Proof. Fix a closed fiber \(F_v\). By [11], its non-uniruledness makes \(K_{F_v}\) pseudo-effective. The companion [40] gives a normal projective model \(M_v\) with semiample \(\mathbf Q\)-Cartier \(K_{M_v}\) and, on a common smooth resolution \(p:W\to F_v\), \(q:W\to M_v\), \[p^*K_{F_v}=q^*K_{M_v}+E,\qquad E\geq0 \quad\text{and $E$ is $q$-exceptional}.\] 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 \[a(P,M_v)=a(P,F_v)+\operatorname{coeff}_P E\geq0\] 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), so [45] gives the dichotomy and the special-base assertion. The case \(\bar\kappa(V)=0\) also follows from the second inequality and \(\mathop{\mathrm{Var}}(f)\geq0\). ◻

Proof. Apply [37]. 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 \(a_i=\mathop{\mathrm{ord}}_{E_0}(\mu^*z_i)\), 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 \[k_{E_0}+1-\sum_i\delta_i a_i \geq\sum_i(1-\delta_i)a_i\geq0.\] 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 \(\mu_*\mathcal O_{X'}(mE)=\mathcal O_X\). 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 6. 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 \(f_j\in\mathcal O(I)\). 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].

For the second claim use \[D+M=b^{-1}(D+bM)+(1-b^{-1})D.\] 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 \(f_*\mathcal O_U=\mathcal O_V\). Thus \(\mathbf C(V)\) is relatively algebraically closed in \(\mathbf C(U)\); 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 \(\mathop{\mathrm{Supp}}(f^*D_Y)\subseteq\mathop{\mathrm{Supp}}(D_X)\). A very general fiber lies over \(V\), has no boundary, and has ordinary Kodaira dimension \(d\). Theorem 4 now applies to these smooth projective varieties, connected fibers, and reduced SNC boundaries. Its conclusion is (4), by the compactification definition of logarithmic Kodaira dimension. ◻

Proof. By (4), \(K_X+D_X\) has nonnegative section dimension. The divisor Iitaka theorem gives a resolved map with base dimension \(\bar\kappa(U)\) and with restriction of the divisor having section dimension zero on a very general fiber. The needed statement is [27]. The stabilized section field is relatively algebraically closed in \(\mathbf C(X)\) by [27]. In the case \(\bar\kappa(U)=0\), 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 (6) 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 (8), including after further compatible resolutions and birational changes of the Iitaka base.

Let \(K=\mathbf C(X)\), \(L_Y=\mathbf C(Y)\), and \(L_Z=\mathbf C(Z)\), 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: \[0\longrightarrow h^*\Omega_Z^1 \longrightarrow\Omega_W^1(\log D_W^0) \longrightarrow\Omega_{W/Z}^1(\log D_W^0) \longrightarrow0.\] 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 \[ \bigwedge^n\Omega_W^1(\log D_W^0) \longrightarrow h^*\!\bigwedge^{p_1}\Omega_Z^1 \otimes\det\Omega_{W/Z}^1(\log D_W^0). \tag{9}\] 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 \(m\)th tensor power of the left side of (9). 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 \[\bigl(\det\Omega_{W/Z}^1(\log D_W^0)|_{W_z}\bigr)^{\otimes m} \simeq\mathcal O_{W_z}\bigl(m(K_{W_z}+D_W^0|_{W_z})\bigr).\] 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 \(Z\): \[x_z:(X'_z,D'|_{X'_z})\longrightarrow (W_z,D_W^0|_{W_z}).\] 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 4 applies. Its conclusion, together with (8) and Lemma 11, is \[0\geq \kappa\bigl(J,K_J+D'|_J\bigr) +\kappa\bigl(W_z,K_{W_z}+D_W^0|_{W_z}\bigr)\geq0.\] 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 \[d\leq\dim W_y+\kappa(J)=\dim W_y.\] Here we use the upper addition inequality for section dimensions with irreducible general fibers, as in [27]; 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 \(\mathbf C(Z)\) and the regular extension \(\mathbf C(W)\). 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] therefore gives a parameter map \[\tau:Z\dashrightarrow\operatorname{Hilb}(Y).\] 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 11 for every \(p_1\)-tuple of basis vectors. By Lemma 12, 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 \[ T_zZ\longrightarrow N_{C_z/Y,y}. \tag{14}\] 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 \(m\)th power of the corresponding \(p_1\)-minor of (14), multiplied by a common nonzero factor. Proportionality of the sections says that the \(m\)th 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 \(\mathbf C\), and hence is constant on an integral complex variety. The Plücker coordinates of (14) 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 \[\mathcal I_z/\mathcal I_z^2\longrightarrow\mathcal O_{C_z}.\] 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 (14). 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 \(\mathbf C(Z)\), 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 (11) are morphisms. Its right square commutes, and the induced map (12) 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 \[ \mathbf C(Y)\subseteq\mathbf C(S)\subseteq\mathbf C(W)\subseteq\mathbf C(X). \tag{15}\] The first extension is algebraic, whereas \(\mathbf C(Y)\) is relatively algebraically closed in \(\mathbf C(X)\). Therefore \(\mathbf C(S)=\mathbf C(Y)\). In particular, the construction embeds \(\mathbf C(T_0)\) in the original base field \(\mathbf C(Y)\).

For completeness, geometric integrality of the generic fiber of \(S\to T_0\) can be checked in the same field tower. The field \(\mathbf C(T_0)\) is relatively algebraically closed in \(\mathbf C(Z)\) by construction, and \(\mathbf C(Z)\) is relatively algebraically closed in \(\mathbf C(W)\). It follows that \(\mathbf C(T_0)\) is relatively algebraically closed in \(\mathbf C(W)\), and hence in its subfield \(\mathbf C(S)\). The extension \(\mathbf C(S)/\mathbf C(T_0)\) 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 \[\dim W=\dim Y+d=\dim Z+r.\] 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 (13). All these final resolutions preserve the section and fiber properties in Proposition 10. 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 \(p\)th root of \(f\). This field extension has degree \(p\). Indeed, since the ground field contains the \(p\)th roots of unity, reducibility of the Kummer polynomial would make \(f\) a \(q\)th power for some prime \(q\mid p\). If \(f=a^q\), then the rational \((p/q)\)-canonical form \(a\tau^{p/q}\) has regular \(q\)th 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 \[ \eta^{\otimes p}=(\pi_0\mu)^*\omega_J \tag{17}\] as pluricanonical differentials. Pullback of a regular pluricanonical differential under a morphism of smooth varieties is regular. Thus (17) 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 \[ \mathop{\mathrm{ord}}(\eta)=\frac{ea}{p}+e-1. \tag{18}\] If \(a=0\), adjoining the root of a unit is unramified at that prime and \(e=1\). If \(a\ge1\), the right side of (18) 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 \[ H^0(\widetilde J,mK_{\widetilde J}) \longrightarrow H^0(J^{\mathrm{nor}},\mathcal O_{J^{\mathrm{nor}}}(2m\pi_0^*E)), \qquad s\longmapsto s/\eta^m. \tag{19}\] Only codimension-one tests on \(J^{\mathrm{nor}}\) enter this injection.

Lemma 6 gives \(\kappa(J^{\mathrm{nor}},\pi_0^*E)=\kappa(J,E)=0\). Since \(\pi_0^*E\) is effective, each space on the right of (19) 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]. 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 \[ r_Q-m_Qt_P+d_Q\ge0. \tag{22}\] 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 \[\frac{1+r_Q}{m_Q}\ge t_P \quad\text{for every }Q\mapsto P, \qquad\text{and hence}\qquad \lambda_P\ge t_P.\] If \(Q\) attains the actual minimum defining \(t_P\), then \[ 0\le\lambda_P-t_P \le \frac{1+r_Q}{m_Q}-\frac{r_Q+d_Q}{m_Q} =\frac{1-d_Q}{m_Q}\le1. \tag{23}\] 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 (23) 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 (22) 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 (20). 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 (20). The equality \(B_P=0\) makes \(\lambda_P-t_P\) in (23) equal to one. Thus \[1\le\frac{1-d_Q}{m_Q}\le1.\] 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 (24). ◻

Proof. On the smooth family locus, finite change of variables gives \[ \|\eta_w\|_{\mathrm{Hdg}}^2 =c_p\int_{J_w}|\omega_{J,w}|^{2/p}, \tag{26}\] 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 (26) 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 \[r_Q-\gamma m_Q.\] 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 \[ \sup\left\{\gamma: |u|^{-2\gamma}\|\eta\|_{\mathrm{Hdg}}^2\in L^1_{\mathrm{loc}} \right\} =\inf_{Q\mapsto P}\frac{1+r_Q}{m_Q} =\lambda_P. \tag{27}\] 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]. Accordingly, a rational section of parabolic order \(a\) has squared norm \[|u|^{2a}\,(-\log|u|)^{O(1)},\] where the notation means two-sided bounds by fixed powers of the logarithm. Its supremal exponent in (27) 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 (27) 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 (27) 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 (26) uses the same normalization. Finally \(t_P=B_P+(\lambda_P-1)\) at every prime, which proves (25). 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 \(K=\mathbf C(T_1)\) has nontrivial restriction \(w\) to \(L=\mathbf C(R)\). 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 \[\mathop{\mathrm{trdeg}}_{\kappa(w)}\kappa(v)\le\mathop{\mathrm{trdeg}}_LK.\] Since \(v\) is divisorial, this gives \[\mathop{\mathrm{trdeg}}_{\mathbf C}\kappa(w) \ge (\dim T_1-1)-(\dim T_1-\dim R)=\dim R-1.\] Conversely, lifts of algebraically independent residue elements for \(w\), together with an element of positive valuation, are algebraically independent over \(\mathbf C\). Thus \(\mathop{\mathrm{trdeg}}_{\mathbf C}\kappa(w)\le\dim R-1\). 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 20 with fixed source \(X_*\) and base \(W\). Retain the maps to \(Y\) and \(Z\), resolve the finite boundary supports, and recompute (20). 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 \[\sigma\in H^0(W_z,m(K_{W_z}+T_z)).\] 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 \[m\bigl(r_Q-m_Qt_P+d_Q\bigr)\ge0\] by (22). 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 \[H^0(X_{*,z},m(K_{X_{*,z}}+D_*|_{X_{*,z}})).\] 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 (16). The identity \(T_z\sim_{\mathbf Q}B|_{W_z}+M|_{W_z}\) from Proposition 19 proves (28). ◻

Proof. This is [37], 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 22 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 \[ \tau^*L\sim_{\mathbf Q}p^*A, \qquad K_R+cA\text{ big}. \tag{30}\]

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 \[\mu:P'\to P, \qquad a:P'\to R.\] On a common smooth modification of \(B'\) and \(P'\), the two maps to \(R\) agree. Pulling back (30) 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 \[\mu^*L\sim_{\mathbf Q}a^*A.\]

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 20 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 \[K_{T_2}+\Delta_2 =\pi^*(K_{T_1}+\Delta)+E_{\mathrm{exc}}, \qquad E_{\mathrm{exc}}\ge0\text{ exceptional over }T_1.\] The section systems in the assertion therefore have the usual pullback and pushdown comparison. For a smooth birational base map \(\rho:R'\to R\), \[K_{R'}+c\rho^*A =\rho^*(K_R+cA)+K_{R'/R}\] 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 \[\mathcal E_m =(a_2)_*\mathcal O_{T_2} \bigl(m(K_{T_2/R}+\Delta_2)\bigr)\] 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] 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 \[\alpha mL-H\quad\text{is big}.\] The definition of weak positivity [19] supplies a positive integer \(\beta\) for which \[ \bigl(\operatorname{Sym}^{\alpha\beta}\mathcal E_m\bigr)^{**} \otimes\mathcal O_R(\beta H) \quad\text{is generically generated by global sections}. \tag{32}\] 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 (32) therefore yields a global section whose evaluation is a nonzero rational section \(u\) of \[\alpha\beta m(K_{T_2/R}+\Delta_2)+\beta a_2^*H.\] 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 \[q\beta(\alpha mL-H).\] The resulting rational sections lie in the class \[\begin{align*} &q\alpha\beta m(K_{T_2/R}+\Delta_2) +q\beta a_2^*H +q\beta a_2^*(\alpha mL-H)\\ &\hspace{25mm} =q\alpha\beta m(K_{T_2}+\Delta_2+cN_2). \end{align*}\] 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 (31). ◻

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 16 therefore identifies \(N\) with the parabolic extension of the restricted variation. Moreover \[\kappa(T_1,K_{T_1}+\Delta)\ge0\] by (16) and \(B\ge D_W^0\). In particular, \(D=K_{T_1}+\Delta\) is rationally equivalent to an effective rational divisor.

Apply Proposition 23 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 \[N\sim_{\mathbf Q}a^*A, \qquad a:T_1\longrightarrow R, \qquad A\text{ nef}, \qquad K_R+cA\text{ big for }c\gg1.\] On the higher source model use strict transform plus reduced exceptional boundary for \(\Delta\). The logarithmic section comparison preserves (28) and the rational effectivity of \(D\); the Hodge line is pulled back. If \(\dim R>0\), choose a sufficiently large rational \(c\ge1\). Lemma 25 gives \[\kappa(T_1,D+cN)\ge\dim R>0.\] But \[ D+N=\frac1c(D+cN)+\left(1-\frac1c\right)D. \tag{34}\] 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 (28). We conclude that \(R\) is a point and \(N\sim_{\mathbf Q}0\). Rational triviality descends through the intervening birational morphisms, since their direct image of the structure sheaf is the structure sheaf. Now (28) 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 \(j\)th 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 \(\alpha-\beta=dd^cu\). Near a puncture, with \(\rho=-\log|z|^2\), the Hodge norm and horizontal Schwarz estimates give \[|u|\le C(1+\log\rho),\qquad 0\le\alpha\le C\frac{i\,dz\wedge d\bar z}{|z|^2\rho^2};\] see [14] and [10]. 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] 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 28 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 26 gives \(N\sim_{\mathbf Q}0\), and the pullback property in Lemma 16 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 27 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 29, 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]. 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], 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 \[B_h(\beta,\gamma)=\mathrm{i}^{n^2} \int_P h\wedge\beta\wedge\overline\gamma.\] 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 \[V\longrightarrow\overline V^{\,*},\qquad \beta\longmapsto B_\ell(\beta,-)\] 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 (35).

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 (36) gives \[ \Theta=\iota_v(dt\wedge\omega). \tag{37}\] 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 \[ 0\longrightarrow\mathcal O_P\longrightarrow\operatorname{At}(H) \longrightarrow T_P\longrightarrow0. \tag{38}\] 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 (38) 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 (37), 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 (35) 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 \[ E=\sum_j e_jD_j,\qquad R_0=E+E_{\mathrm{red}},\qquad S=D_i,\qquad B'=\sum_{j\ne i}D_j. \tag{39}\] 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 \[E\le R_0\le2E,\qquad E\sim K_D,\] for every positive integer \(m\) we have \[1\le h^0(D,mR_0)\le h^0(D,2mE) =h^0(D,2mK_D)=1.\] 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 \[W=Gv,\qquad u=(W(z)/z)|_S.\] The pole bound in Lemma 34 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 (37) identifies its nonvanishing at the general point of \(S\) with the supposed nonvanishing of \(\Theta\) on the relative component. Thus \[ 0\ne u\in H^0(S,\mathcal O_S(R_0)). \tag{40}\] 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 \[q_{k,\lambda}=\frac{\nabla^k a_\lambda}{k!},\qquad b_{k,\lambda}=\frac{G^k\nabla^ka_\lambda}{z}.\] The following recurrence proves in particular that the second expression is holomorphic: \[\begin{align*} b_{k+1,\lambda} &=W(b_{k,\lambda})+ \left(\frac{W(z)}z-k\frac{W(G)}G+Gc\right)b_{k,\lambda}, \tag{41}\\ b_{k,\lambda}|_S &=u^k(a_\lambda/z)|_S \prod_{j=0}^{k-1}\bigl(1-j(e_i+1)\bigr). \tag{42}\end{align*}\] 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 \(\mathcal O_D(kR_0+H-S)\). More explicitly, in a local frame of \(H\), \[q_{k,\lambda}=\frac{z}{G^k}\frac{b_{k,\lambda}}{k!}.\] 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 \(j\)th step of (41), 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 \[ \frac{(Ck)^k}{k!}\le C_1^k. \tag{43}\] This is a bound in the indicated twisted bundle, including across the zero divisor.

Put \(a=e_i+1\ge2\). No factor in (42) is zero, and \[\lim_{k\to\infty} \left|\frac{\prod_{j=0}^{k-1}(1-ja)}{k!}\right|^{1/k}=a>0;\] for example, the ratio of successive absolute values tends to \(a\). On \(D\), take the weights \[\frac1k\log\sum_\lambda \left|\frac{b_{k,\lambda}}{k!}\right|^2 -\frac1k\varphi_{H-S},\] 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 (43). Their regularized upper limit is a semipositive singular weight. Formula (42), 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 \[ \varphi\ge\log|s_{R_0}|^2,\qquad \varphi|_S\ge\log|u|^2-C. \tag{44}\] 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]. 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 \[ \begin{split} F'&=mR_0-(K_D+S)\sim (m-1)R_0+B',\\ \varphi_{F'}&=(m-1)\varphi+\log|s_{B'}|^2,\\ A_0&=R_0-aS=\sum_{j\ne i}(e_j+1)D_j,\\ \varphi_S&=1+\frac{\varphi-\log|s_{A_0}|^2}{a}. \end{split} \tag{45}\] 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 (44) implies \[|s_S|^2e^{-\varphi_S}\le e^{-1}.\] 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 \(T_R=dd^c\varphi\). Directly, \[ \begin{split} dd^c\varphi_{F'}&=(m-1)T_R+[B']\ge0,\\ dd^c\varphi_{F'}-dd^c\varphi_S &=\left(m-1-\frac1a\right)T_R+[B']+\frac1a[A_0]\ge0. \end{split} \tag{46}\] Here \(a\ge2\) and \(m\ge2\). For condition (22), choose a positive number \[\varepsilon_0\le \min\left\{1,\min_{j\ne i}\frac{a}{e_j+1}\right\},\] 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 \[\varphi_{F'}\le \frac{\varepsilon_0}{a}\log|s_{A_0}|^2+C.\] 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 \[u^m\in H^0\bigl(S,\mathcal O_S(mR_0)\bigr) =H^0\bigl(S,K_S+F'|_S\bigr).\] 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 (44): \[ |u|^{2m}e^{-\varphi_{F'}|_S} \le C\frac{|u|^2}{|s_{B'}|_S|^2}. \tag{47}\] 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 \[C|s_{B'}|^{-2m\delta}|w|^{2-2(m-1)\delta}.\] 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 33, 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 35 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 \[\int_P (dd^cT)\wedge\mu =\int_P[\mathcal E]\wedge\mu=0.\] Here and subsequently \(dd^cT\) 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 \[T_b=\max\{T,\log|s|^2+b\}\] is another semipositive weight on the same bundle. The same argument shows that \((dd^cT_b)\wedge\mu=0\).

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 \[\Theta=a(w)\,dw_1\wedge\cdots\wedge dw_n, \qquad a(w)\ne0.\] 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 \(\chi\), \[\bar\partial\bigl(\chi(\psi)\overline\Theta\bigr)=0\] 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 (35) 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 \[\alpha|_{D_t}=a(t)s|_{D_t}\] with \(a(t)\) bounded on the support in question. Dolbeault integration by parts against the closed current just constructed therefore gives \[ \int_P \chi(\psi)\,h\wedge\Theta\wedge\overline\Theta =\int_P\chi(\psi)\,\ell\wedge\alpha\wedge\overline\Theta. \tag{49}\] 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 \[|\omega|^2 e^{(1+\delta)(T-\log|\omega|^2)} =e^{(1+\delta)T}|\omega|^{-2\delta}.\] 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 (49) by \(i^{n^2}\), take absolute values on the right, and let \(M\to\infty\). Monotone convergence proves (48). ◻

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 (50). No extremal interpretation on exceptional fibers is needed.

Set \[\tau_k=\frac{\beta_k-r_k\varphi_H}{k}.\] These are weights on \(K_{P/C}\), and \(dd^c\tau_k\geq-(r_k/k)h\). 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: \[ T=\max\left\{\left(\limsup_{k\to\infty}\tau_k\right)^*, \log|s|^2\right\}. \tag{52}\] 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 \[H^0(D,kK_D+r_kH|_D-j_kS) \longrightarrow H^0\bigl(S,(kK_D+r_kH|_D-j_kS)|_S\bigr).\] Among sections with this prescribed image, minimize the \(2/k\)-integral in (50). 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 (50) holds. All lower coefficients along \(S\) are zero by the definition of \(j_k\).

Choose \(c>0\) so large that, for all sufficiently large \(k\), \[ c\nu-(1+c)\frac{j_k}{k}>1. \tag{53}\] Choose \(p_0>0\) sufficiently small that \[ \int_D |\omega|^2e^{(1+p_0)\psi}<\infty. \tag{54}\] 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 \(g:\mathbf R\to\mathbf R_{\geq0}\) 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 \[ \Phi=(k-1-c-p_0)\varphi_k+cT|_D +p_0\bigl(\varphi_0+g(x_k-b)\bigr)+r_k\varphi_H. \tag{55}\] 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 \(dd^c\varphi_k=-(r_k/k)h\), \(dd^c\varphi_0=0\), and \(dd^cx_k=-(r_k/k)h\). Direct differentiation of (55) gives \[\begin{align*} dd^c\Phi={}&\frac{r_k}{k}(1+c+p_0-p_0g'(x_k-b))h +cdd^c(T|_D) \\ &+p_0g''(x_k-b)\frac{i}{2\pi} \partial x_k\wedge\bar\partial x_k. \tag{56}\end{align*}\] 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, \[u_k=\theta(x_k-b)U_k-V_k\] satisfies \[ \int_D\left|\frac{u_k}{U_k}\right|^2w_k \leq C_{p_0}\int_{\{|x_k-b|\leq1\}}w_k, \qquad w_k=|\omega|^2 e^{(1+c+p_0)x_k-c\psi-p_0g(x_k-b)}. \tag{57}\] 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 \(dd^cT_\ell\geq-\epsilon_\ell h\), 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 (55), obtaining \(\Phi_\ell\). For large \(\ell\), the loss \(c\epsilon_\ell h\) is less than half the ample term in (56). On \[D\setminus\bigl(Z_\ell\cup E\cup\operatorname{div}(U_k)\bigr)\] 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 \[f_k=\theta'(x_k-b)\bar\partial x_k\,U_k.\] 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 \[ \bigl\langle[A,\Lambda_G]^{-1}(U\eta),U\eta\bigr\rangle_G e^{-\Phi}\,dV_G = |U|^2e^{-\Phi}\,\eta^*A^{-1}\eta\,d\lambda, \tag{58}\] 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 (58).

The gradient term in (56) implies, for \(\eta=\bar\partial x_k\), the matrix bound \[\eta^*A_\ell^{-1}\eta \leq\frac{1}{p_0g''(x_k-b)}.\] 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: \[\begin{align*} |U_k|^2e^{-\Phi_\ell} &=\exp\bigl((1+c+p_0)\varphi_k-cT_\ell -p_0(\varphi_0+g(x_k-b))\bigr)\,d\lambda\\ &=|\omega|^2 e^{(1+c+p_0)x_k-c(T_\ell-\varphi_0)-p_0g(x_k-b)}. \end{align*}\] 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 \[\varphi_0+g(\varphi_k-\varphi_0-b) \leq\max\{\varphi_0,\varphi_k-b\}+C_g.\] 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]. 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 (57). 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 (57) 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 37 gives \(x_*\leq\psi\); thus \(\psi\geq b-1\) and \[Bx_*-c\psi-p_0g(x_*-b)\leq(1+p_0)\psi.\] Reverse Fatou therefore yields \[ \limsup_{k\to\infty}\int_{\{|x_k-b|\leq1\}}w_k \leq\int_{\{\psi\geq b-1\}}|\omega|^2e^{(1+p_0)\psi}. \tag{59}\] By (54), 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]. Consequently \[ \Phi\leq\left((k-1-c)\frac{j_k}{k}+c\nu\right) \log|z|^2+O(1). \tag{60}\] The coefficient exceeds \(j_k+1\) by (53). 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 (57). 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 (50) 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 37. In local canonical frames the density is \[w_k=\exp\bigl(B\varphi_k-cT|_D -p_0(\varphi_0+g(x_k-b))\bigr)\,d\lambda.\] 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, \[ w_k\geq C^{-1}\rho_k^B\,dV \tag{61}\] with \(C\) independent of \(k\) and of \(b\) bounded below. Fix \(0<\eta<\min\{1,2/B\}\). Hölder’s inequality gives \[\begin{align*} \int_D q_k^\eta\rho_k\,dV &\leq \left(\int_D q_k^2\rho_k^B\,dV\right)^{\eta/2} \left(\int_D \rho_k^{(2-B\eta)/(2-\eta)}\,dV\right)^{1-\eta/2} \tag{62}\\ &\leq C_\eta \left(\int_D q_k^2w_k\right)^{\eta/2}. \end{align*}\] The exponent in the second factor is positive, so its uniform bound follows from compactness and the upper bound for \(\rho_k\). Equations (57) and (59) 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 (62) 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 \(q\geq0\), \[ (1+q)^p\leq1+C_\eta p q^\eta. \tag{63}\] 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 \[\begin{align*} \int_D |V_k|^{2/k}e^{-r_k\varphi_H/k} &=\int_D |V_k/U_k|^p\rho_k\,dV\\ &\leq1-\frac12(1-2^{-p}) +C_\eta p\int_D q_k^\eta\rho_k\,dV<1 \tag{64}\end{align*}\] 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 \[\frac{r_k}{k}\longrightarrow0, \qquad \frac{j_k}{k}\longrightarrow0,\] 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 37 now constructs one global weight \(T\) from the chosen degrees, before any final fiber is selected.

Apply Lemma 36 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 38 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 \[\Theta=\iota_v(dt\wedge\omega).\] 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 \(\mathop{\mathrm{ord}}_S(\omega)=e_i\), the finite fiber integral from (48) would then dominate a positive constant times \[ \int |z|^{-2-2\delta e_i}e^{(1+\delta)T|_D}\,d\lambda, \qquad S=(z=0). \tag{65}\] 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], 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] then gives \[\frac{1}{\mathop{\mathrm{vol}}(B(a,r))}\int_{B(a,r)}T\,d\lambda =o(|\log r|)\] 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 (65), \[\int_{B(a,r)}F\,d\lambda \geq C^{-1}r^{2n-2-2\delta e_i+(1+\delta)\varepsilon} \qquad(a\in A).\] 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 \[\int F\,d\lambda \geq C^{-1}r^{-2\delta e_i+(1+\delta)\varepsilon}.\] 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 (65) and the Fubini consequence of (48). 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 32. We may assume \(n>0\), and use the preparation and notation of Lemmas 33–35. Proposition 39 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 \(\Delta\sim_{\mathbf Q}\varepsilon H\) 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] 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 \[\chi:P\dashrightarrow Q\] with \(Q\) projective over the curve. Since the adjoint divisor is relatively big, this map is birational. The ample-model comparison [8] 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 \(\mathbf C(P)=\mathbf C(Q)\) 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 \[\partial f\in\bigcap_{\operatorname{ht}\mathfrak p=1} A_{\mathfrak p}=A.\] 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 29, applied anew to this restricted variation, gives finite character. On an algebraic curve in \(T\), Theorem 32 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 33. 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 32. 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 32 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 \(\mathbf C(T)\). 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 \(b_Z=\overline{\mathbf C(Z)}\), 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 15 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 30 makes its projective top line constant. Corollary 40 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 40 to \(W^\circ\to Z\), shrinking \(Z\) if necessary. Its geometric generic fiber is integral by Proposition 10, and the preceding paragraph verifies the fiber and constant-line hypotheses on its very general complex fibers. Over \(b_Z=\overline{\mathbf C(Z)}\) 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. The diagram and dimension assertions follow from Propositions 10 and 13. The latter identifies \(\mathbf C(S)=\mathbf C(Y)\) inside \(\mathbf C(X)\), so the present parameter field is the embedded field \(b(I)=b\,\mathbf C(Y)\) of Remark 14. 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 (70) is the geometric generic form of the corresponding assertion for \(S_t\) in Proposition 13; nonzero divisible sections are preserved by field extension. The second follows from Proposition 17. Finally, (71) is precisely Proposition 26 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 \[a_\eta:H_\eta\longrightarrow (P_0)_{E_0}\] 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 \[H_\eta\dashrightarrow C_1\xrightarrow{\,c\,}(P_0)_{E_0}\] by [8]. 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]. 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 \[ (x_0=0),\qquad (x_j=0),\qquad (x_0+x_j=0) \quad (1\leq j\leq N). \tag{75}\] 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 (75) 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 7, 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 \[\Delta=B_0^-+a_0^*A_{\mathrm{mark}} \quad\text{on }H_0.\] The total degree of \(A_{\mathrm{mark}}\) is chosen sufficiently large that \[K_{C_1}+(B_\eta^-)_{{C_1}}+c^*A_{\mathrm{mark}}\] 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 45 therefore gives restricted adjoint dimension at most zero. The upper addition estimate in Lemma 8 gives \[ \kappa(H_0,K_{H_0}+\Delta)\leq\dim P_0=d. \tag{76}\] 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 \[ K_{\widehat H_\eta}+\widehat\Delta_\eta =\mu_\eta^*(K_{H_\eta}+\Delta_\eta)+E_{\mathrm{exc}}, \qquad E_{\mathrm{exc}}\geq0 \quad\text{$\mu_\eta$-exceptional}. \tag{77}\] For divisible \(m\), normality and testing at codimension-one points give \(\mu_{\eta*}\mathcal O(mE_{\mathrm{exc}})=\mathcal O\). The generic adjoint rings are therefore identical. The generic model pair remains the marked pair on \(C_1\): exceptional divisors added in (77) 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 \[ \widehat\Delta\geq \bigl((p_I\mu)^*D_I\bigr)_{\mathrm{red}}. \tag{78}\] 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 \(P_0\), (76) still yields \[ \kappa(\widehat H,K_{\widehat H}+\widehat\Delta)\leq d. \tag{79}\]

The markings now define a big generic model pair, while the total section dimension still has the upper bound \(d\). We apply Proposition 44 to \[p_I\mu:(\widehat H,\widehat\Delta)\longrightarrow(I,D_I).\] Both varieties are smooth projective over \(b\); the morphism is surjective with connected fibers and relative dimension \(d\); the boundary is effective rational SNC; (78) holds; and the base has nonnegative log Kodaira dimension by (70). Its geometric generic pair is klt and has big adjoint divisor by the construction of the absolute ample model \(C_1\). Thus Proposition 44 and (79) give \[d+\mathop{\mathrm{Var}}(p_I\mu,\widehat\Delta_{\mathrm{hor}}) \leq\kappa(\widehat H,K_{\widehat H}+\widehat\Delta)\leq d.\] The variation here is exactly that of the generic log canonical model pair. It is zero. The finite-extension conclusion of Proposition 44 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 \[\operatorname{div}(x_j/x_0) \quad\text{and}\quad \operatorname{div}(1+x_j/x_0).\] 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 \[ x_j/x_0=c_jf_j, \qquad f_j\in b(C_*),\qquad c_j\in E^*. \tag{80}\] 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 (72).

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 (73). 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 41 applies after extending \(k\) to an algebraic closure: the \(I\)-directions in (72) 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 7, 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 (74); 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 46 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 46 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 \(b=\overline{\mathbf C(T_0)}\) 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 17 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 17. With \(d_Q\) its old source boundary coefficient, that proposition gives \[B_P=1-\lambda_P+t_P=0, \qquad 1=\lambda_P-t_P \leq\frac{1-d_Q}{m_Q}\leq1.\] 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 (81). ◻

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, \[\mathcal B=\mathop{\mathrm{Bir}}(T)_{\mathrm{red}},\] is a group scheme locally of finite type over \(\overline k\). Its identity component \[A_1=\mathcal B^0=\operatorname{Aut}^0(T)\] 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], also stated in [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] and [9]. 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 \[ \begin{split} \mathop{\mathrm{Bir}}((J_0)_{\overline K}) &=A_1(\overline K)\mathop{\mathrm{Bir}}((J_0)_{\overline k}),\\ A_1(\overline K)\cap\mathop{\mathrm{Bir}}((J_0)_{\overline k}) &=A_1(\overline k). \end{split} \tag{87}\] 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 \[\theta:\overline K(J)\xrightarrow{\sim}\overline K(J_0).\] Let \(s_\sigma^J\) and \(s_\sigma^0\) denote the natural coefficient transports on these two fields. The automorphisms \[u_\sigma=\theta s_\sigma^J\theta^{-1}(s_\sigma^0)^{-1} \in\mathop{\mathrm{Bir}}((J_0)_{\overline K})\] satisfy \[ u_{\sigma\tau}=u_\sigma\,\sigma(u_\tau), \qquad \sigma(v)=s_\sigma^0v(s_\sigma^0)^{-1}. \tag{88}\] 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 (87) this gives a locally constant choice \(h_\sigma\) and expressions \[u_\sigma=a_\sigma h_\sigma, \qquad a_\sigma\in A_1(\overline K).\] The component cocycle law and the second equality in (87) imply \[d_{\sigma,\tau} =h_\sigma\sigma(h_\tau)h_{\sigma\tau}^{-1} \in A_1(\overline k).\] Consequently \[\alpha_\sigma(a)=h_\sigma\sigma(a)h_\sigma^{-1}\] 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 \[M_1=A_1(\overline K)/A_1(\overline k)\] 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 \[ c_{\sigma\tau}=c_\sigma+\alpha_\sigma(c_\tau). \tag{89}\]

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, \[A_1[n](\overline K)=A_1[n](\overline k),\] 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 \(\mathbf Q\)-vector space.

Choose an open normal subgroup \(N\subset\Gamma\) contained in the zero set of the continuous cocycle \(c\). Equation (89) gives \(c_{\sigma n}=c_\sigma\) for \(n\in N\). It also gives \[\alpha_n(c_\sigma) =c_{n\sigma} =c_{\sigma(\sigma^{-1}n\sigma)} =c_\sigma.\] Thus \(N\) fixes every cocycle value. The finite sum \[\beta=\frac{1}{[\Gamma:N]} \sum_{\tau\in\Gamma/N}c_\tau\] is well-defined, and summing (89) over the cosets yields \[ c_\sigma=\beta-\alpha_\sigma(\beta). \tag{90}\] Choose \(t\in A_1(\overline K)\) lifting \(\beta\) and replace \(\theta\) by \(\theta'=t^{-1}\theta\). Its cocycle is \[z_\sigma=t^{-1}u_\sigma\sigma(t).\] The class of its \(A_1\) factor is \(-\beta+c_\sigma+\alpha_\sigma(\beta)=0\) by (90). Hence \[z_\sigma\in\mathop{\mathrm{Bir}}((J_0)_{\overline k}) \quad\text{for every }\sigma\in\Gamma.\]

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 \[\Gamma\longrightarrow \operatorname{Aut}_{k'} k'(J_0) \rtimes\operatorname{Gal}(k'/k), \qquad \sigma\longmapsto(z_\sigma,\sigma|_{k'})\] 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 (84), the pairs are faithful by construction, and the fixed-field identity (86) 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 (91) 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 (86) to identify \[\widetilde L=L\widetilde K=\widetilde K(J_0).\] Since \(J/K\) is geometrically integral, \(L/K\) is regular. Hence \(L\) and \(\widetilde K\) are linearly disjoint over \(K\), and restriction identifies \[ \operatorname{Gal}(\widetilde L/L) \xrightarrow{\sim} \operatorname{Gal}(\widetilde K/K)=G. \tag{92}\]

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 \[ m_{Q'}=\frac{e_Qm_Q}{e}, \qquad r_{Q'}=e_Qr_Q+e_Q-1-(e-1)m_{Q'}. \tag{93}\] 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, \[ \frac{1+r_{Q'}}{m_{Q'}} =e\frac{1+r_Q}{m_Q}-(e-1). \tag{94}\] 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 \[ k'\subset R',\qquad k'\hookrightarrow\kappa'. \tag{95}\] Consider the smooth proper constant model \[\mathcal J=J_0\times_{\operatorname{Spec}k}\operatorname{Spec}R'.\] 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 \(\ell=p^{-1}\mathop{\mathrm{ord}}_{P'}(c)\) 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 \[ \frac{1+r_T}{m_T} =\ell+ \frac{A_{\mathcal J}(T)+p^{-1}\mathop{\mathrm{ord}}_T(D_0)}{m_T}, \qquad m_T=\mathop{\mathrm{ord}}_T(S'). \tag{96}\] 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 \(S'\), \[A_{\mathcal J}(T)>\mathop{\mathrm{ord}}_T(S')=m_T.\] 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 \(\mathop{\mathrm{ord}}_T(D_0)\geq0\), Equation (96) proves that \(S'\) is the unique minimizer of the ratio above \(P'\).

Triviality of inertia. Take the multiplicity-one minimizer \(Q\) supplied by (91). 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 (92). Conversely, every divisorial valuation above \(P'\) restricts under the finite extension to a divisorial valuation above \(P\). Equation (94) 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 (93) gives \[ e_Q=e. \tag{97}\]

Let \(\mathcal I_{Q'}\) and \(\mathcal I_{P'}\) be the total and base inertia groups, viewed as subgroups of the common Galois group in (92). The embedding of base residue fields in total residue fields gives \[\mathcal I_{Q'}\subset\mathcal I_{P'}.\] All residue characteristics are zero, so the residue extensions are separable and the orders of these inertia groups are \(e_Q\) and \(e\). Equation (97) 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 (95) 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 (85) 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 (91) 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 \(\Lambda(V')\): \[\mathcal B_\Lambda =\widetilde K\otimes_K\Lambda(V').\] 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 50 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]. Denote this finite étale cover by \[\mathcal C_\Lambda\longrightarrow O_\Lambda.\] 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 \[\mathcal C_{1,\Lambda}\longrightarrow\mathcal T_\Lambda\] 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], since every finite cover is tame in characteristic zero; see also [33] and [22]. 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 \[ E_1/k\ \text{is finite Galois with group }G_1, \qquad k'\subset E_1, \tag{100}\] 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 (100) is the descended one. Define \[ F_b=E_1(J_0)^{G_1}. \tag{101}\] 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 \[ k=b(V')\subset F_b, \tag{102}\] 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 (98), 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 (86) 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 \[\operatorname{Frac}(\Lambda\otimes_b F_b) \simeq \left( \operatorname{Frac}(\Lambda\otimes_b E_1(J_0)) \right)^{G_1} \simeq \operatorname{Frac}(\Lambda\otimes_{L_0}K(J)).\]

Identification with the original fiber. Proposition 13 gives \(\mathbf C(S)=\mathbf C(Y)\) in the original field tower \[\mathbf C(Y)\subset\mathbf C(S)\subset\mathbf C(W)\subset\mathbf C(X).\] In particular, \(\mathbf C(T_0)\subset\mathbf C(Y)\), with its prescribed algebraic closure \(b\subset\Omega\). Before the finite parameter changes the generic parameter field is \[b(I)=b\,\mathbf C(Y)\subset\Omega.\] It is algebraic over \(\mathbf C(Y)\). 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 46, 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 \[\operatorname{Frac}(\Omega\otimes_{L_0}K(J)) \simeq \operatorname{Frac} (\Omega\otimes_{\mathbf C(Y)}\mathbf C(X)).\] 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 (98) says that its base extension to \(\Omega\) is birational to that whole fiber. Since \[\mathop{\mathrm{trdeg}}_{\mathbf C}b=\dim T_0,\] the definition of birational variation gives (99). ◻

Proof of Theorem 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 9, specifically (4), gives \[\bar\kappa(U)\geq d+\bar\kappa(V).\] Equations (7) and (13) identify the dimension of the absolute logarithmic Iitaka base and give \[\bar\kappa(U)=\dim Z=d+\dim T_0.\] Combining this identity with (99) gives \(\bar\kappa(U)\geq d+\mathop{\mathrm{Var}}(f)\). Taking the larger of the two lower bounds proves \[\boxed{\displaystyle \bar\kappa(U)\geq\kappa(F) +\max\{\bar\kappa(V),\mathop{\mathrm{Var}}(f)\}.}\]

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 \[e_y:T_uU\longrightarrow N_{S/Y,y}\] 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 (110); 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 [setup:log-modification]. Thus old coefficients do not change. New horizontal coefficients satisfy the same product calculation, giving SNC horizontal support and coefficients in \([0,1]\). Equation (66) gives Equation (109). ◻

Proof. Write \(C=B_{\mathrm{hor}}\), and let \(V=R_p+B_{\mathrm{vert}}-p^*D_I\geq0\) as in Equation (67). Prescribe the finite extension identifying the chosen generic model with \((F_c,\Delta_c)\), including its Stein field, in Lemma 43. 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 43, 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 \[a_E-\operatorname{ord}_E(t)\geq-1, \qquad \operatorname{ord}_E(t)=1, \qquad a_E\geq0.\] 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 43 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 (112), using Equation (67). 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 \[u^*(m_1L_\eta)=v^*A_c+E_{\mathrm{fix}}, \qquad E_{\mathrm{fix}}\geq0.\] Generation in degree one implies, for every \(a\geq1\), \[|a u^*(m_1L_\eta)|=v^*|aA_c|+aE_{\mathrm{fix}}.\] 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 \(p'(\mathop{\mathrm{Supp}}\rho^*V)\). 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 \(m\in m_0\mathbf N\). Injectivity is already visible on the chosen generic fiber. The last assertion follows from Proposition 44. ◻

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, \[\deg X_m\geq vm^d.\] The pullback linearization gives an action of the Galois group \(\Gamma\) on the complete system and on \(X_m\). It fixes the embedded field \(\mathbf C(V)\), since this field comes from \(H\). The tower \[\mathbf C(V)\subseteq\mathbf C(X_m)\subseteq\mathbf C(F'), \qquad[\mathbf C(F'): \mathbf C(V)]\leq e_0\] 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 \[\dim R_{rk}^{\Gamma} =\frac{r^d\deg(X_m)}{e\,d!}k^d+O(k^{d-1}).\] Normalize in degree \(mrk\) and let \(k\) tend to infinity. Cancelling \(r^d\) gives the lower bound \[\frac{\deg(X_m)}{em^d}\geq\frac{v}{e_0}.\] Thus even an arbitrarily large scalar kernel affects only the divisibility index, not the claimed lower bound. ◻

Proof. Apply Lemma 43, including the Stein field when needed. For a log smooth pair with connected fibers, twisted weak positivity says that \[\mathcal E_m=p'_*\mathcal O_{H'}(m(K_{H'/I'}+C'))\] is weakly positive in divisible degrees [19]. 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 \[\operatorname{Sym}^b\mathcal E_m\longrightarrow p'_*\mathcal O_{H'}(bm(K_{H'/I'}+C')).\] 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 43 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 \(Q'\sim_\mathbf Q2\eta A+E\), 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 43 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]; 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 \[K_{H'/I'}+C'+\rho^*P+\delta A'+\epsilon p'^*A_I\] 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 \[K_{\mathcal V_p}+C+P+ \frac{\delta}{e}\rho_*A'+\frac{\epsilon}{e}\rho_*p'^*A_I\] 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 \(N^1(H)_{\mathbf R}\), 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. The logarithmic Hodge description identifies the map on top pieces with the nonzero inclusion \[H^0(P_t,K_{P_t})\lhook\joinrel\longrightarrow H^0(P_t,K_{P_t}+D_t);\] see [15]. 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]. 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]. 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 \[j_\Gamma=\frac1{|\Gamma|}\sum_{\gamma\in\Gamma} \gamma_{\mathbb H}\,j\,\gamma_{\mathbb Q}^{-1}.\] 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. ◻