

The coefficients e_j are positive integers, E\sim K_D, and the support is simple normal crossing. The components extend relatively over a small disk. In the local computations we use the same letters for these relative extensions and their local equations.

The contradiction will take place in the section spaces of R_0. Since

for every positive integer m we have

Thus every section of mR_0 is a multiple of its divisor section and vanishes on S. We will construct a section whose restriction to S is nonzero.

Suppose that the restriction of \Theta to the horizontal component through S is nonzero. Let z and G be local equations for S and R_0, and put

The pole bound in Lemma 4.3 shows that W is holomorphic and vanishes on the reduced zero divisor. It is consequently logarithmic along that divisor, so that both W(z)/z and W(G)/G are holomorphic. Under changes of the equations, u transforms as a section of R_0|_S. Formula (4.3) identifies its nonvanishing at the general point of S with the supposed nonvanishing of \Theta on the relative component. Thus

Moreover u is divisible by the full reduced divisor B'|_S. To see this at a crossing, the coefficients of sv are holomorphic, while G/s has one factor for each component of E_{\mathrm{red}}. Division by z removes the factor belonging to S and leaves all other reduced factors.

Write the operator in a regular frame of H as \nabla=v+c. Its pole bound also gives Gc|_S=0. Replace H by a sufficiently high tensor power. Relative generation supplies finitely many sections a_\lambda on a neighborhood of the whole fiber, vanishing on the relative S, such that a_\lambda/z generate H(-S) along S. For each \lambda set

The following recurrence proves in particular that the second expression is holomorphic:

For the second equality, W vanishes on S, Gc|_S=0, and (W(G)/G)|_S=(e_i+1)u. The other components of G contribute zero on S; this can be checked away from their intersections and then extended holomorphically. These facts prove both formulas by induction. They also show that the restrictions of q_{k,\lambda} are sections of \cO_D(kR_0+H-S). More explicitly, in a local frame of H,

The meromorphic expression on the left is therefore a holomorphic section of the indicated twisted bundle, with local coefficient b_{k,\lambda}/k!.

We need estimates as k tends to infinity. Cover a compact neighborhood of D by finitely many pairs of nested coordinate charts. For a fixed k, divide each nesting margin into k equal parts. On these charts the coefficients of W, W(z)/z, W(G)/G, and Gc are bounded. At the jth step of (4.7), Cauchy’s estimate for the derivative costs at most a constant times k, and the zeroth-order coefficient costs at most a constant times 1+j\le k+1. It follows that the holomorphic coefficients b_{k,\lambda}/k!, in smooth metrics for kR_0+H-S, are bounded by

This is a bound in the indicated twisted bundle, including across the zero divisor.

Put a=e_i+1\ge2. No factor in (4.8) is zero, and

for example, the ratio of successive absolute values tends to a. On D, take the weights

where \varphi_{H-S} is a smooth weight on H-S in the same local frames. They are weights on R_0 with curvature loss tending to zero and uniform upper bounds by (4.9). Their regularized upper limit is a semipositive singular weight. Formula (4.8), together with generation of H(-S)|_S, gives its restriction a lower bound by \log|u|^2 up to a constant. Finally take the maximum with the divisor weight of R_0. We obtain a semipositive weight \varphi on R_0 satisfying

All weights here are written in compatible local frames; the inequalities are invariant statements about the corresponding metrics.

We now extend a nonzero power of u using [18, Theorem 4.1, conditions (19)–(23)]. We record every hypothesis in this application. The ambient manifold is the smooth projective D, and its smooth hypersurface is S. For any integer m\ge2 put

The first line uses K_D\sim E and E_{\mathrm{red}}=S+B'. Thus F' is an actual line bundle with the indicated weight, after the indicated line-bundle identification.

Condition (19) of the extension theorem holds with its constant equal to one, since (4.10) implies

For condition (20), the weights 1+\varphi/a and a^{-1}\log|s_{A_0}|^2 are semipositive weights on rational line bundles and have difference \varphi_S. If integral bundles are used, choose an integral G_2 sufficiently ample that G_2-A_0/a has a positive smooth rational-bundle metric, and add this metric to both weights. The difference is unchanged and both underlying bundles become integral.

For condition (21), let T_R=\ddc\varphi. Directly,

Here a\ge2 and m\ge2. For condition (22), choose a positive number

omitting the inner minimum if there are no other components. The weight \varphi is locally bounded above. After normalizing local defining functions to have norm at most one, each coefficient of \log|s_{B'}|^2 is at least the corresponding coefficient in \varepsilon_0 a^{-1}\log|s_{A_0}|^2. Therefore

A common smooth metric added to the two weights above changes this inequality only by a bounded local term. This proves precisely the required domination of the subtracted weight’s singularities.

The section to extend is

The lower bound for \varphi|_S and u\ne0 also ensure that the restricted weights are not identically minus infinity. Condition (23) follows from the second bound in (4.10):

The quotient on the right is locally bounded because u is divisible by the full reduced divisor B'|_S. There is in fact integrability with a slightly larger weight exponent. Writing u=s_{B'}w, the density with exponent 1+\delta on \varphi_{F'} is bounded by

It is integrable for 0<\delta<1/m, by the normal crossings of B'|_S and the positive exponent on |w|. This argument requires no normal-crossing assertion about the other zeros of w.

The extension theorem now gives a section of mR_0 with restriction u^m\ne0, contradicting the one-dimensionality of this section space established above: its divisor section vanishes on S. This proves the horizontal vanishing.

For vertical components, use the reduced-fiber model P_0\to C in Lemma 4.2, to which \Theta descends. At a general point of a reduced fiber component the map has local equation t=z. There the vanishing of the relative coefficient s is exactly the vanishing of the restriction of \Theta to that component. Every vertical component on P which is not exceptional over P_0 is tested at the same generic point. An exceptional divisor maps into a subset of P_0 of dimension at most n-1, and the pullback of an n-form to its resolution is zero. These observations prove the remaining vanishing assertions. They also explain why the reduced-fiber model is used: for t=z^r with r>1, an additional differential factor z^{r-1} would not itself imply vanishing of the absolute form.□

Proof. The form \mu is smooth, closed, and positive of type (n,n). Lemma 4.4 implies that [\mathcal E]\wedge\mu=0: the integral over each component can be computed on a resolution of that component, where the pullback of \Theta vanishes. Since the curvature current of \log|s|^2 is [\mathcal E], cohomology and closedness give

Here and subsequently \ddc T denotes the globally defined curvature current of the bundle weight. The measure on the left is positive, so it vanishes. For every real b, the weight

is another semipositive weight on the same bundle. The same argument shows that (\ddc T_b)\wedge\mu=0.

Delete \mathcal E, the zeros of \Theta, and the bad-fibration locus. In a holomorphic flow box for the kernel of \Theta, use coordinates (w_1,\ldots,w_n,\tau), where the leaves vary only in \tau. Closedness gives

The weight \log|s|^2 is pluriharmonic here. Thus wedging the curvature of T, or of T_b, with \mu selects the \tau-Laplacian of \psi, or of \max\{\psi,b\}, against a nonvanishing transverse volume. Slicing these zero positive measures shows that both functions are harmonic on almost every local leaf. Take the intersection of these full-measure sets for rational b. If a harmonic function is nonconstant, there is a small leaf neighborhood on which its differential is nonzero and whose image contains an open real interval. Choose a rational b in that interval. Taking the maximum with b produces a nonzero positive Laplacian on the level crossing in that neighborhood. Hence \psi is constant on almost every local leaf.

In these coordinates \overline\Theta contains all the transverse differentials d\bar w_j. Thus only a derivative in the leaf direction can contribute to \bar\partial(\chi(\psi)\overline\Theta), and leafwise constancy gives, for every bounded continuous function \chi,

off the deleted analytic set, in the sense of distributions. This identity extends across that set. Indeed, the coefficients of this current are locally bounded. Across a smooth analytic stratum of complex codimension q\geq1, a cutoff at distance \epsilon has derivative O(\epsilon^{-1}) on a tube of volume O(\epsilon^{2q}), so its contribution tends to zero. Applying this successively to the smooth strata, and choosing the tubes about the remaining lower-dimensional strata sufficiently small, proves the distributional extension. This argument uses boundedness of \chi; it makes no differentiability assumption on \psi.

We may now pair (4.1) with \chi(\psi)\overline\Theta. Choose the representative \ell there to be the pullback of a nonnegative smooth area form supported in relatively compact good base charts. This does not change its cohomology class. On those charts, after shrinking within the good locus, fiberwise rank one gives

with a(t) bounded on the support in question. Dolbeault integration by parts against the closed current just constructed therefore gives

On a good fiber write \omega=s|_{D_t}. In local frames the absolute value of the right-hand integrand for \chi(x)=\min\{M,e^{(1+\delta)x}\} is bounded by a fixed multiple of

Here |\omega|^2 denotes the canonical volume density. The weights T have local upper bounds. The zero divisor of \omega is relatively simple normal crossing on this compact set, so the last expression is integrable if \delta>0 is small enough that \delta e_j<1 for its finitely many multiplicities. The bound is independent of M. Multiply (4.15) by i^{n^2}, take absolute values on the right, and let M\to\infty. Monotone convergence proves (4.14).□

Proof. Apply the relative k-Bergman construction to \pi:P\to C and the twisting bundle r_kH with its smooth positive metric. The source and base are smooth, the morphism is projective and surjective with connected fibers, and a nonzero fiber section exists generically. Generic base change supplies a nonempty open on which all such sections extend locally. The smooth twisting metric makes every fiber section 2/k-integrable. Thus Theorem 4.2.7 of [41], in the cited arXiv version, supplies a semipositive weight \beta_k on kK_{P/C}+r_kH. Over the smooth extendibility locus, its exponential is the extremum of |U|^2 among sections normalized as in (4.16). No extremal interpretation on exceptional fibers is needed.

Set

These are weights on K_{P/C}, and \ddc\tau_k\geq-(r_k/k)h. On a fixed smaller submersion chart, the mean inequality for the subharmonic function given by the absolute value of a section coefficient to the power 2/k, together with its normalized integral and the bounded factor e^{-r_k\varphi_H/k}, gives an upper bound independent of k. Taking the extremum gives that bound for \tau_k. The bound persists across degree-dependent exceptional base values by plurisubharmonic extension. Equivalently, the ambient estimate in Remark 4.3.1 of [41] gives precisely this uniformity when the divided twisting weights are bounded above.

For completeness, local boundedness on one fixed nonempty open also globalizes as follows. Subtract a fixed smooth reference weight on K_{P/C}. The resulting globally defined potentials have a uniform lower bound for their complex Hessians. If their suprema were unbounded, subtracting those suprema and using compactness of sup-normalized quasi-plurisubharmonic potentials would give a subsequence converging in L^1 to a finite quasi-plurisubharmonic function. On the fixed open, however, the normalized functions would tend uniformly to minus infinity. This is impossible. Hence the weights \tau_k are uniformly bounded above in local smooth frames on P.

These constructions use the relative section spaces, without selecting a final fiber or an individual section. Take their regularized upper limit on P and then its maximum with the divisor weight:

The curvature losses tend to zero, so this is semipositive; if the first term is identically minus infinity, the second term still defines the weight. Its restriction to a good fiber is well defined because it dominates \varphi_0. Every normalized section is bounded above by the corresponding Bergman extremum. Consequently \varphi_k\leq\tau_k|_D, which proves all the assertions.□

Proof. Suppose that coefficient is \nu>0, with Lelong coefficients normalized by \log|z|^2. For each k, choose a nonzero leading section in the image of

Among sections with this prescribed image, minimize the 2/k-integral in (4.16). This minimum exists: the constraint is a nonempty closed affine subset of a finite-dimensional vector space, and the integral is continuous, positive away from zero, and homogeneous of positive degree. It is therefore coercive. Denote a minimizer by U_k, and rescale both it and the prescribed image so that (4.16) holds. All lower coefficients along S are zero by the definition of j_k.

Choose c>0 so large that, for all sufficiently large k,

Choose p_0>0 sufficiently small that

This follows already from the local upper bound for T|_D: the integrand is bounded by a constant times |\omega|^{-2p_0}, which is integrable when p_0e_j<1 for every component of the simple normal crossing divisor E.

Fix a smooth convex function g:\R\to\R_{\geq0} with 0\leq g'\leq1, equal to 0 sufficiently far to the left and to its argument sufficiently far to the right, and with g''>0 on [-1,1]. Choose a smooth function 0\leq\theta\leq1 that is zero on (-\infty,-1], one on [1,\infty), and whose derivative is supported in [-1,1]. For b>0, put

This is a weight on the integral bundle (k-1)K_D+r_kH|_D: the coefficients of its canonical weights sum to k-1, and x_k is a global function off the divisors. We henceforth take k>1+c+p_0. The constants c,p_0 are fixed. For each fixed k and cutoff b, we first remove the regularization used below. We then estimate the upper limit as k\to\infty with b fixed, and finally let b\to\infty. The contradiction will use one sufficiently large b followed by sufficiently large k.

The curvature and the adjoint estimate. Off E and the zero divisor of U_k, one has \ddc\varphi_k=-(r_k/k)h, \ddc\varphi_0=0, and \ddc x_k=-(r_k/k)h. Direct differentiation of (4.21) gives

In particular it dominates (1+c)r_kh/k and retains the displayed gradient term. The resulting adjoint estimate produces V_k\in H^0(D,kK_D+r_kH|_D) such that, on the complement of the divisors,

satisfies

The constant is independent of b and k: the retained gradient term controls the datum on the cutoff shell, even as the ample coefficient tends to zero. We give the singular-weight justification, including the choice of complete metric, before using this inequality.

Subtract a fixed smooth reference metric from T|_D and apply Theorem 14.12(a),(c) and Corollary 14.13(c) of [16]. We obtain weights T_\ell converging almost everywhere to T|_D, uniformly bounded above locally, smooth outside an analytic set Z_\ell, and satisfying \ddc T_\ell\geq-\epsilon_\ell h, where \epsilon_\ell\to0. The cited construction gives pointwise approximation above the original potential. In any event, taking a smooth regularized maximum with \varphi_0, of width tending to zero, ensures T_\ell\geq\varphi_0. The regularized maximum is invariant under simultaneous translation of its arguments, so it gives a weight on the same bundle. Its monotonicity and convexity preserve the lower curvature bound. It is smooth off Z_\ell\cup E; preservation of analytic singularities of the maximum itself is not required.

Keep k,b fixed and replace T|_D by T_\ell in (4.21), obtaining \Phi_\ell. For large \ell, the loss c\epsilon_\ell h is less than half the ample term in (4.22). On

the weight is smooth with positive curvature and the same gradient lower bound. This complement admits a complete Kähler metric. Indeed its deleted analytic set is the zero set of a section of a Hermitian vector bundle: twist its coherent ideal by a sufficiently ample bundle and use finitely many generators. Lemma 12.9 of [16] applies, because D is compact Kähler and the curvature of that smooth bundle has a uniform upper bound.

Write A_\ell=i\partial\bar\partial\Phi_\ell for the full positive curvature form on the complement. Add A_\ell to a complete Kähler metric there, obtaining a complete metric \widehat h_\ell\geq A_\ell. This particular choice also verifies the ordinary L^2 hypothesis for the datum

To see both this and the uniform estimate explicitly, let G be the matrix of an auxiliary Kähler metric and A the positive curvature matrix. For a bundle-valued (n,1)-form with local coefficient U\eta, matrix cancellation gives

up to the fixed convention for the canonical volume form. The determinants from the top holomorphic degree and the volume cancel, and so do the metric factors in the antiholomorphic degree and in the curvature operator. The solution norm in bidegree (n,0) is likewise independent of G. When G\geq A, the ordinary datum norm is bounded by the right side of (4.24).

The gradient term in (4.22) implies, for \eta=\bar\partial x_k, the matrix bound

Since \theta' is supported where g'' has a positive minimum, the inverse-curvature density of f_k is at most a constant times the shell density. That density is integrable. The datum is \bar\partial-closed, being the derivative of \theta(x_k-b)U_k on the complement. The complete Kähler estimate, Theorem 5.1 of [16], consequently gives \bar\partial u_{k,\ell}=f_k with the desired bound for \Phi_\ell.

Here the local adjoint density has precisely the required exponents:

On the fixed shell, it is bounded above by e^{(1+c+p_0)(b+1)}|\omega|^2, because T_\ell-\varphi_0\geq0. Moreover \Phi_\ell is bounded above locally uniformly in \ell, using the positive coefficient k-1-c-p_0 and

The weighted solution estimate therefore bounds the ordinary local L^2 norms of u_{k,\ell}. The holomorphic section V_{k,\ell}=\theta(x_k-b)U_k-u_{k,\ell} extends across its deleted analytic set by the distributional L^2 removable-singularity lemma [16, Lemma 12.10]. In a local bundle frame the right side of the distributional \bar\partial equation is zero, so the extended coefficients have holomorphic representatives. The extensions lie in the fixed finite-dimensional space H^0(D,kK_D+r_kH|_D), with bounded ordinary norm. Take a subsequence converging as global sections. Fatou’s lemma for the left side and dominated convergence on the fixed shell give (4.23). For this passage one may discard the countable union of the sets Z_\ell; no nesting of those sets is needed.

The shell becomes small. Put B=1+c+p_0. For fixed b, the shell densities in (4.23) are bounded by e^{B(b+1)}|\omega|^2. At a point off E, a subsequence contributing to their limsup has a further subsequence with x_k\to x_*\in[b-1,b+1]. Lemma 4.6 gives x_*\leq\psi; thus \psi\geq b-1 and

Reverse Fatou therefore yields

By (4.20), the right side tends to zero as b\to\infty.

The prescribed leading section is preserved. Near a general point of S=(z=0), write U_k=z^{j_k}a_k and \omega=z^{e_i}a_0, with the coefficients nonvanishing in the chosen frames. Since j_k/k<e_i, the function x_k tends to plus infinity as z\to0. Thus \theta(x_k-b)=1 and g(x_k-b)=x_k-b sufficiently near that point. The divisorial decomposition of the positive curvature current of T|_D gives T|_D\leq\nu\log|z|^2+O(1): subtracting \nu[S] leaves a positive current, whose local potential is bounded above. See [17, Chapter III, Proposition 8.16]. Consequently

The coefficient exceeds j_k+1 by (4.19). In this neighborhood u_k=U_k-V_k is holomorphic. If its vanishing order were at most j_k, the transverse integral against e^{-\Phi} would diverge, contradicting (4.23). Hence U_k-V_k vanishes along S to order at least j_k+1. This divisorial statement says exactly that V_k has the prescribed leading section and all the prescribed zero lower coefficients.

A gain of order 1/k. Let \rho_k\,dV be the normalized density in (4.16) relative to a fixed smooth volume on D, and put q_k=|u_k/U_k| off the zero divisor. The functions \rho_k have a uniform upper bound by Lemma 4.6. In local canonical frames the density is

The local upper bounds for T|_D, and the earlier bound \varphi_0+g(x_k-b)\leq\max\{\varphi_0,\varphi_k-b\}+C_g, therefore give, on a finite cover by smooth frames,

with C independent of k and of b bounded below. Fix 0<\eta<\min\{1,2/B\}. Hölder’s inequality gives

The exponent in the second factor is positive, so its uniform bound follows from compactness and the upper bound for \rho_k. Equations (4.23) and (4.25) show that this moment can be made arbitrarily small by first fixing b sufficiently large and then taking k sufficiently large.

The set where \theta(x_k-b)\ne0 is contained in \{\varphi_0\leq C-b+1\}, by the upper bound for \varphi_k. Its smooth volume tends to zero with b, and therefore its \rho_k-mass tends to zero uniformly in k. Markov’s inequality applied to (4.28) now gives a set of \rho_k-mass at least 1/2 on which \theta(x_k-b)=0 and q_k\leq1/2. On this set |V_k/U_k|=q_k\leq1/2; everywhere else, |V_k/U_k|\leq1+q_k.

For 0<p\leq\eta/2 and q\geq0,

For q\leq1, this follows from concavity. For q\geq1, integrate the derivative with respect to the exponent and use (1+q)^{\eta/2}\log(1+q)\leq C_\eta q^\eta. Taking p=2/k, we therefore obtain

for the indicated choices of b and then k. Indeed (1-2^{-p})/p\to\log2, whereas the moment can be made arbitrarily small independently of large k. This is a strict improvement of the minimizing integral for a section with the same prescription, a contradiction. Therefore \nu=0.□

Proof. Suppose a horizontal pole divisor induces S=D_i and has zero limiting fixed order on the geometric generic fiber. Choose rational perturbations tending to zero and degrees in which the normalized minimal orders approach their infima. Passing to powers as necessary gives integers k\to\infty, r_k>0, with

where j_k is the minimal actual order of the whole section space. Use the corresponding relative section spaces and vanishing filtrations on the fixed prepared family, and impose the countably many base-change conditions that preserve these orders. No further field extension or spreading of individual sections is required. Lemma 4.6 now constructs one global weight T from the chosen degrees, before any final fiber is selected.

Apply Lemma 4.5 to this T. Fubini gives a full-measure set of good fibers on which the integral is finite. The excluded algebraic conditions form a countable subset of the complex curve and hence have measure zero. Choose D satisfying both requirements. Lemma 4.7 applies on this same fiber and proves that T|_D has zero generic divisorial coefficient at S. The normalized minimizing sections in that Lemma are chosen only now; they do not change T.

In a smooth chart over the base, contraction gives

The density of i^{n^2}h\wedge\Theta\wedge\overline\Theta relative to base area is comparable, by the smooth positive metric, to |v|_h^2|\omega|^2. This formula includes the mixed components of the metric. At a generic point of a pole of order at least one, shrink the chart so its leading coefficient is bounded below. Since \ord_S(\omega)=e_i, the finite fiber integral from (4.14) would then dominate a positive constant times

We show that this last integral is infinite.

The generic Lelong coefficient is zero. The analyticity of Lelong upper-level sets, or equivalently the generic-Lelong-number statement in [17, Chapter III, Lemma 8.15], implies that the point Lelong number is zero at almost every center on a small smooth patch of S. For a local plurisubharmonic weight T, the logarithmic mean characterization [17, Chapter III, Corollary 6.8 and Example 6.9] then gives

at almost every such center a. The characterization for ball means follows from that for sphere means by radial integration.

Fix \varepsilon>0. For almost every center, the ball mean is at least \varepsilon\log r at all sufficiently small dyadic radii. The sets on which this holds beyond one fixed dyadic index have a union of full divisor measure. Hence there is a measurable set A of positive divisor measure on a compact patch of S, and one common starting index, for which the lower bound holds uniformly. In dimension one the divisor measure here is counting measure.

On B(a,r), |z|\leq Cr. Jensen’s inequality therefore gives, for the nonnegative integrand F in (4.31),

Integrate this inequality in a. For each ambient point, the measure of centers a\in A whose radius-r ball contains it is at most Cr^{2n-2}, by projection to the smooth tangential coordinates of S. Fubini consequently yields

The overlap bound is constant when n=1, so the same formula holds in that case. Since e_i\geq1, choose \varepsilon<2\delta e_i/(1+\delta) and let the dyadic radius tend to zero. The right side diverges, contradicting (4.31) and the Fubini consequence of (4.14). A pole with zero limiting fixed order is therefore impossible.

There are only finitely many horizontal pole divisors. For each, positivity of its limit and monotonicity give a sufficiently small rational perturbation with positive actual asymptotic fixed order. Taking one common smaller rational perturbation proves the last assertion.□

Proof of Theorem 4.1. We may assume n>0, and use the preparation and notation of Lemmas 4.2–4.4. Proposition 4.8 gives one sufficiently small positive rational \varepsilon for which every horizontal pole divisor has positive actual asymptotic fixed order for K_D+\varepsilon H|_D. We fix this same perturbation for the relative model construction. The order comparisons hold on the geometric generic fiber and on simultaneous very general fibers by base change in the countably many divisible section degrees.

Over the generic curve choose an effective rational divisor \Delta\sim_{\Q}\varepsilon H such that the pair is klt. One obtains it by taking a general sufficiently divisible member of the ample system and a small coefficient. In the equivariant case average its translates; the klt condition is preserved because discrepancies are linear in the boundary and the translates are klt. Spread this choice and shrink the curve so that the total pair is klt. The relative adjoint divisor is big: on a good fiber K_D\sim E\ge0, and an ample positive perturbation of an effective divisor is big.

Apply [8, Theorem 1.2(2)] to this projective morphism over the shrunken curve. Its hypotheses are a klt pair and a big relative adjoint divisor. It gives a normal relative log canonical model

with Q projective over the curve. Since the adjoint divisor is relatively big, this map is birational. The ample-model comparison [8, Definitions 3.6.4 and 3.6.6, Lemma 3.6.5(4)] shows that it is a birational contraction, so it extracts no divisors. In a common resolution the pullback adjoint system is the pullback of the ample system on Q plus an effective exceptional fixed part.

Every horizontal pole divisor is contracted by \chi. Indeed a prime which survived would be tested at its corresponding prime on the ample model. Sufficiently divisible ample systems have no fixed component there, and the exceptional section comparison changes no order at that prime. Its asymptotic fixed order would therefore be zero, contrary to the choice of \varepsilon. The canonical ring is invariant under the finite group when \Delta is invariant, and hence the action descends to Q.

Choose a nonvanishing algebraic vector field on a smaller open of the curve. Normalize the kernel line of \Theta against that vector field, and denote the resulting rational derivation of \C(P)=\C(Q) by \partial. It is rational because \Theta is algebraic and its kernel and prescribed base projection are given by algebraic linear equations. This normalization multiplies the previous local v by a nonzero base function, so it creates no horizontal pole. Remove the finitely many vertical pole loci and the zeros or poles of the chosen base vector field.

The derivation is regular at every prime divisor of Q. Because \chi extracts no divisors, each such prime corresponds to a prime on P, with the same discrete valuation ring; it cannot be one of the contracted pole divisors. It remains to pass through codimension two. For any normal affine open with coordinate ring A, and any f\in A, regularity at the height-one points gives

Thus \partial is a regular derivation on the entire normal space. Its projection to the base is still the chosen base vector field, since this equality is an identity of derivations on the function field. Locally reparametrize the base analytically so that \partial t=1.

A regular derivation on a complex analytic space has local holomorphic flows, including at singular points. Here is the local justification. Embed the space in a polydisk with coordinate functions z_1,\ldots,z_N, lift the functions \partial z_j to holomorphic functions on the polydisk, and form the corresponding ambient vector field. It preserves the defining ideal, because its induced derivation on the quotient is \partial. Its flow preserves the analytic subspace: for finitely many generators of the ideal, their values along an integral curve satisfy a homogeneous linear differential system with zero initial value. Uniqueness makes the restrictions of these local flows independent of the ambient lifts and permits them to glue.

Properness now gives a uniform flow time near a compact fiber. More explicitly, the inverse image of a sufficiently small closed base disk is compact; a finite cover by flow neighborhoods provides a common smaller disk of complex times. Since \partial t=1, forward and backward flows identify the fiber over its center with every fiber over a smaller disk. These identifications are biholomorphic, and hence algebraic because the fibers are projective. They commute with \Gamma: the kernel of \Theta is invariant, and its normalization by a base vector field is unique. Shrink the algebraic open as needed so that the original family and the model have birational fibers and the model fibers under consideration are geometrically integral.

Fix one fiber Q_{t_0} in this disk. Graphs of isomorphisms from Q_{t_0} to fibers of Q belong to countably many finite-type loci of relative Hilbert schemes. Requiring equivariance is a closed condition on these graphs. Their images in the algebraic curve are constructible and together contain the disk, by the algebraic flow identifications. If none were dominant, each image would be a finite subset of the curve; their countable union could not contain a disk. Some graph locus is therefore dominant. Its generic fiber has a closed point defined over a finite extension of the curve’s function field. Over that extension the generic fiber of Q is isomorphic to the fixed Q_{t_0}, equivariantly when prescribed.

Take a smooth projective resolution D_0 of Q_{t_0}, equivariant in the finite-group case. The birational comparison with P gives the required birational identification of its generic fiber with D_0 after a finite extension. The preliminary finite changes of the curve compose with this extension. For n=0 the fiber is a point and the conclusion is immediate. This completes the proof.□

Proof. Restrict the integral polarized cohomology variation to T. Its entire highest Hodge step has rank one. Constancy of the projective line makes it a flat rank-one local subsystem, and Corollary 3.15, applied anew to this restricted variation, gives finite character. On an algebraic curve in T, Theorem 4.1 therefore applies, including its equivariant assertion.

We pass from curves to the geometric generic point of T. After shrinking T to an affine open, graphs of birational identifications between pairs of fibers are parametrized by countably many finite-type algebraic loci over T\times T. For example, take relative Hilbert schemes of the product family and their constructible loci of geometrically integral graphs on which both projections are dominant of generic degree one. Geometric integrality and these degree conditions are imposed after flat stratification. Equivariance is expressed by invariance of the graph under the diagonal finite action; a birational action is first regularized as in Lemma 4.2. The images of the graph loci in T\times T are constructible.

For \dim T=0 the conclusion is immediate, and for \dim T=1 it is Theorem 4.1. Suppose \dim T\ge2. Choose a smooth projective compactification and a sufficiently ample complete-intersection family of curves. The incidence of a curve with two distinct points on it has an irreducible open dominating T\times T: sufficiently ample systems separate the two point conditions, and their general members through those points are smooth and connected. Its general fiber over the curve parameter space is the irreducible surface of pairs of points on that curve.

If none of the graph images were dense in T\times T, their closed image closures would pull back to countably many proper closed subsets of this incidence. For a very general curve, each such subset meets its surface of pairs in a proper closed subset; those not dominating the curve parameter space are avoided. The curve can also satisfy the countably many generic section conditions giving \kappa(D)=0 and h^0(D,K_D)=1. Theorem 4.1 and spreading its birational identification give a dense open of pairs of fibers on that curve which are birational, equivariantly when required. A countable union of proper closed subsets of an irreducible complex surface cannot contain this open. This contradiction proves that some graph locus dominates T\times T.

A dense constructible image contains a nonempty open. Specialize the first coordinate to a sufficiently general complex point so that the slice of this open is dense in the second factor. The first fiber is now a fixed smooth projective variety. The graph parameter space over this slice dominates T, and a closed point of its generic fiber is defined over a finite extension of \C(T). It gives the desired birational identification, with the finite action.

For the relative assertion use these same graph loci over B\times_Z B, taking the component that dominates Z. Its geometric generic fiber is integral by hypothesis. If no graph locus dominated this component, the closures of their images would remain proper on every very general fiber over Z, after excluding countably many proper specialization loci. This contradicts the assertion just proved on those complex fibers. Thus one graph locus dominates over the generic point of Z. Over b_Z=\overline{\C(Z)}, its constructible image contains an open in the space of pairs. Choose a b_Z-point in a suitable open for the first projection. Its fiber is a fixed smooth projective b_Z-variety, and the graph locus on the remaining slice supplies the birational identification after a finite extension of the function field of the geometric generic B-fiber. This proves the relative conclusion. Finally, all the graphs, inverse maps, and equivariance identities are algebraic data, so these identifications persist under field extension.□

Proof. On the smooth family locus W^\circ, Lemma 3.1 gives a smooth projective cyclic-cover family with connected fibers, Kodaira dimension zero, and a one-dimensional entire ordinary top-form space. On the connected open of every very general W_z, Corollary 3.16 makes its projective top line constant. Corollary 4.9 therefore makes that restricted cover family geometrically birationally constant, equivariantly for its cyclic deck group. Taking invariant function fields gives the asserted constancy of the original J-family.

For the relative assertion, apply the second part of Corollary 4.9 to W^\circ\to Z, shrinking Z if necessary. Its geometric generic fiber is integral by Proposition 2.7, and the preceding paragraph verifies the fiber and constant-line hypotheses on its very general complex fibers. Over b_Z=\overline{\C(Z)} the reference cover and its finite deck action are consequently defined over b_Z. Their invariant function field has a smooth projective b_Z-model, whose base extension is birational to the original geometric generic x-fiber. This is geometric birational constancy; the descent of a chosen trivialization and of the whole original fiber is addressed in the remaining sections.□

Proof. Test a prime E of H, with uniformizer x. In local boundary coordinates on I, write p^*t_j=x^{a_j}u_j, with u_j a unit. Each logarithmic differential is a_j\,dx/x+du_j/u_j, and a nonzero wedge contains at most one factor dx/x. If a=\operatorname{ord}_E(p^*D_I)>0, the pulled-back ordinary base volume thus vanishes to order at least a-1. This proves the first inequality, also when p(E) has higher codimension. If a=0, use R_p\geq0. The second assertion follows from Equation (5.1) and the coefficient-one boundary along the reduced inverse image of D_I.□

Proof. There are only finitely many prime divisors of H outside a fixed smooth log-family open. They include every multiple fiber component and every divisor whose image on I has codimension at least two. Their restricted valuations can all be extracted on one smooth base model. Indeed, if a divisorial valuation v of k(H) restricts nontrivially to w on k(I), the residue-field inequality gives

The valuation inequality gives the reverse bound. Thus w is a positive multiple of a divisorial valuation. Extract the finitely many such valuations and resolve their common base model. Every original vertical prime now has a divisorial center, which remains divisorial on higher smooth base models.

All multiplicities in the next step are measured over this prepared base. Normalize it in the prescribed field extension. For each valuation above the selected base primes, take a root of a uniformizer of degree equal to the least common multiple of the relevant original multiplicities. Do this for all the finitely many valuations and take a Galois closure. The ramification indices in these towers multiply, so the final base index is divisible by every required multiplicity; compare [32, Lemmas 9.10–9.11].

The local calculation is at discrete valuation rings. If t=ux^e, where u is a unit, base ramification of index n changes the normalized fiber multiplicity to e/\gcd(e,n). After an étale unit-root extension this is the normalization of s^n=x^e; separable residue extensions do not change the indices. Thus e\mid n makes the multiplicity one, and further base ramification preserves it. This also protects every original vertical prime outside the chosen list, whose multiplicity was already one. Resolve the altered base, the induced source map, and the boundary without changing the generic smooth pair. The original divisorial centers and their DVRs retain the properties just established.

At a vertical prime, choose separating relative residue parameters z_1,\ldots,z_r. For the extracted base model, the ordinary relative lattice has generator x^{-(e-1)}dz_1\wedge\cdots\wedge dz_r up to a unit: complete these with base residue parameters and use dt=x^{e-1}(eu\,dx+x\,du). Saturation removes the factor x^{-(e-1)}. The same residue differentials remain separating after finite extension, and the new multiplicity is one. Hence the ordinary relative lattice upstairs is the pullback of the original saturated lattice. At a horizontal prime the DVR contains k(I) as a field, so finite separable base extension is étale there. Its relative lattice and boundary coefficient are unchanged. This proves (i)–(iii).

Every prime of H^a lies over a prime of H, since \tau is finite. Its strict transform is therefore among the primes just tested. Normality of H^a gives the asserted global section. The same tests hold after tensoring by a pulled-back line bundle. Property (ii) proves the final assertion. Poles at the remaining resolution-exceptional primes are harmless because the section is asserted on H^a, not on that resolution.□

Proof. If r=0, the chosen geometric generic component is a point and its variation is zero. The saturated relative tangent sheaf has rank zero, so K_{\mathcal V_p}=0, and there is no horizontal boundary. Equation (5.2) and a nonzero plurisection of K_I+D_I give the required nonnegative Iitaka dimension directly. The model is a point after the Stein extension. We may therefore assume r>0.

First suppose the generic pair is klt. Apply Lemma 5.2, prescribing the algebraic closure of k(I) in k(H) and choosing the corresponding component. The altered geometric generic fiber is integral and has the same klt big pair. The relative form of Kovács–Patakfalvi’s inequality, [32, Theorem 9.9] with its auxiliary base divisor M=0, gives

Its hypotheses are satisfied by the smooth projective SNC pair and its klt big generic log divisor; no condition on \kappa(I') is required. The model-pair convention is [32, Definition 6.16, Remark 6.17, Corollary 6.20, Definition 9.3 and Remark 9.4]; the canonical model exists for klt big pairs by finite generation [8]. Finite extension does not change this variation.

By Lemma 5.2, these systems give sections of m\tau^*(K_{\mathcal V_p}+C) on the finite normalization of the original H. Their ratios are unchanged. Lemma 2.3 therefore gives the same lower bound for K_{\mathcal V_p}+C on H. Choose 0\ne\alpha\in H^0(I,\ell(K_I+D_I)). In common divisible degrees m, multiply by p^*\alpha^{m/\ell} and the canonical section of mV in Equation (5.2). These common factors embed the systems into those of m(K_H+B) without changing their rational-map dimensions. This proves Equation (5.3).

For a generic lc pair, decrease the horizontal coefficients by a sufficiently small rational factor. Its log divisor stays big, while the generic pair becomes klt. The required vertical boundary is unchanged. The resulting systems are contained in those for B, which proves the lower bound r; no assertion about the variation of the unperturbed lc pair is needed.

In the zero-excess klt case, the inequality forces model-pair variation zero. A stable general-type pair has finite automorphism group. The constant moduli point has a representative over k, and its generic isomorphism torsor has a point after a finite field extension. This makes the model pair constant, with its boundary retained.□

Proof. The diagram and dimension assertions follow from Propositions 2.7 and 2.10. The latter identifies \C(S)=\C(Y) inside \C(X), so the present parameter field is the embedded field b(I)=b\,\C(Y) of Remark 2.11. The generic extensions for W\to S and W\to Z are regular. Restriction to the generic point of T_0 and extension to b therefore preserve geometric integrality. Stein factorization over the normal bases also gives connected fibers. Generic smoothness, applied also to the finitely many boundary strata, gives the stated smoothness and SNC properties. The first assertion of (6.2) is the geometric generic form of the corresponding assertion for S_t in Proposition 2.10; nonzero divisible sections are preserved by field extension. The second follows from Proposition 3.3. Finally, (6.3) is precisely Proposition 3.12 after this generic restriction.□

Proof. If d=0, the generic fiber of p_I is geometrically integral and zero-dimensional, so its degree is one. The variety P_0 is a point and the product assertion holds with V'=\operatorname{Spec}b. The construction of J_0 at the end of the proof applies in this case as well. Assume for the marking construction that d>0.

The relative model. Let E_0=b(I) and H_\eta=(H_0)_{E_0}. Decrease every positive coefficient of B_0 horizontal over I slightly, keeping it positive and strictly less than one; leave the vertical coefficients unchanged. Denote the resulting boundary by B_0^-. Since the generic support is SNC, the pair (H_\eta,B_\eta^-) is klt. The morphism

is generically finite. Consequently K_{H_\eta}+B_\eta^- is big relative to a_\eta, since its generic fiber is zero-dimensional. There is therefore a relative log canonical model

by [8, Theorem 1.2(2)]. To apply that theorem over an algebraically closed ground field, one can first shrink I so that the spread of the decreased pair is klt, apply the theorem to the projective morphism to I\times_bP_0, and then restrict to its generic I-fiber. The characteristic-zero field comparison in Section 5 justifies this use of the cited complex formulation over b. The model C_1 is normal, and its adjoint divisor is relatively ample. The canonical rational map is a birational contraction, so it extracts no divisors; see [8, Definition 3.6.6]. The map c is proper, surjective and generically finite, and factors through the normalization of (P_0)_{E_0} in E_0(H_\eta).

Choice of the markings. Choose a very ample line bundle L on P_0 and a general basis x_0,\ldots,x_N of H^0(P_0,L). The coordinate divisors will recover each ratio x_j/x_0 up to a scalar. The divisor of x_0+x_j will determine that scalar, so that the marked model remembers the morphism to this fixed P_0. We therefore use the divisors

Their pullbacks receive distinct positive rational coefficients, chosen sufficiently small that the resulting generic pair is still klt. They are also chosen distinct from every coefficient of the decreased generic strict boundary B_\eta^-. The finite number of dependencies in (6.7) is harmless: on a fixed log resolution, sufficiently small coefficients preserve all the strict klt discrepancy inequalities. Add sufficiently many further general members of |L|, again with small coefficients less than one and with no coefficient equal to one of the displayed labels. These can have arbitrarily large total degree. General members of the free pullback system meet the strata on a resolution transversely, so the generic pair remains klt.

Make the following avoidance choices at the same time. None of the hyperplanes contains the image of any relevant divisorial exceptional component on the fixed generic models, any old generic boundary component, or any branch divisor. In addition, none contains the P_0-image of an I-vertical divisor of H_0 having proper image in P_0. There are only finitely many divisors in the last condition: such a divisor has image of codimension at least two under the generically finite morphism H_0\to I\times_bP_0, so it is one of its divisorial exceptional components. An I-vertical divisor dominating P_0 cannot be a component of the pullback of a hyperplane. Thus the last condition prevents any added marking from overlapping a vertical old coefficient-one component or acquiring a vertical divisorial multiplicity. It also makes the corresponding total boundary a boundary with coefficients at most one.

These finitely many Bertini, avoidance, and klt conditions define a nonempty open in the parameter space of the chosen sections, initially over E_0. Lemma 2.4, applied after extending E_0 to its algebraic closure, shows that this open contains a b-point. Thus all the markings can be chosen over b. The same argument applies to the open set of bases of H^0(P_0,L) and to each additional member.

Write A_{\mathrm{mark}} for the weighted sum of all the chosen divisors on P_0, and set

The total degree of A_{\mathrm{mark}} is chosen sufficiently large that

is ample: a relatively ample divisor plus a sufficiently large pullback of an ample divisor is ample. Twisting by pullback from P_0 tensors each divisible graded piece of the relative adjoint ring by the corresponding power of a line bundle on P_0. It therefore leaves the relative \operatorname{Proj}, and hence the model, unchanged. Equivalently, on a common resolution the exceptional comparison for the old adjoint divisor remains identical after adding this pullback on both sides. Hence C_1 is now the absolute log canonical model of (H_\eta,\Delta_\eta).

The pullbacks of the displayed divisors on C_1 are their complete pullback divisors, each with multiplicity one at every component. A divisorial exceptional component occurs in a pullback only when its image is contained in the hyperplane; merely meeting its image does not produce that component. The avoidance of the branch divisors gives multiplicity one at the other generic points. No old boundary prime is shared with a marking. Thus their distinct coefficients identify the complete displayed divisors on the canonical model, even when a pullback is reducible.

The upper bound and the model-pair estimate. On a general a_0-fiber the marking line bundles restrict trivially, and B_0^-\leq B_0. Lemma 6.1 therefore gives restricted adjoint dimension at most zero. The upper addition estimate in Lemma 2.5 gives

If the restricted section system is empty, the same bound follows because the total system is empty. Only the upper bound is needed.

Resolve the support of \Delta, writing \mu:\widehat H\to H_0. Over the generic point of I, use the crepant boundary and replace its negative exceptional coefficients by zero; keep all strict coefficients. Call the resulting generic boundary \widehat\Delta_\eta. The generic pair is effective and klt, and

For divisible m, normality and testing at codimension-one points give \mu_{\eta*}\cO(mE_{\mathrm{exc}})=\cO. The generic adjoint rings are therefore identical. The generic model pair remains the marked pair on C_1: exceptional divisors added in (6.9) are contracted to the original generic source, and the canonical map from that source extracts no divisors.

On \widehat H, assign coefficient one to each new exceptional divisor vertical over I, and retain the specified coefficients on all other components. This gives an effective rational SNC boundary \widehat\Delta, with generic restriction just described. All strict components above D_I already had coefficient one, and any newly created component above it is vertical over I. Consequently

The total adjoint sections on \widehat H inject into those on H_0: a rational pluriform that is a section upstairs passes the unchanged strict-divisor tests at every prime downstairs. This argument does not require the vertical exceptional coefficients to be crepant. In particular, even if one of these exceptional divisors dominates P_0, (6.8) still yields

The markings now define a big generic model pair, while the total section dimension still has the upper bound d. We apply Proposition 5.3 to

Both varieties are smooth projective over b; the morphism is surjective with connected fibers and relative dimension d; the boundary is effective rational SNC; (6.10) holds; and the base has nonnegative log Kodaira dimension by (6.2). Its geometric generic pair is klt and has big adjoint divisor by the construction of the absolute ample model C_1. Thus Proposition 5.3 and (6.11) give

The variation here is exactly that of the generic log canonical model pair. It is zero. The finite-extension conclusion of Proposition 5.3 makes that model pair isomorphic, after a finite extension E/E_0, to the base extension of a model pair over b. Denote its underlying normal projective variety by C_*. Its coefficient-labeled displayed divisors are now defined over b.

Recovering the map to P_0. View C_1 after extension to E as (C_*)_E. The displayed markings fix both

These are differences of the corresponding complete marking divisors. In particular, the first is the base extension of a divisor over b which is principal after extension to E. Cartierness descends under this faithfully flat extension. Triviality of its associated line bundle descends as well: its space of sections commutes with field extension, and a nonzero descended section becomes a scalar multiple of a nowhere-vanishing section upstairs. We may therefore write

Here the quotient of two functions with the same divisor is a constant, since C_* is proper and geometrically integral.

The sum marking (x_0+x_j=0) has a b-point off the two coordinate markings. Indeed, for a general basis this is a nonempty open part of the pullback of that hyperplane, by surjectivity onto P_0 and d>0; as it is defined over the algebraically closed field b, it has a b-point. Evaluation at such a point in 1+c_jf_j=0 gives c_j\in b. Thus every coordinate ratio of the map to the fixed projective embedding of P_0 belongs to b(C_*). The map itself descends to a morphism C_*\to P_0, either by descent of its graph or by descent of regularity of the resulting rational map.

Let V' be the normalization of P_0 in the finite extension b(C_*)/b(P_0). It is finite over P_0. The map C_*\to P_0 factors properly and birationally through V'. Normalization commutes with the present field extensions: b has characteristic zero and is algebraically closed, so V' is geometrically normal; its base extension is finite normal with the required fraction field and hence has the normalization’s universal property. The original generic source H_\eta after extension to E also maps properly and birationally to V'_E. This proves (6.4).

A proper birational morphism to a normal variety is an isomorphism at the generic point of every target divisor. Consequently a prime on V'_E has its unique strict prime both on the original generic source and on (C_*)_E. If the old boundary coefficient at that prime is positive, its decreased coefficient is still positive and occurs in the constant model pair. Its image on V' is therefore defined over b. This proves the asserted control of old boundary images. Notice that this reasoning uses the old boundary; a newly assigned exceptional coefficient on \widehat H cannot hide such a divisor.

Choosing the reference fiber. Replace I by a smooth projective model of E, and use the notation (6.5). All subsequent parameter extensions will be made relative to this choice. The original x-fiber gives a smooth projective geometrically integral variety J/K with geometric Kodaira dimension zero. Proposition 4.10 applies after extending k to an algebraic closure: the I-directions in (6.4) lie over one geometric generic Z-point. Thus this family is geometrically birationally constant in those directions.

Spread J as a smooth projective family over a nonempty open of I_k. Fix one geometric birational identification with a reference variety over \bar k, together with its inverse. Their finitely many coefficients lie in a finite extension of \bar k(I). Normalize an affine open of I_{\bar k} in that extension. After shrinking, this is a finite surjective cover on which the maps and their inverse identities spread. Shrink again so that their fiberwise domains are dense and the two maps are inverse there. This uses one chosen birational identification and finitely many equations, not a finite-type parameter space for all birational maps.

The image of this comparison cover contains a nonempty open of I_{\bar k}. Intersect it with the smooth family open. By Lemma 2.4, that intersection contains a point of I(b); a point of the finite comparison cover above it exists over \bar k. Specializing the original k-family at this b-point gives a smooth projective variety J_0/k. Its base extension to \bar k is birational to the chosen reference, hence geometrically birational to J. Geometric integrality and geometric Kodaira dimension zero follow. Ordinary birational invariance of pluricanonical sections and their compatibility with field extension give (6.6); in particular a nonzero ordinary p-pluriform is defined over k itself.□

Proof. A moving test is trivial on k because its residue field contains the function field of its dense image in V'; it is trivial on E because it is a divisor on an E-variety. The proper birational maps to the finite normalization used in Proposition 6.2 identify its generic point with a strict divisor of the old generic source H_0. If that divisor had positive old B_0 coefficient, the boundary-image assertion of Proposition 6.2 would place its image in a fixed b-divisor on V'. Such a divisor, after any parameter extension, cannot dominate V'. Thus its old coefficient is zero.

We explain why these are indeed the old tests and orders. A finite separable extension of the parameter field is finite étale as a field morphism. Base change is therefore finite étale on both the base and the source of the generic family. Height-one primes lie over height-one primes, and actual source components persist, possibly splitting. Their pullback multiplicities and relative canonical orders are unchanged. Generic restriction and adjunction cancel the parameter-canonical factors in the relative canonical bundles. The same assertion for the all-valuation infimum follows by base changing a fixed log resolution computing the threshold. Extensions involved in reaching b=\overline{\C(T_0)} are algebraic, and each prime and the finite data just used descend to a finite stage. Consequently the unchanged-divisor and étale-extension compatibility in Proposition 3.3 applies at this strict old prime. No new exceptional boundary coefficient enters the calculation.

Choose an actual component Q attaining the first minimum t_P of Proposition 3.3. With d_Q its old source boundary coefficient, that proposition gives

Since m_Q is a positive integer and 0\leq d_Q\leq1, all these inequalities are equalities. Hence m_Q=1 and d_Q=0. Equality in the first inequality says that this same actual component attains \lambda_P=\inf_{Q'\mapsto P}(1+r_{Q'})/m_{Q'}. This proves (6.13).□

Proof. The birational group. Choose compatible algebraic closures \overline k\subset\overline K. A nonzero ordinary pluricanonical form on J_0 excludes uniruledness: if a smooth projective model of J_0 were dominated generically finitely by a ruled variety, its pullback would be a nonzero pluricanonical form on a resolution of that ruled variety, which has none.

Hanamura’s structure theorem for non-uniruled projective varieties in characteristic zero provides a suitable smooth projective birational model T of (J_0)_{\overline k} for which the reduced scheme of flat graphs of birational maps,

is a group scheme locally of finite type over \overline k. Its identity component

is an abelian variety. This group structure is intrinsic among such birational models, and represents their algebraic birational group actions. These are exactly the conclusions of [25, Theorems 2.1–2.2], also stated in [9, Theorems 3.8–3.9]. The theorem applies to a non-uniruled projective variety over an algebraically closed characteristic-zero field; it does not require a minimal model of J_0.

The scheme of flat graphs is an open subscheme of the Hilbert scheme of T\times T; see [24, Definitions 1.6 and 1.8, Proposition 1.7] and [9, Proposition 3.7]. The Hilbert functor commutes with field extension, and the open graph condition is geometric. Every birational selfmap over the extended field is therefore represented by a point of the base-changed graph scheme. In characteristic zero a reduced locally finite type scheme is geometrically reduced, so passing to its reduction introduces no new field-valued points after extension. Every connected component of \mathcal B contains a \overline k-point, since \mathcal B is locally of finite type over the algebraically closed field \overline k. Translation by such a point identifies that component with A_1. Each of these abelian translates remains connected after extension to \overline K, and the disjoint union of the original open components exhausts the base-changed scheme. Thus, after identifying the birational groups of T and J_0, we have

This argument allows infinitely many components. The intrinsic group structure also shows that coefficient transport and conjugation preserve A_1.

Removing the nonconstant abelian part. Set \Gamma=\operatorname{Gal}(\overline K/K), and choose an isomorphism of \overline K-function fields

Let s_\sigma^J and s_\sigma^0 denote the natural coefficient transports on these two fields. The automorphisms

satisfy

The map \sigma\mapsto u_\sigma is continuous when the algebraic points have the discrete topology: the rational maps defining \theta and its inverse use only finitely many algebraic coefficients over K. Its image is therefore finite.

For every component occurring among these values choose a representative over \overline k, choosing the identity in the identity component. By (7.6) this gives a locally constant choice h_\sigma and expressions

The component cocycle law and the second equality in (7.6) imply

Consequently

defines an action on A_1(\overline K) preserving A_1(\overline k). Indeed the discrepancy between \alpha_\sigma\alpha_\tau and \alpha_{\sigma\tau} is conjugation by d_{\sigma,\tau}, which acts trivially on the commutative group A_1.

Give

the discrete topology. The induced action is continuous: every algebraic point has an open stabilizer for coefficient transport, and h_\sigma is locally constant. In additive notation the classes c_\sigma=[a_\sigma] satisfy

The abelian group M_1 is uniquely divisible. Multiplication by an integer n>0 is surjective on the points of an abelian variety over either algebraically closed field. Moreover,

since the finite torsion scheme is already defined over the algebraically closed smaller field. If na is constant, choose a_0\in A_1(\overline k) with na_0=na; then a-a_0 is constant torsion, so a is constant. This proves injectivity of multiplication on the quotient as well as its surjectivity. Thus M_1 is a \Q-vector space.

Choose an open normal subgroup N\subset\Gamma contained in the zero set of the continuous cocycle c. Equation (7.8) gives c_{\sigma n}=c_\sigma for n\in N. It also gives

Thus N fixes every cocycle value. The finite sum

is well-defined, and summing (7.8) over the cosets yields

Choose t\in A_1(\overline K) lifting \beta and replace \theta by \theta'=t^{-1}\theta. Its cocycle is

The class of its A_1 factor is -\beta+c_\sigma+\alpha_\sigma(\beta)=0 by (7.9). Hence

A faithful finite splitting group. The modified cocycle is still continuous, because the algebraic point t has an open stabilizer. Its image is therefore finite. Choose a common finite Galois extension k'/k defining all these finitely many birational maps and their inverses. The map

is a homomorphism, with the multiplication (z,\gamma)(w,\delta)=(z\gamma(w),\gamma\delta). It has finite image and an open normal kernel N_0. Put \widetilde K=\overline K^{N_0}. Then \widetilde K/K is finite Galois and contains k'. For \sigma\in N_0 we have z_\sigma=1, so \theta' intertwines the natural coefficient actions of N_0. Its rational maps descend to \widetilde K by Galois descent. The quotient group G=\Gamma/N_0 acts by (7.3), the pairs are faithful by construction, and the fixed-field identity (7.5) follows.□

Proof. We compare divisorial orders before and after splitting. On the constant model there is a unique minimizing divisor. The actual multiplicity-one minimizer in (7.10) will force equality of total and base inertia, and faithfulness will make both trivial.

Relative canonical orders. First work over L_0. Put L=K(J) and use (7.5) to identify

Since J/K is geometrically integral, L/K is regular. Hence L and \widetilde K are linearly disjoint over K, and restriction identifies

Fix a moving divisor P and a prolongation P' to \widetilde K. Write e=e(P'/P). If a divisorial valuation Q of L above P prolongs to Q' above P', write e_Q=e(Q'/Q). On smooth models at these generic divisor points, the multiplicities and relative differential orders satisfy

The first identity is the order of a base uniformizer. For the second, the total canonical differential acquires order e_Q-1 under the finite extension, whereas the base canonical differential acquires order e-1 and is pulled back with multiplicity m_{Q'}. These are the tame different orders of characteristic-zero divisorial DVRs. They can also be computed by differentiating uniformizers and completing them by separating residue parameters. Dividing the degree-p calculation by p gives the stated formula.

In particular,

This is an increasing affine transformation independent of Q.

The unique minimizing divisor. Let R' be the valuation ring of P' and let \kappa' be its residue field. The valuation is trivial on k, because P is moving. It is also trivial on the finite algebraic extension k'/k: the valuation of a nonzero element algebraic over a trivially valued field must be zero, as follows immediately from its minimal polynomial. Thus

Consider the smooth proper constant model

Its special fiber S' is smooth and geometrically integral over \kappa'. Choose the ordinary regular degree-p generator \omega_0 on J_0. The pulled-back generator \omega_J is a scalar multiple c\omega_0 over \widetilde K, because the ordinary degree-p space has dimension one. Put \ell=p^{-1}\ord_{P'}(c) and D_0=\operatorname{div}(\omega_0)\geq0.

Every divisorial valuation of \widetilde L above P' has a center on \mathcal J, by properness, and that center lies in S'. For such a valuation T, let A_{\mathcal J}(T) denote its log discrepancy over the smooth model \mathcal J. Removing the common scalar shift gives

The divisor D_0 is pulled back to the constant model in this formula. It has no special-fiber component. Hence S' itself has m_{S'}=1 and ratio \ell+1.

For every other divisorial valuation centered in S',

This is the pure log terminality of a smooth divisor on a smooth scheme. In local parameters it follows by including the equation of S' and at least one further parameter vanishing at the proper center: the log discrepancy is at least the sum of their positive orders. Since \ord_T(D_0)\geq0, Equation (7.15) proves that S' is the unique minimizer of the ratio above P'.

Triviality of inertia. Take the multiplicity-one minimizer Q supplied by (7.10). It has a prolongation to \widetilde L; by Galois transport we may arrange that the restriction of this prolongation to \widetilde K is the specified P'. Indeed G acts transitively on the prolongations of P, and its action on \widetilde L fixes L by (7.11). Conversely, every divisorial valuation above P' restricts under the finite extension to a divisorial valuation above P. Equation (7.13) therefore shows that the chosen prolongation of Q attains the upstairs infimum. The uniqueness just proved identifies it with S'. Consequently m_{Q'}=m_Q=1, and (7.12) gives

Let \mathcal I_{Q'} and \mathcal I_{P'} be the total and base inertia groups, viewed as subgroups of the common Galois group in (7.11). The embedding of base residue fields in total residue fields gives

All residue characteristics are zero, so the residue extensions are separable and the orders of these inertia groups are e_Q and e. Equation (7.16) forces equality of the two inertia groups. Thus every element of \mathcal I_{P'} acts trivially on the special-fiber function field \kappa'(J_0).

For g\in\mathcal I_{P'}, coefficient transport is already trivial on \kappa'. Equation (7.14) also gives g|_{k'}=1. Its induced action on \kappa'(J_0) is therefore the base extension of the constant map z_g. Such an action is faithful: a birational function-field automorphism over k' that becomes the identity after an injective extension of fields was the identity to begin with. Hence z_g=1. The pair (7.4) is faithful, so g=1. This proves \mathcal I_{P'}=1, and hence unramifiedness at P.

Finite parameter extensions. Let E/L_0 be finite. The base change of the finite Galois splitting field is a finite product of fields, with a transitive G-action; each component is Galois over E(V') with group its stabilizer in G. The constant field k' embeds in every component. The same constant maps and the restricted faithful pairs define its semilinear action. Geometric integrality of J and the multiplicity-one conclusion (7.10) both persist. Repeating the preceding argument for this component and its stabilizer proves the stated unramifiedness after E. In particular, no uniform choice of a finite parameter extension is being assumed for all divisors.□

Proof. A fixed open set carrying the cover. Extend the splitting algebra to \Lambda(V'):

It is a finite product of fields and a Galois algebra with group G. We first descend its normalization over a suitable fixed open subset of V'.

Every prime divisor of V'_\Lambda is either the base extension of a prime divisor of V' or is moving. On an affine chart with coordinate ring A, the map A\to A\otimes_b\Lambda is faithfully flat. If a height-one prime \mathfrak q contracts to \mathfrak p, flat going down gives \operatorname{ht}\mathfrak p\leq1. A height-zero contraction is zero and means that the divisor is moving. If \operatorname{ht}\mathfrak p=1, the corresponding integral b-divisor is geometrically integral, since b is algebraically closed. Hence \mathfrak p(A\otimes_b\Lambda) is a height-one prime. Its inclusion in \mathfrak q is equality. No finite-type assumption on the field extension \Lambda/b is used here.

A moving prime divisor of V'_\Lambda descends to a moving prime divisor after some finite extension E/L_0. Indeed, on the above affine chart, choose finitely many generators of its prime ideal. Their coefficients lie in such an E, since \Lambda/L_0 is algebraic. The ideal generated by the same equations over E is prime by faithful-flat descent, and has height one by invariance of codimension under field extension. Its contraction to A is still zero. Thus the descended divisor remains moving, and Proposition 7.2 applies to every field component at this finite stage.

It follows that all divisorial branching of \mathcal B_\Lambda lies on finitely many fixed divisors. The normalization is finite because the varieties over a field are excellent; its branch locus has only finitely many divisorial components. Remove those divisors and the singular locus of V'. Also remove the fixed closed locus over which the normalization in k'/k is not finite étale. This leaves a smooth nonempty open subset O\subset V' over b. The normalization of O_\Lambda in \mathcal B_\Lambda is finite, normal, and unramified at all height-one points. Purity of the branch locus makes it finite étale. Here the base is regular, the source is normal, the morphism is finite and generically separable, and all divisorial ramification has been removed, which are the hypotheses of Zariski–Nagata purity; see [22, Exposé X, Théorème 3.1]. Denote this finite étale cover by

Its generic G-action extends by normalization. It is a G-torsor, possibly disconnected, since the generic Galois algebra is a G-torsor and all these covers are finite étale over the connected base.

Let \mathcal C_{1,\Lambda} be a connected component, and let G_1\subset G be its stabilizer. Then \mathcal C_{1,\Lambda}\to O_\Lambda is a connected finite étale G_1-torsor. Let \mathcal T\to O be the finite étale cover obtained by normalization in k'/k. The inclusion k'\subset\mathcal B_\Lambda gives a morphism

over O_\Lambda, compatible with the action of G_1 through its restriction to k'.

Descent with the coefficient field. Finite étale covers are invariant under an extension of algebraically closed fields in characteristic zero. In the needed form, base extension from b to \Lambda gives an equivalence between the categories of finite étale covers of the connected normal separated finite-type scheme O and of O_\Lambda. This follows from [33, Theorem 1.1], since every finite cover is tame in characteristic zero; see also [33, Remark 1.5] and [22, Exposé XIII, Proposition 4.6]. Full faithfulness descends the morphisms as well as the covers.

Apply this equivalence simultaneously to \mathcal C_{1,\Lambda}, its G_1-action, and its map to the fixed cover \mathcal T_\Lambda. We obtain a connected finite étale G_1-torsor \mathcal C_1\to O and a compatible map \mathcal C_1\to\mathcal T. Let E_1 be the function field of \mathcal C_1. Then

and the chosen component of \mathcal B_\Lambda is \operatorname{Frac}(\Lambda\otimes_bE_1). The embedding of k' and the restriction of the G_1-action to it are part of this descended diagram.

The invariant field. For g\in G_1, the birational map z_g is defined over k', and k'\subset E_1. Hence the semilinear action z_gs_g descends to the field E_1(J_0). Its multiplication law follows from the cocycle law, because the action on k' in (7.19) is the descended one. Define

This is a finitely generated field over b. Indeed E_1(J_0) is finitely generated over b and finite over its invariant field. Also

since both the coefficient action of G_1 and the constant birational maps fix k. This inclusion retains the coordinates of the entire x-base in the descended field.

The coefficient field and its action have both descended. We now verify (7.17), including its retained k-coordinates. Invariants on a finite product of fields permuted transitively by a finite group are identified with invariants on a chosen factor under its stabilizer: the other coordinates of an invariant element are its translates. Thus after extension to \Lambda, Equation (7.5) may be computed on the chosen component with group G_1.

Invariants also commute with extension from b to \Lambda. For a ring with a finite G_1-action this follows by applying the averaging projector |G_1|^{-1}\sum_{g\in G_1}g. Passage to fraction fields causes no change: if an invariant fraction has denominator a, multiplication by the invariant denominator \prod_{g\in G_1}g(a) puts its numerator in the invariant ring. These arguments apply to the present tensor products, which are domains on the chosen component; the finitely generated fields over the algebraically closed field b are geometrically integral. Consequently

Identification with the original fiber. Proposition 2.10 gives \C(S)=\C(Y) in the original field tower

In particular, \C(T_0)\subset\C(Y), with its prescribed algebraic closure b\subset\Omega. Before the finite parameter changes the generic parameter field is

It is algebraic over \C(Y). Each subsequent finite parameter extension can be embedded in \Omega, and an algebraic closure of the resulting L_0 is still \Omega. We may therefore take \Lambda=\Omega.

By Proposition 6.2, after this parameter change the function field of the generic x-base is L_0(V')=K. The field K(J) is its total field in the original fibration. Birational model changes preserve these fields, so

The right-hand side is the function field of the whole smooth geometric generic fiber of f, not just that of J over an algebraic closure of k.

Finally, take a smooth projective b-model of F_b, by a projective compactification and resolution of singularities in characteristic zero. Equation (7.17) says that its base extension to \Omega is birational to that whole fiber. Since

the definition of birational variation gives (7.18).□

Proof of Theorem 1.1. If \kappa(F)=-\infty, the assertion holds by the stated convention. Otherwise set d=\kappa(F)\geq0. The compactification and logarithmic subadditivity argument in Lemma 2.6, specifically (2.1), gives

Equations (2.4) and (2.10) identify the dimension of the absolute logarithmic Iitaka base and give

Combining this identity with (7.18) gives \kbar(U)\geq d+\Var(f). Taking the larger of the two lower bounds proves

The dimension-zero cases fit the same field argument. If d=0, then V' is a point over b, so k=b and the geometric constancy already gives the required descent. If I is a point, there are no parameter directions and \Lambda=b. If J_0 is a point, its birational group is trivial and the descended field retains precisely the fixed V' coordinates. For a point base or a zero-dimensional fiber the same conventions give the asserted inequality. Throughout, the Kodaira dimension used for the original compact generic fiber is its ordinary Kodaira dimension, and every auxiliary marking has been removed from the final invariant field.□

Proof. Put r=\dim G and k=\dim Y-r. On a smooth parameter open, the image family is flat. At a general y\in S=g(G) the normal evaluation map

has rank k, since the family covers Y. Its nonzero maximal minors are obtained from transverse k-dimensional parameter slices. Pullback of \alpha to each sliced family and adjunction produce sections of \ell(K_G+B_G). Their local coefficients are the \ell-th powers of the minors, multiplied by the same tangential Jacobian and the coefficient of \alpha. The reduced inverse-boundary inclusion makes every logarithmic pole allowed by B_G, proving global regularity in the indicated logarithmic bundle. Changing lifts of parameter vectors by vectors tangent to G does not change their wedge determinant.

Every nonzero section space of a divisor with Iitaka dimension zero is one-dimensional. Ratios of the resulting sections are therefore constant on G. The common factors cancel, so all nonzero minor ratios are constant as well: their \ell-th powers are constant, and a rational function on an integral complex variety with a constant positive power is itself constant. Thus the Plücker point of the rank-k quotient e_y, and hence \ker(e_y), is independent of general y. A parameter vector in that kernel induces an embedded deformation of S zero at its generic point. Such a deformation is a homomorphism from the conormal sheaf to \mathcal O_S; since S is integral, it is detected at the generic point and must be zero. The common kernel is therefore the kernel of the image-cycle map. In characteristic zero its generic differential rank is its image dimension, namely k.□

Proof. First check specialization. Fix rational frames and include their divisors, the combined form, B, and all divisorial nonsmooth data in one finite support. Record separately their codimension-two images on U. At very general t, vertical members are absent and horizontal strata meet the fiber transversely. Their coefficients, orders, and multiplicities are unchanged. Every minimizing divisor E\to P remains surjective after base change, so every component of P_t receives an unchanged minimizing value. The complements of the order-computation loci have smaller fiber dimension and contain no such component. Thus the minimum remains c_P.

A listed divisor dominating U cannot create an extra vertical test: over the geometric generic point of T_0, its components are conjugate, and one dominates the geometrically integral U_t, hence all do. This spreads after shrinking the base. A codimension-two image stays of codimension at least two, or is avoided. Outside the finite support the frames are units and the generic divisor fibers are smooth and reduced. The fiber minimum is therefore exactly C_h|_{U_t}. Division by the rank-one fiber form proves Equation (8.8); no extension of a fiber section to W is asserted.

For a horizontal P, work on such a fiber. A regular top form on U_t wedged with the logarithmic pluriform on S_t pulls back with only the poles allowed by B_{W_t}, so c_P\geq0. In the finite normalization of S_t\times U_t, choose a divisor over S_t\times P_t. Its ramification index e gives the value (e-1+b_E)/e\leq1, because the form from S_t is a unit at its generic point. Thus c_P\leq1.

For a vertical prime P, work at its discrete valuation ring, with uniformizer x, and choose a local volume form \theta on T_0. If r_P is the coefficient of R_s, write s^*\theta=x^{r_P}\beta with \beta primitive. The saturated subspace defined by this decomposable form is a direct summand over the discrete valuation ring. There is consequently a regular complementary d-form \lambda with \beta\wedge\lambda a unit top form. As a relative form, \lambda represents x^{r_P}\tau, where \tau generates K_{U/T_0}. Wedging q^*\lambda with the form from S shows that (q^*x)^{\ell r_P}\eta_P is allowed by B: its only possible logarithmic poles lie in the reduced inverse boundary already contained in B. Therefore c_P\geq-r_P. This argument also applies when the image of P has codimension greater than one on T_0.

Finally resolve the finite support on U, resolve the source map, and recompute the minimum. At an old prime, old minimizing valuations persist and pullback of a log-regular form stays regular with the boundary rule of Lemma 2.2. Thus old coefficients do not change. New horizontal coefficients satisfy the same product calculation, giving SNC horizontal support and coefficients in [0,1]. Equation (5.1) gives Equation (8.7).□

Proof. Write C=B_{\mathrm{hor}}, and let V=R_p+B_{\mathrm{vert}}-p^*D_I\geq0 as in Equation (5.2). Prescribe the finite extension identifying the chosen generic model with (F_c,\Delta_c), including its Stein field, in Lemma 5.2. A model section defines a relative rational pluriform on H'. At horizontal primes it is allowed by the generic canonical-ring comparison. At a vertical prime tested by Lemma 5.2, work over R=\mathcal O_{I',D} and compare with (Z,\Delta_Z)=(F_c,\Delta_c)\times\operatorname{Spec}R. Its reduced special fiber Z_0 makes (Z,\Delta_Z+Z_0) log canonical. On a common graph resolution, the discrepancy a_E of the tested valuation satisfies

Use the same base canonical divisor on both sides of the comparison, so its pullbacks cancel and these are also relative discrepancies. In a degree clearing the model index, the zero divisor of a model section is effective Cartier. Its pullback is effective, and the remaining relative-form order is ma_E\geq0. Thus every model section is regular at every tested prime, including those centered on the boundary or singular locus of the fixed model.

Lemma 5.2 gives sections of m\tau^*(K_{\mathcal V_p}+C). Fix 0\ne\alpha\in H^0(I,\ell(K_I+D_I)). Multiplication by its pulled-back m/\ell-th power and by the section of m\tau^*V gives Equation (8.10), using Equation (5.2). All factors are independent of the chosen model section and nonzero on a general fiber.

To fix one open for every degree, use finite generation of the generic log canonical ring [8]. Choose a divisible index m_1 for which its Veronese is generated in degree one and A_c=m_1(K_{F_c}+\Delta_c) is basepoint-free Cartier. Write L_\eta=K_{H'_\eta}+C'_\eta on the generic fiber of p'. Resolve the base ideal of |m_1L_\eta| and its model map on one common graph, with maps u to H'_\eta and v to the fixed model. The free and fixed parts give

Generation in degree one implies, for every a\geq1,

Spread these maps and this divisor identity over one dense open of I'. Remove the images of its finitely many vertical terms and the zeros of the base multiplier, as well as p'(\Supp\rho^*V). Increase m_0 to a common multiple of m_1 and the earlier indices. Every restricted comparison is then a tensor power of the same free-and-fixed-part identity, so this open and index work for all m\in m_0\N. Injectivity is already visible on the chosen generic fiber. The last assertion follows from Proposition 5.3.□

Proof. Choose one sufficiently divisible m for which the ample-model system is an embedding, and let X_m be the complete-system image. Finite pullback preserves Iitaka dimension, so \dim X_m=d. Because F' dominates X_m, restriction of the global sections defining X_m is injective on their image: a section whose restriction vanishes gives a hyperplane containing the dense image of F', and hence vanishes on X_m. The subspace in (3) therefore lifts to a linear subsystem on X_m. Its rational linear projection has as its image precisely the embedded ample model. A general linear section of the projected image lifts to a linear section of X_m; the points in a general finite fiber are counted with their positive multiplicities. Hence \deg X_m is at least the product of the projection degree and the degree of that model. In particular,

The pullback linearization gives an action of the Galois group \Gamma on the complete system and on X_m. It fixes the embedded field \C(V), since this field comes from H. The tower

shows that its effective image on X_m has order e\leq e_0.

Let R be the homogeneous coordinate ring of X_m. Evaluation embeds R_k equivariantly into H^0(H',km\pi^*D). Its invariants descend to H^0(H,kmD), since (\pi_*\mathcal O_{H'})^\Gamma=\mathcal O_H. Choose a positive integer r divisible by the orders of the characters with which every stabilizer in the full group \Gamma acts on the fibers of \mathcal O_{X_m}(1); for example, |\Gamma| suffices. Thus the line bundle \mathcal O_{X_m}(r) with its given linearization descends to an ample line bundle A_m on the finite quotient X_m/\Gamma. The quotient has generic degree e, the order of the effective action, even when the kernel acts nontrivially by scalars on the original line bundle. It follows that A_m^d=r^d\deg(X_m)/e.

For sufficiently large n, the coordinate-ring piece R_n agrees with H^0(X_m,\mathcal O_{X_m}(n)): this is the restriction sequence in projective space and Serre vanishing for the ideal sheaf of X_m. Finite quotient descent identifies the invariant sections in degree rk with H^0(X_m/\Gamma,A_m^k). The Hilbert polynomial therefore gives

Normalize in degree mrk and let k tend to infinity. Cancelling r^d gives the lower bound

Thus even an arbitrarily large scalar kernel affects only the divisibility index, not the claimed lower bound.□

Proof. Apply Lemma 5.2, including the Stein field when needed. For a log smooth pair with connected fibers, twisted weak positivity says that

is weakly positive in divisible degrees [19, Theorem 1.1]. Its reflexive symmetric powers are generically generated after arbitrarily small positive ample twists.

Here is the extension step needed when p' is not equidimensional. Since \mathcal E_m is torsion free, it is locally free on an open J\subset I' containing every codimension-one point. On J, the reflexive symmetric power agrees with the ordinary symmetric power, and multiplication of sections defines the regular evaluation map

Thus a global section of the reflexive symmetric power with a base line-bundle twist evaluates to a rational pluriform regular over J. Every possible divisorial pole lies over I'\setminus J. Lemma 5.2 makes such a divisor exceptional over the original H; its order test at all other primes and normality give a section on H^a. This argument is the replacement for a global reflexivity assertion about the direct image.

For (i), the assertion is automatic if the fiber Iitaka dimension is -\infty. Otherwise discard the vertical boundary and write the pulled-back big divisor as Q'\sim_\Q2\eta A+E, with A ample, \eta>0 rational, and E\geq0. Choose a fiber degree m realizing the fiber Iitaka dimension k. Weak positivity with the first \eta A supplies systems whose restrictions include all products of those degree-m fiber sections, so their fiber image has dimension k. Multiplying by the sections of the second \eta A and by the fixed section of E makes the image field contain the base field. The resulting image has dimension \dim I+k. Lemma 5.2 descends these systems to the finite normalization of H; adding R_p and the discarded vertical boundary embeds them in the required systems. Finite pullback preserves Iitaka dimension. This proves the big-base comparison by the weak-positivity method of [19, Proposition 9.1]; the divisorial tests above supply the extension step that equidimensionality supplies in that formulation.

For (ii), first assume P is rational, and take all perturbation parameters positive and rational. Fix ample divisors A' on H' and A_I on I'. Represent \rho^*P+\delta A' by a general small-coefficient SNC boundary. Its addition preserves the big generic log divisor. The same weak-positivity argument, with a twist \epsilon A_I, produces a nonzero section of a sufficiently divisible positive multiple of

regular at every prime tested over H. Push its divisor to H: any negative components are exceptional over H, so the pushforward is effective. Divide by e=\deg\rho. The tested lattice comparison and the norm for principal divisors show that

is pseudoeffective. These two error classes are fixed. Let \delta,\epsilon decrease to zero and use closedness of the pseudoeffective cone.

For a nef real divisor P, choose ample rational divisor classes Q_j tending to [P] in N^1(H)_{\R}, by approximating the ample classes P+\epsilon_j A_H with \epsilon_j\downarrow0 and A_H ample. Their restrictions tend to [P|_{H_i}] on each geometric generic fiber component. Bigness is open, so K_H+C+Q_j is big over I for all sufficiently large j. The rational case makes K_{\mathcal V_p}+C+Q_j pseudoeffective. Closedness of the pseudoeffective cone gives the assertion for P.□

Proof. Take the relative Iitaka fibration of K_X over Y, using the relative algebraic closure of its section field, and resolve its maps. The Iitaka fibration theorem gives relative dimension d and Kodaira dimension zero on its geometric generic fiber [27, Section 6]. Give X' the strict transform of the zero boundary plus the reduced exceptional divisor. The multiplicative section-ring identity in Lemma 2.2, also relative over Y, identifies its relative log Iitaka fibration with the one just chosen. Consequently (J,D_J) has logarithmic Kodaira dimension zero.

Apply the rank-one comparison of [37, Proposition 2.7], with the original X as its reference model. The relative section comparison gives K_W+B+M the full section field of dimension d on the generic fiber over Y; hence it is relatively big. It also gives (iii). The normalization lemma in [37, Lemma 7.4] identifies its initial boundary using two different order tests. The coefficient defining sections is the minimum over the actual components of a fiber, whereas the discriminant threshold also allows divisorial valuations on higher models. The stated normalization compares these quantities on the same SNC model; we do not replace one by the other. The reduced-boundary results [37, Lemma 7.1] and [37, Corollary 7.2] give (iv)–(v), including the normalization of M without an extra cyclic-index factor.

The entire highest line in (v), with its actual parabolic extension, satisfies the hypotheses of Proposition 3.9. That proposition gives h_0, Z and A after smooth projective modifications. Its positive value of a suffices for all larger a, since A is nef. Use Lemma 2.2, with the pulled-back rational twist, for the boundary on the modified W. Its section comparison, and the original-reference convention in [37, Proposition 2.7], preserve (i) and (iii). The parabolic line pulls back functorially, so (v) is preserved as well. The modifications and the relative Iitaka construction do not change geometric generic integrality.□

Proof. Take M=h_0^*A in Theorem 8.1; rational linear equivalence suffices there. Its conclusions, with its product diagram (8.3) and dimension identity (8.1), give the asserted diagram and dimension formula. Its hypotheses are exactly (i)–(ii) of Proposition 9.1, the effective SNC boundary, and \kappa(Y)\geq0, with D_Y=0 in that theorem. The section-system identification in Proposition 9.1(iii) transfers the resulting dimension to X. This retains the actual image-cycle parameter space, before its dimension is compared with \kappa(Y).□

Proof. Choose a nonzero ordinary pluricanonical form on the generic fiber of f, which has Kodaira dimension d\geq0. Pull it to the induced smooth birational model and restrict to the generic fiber of its map to W_y. Adjunction, after dividing by a local determinant frame on W_y, gives a nonzero ordinary form \theta\in H^0(J,mK_J) for some m>0. The restriction is nonzero because a proper zero divisor cannot contain the generic fiber. The logarithmic section space in degree mp is one-dimensional, so

Testing divisorial orders shows that \omega has no poles. Its index remains the original minimal p; the common degree is used only to compare the forms.

In the reduced cyclic-cover lemma of [37, Lemma 7.1], this gives

Thus \Delta_J^{=1} and the specified pole boundary on the root cover are empty. That lemma proves the displayed equality of ordinary pluricanonical spaces. Moreover, [37, Corollary 7.2] now identifies M directly with the rational parabolic extension of the entire compact top Hodge line. There is no open-fiber boundary to compare.

Spread an equivariant smooth projective model of the cover over a nonempty smooth open W^\circ\subset W, shrinking it to remove any vertical remnants of the spread pole boundary. Let G be a very general smooth q-fiber meeting this open. Since h_0=j\circ q and M\sim_{\Q}h_0^*A, the degree of M on every complete test curve in G is zero. Lemma 3.2 and Corollary 3.13 therefore make the projective compact top line constant on each such curve. Complete-intersection curves through a general point with tangent directions spanning T_G show that the line is constant on G\cap W^\circ. Corollary 3.15, applied anew on this restricted locus, gives finite character.

Apply Corollary 4.9 to W^\circ\to U, using its relative assertion and the finite deck action. Its hypotheses on geometric generic integrality follow from Corollary 9.2. This proves the stated constancy of the covers, including over b_U. Taking invariant function fields gives the assertion for J: invariants under a finite group commute with field extension in characteristic zero.□

Proof. We identify first the absolute section field, then the intermediate field of the combined map, and finally the intersection of the original base field with the absolute section field. All these identifications take place inside \C(X).

Put L=K_W+B+M, and let E\subset\C(W) be its stabilized section field. Since L|_G=K_G+B_G has Iitaka dimension zero, every section ratio is constant on the general q-fiber G. Connectedness of q gives E\subseteq\C(U). The equality \kappa(W,L)=\dim U makes \C(U)/E a finite extension. The stabilized section field is relatively algebraically closed in \C(W), by the Iitaka theorem as used in Proposition 2.7. Hence E=\C(U). The multiplicative section comparison in Proposition 9.1 identifies this field with the absolute canonical section field of X.

The product diagram in Corollary 9.2, with S\to Y birational, shows that \C(W) is finite over \C(Y)\C(U). It is relatively algebraically closed in \C(X) because x has connected fibers. These two facts characterize the relative algebraic closure used in Proposition 2.7, so the intermediate fields coincide.

It remains to identify the parameter field with its embedding, rather than only its transcendence degree. In either construction, the product diagram and geometric generic integrality give an injection

For u\in\C(Y) and v\in\C(U), linear independence over \C(T_0) shows that u\otimes1=1\otimes v forces u=v\in\C(T_0). Consequently

The two larger fields have already been identified, so the parameter fields coincide with their embeddings in \C(Y). Proposition 7.3 now gives Equation (9.2) over this exact field b. The variation bound follows from its definition.□

Proof. The logarithmic Hodge description identifies the map on top pieces with the nonzero inclusion

see [15, Corollaries 3.2.13(ii) and 3.2.14]. The dimensions make it an isomorphism. Degree-r open cohomology has weights at least r, and its compact image is precisely W_r\mathbb V [15, Corollary 3.2.17]. Strictness therefore identifies the three top lines. In particular, the only weight grade carrying the logarithmic top line is this pure weight-r image.

The kernel of \mathbb H\twoheadrightarrow\mathbb Q is a pure Hodge subvariation. Its orthogonal complement for a polarization of \mathbb H is horizontal and is a Hodge subvariation, and maps isomorphically to \mathbb Q; this is the polarized splitting argument in [15, Lemma 4.2.3(i)]. It supplies a horizontal Hodge right inverse j. Since the kernel has zero F^r piece, F^r\mathbb H=j(F^r\mathbb Q). Thus flatness of the latter line gives flatness of the former.

For a finite group \Gamma, replace j by

Every summand is a horizontal Hodge right inverse, so their average is one as well, and reindexing the sum proves equivariance. The rank-one top identification is unchanged. This proves the comparison with the finite action retained.□