Proof. At a general point of a prime of \(W\), the variety is smooth and the Jacobian of the morphism to \(X\) is holomorphic. Hence \(a_E\geq0\). At a prime that is not exceptional over \(X\), the morphism is an isomorphism at its general point, so \(a_E=0\).

Every vertical prime of \(W\) dominates a prime of \(Y\). Indeed, an equidimensional map of relative dimension \(\dim W-\dim Y\) cannot map a divisor into a set of codimension at least two: its dimension would then be at most \(\dim W-2\). If \(a_Q>0\), the minimum in (2) is positive, so every prime over \(Q\) has \(a_E>0\) and is \(p\)-exceptional. This proves the asserted support of \(f^*A_0\). It also proves finite support, since there are only finitely many \(p\)-exceptional prime divisors. Rationality follows from the integer orders in (2).

If \(E\) dominates \(Q\) with \(a_Q>0\), its coefficient in \(\Theta\) satisfies \[-a_E+e_Ea_Q\leq 1-e_E\leq0.\] At a prime above a \(Q\) with \(a_Q=0\), and at any horizontal prime, the coefficient is \(-a_E\leq0\). Thus \(\Theta\leq0\).

Let \(D_W\) be the full reduced toroidal boundary. The toroidal pair \((W,D_W)\) is log canonical, \(K_W+D_W\) is rationally Cartier, and \(W\setminus D_W\) is smooth. These assertions can be checked on a regular subdivision of each toric chart: the logarithmic top form gives \(K_{\widetilde W}+D_{\widetilde W}=\tau^*(K_W+D_W)\), and the reduced boundary upstairs has simple normal crossings. We also have \[K_W+\Theta=p^*K_X+f^*A_0.\] Consequently \(B=D_W-\Theta\) is an effective rational Cartier divisor whose support contains \(D_W\). For any divisorial valuation \(v\) over \(W\), the log discrepancies satisfy \[ A_{W,\Theta}(v)=A_{W,D_W}(v)+v(B). \tag{5}\] The full toroidal boundary gives nonnegative log discrepancies; the effective rational Cartier difference adds a strictly positive term precisely where that discrepancy could vanish. Both terms on the right are nonnegative. If the first is zero, the center of \(v\) lies in \(D_W\), since the complement is smooth with zero boundary. Its center is then contained in the support of the effective rational Cartier divisor \(B\), so \(v(B)>0\). If the first term is positive there is nothing more to check. This proves \(A_{W,\Theta}(v)>0\) for every divisorial valuation, which is the sub-klt assertion. This argument does not require the support of \(K_{W/X}\) to lie in the toroidal boundary.

Finally the crepant pullback of \(\Theta\) to \(Z'\) is \[\Theta_{Z'}=(f\tau)^*A_0-K_{Z'/X}.\] Its coefficients are less than one by the sub-klt assertion, giving (4). ◻

Proof. A section divided by \(s_{mA_0}\) is a rational function \(q\) on \(Y\) with poles bounded by \(mA_0\). The rational function \(f^*q\) on \(W\) has only \(p\)-exceptional poles. Viewed in \(\mathbb C(W)=\mathbb C(X)\), it consequently has no pole along any prime of the smooth variety \(X\). Normality extends it regularly across the remaining subset of codimension at least two. It is constant because \(X\) is connected and projective. Dominance of \(f\) makes \(q\) constant as well. The canonical section exists since \(A_0\geq0\), giving the asserted one-dimensional space and Iitaka dimension. ◻

Proof. We first normalize the section, obtaining both the regular map and its bounded base metric. We then correct the pushforward measure along base primes whose poles disappear on return to \(V\) or remain within \(D\).

For a prime \(T\) of \(Y\) set \[d_T=\min_{Q\mapsto T} \frac{\operatorname{ord}_Q(\bar p^*\rho)} {\operatorname{ord}_Q(\bar f^*T)}, \qquad B_0=\sum_Td_TT.\] The orders in this expression are taken on \(\bar V\). Only finitely many \(d_T\) are nonzero. Equidimensionality and the absence of horizontal poles give \[\operatorname{div}_{\bar V}(\rho)-\bar f^*B_0\geq0.\] This is an effective \(\mathbb Q\)-Cartier divisor. Choose \(b>0\) clearing the coefficients of \(B_0\) and set \(B=bB_0\). Normality extends the normalized section in codimension one to a regular morphism \(\bar f^*B\to bk\bar p^*L\), and pullback gives the displayed map. Over each prime \(T\) a component realizing the minimum is absent from its vanishing divisor.

For a local meromorphic equation \(l\) of \(B\), define the weight \[-b\log S_\rho+\log|l|^2.\] It is psh on the initial open set. Its negative exponential is the fibre supremum of the squared norm of the regular normalized section \((p^*\rho)^b/f^*l\). Properness and bounded weights of \(h_L\) bound this norm above locally over every base point, giving a lower bound for the weight even near the omitted locus. At a general point of each omitted base prime \(T\) choose a minimizing component. The normalized section is nonzero there. At general points of this component and \(T\), the transverse map has coordinates \(y_1=z_1^e\), \(y_i=z_i\) for \(2\leq i\leq\dim Y\). Thus every nearby fibre meets a patch where the normalized section norm is bounded below. The fibre measure has positive density almost everywhere on such a patch, so the essential supremum has a fixed positive lower bound. This gives an upper bound for the weight near a general point of that prime. Psh removable singularities extend the weight there and then across the remaining analytic set of codimension at least two. The lower bound persists, while the extended psh weight is locally bounded above. These weights glue and give the bounded metric on \(B\).

The bounded metric is now constructed. For the finite-volume metric, call a base prime \(T\) invisible if every component of \(\bar f^*T\) is exceptional over \(V\) or maps into \(D\). There are finitely many such primes, since they are images of the finitely many exceptional or boundary primes. Choose a component above \(T\) on \(\bar V\) and denote its strict transform on \(W\) by \(Q\). At its general point write \[e=\operatorname{ord}_Q(f^*T),\qquad a=\operatorname{ord}_Q(J+p^*D).\] The local psh coefficient weight of \(h_A\) is bounded above, so its negative exponential is bounded below by a positive constant. The Jacobian and boundary section therefore bound the pulled-back measure below by a positive constant times \(|z_1|^{2a}\) times coordinate volume. Integrating over a fixed disc in the free variables and using \(y_1=z_1^e\) gives \[ g_0(y)\geq C|y_1|^{2(a-e+1)/e} \tag{12}\] almost everywhere near the general base point. If \(T\) is visible, choose a component with \(a=0\); then \(-\log g_0\) is bounded above there. For the invisible primes choose sufficiently large integral coefficients of \(D_Y\). Equation (12) makes \[-\log g_0+\log|s_{D_Y}|^2\] locally bounded above at their general points. It extends psh in codimension one and then across codimension two, defining a metric on \(-K_Y+D_Y\). Pairing it with \(s_{D_Y}\) cancels the added factor. Its adjoint measure is \(g_0d\lambda_y=f_*p^*\mu\) almost everywhere and is therefore finite.

A section of \(aB+cD_Y\) is a rational section of \(aB\) with poles only on \(D_Y\). Pull it back and compose with the \(a\)-th power of the normalized map. On \(\bar V\) all possible new poles lie above invisible primes. They are thus exceptional over \(V\) or map into \(D\). The normal intermediate model excludes nonexceptional divisors over base subsets of codimension at least two. Birational pushdown therefore leaves poles only on \(D\), and a sufficiently large multiple of \(D\) clears them. The rational section remains nonzero. The normalization uses only base factors, so it preserves invariance when present. ◻

Proof. On a base open set where \(\rho\) is regular, define \[g_m(y)\,d\lambda_y =f_*\bigl(|p^*\rho|_{h_L^k}^{2m}p^*\mu\bigr).\] These are finite positive densities almost everywhere, since the section norms are bounded on compact preimages and \(p^*\mu\) has finite mass. Give \(p^*(A+mkL)\) the coefficient metric \(h_m=p^*(h_Ah_L^{mk})\), and consider the integrable direct image \[\mathcal F_m=f_*\bigl(\mathcal O_W(K_{W/Y}+p^*(A+mkL)) \otimes\mathcal I(h_m)\bigr).\] For a local canonical frame \(\eta_Y\) on the base, the section \[u_m=\frac{s_{J+p^*D}(p^*\rho)^m}{f^*\eta_Y}\] is locally integrable on the total space and therefore belongs to \(\mathcal F_m\). It is nonzero and generates the ordinary direct image generically by the rank-one hypothesis. Consequently \(\mathcal F_m\) also has rank one. On a dense open set where \(u_m\) is its frame, the squared direct-image norm is \(g_m\).

Singular direct-image positivity gives \(-\log g_m\) a psh representative there. We use the formulation with the actual multiplier ideal in (Hacon et al. 2018, Theorem 21.1), based on (Păun and Takayama 2018, Theorem 3.3.5). Equivalently, in the relative Bergman description, the weight is the logarithm of the generator’s squared holomorphic coefficient minus \(\log g_m\) pulled back from the base (Berndtsson and Păun 2008, Theorem 0.1). In the invariant case, use smooth direct-image positivity (Berndtsson 2009, Theorem 1.2). Averaging over the compact torus gives a holomorphic projection onto the invariant line. Distinct weight spaces are orthogonal, so its subspace metric equals the quotient metric and is semipositive.

There is a fixed dense open set where \(f\) is a submersion and each fibre has a patch avoiding the zeros of \(s_{J+p^*D}\) and \(\rho\). Integrating over such a patch gives \(g_m\geq Cc^m\) locally, with \(C,c>0\). Hence the psh representatives of \(-m^{-1}\log g_m\) have uniform local upper bounds, independent of possible extra base-change exceptions for individual \(m\); psh extension fills those exceptions on the fixed open set. Finite fibre measure and its positive density almost everywhere imply \[-\frac1m\log g_m(y)\longrightarrow -\log\operatorname*{ess\,sup}_{W_y}|p^*\rho|_{h_L^k}^2\] almost everywhere. The limit is finite there by the bounded metric and the regularity of \(\rho\). Local compactness for psh functions now gives the required representative. Indeed, a subsequence tending locally uniformly to \(-\infty\) would contradict this finite limit. Every \(L^1_{\mathrm{loc}}\)-convergent subsequence has a further subsequence converging almost everywhere, so its psh limit agrees almost everywhere with the displayed power-mean limit. The direct-image argument at \(m=0\) supplies \(-\log g_0\). ◻

Proof. Replacing \(L\) at the outset by a prescribed positive multiple gives the divisibility assertion; it is therefore enough to prove the statement without that prescription. Hodge symmetry gives \(\chi(\mathcal O_V)=1\), so Serre duality gives \(\chi(K_V)=(-1)^n\). Riemann–Roch makes \(\chi(K_V+mL)\) a polynomial in \(m\) with nonzero constant term. It is nonzero for all sufficiently large integers. A fixed cohomological degree \(q\) therefore has \(H^q(V,K_V+mL)\ne0\) for unbounded \(m>0\). The metric on \(mL\) has trivial multiplier ideal. Hard Lefschetz with pseudoeffective coefficients (Demailly et al. 2001, Theorem 0.1) gives nonzero sections of \(\Omega_V^{n-q}\otimes mL\) at these exponents.

Take the saturated subsheaf \(\mathcal E\subset\Omega_V^{n-q}\) defined by the span of their untwisted generic images. Its determinant \(M=(\bigwedge^{\operatorname{rank}\mathcal E}\mathcal E)^{**}\) is a line bundle and maps to a cotangent tensor power. This map is first defined outside codimension two and extends because the ambient tensor bundle is locally free. Every nonzero vector from the family can be completed to a generic basis using vectors of the same family. Wedging that basis gives a nonzero section of \(M\) twisted by the sum of its positive exponents. These sums are unbounded because one basis vector can have an arbitrarily large exponent. Saturation ensures that the maps factor through \(\mathcal E\), and extension across codimension two gives global determinant sections. Their zero divisors are the required \(N_i\), after taking a strictly increasing subsequence. This is the determinant method of (Lazić and Peternell 2018, Lemma 4.1), used for anticanonical nonvanishing in (Lazić et al. 2023, Lemma 5.1). ◻

Proof. The divisor \[ K_{Y_0}+\Sigma-\pi_0^*K_X-\sum_i c_i f_0^*P_i \tag{20}\] is effective and \(\mathbb Q\)-Cartier. Indeed, no prime contributes to the pullbacks of two distinct \(P_i\), by equidimensionality. At a component above \(P_i\), its coefficient is \(1+e_D-c_i b_{iD}\geq0\). At the other boundary components it is \(1+e_D\), and away from the boundary it is \(e_D\). The divisor \(K_{Y_0}+\Sigma\) is Cartier and pulls back log crepantly under toroidal subdivisions: the local top logarithmic form of a toric model trivializes it and pulls back to the top logarithmic form on the subdivision. Consequently, on a smooth toroidal resolution, if \(x_j=0\) are the local source boundary branches, then \[ a_j:=1+e_j>0,\qquad a_j\geq\sum_i c_i b_{ij}, \tag{21}\] where \(e_j\) is the Jacobian order over \(X\) and \(b_{ij}\) is the order of \(y_i\) along \(x_j=0\). These inequalities include every new ray on every toroidal subdivision, in particular the exceptional rays over intersections of boundary components. They follow from pullback of an effective log-Cartier divisor, so this is more than an estimate at the original boundary primes.

For use in analytic integration we make explicit why formal toroidal charts can be replaced by analytic monomial charts. In a formal chart write \[ f^*y_i=\prod_jx_j^{A_{ij}},\qquad x_j=X_jU_j,\quad y_i=Y_iV_i. \tag{22}\] In (22), the indices include all source and target toric coordinates, including unit coordinates; thus \(A\) is a \(d\times\dim X\) integer matrix. Here \(X_j\) and \(Y_i\) are analytic boundary equations, or the constant \(1\) for unit coordinates; \(U_j,V_i\) are formal units. The coordinates minus their residues are regular parameters. Dominance implies that \(A\) has full row rank, since the logarithmic monomial map has generic rank equal to the target dimension. Factoring the analytic boundary equations gives analytic units \(R_i\) with \[f^*Y_i=\Bigl(\prod_jX_j^{A_{ij}}\Bigr)R_i.\] Choose analytic units \(V_i'\) having the same first jets as \(V_i\). We seek analytic units \(U_j'\) satisfying \[ \prod_j(U_j')^{A_{ij}}=R_i f^*V_i'. \tag{23}\] Use the constants of the formal solution, divide by them, and take logarithms near \(1\). The resulting equations are linear equations with matrix \(A\) and holomorphic right hand sides. A linear right inverse gives a holomorphic solution. Its first jet differs from the prescribed formal first jet by a vector in \(\ker A\); a kernel-valued affine correction removes that difference without changing the equations. Exponentiating gives the desired \(U_j'\). Thus \(X_jU_j'\) and \(Y_iV_i'\) have the original first jets and remain coordinate systems, while the monomial identities hold exactly analytically. Negative exponents can occur only on unit coordinates and cause no difficulty.

We can now estimate by coarea in logarithmic coordinates. Cut arguments into bounded sectors and work on relatively compact pieces of these charts, away from the boundary itself. Relative to Lebesgue measure in the logarithmic variables, the pullback of \(d\mu_X\) is bounded above by a constant times \[\prod_j|x_j|^{2a_j}\] over the vanishing coordinates; all unit moduli are bounded above and away from zero. The factor \(|x_j|^2\) in this formula is the Jacobian of the logarithmic change of variables. The analogous change on the base introduces \(\prod_i|y_i|^2\), up to bounded positive factors from its unit coordinates. The logarithmic map is the fixed full-rank complex linear map \(A\), up to boundedly many argument translates. Its pushforward density is therefore an integral over affine real planes with fixed coarea factors.

In the following products, \(i\) ranges only over vanishing target coordinates and \(j\) only over vanishing source coordinates; \(b_{ij}\) is the corresponding boundary-order block of \(A\). To estimate \(\rho\prod_i|y_i|^{2(1-c_i+\eta)}\), divide the logarithmic pushforward density by \(\prod_i|y_i|^{2(c_i-\eta)}\). On an affine fiber, this factor is constant and can be brought into the integrand. The resulting exponent on a vanishing source coordinate is \[ 2\Bigl(a_j-\sum_i(c_i-\eta)b_{ij}\Bigr)>0. \tag{24}\] If some \(b_{ij}>0\), strict positivity follows from (21) and \(\eta>0\); otherwise it follows from \(a_j>0\). The integrand thus decays exponentially as the real logarithms of vanishing coordinates tend to \(-\infty\). Their upper bounds, and the bounds on all remaining logarithmic coordinates, allow us to dominate it by \(C\exp(-\varepsilon\|u\|)\) on the whole chart piece in real logarithmic variables. Integrals of this bound over affine planes of a fixed direction are uniformly bounded independently of the translation. For example, write such a plane as \(v+\ker A\) with \(v\) orthogonal to \(\ker A\) and use \(\|v+w\|\geq\|w\|\). Properness supplies finitely many chart pieces over a sufficiently small base neighborhood. The coordinate equations of each boundary prime differ only by bounded units on overlaps. Summing the bounds proves (18) almost everywhere, and hence on the general locus where the density is continuous.

For (19), take a general point of a component which realizes the minimum in (17). At that point take coordinates \[ y_1=x_1^{b_{iD}},\qquad y_j=x_j\quad(2\leq j\leq d). \tag{25}\] The pulled-back volume is comparable there to \(|x_1|^{2e_D}\) times ordinary coordinate volume. The change \(y_1=x_1^{b_{iD}}\), followed by integration over a fixed patch in the remaining fiber variables, contributes \[|y_1|^{2((1+e_D)/b_{iD}-1)} =|y_1|^{-2(1-c_i)}.\] This is a lower bound for the full pushforward density.

At a general point of an excluded prime outside the base boundary, the density also has a positive constant lower bound. In fact there is a nonexceptional component over that prime. An exceptional component would lie in the source boundary, whereas a boundary component meeting the inverse image of the base toroidal open cannot be vertical there: all target coordinates are units, so the full-rank monomial matrix uses only source unit coordinates and the map restricted to the component is dominant. At a nonexceptional component the Jacobian order is zero. If \(b\) is its pullback multiplicity over the base prime, the same power-coordinate calculation gives a lower bound, since its exponent is \(2(1/b-1)\leq0\). ◻

Proof. We first match the data with the smooth invariant version of Lemmas 5 and 4. Their original variety is \(X\), their allowed-pole divisor is zero, and both input bundles are \(A=L=-K_X\) with the compact-invariant smooth metric \(h\). The finite measure is \(d\mu_X\). Take exponent \(r\) and the rational section \(\widetilde s\) of \(rL\) whose pullback has divisor \(rH+f_0^*(rV)\). It is the power of the original generic section used to construct \(\sigma\). It is invariant and has no horizontal poles. On an equivariant resolution the invariant adjoint ranks equal one for every multiple of \(r\), including zero by the argument above. All hypotheses of Lemma 5 are therefore satisfied.

The zeroth density is \(\rho\) from (16); hence \(-\log\rho\) is psh on the smooth base-change locus. The normalized base divisor in Lemma 4 is \(rV\), by the full-component minimum in (14). Its bounded metric, divided by \(r\), gives the asserted metric on \(V\). More concretely, for a local frame \(v\) of \(rV\), it is determined by \[|v|^2=\left(\max_{f_0^{-1}(z)} |\sigma f_0^*v|_{(\pi_0^*h)^r}^{2/r}\right)^r.\] The essential supremum in the transfer lemma is this maximum: on a smooth connected general fiber the smooth measure has full support. The invariance step is the orthogonal quotient in the smooth Berndtsson theorem, as proved in Lemma 5; no singular invariant-subbundle positivity is invoked.

We now use Lemma 8 to construct the signed log-anticanonical metric. Put \(M=-K_Z-B\). On the general open its local metric weight is \[ \varphi_M=-\log\rho-\sum_i(1-c_i)\log|y_i|^2. \tag{27}\] This has the transition law of \(M\): a frame of \(-K_Z\) is multiplied by the boundary factor \(\prod_i y_i^{1-c_i}\), after clearing denominators. It is plurisubharmonic away from the boundary. The lower density bounds give the upper bound needed for extension across a dense open of every excluded divisor. Hartogs extension treats the remaining codimension-two set. The upper density estimate implies \[ \varphi_M\geq\eta\sum_i\log|y_i|^2-O_\eta(1) \qquad\text{for every }\eta>0. \tag{28}\] In particular this metric has zero Lelong numbers: the lower estimate holds with arbitrarily small logarithmic coefficients. For every fixed divisible \(k>0\), choose \(\eta<1/k\). Normal-crossing integrability shows that \(\mathcal I(k\varphi_M)=\mathcal O_Z\). This assertion is about every such power, not just the first multiplier ideal.

Choose a very ample divisor \(A\) on \(Z\). Nadel vanishing (Nadel 1990), applied after adding the smooth positive metric on \((d+1-i)A\), gives \[H^i\bigl(Z,K_Z+kM+(d+1-i)A\bigr)=0 \qquad(1\leq i\leq d).\] Castelnuovo–Mumford regularity makes \(K_Z+kM+(d+1)A\) globally generated. Its class divided by \(k\) is nef, and the limit as \(k\) tends to infinity proves that \(M\) is nef. The identical argument for the bounded metric on \(V\), whose every power has trivial multiplier ideal, also shows that \(V\) is nef. Equation (26) follows from the definition of \(M\). ◻

Proof. For \(s=0\), the inclusion into \(\mathcal O_Z\) gives the assertion directly. Let \(s>0\) and let \(t>0\) be rational. Since \(M\) is nef, \(M+tA\) is ample. Choose a sufficiently divisible very ample multiple and a general member; divide that member by the multiple and add it to \(B^+\) to obtain a rational simple normal crossings boundary \(\Delta_t\), all of whose coefficients lie in \([0,1)\). Then \[K_Z+\Delta_t\sim_{\mathbb Q}B^-+tA\] is pseudoeffective.

Apply the orbifold cotangent theorem (Campana and Păun 2019, Theorem 1.3 and Section 5) to this smooth projective log-canonical pair. On a finite adapted cover \(\tau:\widetilde Z\to Z\), the adapted orbifold cotangent bundle \(\mathcal E_t\) has determinant class \(\tau^*(K_Z+\Delta_t)\). Every torsion-free quotient of each \(\mathcal E_t^{\otimes s}\) has nonnegative degree against \(\tau^*\beta\). This holds for classes on the boundary of the movable cone as well.

There is an inclusion \(\tau^*\Omega_Z^1\subseteq\mathcal E_t\). In codimension one, at a branch \(z=w^b\) corresponding to boundary coefficient \(\gamma\), the orbifold generator is \(w^{-b\gamma}dz\); hence the ordinary generator \(dz\) belongs to the adapted bundle. Additional branch divisors have \(\gamma=0\). Let \(\mathcal N\) be the saturation of the pulled-back line \(\tau^*N\) in \(\mathcal E_t^{\otimes s}\). Saturation adds an effective divisorial class, so \[\deg(\tau)\,c_1(N)\cdot\beta \leq c_1(\mathcal N)\cdot\tau^*\beta.\] The quotient by \(\mathcal N\) is torsion-free. Its determinant has nonnegative degree, and the tensor determinant identity for a bundle of rank \(d\) is \[c_1(\mathcal E_t^{\otimes s}) =s d^{s-1}c_1(\mathcal E_t).\] If the quotient has rank zero, it is zero and the same inequality holds with determinant degree zero. Dividing by the cover degree and using (29) gives \[ c_1(N)\cdot\beta \leq s d^{s-1}(K_Z+\Delta_t)\cdot\beta =s d^{s-1}tA\cdot\beta. \tag{30}\] Letting \(t\) decrease to zero proves the assertion. ◻

Proof. Choose \(b>0\) with \(bV\) Cartier. Lemma 9 gives \(bV\) bounded semipositive weights, and the base has no positive-degree forms by the argument preceding (13). Apply Lemma 6 to \(bV\), prescribing any further divisibility required for the construction. It gives a cotangent line \(N\) and effective divisors \(D_j\sim N+n_j(bV)\) with \(n_j\to\infty\). Set \(m_j=bn_j\). Lemma 10 gives \(N\cdot\beta\leq0\), while \(V\cdot\beta=0\). Effectivity gives the opposite inequality for \(D_j\cdot\beta\), so every degree is zero. ◻

Proof. Multiply the sections of that system by those defining the chosen maximal system. Their products form a system of degree zero, and its map is the product of the two maps followed by the Segre embedding. By maximality it has the same image dimension as the chosen map. The added section ratios are therefore algebraic over the field of that image. The Stein factorization identifies \(\mathbb C(W)\) with the relative algebraic closure of the image field in \(\mathbb C(Z)\). This proves the assertion. ◻

Proof. Use the sequence in (31). For \(j<k<l\), linear equivalence gives \[ (m_l-m_j)D_k \sim (m_l-m_k)D_j+(m_k-m_j)D_l. \tag{32}\] Both sides are effective divisors in the same zero-degree system. By Lemma 12, their ratio is a rational function from \(W\). Its pullback has no horizontal divisor over \(W\). Thus (32), after pullback by \(h\), is an equality of actual horizontal divisors.

Fix the first two indices. For every horizontal prime \(Q\) on \(\widehat Z\), the coefficients of \(h^*D_j\) have the form \(a_Q+m_jb_Q\). The preceding identity proves this simultaneously for every index. Since all \(D_j\) are effective and \(m_j\to\infty\), every \(b_Q\) is nonnegative. These slopes have finite support: they are the coefficients of the horizontal part of \[\frac{h^*(D_2-D_1)}{m_2-m_1} \sim_{\mathbb Q}h^*V.\] This representative of \(h^*V\) has effective horizontal part. Its vertical components disappear on the generic fiber, yielding the stated linear effectivity there. The argument uses actual divisors and rational functions, rather than only numerical equivalence. ◻

Proof. Choose a very ample \(H\) and an ample \(A\) such that \(A-K_Y-iH\) is ample for \(1\leq i\leq d\). Nadel vanishing and Castelnuovo–Mumford regularity give global generation of \[\mathcal O_Y(mJ+A)\otimes\mathcal I(m\psi).\] Indeed its twist by \(-iH\) is an adjoint sheaf with coefficient metric of strictly positive curvature, supplied by \(A-K_Y-iH\), and multiplier ideal \(\mathcal I(m\psi)\). The ideal is coherent and nonzero; local point extension where \(\psi\) is finite gives a nonzero integrable germ (Nadel 1990; Ohsawa and Takegoshi 1987; Demailly 2011).

Let \(u:Y'\to Y\) resolve this ideal and \(P\), and write \(\mathcal I(m\psi)\mathcal O_{Y'}=\mathcal O_{Y'}(-F_m)\). For a prime \(E\) on \(Y'\) put \[b_E=\operatorname{ord}_E F_m,\qquad p_E=\operatorname{ord}_E u^*P,\qquad a_E=\operatorname{ord}_E K_{Y'/Y}.\] We claim that \[ b_E/m-p_E<1+a_E. \tag{35}\] The local point-extension estimate for \(m\psi\) supplies holomorphic functions \(v_x\) with \(|v_x(x)|^2=e^{m\psi(x)}\) and uniformly bounded \(L^2(e^{-m\psi})\) norm on fixed nested coordinate balls. The local upper bound for \(\psi\) bounds their ordinary norms, hence their suprema on a smaller ball. Each \(v_x\) belongs to \(\mathcal I(m\psi)\). At a general point of \(E=(z_1=0)\), Cauchy estimates after division by \(z_1^{b_E}\) give \[e^{m\psi\circ u}\leq C_m|z_1|^{2b_E}.\] The change-of-variables formula bounds the pulled-back integrable density below by a positive constant times \(|z_1|^{2(p_E+a_E-b_E/m)}\). Its transverse integral is finite only if (35) holds.

Global generation identifies the fixed divisor of the pulled-back system with \(F_m\). Choose a general member of the free residual system and divide the corresponding divisor downstairs by \(m\). The crepant boundary of \(\Delta_m-P\) has coefficient \(b_E/m-p_E-a_E<1\) on each fixed component, and coefficient \(1/m<1\) on new moving components. Bertini, applied also to the strata of the resolved divisor, makes their union simple normal crossings. Decreasing coefficients from the positive part preserves positive log discrepancies, proving sub-klt on every higher model.

There is a second way to choose the member. The global Bergman lower estimate for the same sufficiently ample \(A\) gives local expressions \(s_{i,m}\) of an orthonormal basis with \[\sum_i|s_{i,m}|^2\geq C_m e^{m\psi}.\] This globalization by a fixed ample twist is the point-extension construction in (Demailly 1993, sec. 9, proof of Proposition 9.1). One obtains the estimate by extending from a point, cutting off, and solving \(\bar\partial\) globally with a logarithmic pole of order \(d\) at that point. The fixed positivity of \(A-K_Y\) pays the uniformly bounded negative curvature of the cutoff pole; integrability forces the correction to vanish at the point. Singular weights are handled by decreasing regularization and weak limits in the \(L^2\) estimate (Demailly 2011, Corollary 13.9, Remark 13.10, and Theorem 14.2). For independent standard complex Gaussians \(c_i\) and \(m>1\), \[\mathbb E\left|\sum_i c_i s_{i,m}\right|^{-2/m} =C'_m\left(\sum_i|s_{i,m}|^2\right)^{-1/m},\] where \(C'_m<\infty\) because \(\mathbb E|c_1|^{-2/m}<\infty\). Multiplication by \(|s_P|^2\) and Fubini show that almost every such member has the weighted integrability required for (35). ◻

Proof. The assertion on a point is immediate. For positive dimension, fix a nonzero movable class \(\alpha\) and put \(t=\alpha\cdot P\geq0\). The total degree of \(T_Y\) is at least \(-t\). If its minimum Harder–Narasimhan slope were less than \(-t\), its positive-slope part \(\mathcal F\) would be nonzero and proper: the last negative graded piece already has degree less than \(-t\), whereas the whole sheaf does not. The quotient by the positive part has only nonpositive slopes and degree less than \(-t\). In particular \[\mu_{\alpha,\min}(\mathcal F)>0,\qquad 2\mu_{\alpha,\min}(\mathcal F)> \mu_{\alpha,\max}(T_Y/\mathcal F).\] The bracket map vanishes by the slope inequality, and the Campana–Păun algebraicity theorem makes \(\mathcal F\) algebraically integrable (Ou 2023, Proposition 2.2). But \(c_1(T_Y/\mathcal F)=K_{\mathcal F}-K_Y\) has degree less than \(-t\), contradicting the hypothesis. Thus \(\mu_{\alpha,\min}(T_Y)\geq-t\). Duality and the tensor-product rule for movable slopes give \(c_1(M)\cdot\alpha\leq ht\) (Greb et al. 2016; Campana and Păun 2019). Movable-cone duality proves the assertion (Boucksom et al. 2013). For \(h=0\), the inclusion in \(\mathcal O_Y\) proves it directly. ◻

Proof. The assertion on a point is immediate. Suppose \(\dim Y>0\). Lemma 14 gives effective \(\Delta_m\sim_{\mathbb Q}-K_Y+P+A/m\) with \((Y,\Delta_m-P)\) sub-klt. Resolve a first-integral map for an algebraic foliation \(\mathcal F\) together with that pair, by \(u:Y'\to Y\). Write its crepant boundary as \(B_m^+-B_m^-\), with disjoint effective parts. Then \[K_{Y'}+B_m^+\sim_{\mathbb Q}u^*A/m+B_m^-, \qquad 0\leq u_*B_m^-\leq P.\] The positive boundary is klt. A general fibre of the resolved first-integral map meets the isomorphism locus of \(u\); hence \(u^*A\) restricts to a big divisor there. A divisible positive multiple of \(K_{Y'}+B_m^+\) consequently has a nonzero section on that fibre. Druel’s relative pseudoeffectivity theorem, in the form recalled by Ou, applies to the effective boundary and its log canonical general fibre (Druel 2017, Proposition 4.1), (Ou 2023, Proposition 3.1). It yields pseudoeffectivity of \(K_{\mathcal F'}+B_m^+\). Push down, add \(P-u_*B_m^-\geq0\), and let \(m\) tend to infinity. This gives \(K_{\mathcal F}-K_Y+P\) pseudoeffective. The metric already gives \(-K_Y+P\) pseudoeffective, so Lemma 15 applies. This is the signed finite-volume adaptation of Ou’s tangent-slope argument; its relative positivity input is due to Druel. ◻

Proof. Use Lemma 14 with \(P=0\) to obtain klt \(\Delta_m\sim_{\mathbb Q}J+A/m\). On a log resolution \(u:Y'\to Y\) truncate the negative coefficients of the crepant boundary. This gives an effective simple normal crossing boundary \(\Delta'_m\), with coefficients less than one, and \[K_{Y'}+\Delta'_m=u^*(K_Y+\Delta_m)+E_m,\qquad E_m\geq0\quad\hbox{$u$-exceptional}.\] The left side is pseudoeffective. The numerical pullback \(\alpha'\), defined by \(\alpha'\cdot D=\alpha\cdot u_*D\), is movable and annihilates exceptional divisors.

On an adapted finite cover \(v:\widehat Y\to Y'\), the orbifold cotangent sheaf \(\mathcal E\) contains \(v^*\Omega^1_{Y'}\) and satisfies \(c_1(\mathcal E)=v^*(K_{Y'}+\Delta'_m)\). In a boundary chart its generators are pullbacks of \(dx/x^\delta\), where \(\delta\) is the boundary coefficient and the cover clears its denominator. Campana–Păun tensor-quotient semipositivity gives nonnegative \(v^*\alpha'\)-degree to torsion-free quotients of \(\mathcal E^{\otimes h}\) (Campana and Păun 2019, Theorem 1.3). The line \(v^*u^*M\) injects through ordinary cotangent pullback. Saturating increases its determinant by an effective divisor. The quotient has nonnegative degree; if its rank is zero, the same inequality follows directly from the effective saturation divisor. Since \(c_1(\mathcal E^{\otimes h})=h d^{h-1}c_1(\mathcal E)\), cancellation of the finite covering degree gives \[c_1(M)\cdot\alpha\leq h d^{h-1}(K_Y+J+A/m)\cdot\alpha.\] Let \(m\) increase to obtain (36). ◻

Proof. In a local frame of \(N\), write \(\sigma\) for the regular normalized section. On the smooth open set, the moments have limit \[-m^{-1}\log\int_{Z_y}|\sigma|^{2m} \frac{\pi^*d\mu}{g^*d\lambda_Y} \longrightarrow-\log\max_{Z_y}|\sigma|^2.\] The fibre measure is finite and has dense positive locus. After normalization to total mass one, its power means increase to the maximum. Proper smoothness makes that maximum continuous. Integration over a fixed patch on which both factors are nonzero gives local upper bounds for the rescaled psh weights. The limit is therefore psh. No positivity of the zeroth moment is needed.

Regularity of \(\sigma\) on the proper model gives a local lower bound for the limiting weight up to the boundary. Above each base prime a component attaining the vertical minimum has \(\sigma\ne0\) at its general point. In transverse coordinates the map is \(y_1=x_1^{e_E}\), with the other base coordinates free. A neighbourhood there meets every nearby fibre and bounds its maximum away from zero. This gives the upper bound needed for psh extension at general points of every omitted divisor. Psh extension across the remaining codimension-two set then applies; local upper boundedness there follows from transverse discs with boundary off the removed set. The lower bound persists. The extended weights are locally bounded and obey the frame transformation rule for \(N\). ◻

Proof. If \(Y\) is a point, all the base divisors vanish and their metrics are positive constants. The section \(s\) is then regular on \(Z\) and pushes down to the required section on \(X\). Assume henceforth that \(\dim Y>0\).

For every \(m\geq0\), (37) applied to \(D=K_{Z/X}\) gives invariant rank one for \(g_*(K_{Z/Y}+(ma+1)\pi^*L)\). Its generator is the product of \(s^m\), the invariant Jacobian section, and a local inverse canonical frame of \(Y\). In a local frame of \(N\), let \(\sigma\) be the normalized section from (38). The generator has squared norm \[I_m(y)=\int_{Z_y}|\sigma|^{2m}\, \frac{\pi^*d\mu}{g^*d\lambda_Y},\] where \(d\mu\) is the smooth volume defined by the metric of \(L\). Lemma 5 applies: Berndtsson positivity is used on the smooth base-change locus and the invariant line is a holomorphic orthogonal summand. Lemma 18 then gives the bounded metric of \(N\).

For the second metric use \(\rho=I_0\). Its negative logarithm is psh on the smooth locus, and \(\rho\,d\lambda_Y=g_*(\pi^*d\mu)\) has finite mass. At a general point of a prime \(E\mapsto Q\), choose transverse coordinates with \(y_1=x_1^{e_E}\) and the other base coordinates free. If \(a_E=\operatorname{ord}_E K_{Z/X}\), integration on a small patch gives \[\rho(y)\geq C|y_1|^{2((1+a_E)/e_E-1)}.\] Outside \(B\) choose a nonexceptional component, so \(a_E=0\). On the finitely many components of \(B\) choose \(c\) large enough that \[\psi=-\log\rho+c\log|s_B|^2\] is locally bounded above at their general points. The same bound is already available at every other omitted divisor. Psh extension across these divisors, then across the remaining set of codimension at least two, gives the metric on \(-K_Y+cB\). Its weighted integral is exactly the original finite pushforward mass.

Finally, represent a section of a positive divisible multiple of \(j(N+R)+lB\) by a rational function \(v\) on \(Y\). After incorporating that multiple in \(j,l\), its divisor satisfies \[\operatorname{div}(v)+j(N+R)+lB\geq0.\] Return the rational section \(s^jf^*v\). At a nonexceptional component \(E\) above \(Q\not\subset B\), its order is at least \(j(h_E-e_En'_Q)\geq0\), where \(h_E=\operatorname{ord}_Es\). There are no nonexceptional components above \(B\), and equidimensionality excludes vertical primes over higher-codimension sets. Horizontal orders are nonnegative by the generic regularity of \(s\). Thus every pole is exceptional over \(X\) and disappears on pushdown. All factors are invariant. ◻

Proof. For a point base, \(N=\Theta=0\) and both metrics are positive constants. We may therefore assume \(\dim Y>0\).

Lemma 5 supplies the psh moment weights, and Lemma 18 gives the metric of \(N\). The zeroth moment is the pushed-forward smooth volume density \(\rho\). It can be computed on a toroidal resolution of \(W\), even when that model is not equivariant: change of variables identifies its density with the one on an equivariant resolution. Put \(\theta_Q=\max\{c_Q-1,0\}\) and use \[\psi=-\log\rho+ \sum_Q\theta_Q\log|s_Q|^2.\] At a minimizing component the transverse calculation in the preceding proof gives \(\rho\geq C|s_Q|^{2(c_Q-1)}\). This is the local upper bound needed to extend \(\psi\) at general divisor points. Off the boundary the morphism is smooth and gives a positive lower bound. Extend over the remaining codimension-two set by psh removability.

The correction \(\Theta\) is the divisor \(A_0\) in Lemma 2. To compare the two definitions, it remains only to check primes outside the base boundary. A vertical prime above such a prime cannot belong to the source boundary of a toroidal morphism. Since that boundary contains the exceptional locus, the prime is nonexceptional over \(X\); the morphism is smooth at its general point. Its Jacobian order is zero and its pullback multiplicity is one, so the corresponding coefficient of \(A_0\) is zero.

The rank-free discrepancy lemma now gives both the exceptional support of \(f^*\Theta\) and, on every smooth model \(Z'\to W\) with maps \(\pi':Z'\to X\) and \(g':Z'\to Y\), \[\operatorname{coeff}_F(K_{Z'/X}-(g')^*\Theta)>-1 \qquad\text{for every prime }F\subset Z'.\] On a simultaneous normal-crossing resolution, these are precisely the exponents of the corrected source measure in the projection formula \[e^{-\psi}\,d\lambda_Y =g'_*\left(\prod_Q|s_Q\circ g'|^{-2\theta_Q} (\pi')^*d\mu\right).\] They make that measure locally integrable. Proper pushforward proves the required unweighted integrability on \(Y\), including at boundary intersections. Curvature here came from the invariant zeroth moment; the discrepancy argument used no adjoint-rank assumption. ◻

Proof. At a component with \(D_E=0\), the definition of \(r_Q\) gives \(a_E-e_Er_Q\geq0\). At the finitely many other relevant components, increase \(k\) to absorb \(e_Er_Q\) by \(kD_E\). At horizontal components there is no subtraction. This proves \(\Xi\geq0\), and over every \(Q\) some component has \(\Xi_E=0\).

On a smooth equivariant model over \(Y\) define \[I_m\,d\lambda_Y=g_*\bigl(|\pi^*s|^{2m}\pi^*d\mu\bigr).\] For all sufficiently large \(m\), (41) identifies the invariant adjoint line generated by the Jacobian times \(s^m\). Smooth direct-image positivity and orthogonal compact averaging imply that \(-\log I_m\) is psh on a common smooth open. The densities are smooth there, so the base-change locus for an individual \(m\) can be removed by continuity. High moments give \[I_m^{1/m}\longrightarrow\beta(y)=\max_{Z_y}|\pi^*s|^2.\] For this limit one may divide by \(I_0\) as a positive density on the smooth open; no curvature assertion about \(I_0\) is used. The weights of the required lines are \[ \begin{split} \varphi_N&=-\log\beta+\sum_Q n_Q\log|s_Q|^2,\\ \psi&=-\log I_k+\sum_Q(kn_Q+r_Q)\log|s_Q|^2. \end{split} \tag{42}\] The first extends with locally bounded weights by regularity of the normalized section and a minimizing component above every divisor, by Lemma 18.

For the second, choose a component above \(Q\) with \(\Xi_E=0\). In coordinates \(y_1=x_1^{e_E}\) at its general point, the corrected source density has exponent \[a_E+kh_E-e_E(kn_Q+r_Q)=0.\] Its pushforward is bounded below by \(C|y_1|^{-2(1-1/e_E)}\). Thus \(\psi\) is bounded above at general points of every omitted divisor and extends psh there, then across codimension two. On a log resolution \(v:\widehat W\to W\) the exponents in the corrected source density are the coefficients of \[K_{\widehat W/X}+kv^*D-v^*f^*R =K_{\widehat W/W}+v^*\Xi.\] They are all greater than \(-1\), since \(W\) is klt and \(\Xi\) is effective and \(\mathbb Q\)-Cartier. Normal crossing integration and properness give \(e^{-\psi}\in L^1_{\mathrm{loc}}\) on the entire base.

Finally \(D-f^*R/k\) is nonnegative at each component nonexceptional over \(X\), since \(a_E=0\) there and \(\Xi\geq0\). Pulling back a section of \(N+R/k\) and multiplying by a power of \(s\) leaves only exceptional poles, removed on pushdown. ◻

Proof. The class \(-K_Y+B^-=-(K_Y+B)+B^+\) is pseudoeffective. For an algebraically integrable foliation \(\mathcal F\) of positive rank and corank, resolve its rational first-integral map together with \(B\) by \(u:Y'\to Y\), and write \[K_{Y'}+C=u^*(K_Y+B),\qquad C=C^+-C^-.\] Every coefficient of \(C\) is less than one: the positive part of the original simple normal crossing boundary is klt, and decreasing its coefficients can only increase discrepancies. In particular \((Y',C^+)\) is klt. For an ample Cartier \(H\) on \(Y'\) and rational \(\delta>0\), choose a general effective rational divisor \[A_\delta\sim_{\mathbb Q} -u^*(K_Y+B)+\delta H\] such that \((Y',C^++A_\delta)\) is klt. The class is ample, so a sufficiently divisible multiple supplies such a member by Bertini, also on a general fibre of the resolved map. We have \[K_{Y'}+C^++A_\delta\sim_{\mathbb Q} C^-+\delta H.\] It is big on that fibre and hence has a nonzero section after taking a positive divisible multiple. Apply Druel’s relative pseudoeffectivity theorem (Druel 2017, Proposition 4.1), in the form of (Ou 2023, Proposition 3.1), to conclude that \(K_{\mathcal F'}+C^++A_\delta\) is pseudoeffective. Its pushforward is numerically \[K_{\mathcal F}-K_Y+B^-+\delta u_*H.\] Letting \(\delta\) decrease to zero proves \(K_{\mathcal F}-K_Y+B^-\) pseudoeffective. Lemma 15 now proves the assertion for every movable class, and therefore in the pseudoeffective cone. ◻

First proof of Theorem [rb:allowed]. Clear the denominator of \(L\). We induct on \(\dim V\); a point is immediate. Call a nonempty linear system in a line bundle \(P\) admissible if \[aL+cD-P\quad\hbox{is pseudoeffective}\] for some integers \(a>0,c\geq0\). The constant system is admissible, and products of admissible systems are admissible. Choose one whose image dimension is maximal. If it is generically finite, then \(P\) is big. Consequently \(aL+cD\) is big, which supplies a section after a positive multiple. Otherwise, resolve its map and take the relative algebraic closure of its image field to obtain \(f:W\to Y\) with connected general fibres and \(\dim Y<\dim V\).

Every ratio of sections in every admissible system belongs to \(\mathbb C(Y)\). Indeed adjoining such ratios cannot increase image dimension, since products of the systems remain admissible. The ratios are therefore algebraic over the chosen image field and belong to its relative algebraic closure. This argument also covers subsystems and pencils.

Lemma [rb:determinant] gives \(N_i\sim M+m_iL\), while Proposition [rb:cotangent] gives \(-M+kD\) pseudoeffective for some \(k\geq0\). Systems in all positive combinations of the \(N_i\) are therefore admissible. For \(i<j<l\) the two effective divisors \[(m_l-m_j)N_i+(m_j-m_i)N_l, \qquad (m_l-m_i)N_j\] are linearly equivalent, so the rational function relating them lies in \(\mathbb C(Y)\). After pullback, their horizontal parts are equal. It follows that the horizontal multiplicity at every prime is affine in \(m_i\). Fixing \(i<j\) and letting \(l\) tend to infinity shows that its slope is nonnegative. Thus \(N_j-N_i\) defines a rational section \(\rho\) of \((m_j-m_i)L\) with no horizontal poles on \(W\), and hence on \(\bar V\). If \(Y\) is a point, it has no poles and already proves the theorem.

For a positive-dimensional base we verify the rank-one condition. Let \(P\) denote the original maximal system and let \(A_0\) be the pullback to \(Y\) of its image hyperplane class. It is big and \(p^*P-f^*A_0\) has an effective representative. Given a sufficiently ample base divisor \(H\), a positive multiple of \(A_0\) dominates \(H\) up to an effective divisor. Therefore \(p_*f^*H\) is dominated in the pseudoeffective order by a positive multiple of \(L\) plus a nonnegative multiple of \(D\).

If \(J+p^*D+mp^*L\) has two independent sections on a generic fibre, spread them out and clear their vertical poles using \(f^*H\) with \(H\) sufficiently ample. Their pushdowns are sections of \(D+mL+p_*f^*H\); the exceptional Jacobian \(J\) contributes zero under pushdown. This system is admissible, but its ratio is nonconstant on the generic fibre, contrary to the maximality property. Hence its generic dimension is at most one for every \(m\geq0\). For the multiples of the exponent just constructed, \(s_{J+p^*D}\rho^l\) supplies a nonzero generic section, including \(l=0\). Lemmas [rb:rank-one] and [rb:transfer] apply.

The base has no positive-degree holomorphic forms: pullback by the dominant map is injective, and forms on the smooth birational model identify with forms on \(V\). The descended data thus satisfy exactly the hypotheses of the theorem in lower dimension. Induction gives a section with poles on \(D_Y\), and Lemma [rb:transfer] carries it to a section with poles only on \(D\). An integral multiple of \(D\) clears those finitely many poles. ◻

Second proof of Theorem [rb:allowed]. Again induct on dimension and clear denominators. If \(L+D\) is big, a positive multiple has a section. Otherwise \(L+D\) lies on the boundary of the pseudoeffective cone. Cone duality gives a nonzero movable class \(\alpha\) such that \[L\cdot\alpha=D\cdot\alpha=0.\] The zero-degree part of Proposition [rb:cotangent] and the determinant construction give \(N_i\sim M+m_iL\) with \(M\cdot\alpha\leq0\). Effectivity forces \(N_i\cdot\alpha=M\cdot\alpha=0\). No nonnegative integral combination of \(D,N_1,N_2,\ldots\) is big.

Choose such a combination \(G\) of maximal Iitaka dimension \(d<\dim V\). Take a multiple realizing dimension \(d\), resolve its map, and take the connected-fibre base \(Y\). Every ratio from the linear system of any allowed combination lies in \(\mathbb C(Y)\), since adding systems and multiplying sections cannot increase \(d\). If \(H\) is any base divisor, then \[ ap^*G-f^*H\sim E_H\geq0 \tag{43}\] for some sufficiently large \(a\): the resolved system contains the pullback of the image hyperplane class, whose transform on \(Y\) is big.

We need the following stronger form of the ratio assertion. A rational function whose horizontal poles on \(W\) are bounded by \(p^*G'+E\), where \(G'\) is an allowed combination and \(E\) is effective and \(p\)-exceptional, belongs to \(\mathbb C(Y)\). Its remaining poles are vertical. Choose an effective sufficiently ample \(H\) whose pullback bounds them; a base divisor can contain the images of all these finitely many poles, including images of codimension at least two, to the necessary orders. By (43), the function is a ratio of sections of \(p^*(G'+aG)+E\). Normality gives \(p_*\mathcal O_W(E)=\mathcal O_V\), so the exceptional term adds no sections. The original ratio assertion proves the claim.

The three-index relations among the \(N_i\) now imply affine horizontal multiplicities exactly as in the first proof. Let \(\rho\) have divisor \(N_j-N_i\) and exponent \(k=m_j-m_i>0\). It has no horizontal poles, and its horizontal zeros are bounded by \(p^*N_j\). On a generic fibre \(s_{J+p^*D}\rho^l\) is a nonzero section of \(J+p^*D+lkp^*L\). The ratio of any other section to this one has horizontal poles bounded by \[J+p^*D+l p^*N_j.\] The stronger ratio assertion makes that ratio a base function. Thus the generic section space is one-dimensional for every \(l\geq0\). If \(d=0\), \(\rho\) is already regular. Otherwise Lemmas [rb:rank-one] and [rb:transfer] give the two required metrics and the controlled boundary on the smaller base. Absence of holomorphic forms descends, so dimension induction completes the proof. ◻

Proof. Suppose no positive multiple of \(L\) has a section with poles only on \(D\). In particular no \(aL+cD\) with \(a>0,c\geq0\) is big. Rational connectedness gives absence of holomorphic forms, so take the determinant divisors \(N_i\sim M+m_iL\) and the inequality \(-M+kD\) pseudoeffective. Let \(K\) be the relative algebraic closure in \(\mathbb C(V)\) of the field generated by functions with divisors \[(m_l-m_j)N_i+(m_j-m_i)N_l-(m_l-m_i)N_j, \qquad i<j<l.\] These functions exist because the displayed divisor class is zero. A finite selection gives the same transcendence degree, and the relative algebraic closure is a finite extension of a finitely generated subfield. Thus \(K\) defines a smooth projective base after resolution.

Each of those functions is represented by a pencil whose total bundle has class \(uL+vM\), with \(u,v>0\). Finitely many such pencils give a generically finite map from the base to their combined image; the sum of the pulled-back hyperplane classes is therefore big on the base. The resolved pencils are bounded above by their total bundles. Pushing down and using \(-M+kD\) pseudoeffective proves the first condition of the proposition. A large multiple of the big base class dominates any chosen ample divisor, so the same condition holds with an ample divisor.

On the generic fibre all the displayed principal divisors vanish. The effective restrictions of the \(N_i\) therefore have affine multiplicities in \(m_i\). Their slope is nonnegative by the unbounded sequence, so \(N_j-N_i\) supplies the second condition. A point base would already give a regular section, contrary to our supposition. A base of full dimension would make the pushdown of an ample base class big, hence would make some \(aL+cD\) big, also impossible. We have found a base of dimension strictly between zero and \(\dim V\).

Choose, among fibre spaces with these two properties, one with maximal base dimension. The properties survive higher models. For the first, pullback of an ample class to a birational base model is big and a large multiple dominates every fixed divisor; pulling and pushing preserve effectivity. The second is the regularity of a fixed rational section over the generic fibre, which survives resolution.

Let \(\rho\) be the supplied generic section of \(kp^*L\). If \(J+p^*D+mkp^*L\) had two independent generic sections, their ratio would be transcendental over the relatively algebraically closed base field. Adjoining it and taking relative algebraic closure enlarges the base dimension. Clear the two sections’ vertical poles by a base twist \(f^*H'\). Their ratio pencil has class \[J+p^*D+mkp^*L+f^*H'.\] Add an ample system from the old base. The product map supplies a big class on the enlarged base. On pushdown \(J\) vanishes, and the first property for the old map bounds the remaining class by a multiple of \(L\) plus a multiple of \(D\). Thus the first property holds for the enlarged map. Restricting \(\rho\) to its new generic fibre gives the second property and stays nonzero at the generic point. This contradicts maximality.

The spaces in Lemma [rb:rank-one] consequently have dimension at most one and have a nonzero element \(s_{J+p^*D}\rho^m\), also at \(m=0\). Transfer gives the required data on the smaller base, which is rationally connected because it is a smooth projective model of a rational image of \(V\). Induction gives its permitted section and transfer gives one on \(V\), the required contradiction. ◻

Proof. Average local weights, with their line-bundle transition rule, over the compact form of \(T\) using the natural linearization. This gives a smooth semipositive metric invariant under that compact group. If the action is trivial, Theorem [rb:allowed] applies with \(D=0\) and \(A=L=-K_V\). Otherwise take the rational quotient with field \(\mathbb C(V)^T\) (Rosenlicht 1956; Bell et al. 2017), and its connected-fibre smooth projective model \(Y\). The invariant field is relatively algebraically closed in \(\mathbb C(V)\): an element algebraic over it has a finite orbit, and the connected torus acts trivially on that orbit. General fibres contain a dense orbit.

Choose the model diagram equivariantly, by modifying the trivially acted-on base, normalizing the dominating graph, and taking an equivariant resolution. Write \(p:W\to V\) and \(f:W\to Y\) for the smooth resolution maps, \(J=K_{W/V}\), and \(\bar V\) for the normal equidimensional intermediate model of Section 3.1. Let \(r\) be the orbit dimension. A wedge of \(r\) independent infinitesimal action fields is an invariant regular relative anticanonical section on the generic fibre; commutativity of \(T\) gives invariance. A rational frame of \(-K_Y\) and the canonical Jacobian section \(s_J\) put it in \[p^*(-K_V)=-K_{W/Y}+f^*(-K_Y)+J.\] It has no horizontal poles. Regard this section of \(p^*(-K_V)\) birationally as a rational section \(\rho\) on \(V\). The Jacobian section is invariant for the induced canonical actions. The ratio of any two invariant sections of \(J+mp^*(-K_V)\) is a rational invariant function, hence belongs to the base field. The sections \(s_J(p^*\rho)^m\) supply the nonzero element at every \(m\geq0\), including \(m=0\). Thus the invariant rank-one condition holds.

If \(Y\) is a point, the full wedge of infinitesimal fields on \(V\) already gives a global invariant anticanonical section. Otherwise the invariant case of Lemma [rb:rank-one] and Lemma [rb:transfer] gives a bounded semipositive base bundle and a finite-volume donor metric. The base has no holomorphic forms by dominant pullback, and Theorem [rb:allowed] gives a section with permitted base poles. Its transferred section is invariant, and its only permitted poles on \(V\) lie in \(D=0\). It is therefore regular. ◻

Proof. For \(r=0\), give \(\mathbb R^0\) its unit mass. Both expressions equal \(\Phi(z)\), so the assertion is immediate. Assume \(r>0\). Plurisubharmonicity implies convexity of \(x\mapsto\Phi(z,x)\). An integrable positive smooth log-concave function on \(\mathbb R^r\) is bounded. To see this, choose a fixed ball on which its value is at least \(c>0\). If values \(a_j\) at points \(x_j\) tended to infinity, log-concavity on the midpoint ball between \(x_j\) and that fixed ball would give a lower bound \((ca_j)^{1/2}\) on a ball of a fixed positive volume. Its integral would tend to infinity, a contradiction. This proves the last assertion.

Fix a bounded real ball \(D\) and first replace \(\Phi\) by \(\Phi+\epsilon|x|^2\), so the Hessian in \(x\) is positive definite. Until \(\epsilon\) is sent to zero, write \(\Phi\) for this regularized weight. Restrict the base variable to a complex disc. Let \(\mathbb E\) denote expectation for the probability density proportional to \(e^{-\Phi}\) on \(D\). Differentiating under the integral gives \[ \frac{\partial^2}{\partial z\partial\bar z} \left(-\log\int_D e^{-\Phi}\,dx\right) =\mathbb E\Phi_{z\bar z} -\mathbb E|\Phi_z-\mathbb E\Phi_z|^2. \tag{44}\] For a smooth real weight \(b\) whose Hessian \(b_{xx}\) is positive definite on \(\bar D\), put \[\mathbb E_b F=\frac{\int_D F e^{-b}dx}{\int_D e^{-b}dx}.\] The Brascamp–Lieb bound is \[ \mathbb E_b|v-\mathbb E_bv|^2 \leq\mathbb E_b\bigl(b_{xx}^{-1}(\nabla v,\nabla\bar v)\bigr). \tag{45}\] Here it is enough to prove it for smooth real mean-zero \(v\) and then add the estimates for real and imaginary parts. The bounded-domain form and its Neumann identity are also proved in (Kolesnikov and Milman 2017, Theorems 1.1 and 1.2(1), and Section 3.3). The preceding addition of \(\epsilon|x|^2\) provides the stated positive definiteness; the limit \(\epsilon\downarrow0\) is taken afterwards.

Solve the weighted Neumann problem \[(-\Delta+\nabla b\cdot\nabla)u=v, \qquad \partial_{\mathbf n}u=0.\] The mean-zero condition is precisely its solvability condition on the ball. Weighted integration by parts and the differentiated equation give \[\int_Dv^2e^{-b} =\int_D|\nabla^2u|^2e^{-b} +\int_Db_{xx}(\nabla u,\nabla u)e^{-b} +\int_{\partial D}\mathrm{II}(\nabla u,\nabla u)e^{-b}.\] The boundary second fundamental form \(\mathrm{II}\) is nonnegative because \(D\) is convex, and \(\nabla u\) is tangential there. Also \(\int_Dv^2e^{-b}=\int_D\nabla v\cdot\nabla u\,e^{-b}\). Cauchy–Schwarz for the quadratic form \(b_{xx}\) bounds the latter integral by \[\left(\int_Db_{xx}^{-1}(\nabla v,\nabla v)e^{-b}\right)^{1/2} \left(\int_Db_{xx}(\nabla u,\nabla u)e^{-b}\right)^{1/2}.\] Combining the two identities gives (45), including the case of zero variance by continuity.

Apply (45) with \(b(x)=\Phi(z,x)\) and \(v=\Phi_z\). The complex Hessian of \(\Phi(z,\operatorname{Re}w)\) is positive semidefinite. Its Schur complement gives \[\Phi_{z\bar z}\geq \Phi_{zx}(\Phi_{xx})^{-1}\Phi_{x\bar z}.\] The factors \(1/2\) in complex vertical derivatives cancel against the factor \(1/4\) in the vertical Hessian. Equations (44) and (45) prove plurisubharmonicity of \(-\log\int_D e^{-\Phi}dx\). Decreasing \(\epsilon\) to zero preserves this conclusion. Exhausting \(\mathbb R^r\) by balls gives a decreasing sequence of psh functions; its almost-everywhere finite limit is the first required representative.

For the supremum, use the high-power integral argument of (Berndtsson 2006, 1635, following Theorem 1.2): take the probability-normalized Lebesgue measure on \(D\) and replace \(\Phi\) by \(l\Phi\), with integers \(l\geq1\). The functions \[-\frac1l\log\left(\frac1{|D|}\int_D e^{-l\Phi}dx\right)\] are psh and decrease with \(l\), by monotonicity of \(L^l\) norms. Their finite limit is \(-\log\sup_D e^{-\Phi}\). Increasing the balls again gives a decreasing psh limit, finite almost everywhere by the first paragraph. This is the second representative. ◻

Orbit-integral proof of Theorem 25. If the torus acts trivially, apply Theorem 23 with \(D=0\) and \(A=L=-K_V\). Assume the action is nontrivial. Give \(L=-K_V\) the same compact-torus-invariant smooth semipositive metric and let \(\mu\) be its smooth positive volume. Use the maps \(p:W\to V\), \(f:W\to Y\) and intermediate model \(\bar V\) recalled in the first proof. Divide out the ineffective kernel of the torus action. The effective torus has trivial generic stabilizer. Indeed a subgroup fixing the generic point would act trivially on the function field and hence on \(V\); subgroups of a split torus are determined by their character lattices, so the geometric generic stabilizer is such a fixed subgroup. The generic orbit over the rational quotient field is consequently a torsor under a split torus. Hilbert’s Theorem 90 trivializes it. Spreading out gives a dense orbit family \(T\times Y^0\) over a nonempty base open set. These quotient and triviality statements use (Rosenlicht 1956; Bell et al. 2017) for the quotient and (Milne 2017, Corollary 3.47 and Proposition 3.50) for split-torus torsor triviality.

On this product take logarithmic coordinates \(t_j=e^{x_j+i\theta_j}\) on \(T\), with \(x\in\mathbb R^r\) and \(\theta\in(\mathbb R/2\pi\mathbb Z)^r\). Wedge the logarithmic action fields with a base coordinate anticanonical frame to obtain a frame \(\sigma\) of \(L\) along the product. Its weight \[\Phi(y,x)=-\log|\sigma|^2\] is smooth and psh in \((y,x+i\theta)\), and is independent of \(\theta\) by compact invariance. In one coordinate, \[\frac{i}{2}\frac{dt_j}{t_j}\wedge \frac{d\bar t_j}{\bar t_j}=dx_j\wedge d\theta_j.\] The top form dual to the logarithmic anticanonical frame therefore identifies the volume with a fixed positive constant times \(e^{-\Phi}\,dx\,d\theta\,d\lambda_y\). Angular integration contributes a positive constant independent of \(y\). The complement of the orbit family is algebraic and has measure zero. Consequently the pushforward density of \(\mu\) is \[g_0(y)=C\int_{\mathbb R^r}e^{-\Phi(y,x)}dx\] almost everywhere. It is finite almost everywhere by Fubini.

Choose a nonzero rational base anticanonical frame. It differs from the coordinate frame by a base factor and gives an invariant rational section \(\rho\) of \(L\). For almost every general fibre, the supremum of \(|\rho|^2\) on its dense orbit is finite by Lemma [rb:prekopa]. It follows that \(\rho\) has no horizontal pole on the normal intermediate model \(\bar V\): a pole along a horizontal prime would make its norm unbounded near that prime on a general compactified fibre, and the dense orbit approaches its general points. The smooth metric is comparable to a positive smooth frame there, so such unboundedness cannot be hidden in the norm. Normality therefore makes the section regular on the complete generic fiber of \(\bar V\to Y\), and its pullback to \(W\) remains regular.

The supremum on the dense orbit equals the essential supremum for transfer, since the rational section is now regular on the generic fibre and the metric is continuous. Lemma [rb:prekopa] proves both \(-\log g_0\) and \(-\log S_\rho\) psh on the quotient open set; nonzero base frame factors contribute only pluriharmonic terms. If the quotient is a point, the same no-horizontal-pole argument already says that \(\rho\) is a regular invariant section. Assume now \(\dim Y>0\). Apply Lemma 4 with \(D=0\), \(A=L=-K_V\), \(k=1\), and this rational section \(\rho\). The preceding calculation supplies its two psh weights. Vertical normalization, properness, and the bounded input metric give the local upper bound for the normalized section norm; a component attaining the vertical minimum gives the complementary lower bound. Thus the transfer produces a bounded metric on its line \(B\) and a finite-volume metric on the different line \(-K_Y+D_Y\). Theorem 23 applies on \(Y\), which inherits the absence of positive-degree holomorphic forms by dominant pullback. The output has the explicit form \((\psi\circ f)(p^*\rho)^m\) on \(W\) for a rational base function \(\psi\). Its birational pushdown to \(V\) is therefore invariant. Since the initial boundary was \(D=0\), the support conclusion of transfer removes all divisorial poles on \(V\). This completes the orbit-integral proof. ◻

Proof. The second assertion implies the first by comparing \(u\) with \(s^j\); the regularity hypothesis on \(s\) guarantees that this comparison is permitted. Conversely, let \(u_1,u_2\) have degree \(b\). The first assertion applied to \(u_i^a\) gives \(u_i^a/s^b\in K\). Hence \((u_1/u_2)^a\in K\), and relative algebraic closedness puts \(u_1/u_2\) in \(K\). This argument also applies when \(b=0\). ◻

Proof. Products and inverses preserve the support condition. The poles of a nonzero sum remain on null primes. Since a principal divisor has degree zero, its effective zero divisor then has degree zero as well; movability makes each of its components null. Thus \(K_\alpha\) is a field.

Equivalently, every valuation at a non-null prime is trivial on \(K_\alpha\). Such a valuation remains trivial on an element algebraic over that field. An element of positive valuation cannot satisfy a polynomial with nonzero constant term over the field, since that term would uniquely have minimal valuation. Applying the same argument to the inverse excludes negative valuation. This proves relative algebraic closedness.

If \(u_1,\ldots,u_d\in K_\alpha\) were algebraically independent, the sum \(P\) of their polar divisors would be null. The monomials of total degree at most \(m\) in the \(u_i\) are linearly independent sections of \(mP\), so \(P\) would be big. A nonzero functional nonnegative on the pseudoeffective cone is strictly positive on its interior. Thus \(\alpha\cdot P>0\), a contradiction.

Every ratio in a zero-degree system belongs to \(K_\alpha\), since the zero divisors of its sections are effective of degree zero. Conversely, if \(u\in K_\alpha^*\) and \(P=(\operatorname{div}u)^-\), then \(1\) and \(u\) are sections of \(P\) and \(\alpha\cdot P=0\). Choose a maximal-image zero-degree system and write \(K_0\) for the field of its Stein base. The product-system argument of Lemma 12 puts all zero-degree section ratios in \(K_0\), hence \(K_\alpha\subseteq K_0\). The generating ratios of the chosen system lie in \(K_\alpha\); its relative algebraic closedness gives the reverse inclusion. This proves the claimed equality, rather than just a comparison of dimensions.

A non-null prime is horizontal over \(K_\alpha\) by definition. Finally, if there were infinitely many horizontal null primes, finite generation of \(\operatorname{Pic}(Y)\) would give a nonzero integral relation among finitely many of their classes (multiply to remove torsion). The resulting nonzero divisor would be principal. Its defining function would lie in \(K_\alpha\) but have nonzero order at one of these horizontal primes, which is impossible. ◻

Proof. If \(Y\) is a point, then \(D=0\); take \(P_0=0\) and positive Hermitian norms on the two trivial lines. Their weights are bounded and their adjoint measure is finite. We may therefore assume \(\dim Y>0\). Apply Lemmas 5 and 4 with original variety \(X\), allowed-pole divisor zero, \(A=L=-K_X\), and both metrics equal to the compact-averaged smooth metric on \(L\). The section is \(s\) of degree \(a\). It has no horizontal poles on the normal equidimensional graph. The rank computation following Lemma 27 verifies the smooth invariant adjoint rank condition for every \(j\geq0\). Thus the two required psh functions are supplied by Lemma 5.

The full-component normalization in Lemma 4 is exactly \(D\); a denominator-clearing multiple of \(D\) receives its bounded metric, which scales back to the rational divisor. Since the allowed-pole divisor on \(X\) is zero, the invisible primes in that lemma are precisely our omitted primes. Its correction is therefore an effective integral \(P_0\) with the asserted support. The distinguished measure on \(-K_Y+P_0\) equals the pushforward of the smooth anticanonical volume almost everywhere, so its local integral is finite, including at boundary intersections. This proves both assertions on the flattened graph, without requiring toroidalization. ◻

Proof. The bounded psh metric makes \(D\) pseudoeffective. Alternatively, apply Nadel vanishing and regularity to its divisible multiples: local boundedness makes their multiplier ideals trivial, and \(mD+A\) is effective for fixed sufficiently positive \(A\).

The divisor \(D^*\) cannot be rationally linearly equivalent to an effective rational divisor modulo the span of omitted primes. Otherwise, after clearing denominators, there would be \(N>0\) and \(g\in K^*\) with \[N d^*_\Gamma+\operatorname{ord}_\Gamma(g)\geq0 \quad\text{at every nonomitted prime}.\] Then \(s^Ng\) would have no pole along any prime divisor on \(X\). For horizontal divisors this follows from the generic regularity of \(s\). For a vertical divisor, its nonexceptional strict transform on \(W\) dominates a nonomitted prime of \(Y\), and the displayed inequality and the minimum defining \(d^*_\Gamma\) give the result. Thus \(s^Ng\) would be a nonzero invariant global section.

Let \(\mathcal S\subset N^1(Y)_{\mathbb R}\) be the span of omitted primes, and project the cone of effective divisor classes to \(N^1(Y)_{\mathbb R}/\mathcal S\). Denote this convex cone by \(C\). Its closure has nonempty interior, since the ample cone already has nonempty interior before projection. The image of \([D^*]\) belongs to \(\overline C\), by pseudoeffectivity of \(D\) and effectivity of \(D^*-D\). It cannot lie in its interior. Indeed, the interior of a convex cone agrees with that of its closure and is contained in the cone itself. Membership of this rational point in \(C\) would express it as a finite nonnegative real combination of rational effective classes. The corresponding finite rational linear system then has a nonnegative rational solution. Since numerical and rational linear equivalence agree on \(Y\), that would give the rational effectivity just excluded.

A supporting functional at this boundary point is nonzero, nonnegative on \(C\), and zero on the image of \([D^*]\). Pull it back to \(N^1(Y)_{\mathbb R}\). It is nonnegative on effective classes and annihilates \(\mathcal S\); by cone duality (Boucksom et al. 2013), it is a nonzero movable class \(\alpha\). Pseudoeffectivity of \(D\) and effectivity of \(D^*-D\) give \(0\leq\alpha\cdot D\leq\alpha\cdot D^*=0\). The effective difference therefore has zero degree; each of its components has zero degree. This proves the final assertion as well. ◻

Proof. Choose one higher equivariant resolution \(\widehat Z\to Z\) on which the rational map for \(K_\alpha\) is also a morphism. Pull back the divisors \(N_m\). Their parts horizontal for \(K_\alpha\) have support in a fixed finite set of prime divisors on \(\widehat Z\). Indeed, a horizontal component mapping onto a prime of \(Y\) must lie over one of the finitely many horizontal null primes in Lemma 28. There are only finitely many components over those primes. A prime whose image in \(Y\) has codimension at least two must be exceptional over \(W\), by the fibre dimension bound, and there are only finitely many such divisors on this fixed resolution.

List the horizontal multiplicities of the pulled-back \(N_m\) as vectors in \(\mathbb Z_{\geq0}^r\), using this fixed finite support. Order the indices \(m\) increasingly. Dickson’s lemma, in its sequence form (see (Aichinger and Aichinger 2018, Theorem 1.1(1) and Lemma 3.2)), gives \(m_2>m_1\) such that the earlier vector is componentwise at most the later one. For completeness, the needed sequence form follows by induction on \(r\): a sequence of nonnegative integers either has an infinitely repeated value or has a strictly increasing subsequence; apply this successively to each coordinate. This also covers empty horizontal support. Hence \(N_{m_2}-N_{m_1}\), pulled back to \(\widehat Z\), is effective on every \(K_\alpha\)-horizontal divisor.

Put \(b=m_2-m_1>0\). From (46), choose \(g_0\in K^*\) satisfying \[N_{m_2}-N_{m_1}=bD+\operatorname{div}_Y(g_0).\] Set \[ a'=ba,\qquad s'=s^bg_0. \tag{47}\] The section \(s'\) is invariant. On \(\widehat Z\), its divisor is the pullback of \(b(\operatorname{div}_Z(s)-g_Z^*D)\) plus the pullback of \(N_{m_2}-N_{m_1}\). It is therefore nonnegative along all \(K_\alpha\)-horizontal divisors. The finitely many remaining poles lie over proper closed subsets of the new base, so a dense open avoids them. This proves generic regularity of the new section.

It remains to verify the ratio requirement, which is essential for another iteration. Let \(k\geq0\), and let \(t\) be an invariant nonzero rational section of \(ka'L\) without poles along prime divisors on \(X\) horizontal for \(K_\alpha\). Since \(K_\alpha\subseteq K\), every \(K\)-horizontal valuation is \(K_\alpha\)-horizontal. The old fixed-section uniqueness condition therefore gives \[g=\frac{t}{s^{kb}}\in K.\] Let \(\Gamma\) be a non-null prime of \(Y\). It is nonomitted, and \(d^*_\Gamma=d_\Gamma\) by Lemma 30. Each nonexceptional component \(Q\) over it on \(W\) is horizontal for \(K_\alpha\): its restriction to \(K_\alpha\) is \(e_Q\operatorname{ord}_\Gamma=0\). Its valuation is the valuation of a prime divisor on \(X\). Consequently \[e_Q\operatorname{ord}_\Gamma(g)+kb\operatorname{ord}_Q(s) =\operatorname{ord}_Q(t)\geq0.\] Taking the minimum over these nonexceptional components proves \[ \operatorname{ord}_\Gamma(g)+kb\,d_\Gamma \geq0. \tag{48}\] The rational divisor \(\operatorname{div}_Y(g)+kbD\) has total \(\alpha\)-degree zero. Its coefficients at every non-null prime are nonnegative by (48), and the remaining primes have degree zero. Each of those nonnegative coefficients must therefore be zero. Moreover, \(\operatorname{div}_Y(g_0)+bD=N_{m_2}-N_{m_1}\) is supported on null primes. Subtracting \(k\) times this identity shows that \[\frac{t}{(s')^k}=\frac{g}{g_0^k}\in K_\alpha.\] This includes \(k=0\), so fixed-section uniqueness is preserved in its full stated range.

Strict transcendence-degree decrease follows from Lemma 28. The regularity just proved holds on the complete generic fiber and hence survives further modification. This completes the new data, including the original-prime uniqueness test. ◻

Proof. The invariant field is relatively algebraically closed in \(\mathbb C(X)\): an element algebraic over it has a finite orbit, and the divisible abelian group \(T(\mathbb C)\) has no nontrivial finite quotient. Rosenlicht’s quotient theorem gives a dense orbit on the smooth connected general fiber \(F\) of \(g\) (Rosenlicht 1956). The stabilizer of that orbit fixes its closure, so \(F\) is toric for an effective quotient of \(T\). The nef bundle \(\pi^*L|_F\) has a basepoint-free positive multiple by the toric basepoint-freeness theorem (Cox et al. 2011, Theorem 6.3.12).

For any one-parameter subgroup \(\lambda\), the Białynicki-Birula decomposition of \(X\) has an open attracting stratum as \(t\to0\), whose fixed component has nonnegative tangent weights, and an open attracting stratum as \(t\to\infty\), whose tangent weights are nonpositive (Białynicki-Birula 1973). Choose \(F\) very general so that its image meets both strata for every \(\lambda\); there are only countably many integral cocharacters. The corresponding orbit limits on the proper \(F\) give fixed points with the two weak signs for the fiber weights of \(\pi^*L\), since \(L=\det T_X\) has its natural action. Evaluation of a basepoint-free multiple at either point is equivariant and surjective. Thus its section space has weights of both weak signs for every \(\lambda\).

It follows that zero is in the convex hull of the torus characters in this section space: otherwise a strict separating functional could be chosen rational and then integral. Choose a rational convex combination with sum zero and clear denominators. The product of the corresponding nonzero eigensections is a nonzero invariant section of a positive power; nonzeroness uses integrality of \(F\). This is the invariant-sections construction used in the toric setting of (Müller 2025, sec. 3).

For each positive integer \(m\), cohomology and base change and torus weight splitting identify the invariant section space of \(m\pi^*L\) on fibers over a dense open with a vector bundle of its generic rank. Choose the very general fiber also in all these countably many opens. The invariant section just found therefore implies a nonzero invariant generic rank for its particular exponent. This yields the required section over \(\mathbb C(Y)\). The empty-dimensional fiber and the trivial torus are included in the argument. ◻

Proof. Average the metric on \(L\) over the compact form of \(T\), so that it is invariant. When \(U\) is a point the determinant assertion is immediate. Otherwise replace \(U\) birationally, without changing \(\mathcal F\), so that every \(g\)-vertical prime \(P\subset Y\) maps onto a prime \(Q\subset U\). To arrange this, flatten the dominating graph and resolve the base (Raynaud and Gruson 1971). Each \(g\)-vertical prime of \(Y\) has a strict transform on the resulting equidimensional source. Properness defines the rational lift from \(Y\) at its generic point, so the resulting map sends that prime onto a base prime. Put \(e_P=\operatorname{ord}_P(g^*Q)\). At the general point of \(P\), the map has coordinates \[(x,y_2,\ldots,y_{\dim Y}) \longmapsto (x^{e_P},y_2,\ldots,y_{\dim U}).\] The determinant of the differential, after quotienting by its saturated kernel, therefore has order \(e_P-1\). Along a horizontal prime it has order zero. Thus, in codimension one and hence as line bundles on \(Y\), \[ \det(T_Y/\mathcal F)=-g^*K_U-R_g, \qquad R_g=\sum_{P\text{ vertical}}(e_P-1)P. \tag{49}\] The pullback here is defined on the domain of the rational map and extended across its codimension-two complement. The sum is finite, since its coefficients are the divisorial orders of one determinant map.

Take a smooth projective equivariant resolution dominating \(W\) and resolving both rational maps, with morphisms \[\pi:Z\longrightarrow X,\qquad q:Z\longrightarrow Y, \qquad r:Z\longrightarrow U.\] Write \(e_\pi\) for the invariant Jacobian section of \(K_Z+\pi^*L\). On a dense open of \(U\), the sheaf \[\mathcal E=\bigl(r_*(K_{Z/U}+\pi^*L)\bigr)^T\] is locally free, commutes with base change, and has its adjoint \(L^2\) metric. Berndtsson’s theorem gives Griffiths semipositivity (Berndtsson 2009). The same is true of the invariant summand: the weight-zero projection is holomorphic and orthogonal, because the compact torus preserves the coefficient metric and the fiber integral. Its metric is therefore the quotient metric. This argument concerns a smooth coefficient metric on the smooth-family open.

Choose a rational frame for \(\mathcal E\) and a rational frame \(\eta\) of \(K_U\). Its sections on \(Z\) have the form \[ (q^*\phi_i)e_\pi/(r^*\eta),\qquad \phi_i\in\mathbb C(Y). \tag{50}\] Indeed, dividing first by the Jacobian and the inverse base frame gives invariant rational functions on \(X\). These have no poles at the primes dominating \(Y\): such primes also dominate \(U\), and the Jacobian is a unit at their strict transforms. The assumed degree-zero uniqueness places the functions in \(\mathbb C(Y)\).

After shrinking dense opens, evaluation of the coefficients in (50) defines a holomorphic functional on \(g^*\mathcal E\). For \(v\in\mathcal E_{g(y)}\), denote its coefficient at \(y\) by \(\phi_v(y)\). In the frame \(g^*\eta^{-1}\), set \[ \psi(y)=\log\sup_{\|v\|_{g(y)}=1}|\phi_v(y)|^2. \tag{51}\] This is psh, being the log squared norm of a holomorphic section of the dual of a Griffiths-semipositive bundle. Replacing \(\eta\) by \(a\eta\) adds \(\log|g^*a|^2\) to \(\psi\), exactly the transformation law for a weight on \(-g^*K_U\). The section \(e_\pi/(r^*\eta)\) has coefficient \(1\), so this weight is not identically \(-\infty\).

It remains to account for ramification and for the Jacobian zeros above the boundary. Fix a prime \(P\subset Y\), choose a prime \(D\subset Z\) mapping onto it, and put \[l=\operatorname{ord}_D(q^*P),\qquad d_\pi=\operatorname{ord}_D(e_\pi).\] If \(P\not\subset N\), choose \(D\) nonexceptional over \(X\), so that \(d_\pi=0\). Suppose first that \(P\) maps onto a prime of \(U\). With \(j=le_P\), suitable coordinates around a general point of \(D\) express \(r\) as \[(\tau,z_2,\ldots,z_n)\longmapsto (\tau^j,z_2,\ldots,z_h), \qquad h=\dim U,\quad D=(\tau=0).\] Take \(\eta\) regular and nonvanishing on this base patch. If a fiber section \(v\) has coefficient \(a_v\) in a regular frame of \(K_{Z/U}+\pi^*L\), its adjoint fiber norm contains the density \[\frac{|a_v|^2}{|j\tau^{j-1}|^2} |dz_{h+1}\cdots dz_n|^2\] up to positive bounded factors with bounded inverse. The denominator comes from dividing the total coordinate top form by the pulled-back base form. For \(\|v\|=1\), holomorphic submean on a smaller fiber polydisc gives \(|a_v|\leq C|\tau|^{j-1}\). When the fiber dimension is zero, its point contribution to the norm gives the same bound. Dividing by the Jacobian coefficient, which is \(\tau^{d_\pi}\) times a unit, yields \[ |\phi_v(q(z))|\leq C'|\tau(z)|^{j-1-d_\pi}. \tag{52}\] If \(P\) dominates \(U\), then \(r\) is submersive at a general point of \(D\). Submean gives instead \(|\phi_v(q(z))|\leq C'|\tau(z)|^{-d_\pi}\). In both cases the constants are uniform for unit-norm sections.

Let \(x\) define \(P\) near a general point. On the chosen patch, \(q^*x\) is a unit times \(\tau^l\). Generic submersivity along \(D\to P\), together with this transverse power, shows that the patch maps onto a neighborhood of that point of \(P\). We can therefore apply the preceding estimates over each nearby \(y\), with \(|\tau|\) comparable to \(|x(y)|^{1/l}\). They give \[ \psi\leq O(1)+ \begin{cases} \displaystyle\frac{le_P-1-d_\pi}{l}\log|x|^2, &P\text{ vertical over }U,\\[4pt] \displaystyle-\frac{d_\pi}{l}\log|x|^2, &P\text{ horizontal over }U. \end{cases} \tag{53}\] For \(P\not\subset N\), subtracting the weight of \(R_g\) leaves a locally bounded-above function: in the vertical case the remaining coefficient is \(1-1/l\geq0\), and in the horizontal case it is zero. At a component of \(N\), adding \(c\log|x|^2\) makes the remaining coefficient nonnegative once \[c\geq (1+d_\pi)/l-1\quad\text{in the vertical case}, \qquad c\geq d_\pi/l\quad\text{in the horizontal case}.\] There are finitely many omitted primes, so one nonnegative integer \(c\) works for all of them. The adjusted weights are psh off the divisors, since the logarithms there are pluriharmonic. Their upper bounds extend them across the general points of every boundary divisor. The remaining exceptional sets have codimension at least two; psh removability extends the weights there as well. The frame changes identify the resulting metric as one on \(-g^*K_U-R_g+cN\). This proves pseudoeffectivity and hence \[ \alpha\cdot c_1(T_Y/\mathcal F)\geq0 \qquad(\alpha\text{ movable},\ \alpha\cdot N=0). \tag{54}\]

We finish with the movable-slope argument of Ou, using the tensor slope theorem of Greb–Kebekus–Peternell and the algebraicity criterion of Campana–Păun (Greb et al. 2016; Campana and Păun 2019; Ou 2023; Lazić et al. 2023). Suppose that a line in \((\Omega_Y^1)^{\otimes j}\) has positive \(\alpha\)-degree. For \(j=0\) this is impossible, since a line subsheaf of \(\mathcal O_Y\) has an effective negative. For \(j>0\), tensor slopes and duality imply \(\mu_{\alpha,\min}(T_Y)<0\). Applying (54) to the identity of \(Y\) gives \(\alpha\cdot c_1(T_Y)\geq0\). Thus the positive-slope part \(\mathcal F_+\) of its Harder–Narasimhan filtration is nonzero and proper. Its minimal slope is positive, whereas \(T_Y/\mathcal F_+\) has nonpositive maximal slope and strictly negative degree.

The bracket map \(\mathcal F_+\otimes\mathcal F_+\to T_Y/\mathcal F_+\) vanishes by the slope gap, so \(\mathcal F_+\) is a foliation. The positive-minimal-slope criterion makes it algebraically integrable. Its rational quotient contradicts (54). These uses of slope theory also apply to a boundary movable class. Equivalently, perturb toward a rational interior movable class: the strict positive minimal slope and the bracket slope gap persist. The tensor inequality at the original class follows by taking limits from the interior, using \[\mu_{\alpha+\epsilon\beta,\min}(\mathcal G) \geq\mu_{\alpha,\min}(\mathcal G) +\epsilon\mu_{\beta,\min}(\mathcal G)\] and the corresponding upper bound for maximal slopes. This proves the cotangent assertion. ◻

Proof. The divisor \(D+G+N=D^*+N\) cannot be big. A section of a positive multiple, represented by \(\gamma\in\mathbb C(Y)^*\), would satisfy \(\operatorname{ord}_P\gamma+j d_P^*\geq0\) at every nonomitted prime. The minimum defining \(d_P^*\) then makes \(s^j\gamma\) regular at every vertical prime of \(X\); its horizontal regularity is already known. This would be an invariant global section. Since \(D\) is pseudoeffective and \(G,N\) are effective, cone duality supplies \(\alpha\ne0\) movable with \[\alpha\cdot D=\alpha\cdot G=\alpha\cdot N=0.\] The assumed cotangent estimate and the determinant proof in Lemma 11 produce \(M\) and effective \(C_j\sim M+m_jD\); effectivity forces \(\alpha\cdot M=0\). Thus every displayed system has degree zero, and none gives a generically finite map on \(Y\).

Choose one nonempty displayed system \(|\Lambda|\) of maximal image dimension, and let \(Y'\) be a smooth model of its Stein field. Products remain in the displayed collection. Exactly the product-map argument in Lemma 12 therefore shows that ratios in any one of these systems lie in \(\mathbb C(Y')\). Moreover \(\dim Y'<\dim Y\). The interpolation of Lemma 13 uses only products of the sections of \(M+m_jD\), so it applies within this collection. For \(m_1>m_0\), the ratio of those sections gives a rational section of \((m_1-m_0)D\) regular on the complete generic fiber over \(Y'\). Write \(\delta=m_1-m_0\) and let \(u\) be the normalized section of \(aL-f^*D\). If \(\xi\) is the generic section of \(\delta D\) just obtained, then \(u^\delta f^*\xi\) is the new invariant generic section of \(a\delta L\).

We check all-degree original-prime uniqueness directly. Let \(w_1,w_2\) be invariant rational sections of \(bL\), \(b\geq0\), regular at the primes of \(X\) horizontal over \(Y'\). Old uniqueness gives \[w_i^a=s^b f^*\gamma_i,\qquad \gamma_i\in\mathbb C(Y)^*.\] At a nonomitted prime \(P\) of \(Y\) horizontal over \(Y'\), every nonexceptional component above it is horizontal over \(Y'\) too. Testing regularity there and taking the nonexceptional minimum gives \[\operatorname{ord}_P\gamma_i+b(d_P+\operatorname{coeff}_P G)\geq0.\] Hence, for one sufficiently large \(l\), the \(\gamma_i\) are rational sections of \(bD+bG+lN\) with poles only at primes vertical over \(Y'\).

These finitely many poles can be cleared by the same section of some \(n\Lambda\); if there are no poles, use \(n=0\) and the constant section. Resolve \(|\Lambda|\). Each such prime maps into a proper closed subset of its image: a valuation trivial on the image field is trivial on its algebraic closure, so verticality persists for the image field. Choose a projective hypersurface containing those images but not the whole system image. Substituting the defining sections of \(\Lambda\) gives a nonzero section vanishing at each pole prime; its sufficiently high power gives the required orders. The fixed divisor of the resolved system only adds vanishing. Multiplication by this common section places both \(\gamma_i\) in one system of the displayed semigroup. Maximality puts their ratio in \(\mathbb C(Y')\). Thus \((w_1/w_2)^a\in\mathbb C(Y')\), and relative algebraic closedness inside \(\mathbb C(X)\) gives \(w_1/w_2\in\mathbb C(Y')\). Lemma 27 now restores fixed-section uniqueness, including degree zero. This proves the proposition. ◻

Proof. Fix \(a\in\mathcal M\). For \(t\in\mathbb Q\), \(t\geq a\), and \(h\in\mathbb Z_{>0}\), define a vector space of rational functions by \[ V_{t,h}=\{0\}\cup \left\{v\in\mathbb C(Y)^*: \operatorname{div}(v)+h(tB^\sharp-R)\geq0 \text{ away from the }C_i\right\}. \tag{58}\] Rational coefficients cause no ambiguity: this is an inequality of rational divisors. These spaces need not be finite-dimensional. Their multiplication law is \[ V_{t_1,h_1}V_{t_2,h_2} \subset V_{(h_1t_1+h_2t_2)/(h_1+h_2),\,h_1+h_2}. \tag{59}\] By (57) and \(B^\sharp\geq B\) away from the omitted primes, \(q_m\in V_{m,1}\). Every rational \(t\geq a\) is a rational convex combination of \(a\) and some \(m\in\mathcal M\) with \(m\geq t\). Products of powers of \(q_a,q_m\) therefore show that \(V_{t,h}\ne0\) for some \(h>0\).

At a fixed \(t\), let \(F_t\) be the set of ratios \(v/w\) with \(v,w\in V_{t,h}\), \(w\ne0\), allowing all positive \(h\). It is a field: multiplication and cross-multiplication for addition use (59), and inverses interchange numerator and denominator. Its relative algebraic closure in \(\mathbb C(Y)\) is independent of \(t>a\). To see this, take \(a<t_1<t_2\) and a ratio at \(t_1\). Pad equal powers of its numerator and denominator with a common power of \(q_m\), where \(m>t_2\). Choose the powers so that their weighted-average parameter is \(t_2\). A positive power of the original ratio then belongs to \(F_{t_2}\). Conversely, pad a ratio at \(t_2\) with \(q_a\) to reach \(t_1\). Clearing the rational weights makes all the powers integral. Each field is therefore contained in the relative algebraic closure of the other.

Write \(K\) for the common relatively closed field, and choose a smooth projective model \(Y'\) with \(\mathbb C(Y')=K\). This field is finitely generated: choose a transcendence basis of \(K/\mathbb C\); the algebraic closure of the resulting rational function field inside the finitely generated field \(\mathbb C(Y)\) is finite over it. Since \(\mathbb C(Y)\) is relatively algebraically closed in \(\mathbb C(X)\), the same is true of \(K\). We now distinguish whether this field has lost dimension.

The full-dimensional case. Suppose \(\operatorname{trdeg}_{\mathbb C}K=d=\dim Y\). At one fixed rational \(t>a\), finitely many ratios in \(F_t\) are algebraically independent of cardinality \(d\). Use common products and powers to place their numerator and denominator sections in one degree. A sufficiently large rational divisor \(C\) supported on the \(C_i\) compensates their finitely many omitted-prime poles. After clearing denominators, these sections belong to a multiple of \(tB^\sharp-R+C\) and give a rational map with \(d\)-dimensional image. Hence \[ tB^\sharp-R+C\quad\text{is big}. \tag{60}\]

The closure in the first condition of (57) is sufficient to remove \(R\) from this assertion. Choose pseudoeffective classes \(P_j\) and real divisors \(E_j\) supported on the \(C_i\) such that \(P_j+[E_j]\to[R]\). Then \[[tB^\sharp+C-E_j]-P_j \longrightarrow [tB^\sharp-R+C].\] The classes on the left are big for large \(j\), by openness of the big cone. Adding \(P_j\) preserves bigness, so \(tB^\sharp+C-E_j\) is big. Rational approximation of the coefficients on the finite support yields a rational divisor \(C'\), supported on that same list, with \(tB^\sharp+C'\) big. Consequently, for some positive integer \(h\) clearing all denominators and some \(v\in\mathbb C(Y)^*\), \[\operatorname{div}(v)+h(tB^\sharp+C')\geq0.\] Away from the omitted primes this is precisely the hypothesis of (56), with the positive integer \(l=ht\). It produces the asserted global invariant section.

The smaller-base case. Suppose now that \(\dim Y'<d\). Choose \(b\in\mathcal M\) with \(b>a\), write \(\delta=b-a\), and set \[ f':X\dashrightarrow Y',\qquad k'=k\delta,\qquad s'=s^\delta f^*(q_b/q_a), \tag{61}\] where \(f'\) is the composite rational map. Both the map and section are invariant. We verify separately that the section is regular over the generic point and that it restores uniqueness.

For every \(m\in\mathcal M\) with \(m>b\), the numerator and denominator in \[ \frac{q_m^{\delta}q_a^{m-b}}{q_b^{m-a}} \tag{62}\] belong to \(V_{b,m-a}\), since \(\delta m+(m-b)a=b(m-a)\). Their ratio therefore lies in \(K\). On a common smooth resolution mapping to \(W,Y,Y'\), let \(n_m\) be the coefficient of the pullback of \(N_m\) at a prime horizontal over \(Y'\). Its valuation is trivial on \(K\), so (62) implies \[\delta(n_m-n_a)=(m-a)(n_b-n_a).\] Every \(n_m\geq0\), and the indices \(m\) are unbounded. It follows that \(n_b-n_a\geq0\). The divisor of the pullback of \(s'\) is \[\delta D_s+\varphi^*(N_b-N_a),\] with pullbacks to the common resolution understood. Its horizontal coefficients are nonnegative by (55) and the preceding inequality. Thus \(s'\) has no poles over the generic point of \(Y'\), including at exceptional horizontal primes.

To prove uniqueness, take \(r\geq0\) and a nonzero invariant rational section \(\tau\) of \(rk'L\) having no poles on the prime divisors of \(X\) horizontal over \(Y'\). These include the primes horizontal over \(Y\). The old fixed-section condition writes \[\tau=s^{\delta r}f^*u,\qquad (s')^r=s^{\delta r}f^*u_s, \qquad u_s=(q_b/q_a)^r,\] with \(u,u_s\in\mathbb C(Y)^*\). At every non-omitted prime \(\Gamma\) whose valuation is trivial on \(K\), the nonexceptional primes above it are horizontal over \(Y'\). Regularity of both sections there gives \[ \operatorname{ord}_\Gamma(u)+\delta r b^\sharp_\Gamma\geq0, \qquad \operatorname{ord}_\Gamma(u_s)+\delta r b^\sharp_\Gamma\geq0. \tag{63}\] Only finitely many non-omitted primes fail either inequality. Every such prime has valuation nontrivial on \(K\).

We can compensate all these deficits with a common element of one \(V_{b,h}\). For a non-omitted prime \(\Gamma\), define the excess of \(v\in V_{b,h}\setminus\{0\}\) by \[\epsilon_\Gamma(v)= \operatorname{ord}_\Gamma(v) +h(bb^\sharp_\Gamma-\operatorname{coeff}_\Gamma R) \geq0.\] If \(\operatorname{ord}_\Gamma\) is nontrivial on \(K\), it is nontrivial on \(F_b\): a valuation trivial on a field remains trivial on an algebraic extension. Some ratio \(v_1/v_2\), with \(v_1,v_2\in V_{b,h_\Gamma}\), therefore has nonzero order. The two nonnegative excesses are different, so at least one is strictly positive. Choose that element. Products of sufficiently large powers of these chosen elements, one for each deficit prime, give \(v\in V_{b,h}\setminus\{0\}\) whose excess compensates every deficit of both \(u\) and \(u_s\). At all other non-omitted primes, the excesses remain nonnegative. If there are no deficits, any nonzero element of \(V_{b,h}\) suffices. Thus \[uv,\ u_s v\in V_{\,b+\delta r/h,\,h}.\] The parameter is still greater than \(a\), also when \(r=0\). It follows that \[\tau/(s')^r=u/u_s\in F_{b+\delta r/h}\subset K.\] This is the required fixed-section uniqueness for the smaller base.

These verifications also retain the adjoint rank conditions: on a smooth resolution, multiplying powers of \(s'\) by the Jacobian gives a nonzero invariant adjoint section in every nonnegative multiple of \(k'\). Dividing any other such section by the Jacobian gives a rational section regular at the original horizontal primes; the uniqueness just proved makes it a multiple over \(K\) of that generator. Thus the contraction returns precisely the section and rank data needed for the next step, including degree zero. ◻

Proof. Choose the Iitaka map from a multiple of \(Q\) attaining the maximal image dimension and take its Stein factorization. A multiple of \(u^*Q\) dominates the pullback of a big divisor on \(Y_2\); after taking a higher multiple and replacing that divisor by an ample divisor plus an effective divisor, it dominates a pulled-back ample divisor. The generic fibre is geometrically integral. Adding an effective exceptional divisor changes no global space of sections, by normality, and hence no Iitaka dimension. If a multiple of \((u^*Q+A)|_S\) had two independent sections, extend them rationally over \(Y'\) and clear their vertical poles with a sufficiently high pulled-back ample twist. The preceding domination absorbs that twist into a multiple of \(u^*Q+A\). Their ratio remains nonconstant on \(S\). Combined with ratios defining the base, it increases the transcendence degree of a global section-ratio field beyond \(\kappa(Q)\), a contradiction. The canonical effective divisor supplies the nonzero section. ◻

Proof. Use the divisors \(N,R,B\) of Proposition 19. If any \(j(N+R)+lB\), \(j>0\), \(l\geq0\), is rationally effective, that proposition gives the section on \(X\). Assume otherwise. The weighted cotangent bound gives \(hcB-c_1(M)\) pseudoeffective for a line \(M\) in a cotangent tensor power of order \(h\). Consequently \[M+j(N+R)+B \quad\hbox{is not big}\qquad(j>0);\] adding the pseudoeffective class \(hcB-M\) to a big class would contradict the preceding noneffectivity.

The bounded metric of \(N\), hard Lefschetz and the nonzero constant term of \(\chi(K_Y+mN)\) give, by the determinant construction of Lemma 6, a fixed cotangent line \(M\) and effective \(D_j\sim M+jN\) for unbounded \(j>0\). Fix two indices \(i<b\) and put \(Q=D_b+bR+B\). This divisor is effective and not big. Resolve its Iitaka map by \(u:Y'\to Y\), \(h:Y'\to Y_2\). Then \(\dim Y_2<\dim Y\).

Restrict pullbacks to the generic fibre \(S\) of \(h\). The interpolation equivalence (32), applied at the twist exponents \(i<b<j\), gives two effective representatives of the class \((j-i)D_{b,S}\). Add the same effective complement \((j-i)(Q_S-D_{b,S})\) to both. Lemma 37 makes the resulting representatives of \((j-i)Q_S\) equal, and cancellation gives \[(j-i)D_{b,S}=(b-i)D_{j,S}+(j-b)D_{i,S} \qquad(j>b).\] In particular all the later \(D_{j,S}\) are supported in \(D_{b,S}\). Since \(j\) is unbounded and \(D_{j,S}\geq0\), the divisor \[V_S=\frac{D_{b,S}-D_{i,S}}{b-i}\] is effective and rationally equivalent to \(N_S\). Its support lies in \(D_{b,S}\). For every effective \(u\)-exceptional \(A\) and integers \(n,l\geq0\) this also gives \[ h^0\bigl(S,n(N+R)_S+lB_S+A|_S\bigr)\leq1. \tag{64}\] To see this, represent \(N_S\) by \(V_S\). The indicated effective representative is supported in \((u^*Q+A)|_S\) and therefore dominated by a multiple of it. Two independent sections would remain independent after taking powers and multiplying by a fixed section, contrary to Lemma 37.

On a common resolution over \(Y'\), lift a denominator-clearing multiple of \(V_S\) through \(f^*N\to ap^*L\). It gives the new invariant generic section. For the rank assertion, let \(F_{\mathrm{exc}}\) be any effective invariant divisor exceptional over \(X\) on the common resolution. A section with this twist over the new generic fibre is, by the old rank condition, rationally \(\lambda s^n s_{F_{\mathrm{exc}}}\) with \(\lambda\in\mathbb C(Y)\), first for exponents divisible by \(a\). At a prime of \(S\) not exceptional over \(Y\) and not in \(B_S\), choose a nonexceptional component realizing the restricted minimum \(n'_Q=(N+R)_Q\). The exceptional twist has coefficient zero there. Regularity gives \[\operatorname{ord}_Q\lambda \geq-n(N+R)_Q.\] All remaining poles occur on \(B_S\) or on exceptional divisors of \(u\). For any proposed pair of sections, choose an effective \(u\)-exceptional divisor \(A_Y\) on \(Y'\) and \(l\geq0\) so that \(lB_S+A_Y|_S\) absorbs them. Equation (64) then forces dependence. Other exponents follow by taking powers to clear divisibility; independent sections cannot become dependent under a power on a geometrically integral fibre. ◻

Proof. Take the prepared toroidal model of Proposition 20. Let \(B\) now be the reduced sum of primes \(Q\) for which no component attaining the minimum \(n_Q\) is nonexceptional over \(X\). For small rational \(\delta>0\), the normalized divisor \(D_s\) absorbs \(\delta f^*B\) at every nonexceptional prime. Thus effectivity of \(N+\delta B\) supplies a section on \(X\). Otherwise this pseudoeffective divisor is not big, so a nonzero movable class \(\alpha\) annihilates both \(N\) and \(B\). The correction \(\Theta\) of Proposition 20 is supported on \(B\). Proposition 17 gives \(\alpha\cdot M\leq0\) for every cotangent line \(M\).

The determinant argument produces effective \(D_j\sim M+jN+B\), \(D_j\geq B\), for unbounded \(j\); their degree against \(\alpha\) is zero. Resolve the Iitaka map of one \(D_b\) after fixing an earlier index \(i<b\). The same interpolation identity as above puts every later restricted divisor in the support of \(D_{b,S}\). Include the finitely many exceptional divisors of the chosen resolution in an auxiliary effective divisor \(A\) and apply Lemma 37 to \(u^*D_b+A\). Choose comparable indices \(j_1<j_2\) in the tail beyond \(b\), using the ascending-pair property in a finite product of \(\mathbb N\). Put \(v=j_2-j_1>0\). Then \[C=(D_{j_2}-D_{j_1})|_S\geq0,\qquad C\sim vN_S, \qquad \operatorname{Supp}C\subseteq\operatorname{Supp}D_{b,S}.\] Let \(\zeta\) be the corresponding section of \(vN_S\), and let \(\sigma\) denote the normalized regular section of \(a\pi^*L-g^*N\) on a common resolution. The section \(\sigma^v g^*\zeta\) gives a new invariant generic section \(s'\) of degree \(avL\).

We verify the new ranks at exactly these multiples. Let \(q\geq0\) and let \(t\) be an invariant section of \(K_{Z'/X}+qav\pi'^*L\) on the new generic fibre, where \(Z'\) is a common smooth resolution. The old rank condition, applied over \(K\), says that division by the Jacobian section and \(\sigma^{qv}\) leaves the pullback of a rational section \(r\) of \(qvN_S\). At a prime of \(S\) outside \(B_S\) and the \(u\)-exceptional locus, a nonexceptional minimizing component has both normalized section order and Jacobian order zero. Regularity of \(t\) therefore makes \(r\) regular at that prime.

The rational function \(r/\zeta^q\) has poles only in \(\operatorname{Supp}(u^*D_b+A)|_S\): the divisor \(D_b\) contains \(B\), the zeros of \(\zeta\) lie in \(D_{b,S}\), and \(A\) contains every \(u\)-exceptional prime of the fixed resolution. Every multiple of this effective divisor is rigid by Lemma 37, so the function is constant over \(\mathbb C(Y_2)\). Thus \(t\) is a base multiple of the Jacobian times \((s')^q\). This also treats \(q=0\), when \(\zeta^q=1\). Those Jacobian products supply nonzero sections in every tested degree, proving rank exactly one. The argument permits extra resolutions without assuming that their exceptional centres lie in the original base locus. ◻

Proof. Use Proposition 21 and put \(N^+=N+R/k\). If \(N^+\) is rationally effective, its section returns to \(X\). Otherwise \(N^+\) is pseudoeffective and not big. Choose a nonzero movable \(\alpha\) annihilating it. Since \(N,R\) are pseudoeffective, \[\alpha\cdot N=\alpha\cdot R=0.\] The unweighted metric of \(J=-K_Y+kN+R\) and Proposition 17 give the cotangent bound for \(\alpha\). The determinant construction supplies effective \(D_j\sim M+m_jN\) with \(m_j\) increasing and \(\alpha\cdot D_j=0\).

Let \(K'\) be the field of functions with poles only on \(\alpha\)-null primes. Lemma 28 gives finite generation, relative algebraic closure, and strict transcendence-degree decrease. On a resolution of the resulting map there are only finitely many possible horizontal components of the \(D_j\): the null horizontal primes are finite because \(\operatorname{Pic}(Y)\otimes\mathbb Q\) is finite dimensional, and the resolution has only finitely many exceptional primes. The finite-support argument in the proof of Proposition 31 therefore gives \(j>i\), \(v=m_j-m_i>0\) and \(c\in K^*\) with \[D_j-D_i=vN+\operatorname{div}(c)\] horizontally effective over \(K'\). Set \(s'=s^vc\). Its divisor upstairs is the sum of the pullback of this difference and \(vD\geq0\). Hence it is regular at every valuation trivial on \(K'\), proving (40).

For all sufficiently large \(m\), a section \(t\) satisfying the new discrepancy inequalities satisfies the old ones, so \(t=s^{mv}u\) with \(u\in K\). If \(Q\) has positive \(\alpha\)-degree, \(r_Q=0\). A component attaining the two minima in the definition of \(R\) consequently has \(a_E=0\) and \(h_E=e_En_Q\). Its valuation is trivial on \(K'\), and the inequality for \(t\) gives \[\operatorname{ord}_Q u+mv n_Q\geq0.\] The null support of \(D_j-D_i\) gives \(\operatorname{ord}_Q c+v n_Q=0\). Thus \(u/c^m\) has no pole on any positive-degree prime and belongs to \(K'\). We obtain \(t\in K'(s')^m\). The opposite inclusion follows from the regularity of \(s'\) and the nonnegative discrepancies over smooth \(X\). ◻

Proof. On a prepared equidimensional toroidal model \(p:W\to X\), \(f:W\to Y\), normalize \(s\) by vertical minima: \[D_s=\operatorname{div}(s)-f^*N\geq0.\] The hypothesis is the fixed-section uniqueness condition of Lemma 27. It gives invariant adjoint rank one in every nonnegative multiple of \(a\): divide by the Jacobian and test the nonexceptional horizontal primes. Hence Lemma 9 supplies a bounded semipositive metric on \(N\) and a signed boundary \(B_{\mathrm{sgn}}\) with simple normal crossing support and coefficients less than one, such that \(-(K_Y+B_{\mathrm{sgn}})\) is nef.

We compare the two choices of permissible base poles before choosing the contraction. Let \(S_0\) be the reduced sum of omitted primes, above which every component is exceptional over \(X\). Define \(N'\) by taking the minimum only over nonexceptional components off \(S_0\), and retaining the coefficient of \(N\) on \(S_0\). Put \(G=N'-N\geq0\). Let \(B_{\min}\) be the reduced sum of primes at which no minimizing component is nonexceptional. Then \[\operatorname{Supp}B_{\min} =\operatorname{Supp}S_0\ \sqcup\ \operatorname{Supp}G, \qquad \operatorname{Supp}B_{\mathrm{sgn}}^- \subseteq\operatorname{Supp}S_0.\] Indeed, away from the omitted primes, failure to attain the full minimum on a nonexceptional component is exactly the strict inequality \(n'_Q>n_Q\).

There are two effectivity tests. If \(N'+S_0\) is rationally effective, its section returns to \(X\) by the nonexceptional minimum. Alternatively, for sufficiently small rational \(\epsilon>0\), \(D_s-\epsilon f^*B_{\min}\) has no negative coefficient at any nonexceptional prime; effectivity of \(N+\epsilon B_{\min}\) therefore also returns a section. If neither test succeeds, the chosen pseudoeffective class is not big. Cone duality gives a nonzero movable class \(\alpha\). Either choice yields the same vanishing conditions: \[\alpha\cdot N=\alpha\cdot G=\alpha\cdot S_0=0, \quad\text{equivalently}\quad \alpha\cdot N=\alpha\cdot B_{\min}=0.\] In particular \(\alpha\cdot B_{\mathrm{sgn}}^-=0\), so Proposition 22 supplies the cotangent bound. The determinant construction gives effective \(D_j\sim M+jN\) at unbounded divisible indices, with \(\alpha\cdot D_j=0\).

Let \(K'\) be the null-prime field of Lemma 28; it has smaller transcendence degree. On a common resolution, the affine interpolation of Lemma 13 gives \(j>i\) and \[D_j-D_i=vN+\operatorname{div}(c),\qquad v=j-i>0,\quad c\in K^*,\] with effective horizontal part over \(K'\). One may instead use the finite-support extraction of Proposition 31 to choose a comparable pair of horizontal multiplicity vectors. Both choices give the same required property. Since the divisor of \(s'=s^vc\) is \(vD_s+f^*(D_j-D_i)\), this section is regular on the new complete generic fibre, including at exceptional horizontal primes.

For uniqueness, suppose that \(t=(s')^kg\) has no poles at original primes horizontal over \(K'\), where \(k\geq0\) and \(g\in\mathbb C(X)^T\). Since these include the \(K\)-horizontal primes, the old test applied to \(t=s^{vk}c^kg\) gives \(c^kg\in K\) and hence \(g\in K\). If a base prime \(Q\) has positive \(\alpha\)-degree, then \(Q\) is not omitted, \(n'_Q=n_Q\), and its valuation is trivial on \(K'\). Its nonexceptional components are therefore original primes horizontal over \(K'\). Testing \(t\) on all those components gives \[\operatorname{ord}_Q g+ k(vn'_Q+\operatorname{ord}_Q c)\geq0.\] The null support of \(D_j-D_i\) gives \(vn_Q+\operatorname{ord}_Q c=0\). Thus \(g\) has no pole at any positive-degree prime. Its pole divisor is null, and the field criterion in Lemma 28 gives \(g\in K'\). This includes \(k=0\) and proves the required uniqueness.

This proof uses the whole bad-minimum set through \(G\) and \(S_0\). Annihilating only \(B_{\mathrm{sgn}}^-\) would not provide the nonexceptional minimizing components needed in the last pole test. ◻

Proof. Smooth semipositivity implies nefness, so Lemma 32 supplies the invariant section. Ratios of invariant rational tensors of the same power of \(L\) lie in \(\mathbb C(X)^T=K\), which proves (65). ◻

Proof. On a dense smooth base open, use coordinates \(y\) and write \(I_p(y)\) for the density of \[f_*\bigl(|\tau|_{(\pi^*h_L)^a}^{2p}\,\pi^*\mu\bigr).\] The pullback measure includes the Jacobian of \(\pi\). The adjoint section is \(s_Es_{c\pi^*D}\tau^p\); its fiberwise squared norm is \(I_p\), after trivializing the canonical base factor. Lemma 5 applies to (69), including \(p=0\). It gives psh representatives of \(-\log I_0\) and \[ u=-\log S,\qquad S(y)=\operatorname*{ess\,sup}_{V_y}|\tau|^2_{(\pi^*h_L)^a}. \tag{72}\] The essential supremum uses the fiber density of \(\pi^*\mu\). For almost every smooth fiber that density is positive almost everywhere. Since \(h_L\) has bounded weights, \(S\) is comparable to the ordinary supremum computed with a fixed smooth reference metric. The comparison constants are uniform on the compact total space wherever the rational section is defined. Fix this smooth reference metric on \(L\), and use its pullbacks on every model below. In the invariant case the same lemma uses the holomorphic orthogonal projection to the invariant direct summand for the smooth coefficient metric. Positivity of that quotient metric is the additional input required in this case.

Constructing a line on an alteration. Apply weak semistable reduction of Abramovich–Karu to \(f\) (Abramovich and Karu 2000, Theorem 0.3 and Definition 0.1, arXiv version 1). After a projective generically finite alteration \(T\to Y\) and modifications, it gives a projective family \(g:W\to T\) with geometrically reduced fibers, with \(T\) smooth projective, and a morphism \(W\to V\) over \(Y\). Its equidimensional toroidal source is Cohen–Macaulay, so it is flat over the smooth base. The source \(W\) need not be smooth. We keep this flat family when making further base modifications; resolving its source would not preserve the flatness needed below. The generic fiber of \(f\) is geometrically integral, because it is smooth and connected in characteristic zero, as required for this application of reduction. The two families fit into the diagram \[\begin{array}{ccc} W&\longrightarrow&V\\ {\scriptstyle g}\downarrow&&\downarrow{\scriptstyle f}\\ T&\longrightarrow&Y. \end{array}\]

Let \(P\) be the pullback to \(W\) of \(a\pi^*L\). Lemma 44 gives \(J\), \(\eta_J\), and \(g^*J\to P\) on a further modified base. The family remains flat with reduced fibers. On the generic base open the map \(W\to V\) is surjective on the fibers under consideration. Give \(J\) a continuous positive reference metric and let \(c(t)\) be the supremum norm of the map \(g^*J\to P\), measured in these reference metrics. Since this map sends \(g^*\eta_J\) to \(\tau\), we have \[\sup_{W_t}|\tau|_{\mathrm{ref}}^2 =|\eta_J(t)|_{\mathrm{ref}}^2c(t), \qquad c_-\leq c(t)\leq c_+\] by Lemma 45. Surjectivity onto the original fiber identifies the left side with \(\sup_{V_y}|\tau|_{\mathrm{ref}}^2\). Comparison with the bounded metric \(h_L\) therefore gives \[ |\eta_J(t)|^2\asymp S(y),\qquad t\mapsto y, \tag{73}\] for almost every parameter in that open. Here and below comparison means upper and lower positive constant bounds in local reference metrics, with constants that remain bounded toward the omitted locus. The fixed reference metrics are pulled back under all subsequent base changes, and the same bounds on \(c(t)\) continue to hold. No rational connectedness of \(T\) is asserted or needed.

Returning to the original function field. By flattening by modification, choose a smooth projective modification \(Y'\to Y\) on which both the main transform \(V_0\to Y'\) of \(V\) and the main transform \(q:T'\to Y'\) of \(T\) are flat (Raynaud and Gruson 1971). The second map is finite: it is projective, flat and has zero-dimensional fibers. Its degree \[d_0=\operatorname{rk}q_*\mathcal O_{T'}\] is positive and constant. The first map is equidimensional of the original relative dimension. Subsequent smooth modifications of the base preserve these properties by flat base change. The main transforms map to \(T\) and \(V\), respectively, so \(J\) and its rational section pull back to \(T'\). The base \(Y'\) still has the function field \(\mathbb C(Y)\). The resulting diagram is \[\begin{array}{ccccc} T'&\xrightarrow{\ q\ }&Y'&\longleftarrow&V_0\\ \downarrow&&\downarrow&&\downarrow\\ T&\longrightarrow&Y&\longleftarrow&V. \end{array}\] Here both maps to \(Y'\) are flat, and \(q\) is finite. The line \(J\) lives on \(T'\); its norm will live on \(Y'\).

Define the actual norm line and its rational section by \[ B=\operatorname{Nm}_q(J) :=\det(q_*J)\otimes\det(q_*\mathcal O_{T'})^{-1}, \qquad \beta=\operatorname{Nm}_q(\eta_J). \tag{74}\] Both direct images in this formula are locally free of rank \(d_0\). To describe the section, view multiplication by \(\eta_J\) as a rational map \(q_*\mathcal O_{T'}\to q_*J\) and take its determinant. Locally a line bundle can be trivialized near a finite inverse fiber; changing the frame by a unit changes its norm frame by the determinant of multiplication by that unit. This proves the compatibility of (74) with changes of frame and arbitrary base change. The construction uses finite local freeness, and does not require \(T'\) to be smooth or \(q\) to be unramified (The Stacks Project Authors 2026, Tags 0BUT, 0BCX and 0BD2).

Over the generically unramified locus the norm is the product over the \(d_0\) sheets. Applying (73) on those sheets gives \[ |\beta(y)|^2\asymp S(y)^{d_0}. \tag{75}\] These constants remain bounded at the branch locus. Indeed, choose a frame for \(J\) near the finite inverse fiber. Its continuous reference norm is bounded above and away from zero there, and its norm is a nonvanishing frame of \(B\). Products of the finitely many reference norms are therefore comparable to any fixed reference norm on \(B\). This also explains why the exponent is multiplied by \(d_0\).

Define a metric on \(B\), initially where \(\beta\) is a nonzero regular frame, by \(|\beta|^2=S^{d_0}\). If \(\beta=b e\) in a local holomorphic frame \(e\), its weight is \[\psi=d_0u+\log|b|^2.\] It is psh on that open. Equation (75) bounds \(\psi\) above and below locally even toward the omitted locus. The bounds hold for the psh representatives, by subaveraging and upper semicontinuity. Removable singularities across analytic sets therefore extend \(\psi\) to a locally bounded psh weight on all of \(Y'\).

Resolve \(V_0\) to \(V'\), with maps \(\pi'\) and \(f'\) as in the statement. The rule \[ \beta\longmapsto\tau^{d_0} \tag{76}\] defines a rational homomorphism in (70) over the original field. It extends regularly. Indeed, write \(\beta=b e_B\) and \(\tau=t e_L^{\otimes a}\) in local holomorphic frames, with the fixed reference norms bounded above and below. The coefficient of this map is \(t^{d_0}/(b\circ f')\). On almost every fiber in the good open, (75) and the definition of the essential supremum give \[\left|\frac{t^{d_0}}{b\circ f'}\right|^2 \leq C\frac{S(f'(\,\cdot\,))^{d_0}} {|\beta\circ f'|_{\mathrm{ref}}^2} \leq C'.\] The bound first holds almost everywhere on these fibers and then everywhere on their good locus by continuity of the rational coefficient. A rational coefficient with a pole along a prime divisor would be unbounded on open subsets approaching its general point; those subsets meet these fibers. This excludes every divisorial pole, and normality extends the map across codimension two. In the invariant case the rule (76) is invariant as a rational identity on \(X\). The auxiliary alterations and final resolution need not be equivariant.

We have constructed the bounded base metric and its map back to \(X\). On the new base, \(S\) denotes the same generic fiber supremum, whereas \(I_0\) now denotes the density of \((f')_*(\pi')^*\mu\) in the chosen coordinates on \(Y'\). The latter transforms as a volume density, including the base Jacobian. Off the exceptional locus of the base modification, this changes \(-\log I_0\) by a pluriharmonic coordinate term and retains its plurisubharmonicity. It remains to extend the volume metric and ensure that all its correction primes disappear, or land in \(D\), upon return to \(X\).

The correction divisor and every component of its pullback. Among the boundary prime divisors of the good open in \(Y'\), let \(D'\) be the reduced sum of those primes \(Q\) for which every prime of \(V'\) dominating \(Q\) is either \(\pi'\)-exceptional or maps into \(D\). This definition first addresses components that dominate base primes. It gives (71) for all components of the total pullback because of the equidimensional model \(V_0\). To see this, put \(n=\dim X\) and \(r=n-\dim Y'\). If a prime \(R\subset V'\) maps into a set \(A\subset Y'\) of codimension at least two, then \[\dim(V_0\times_{Y'}A)\le\dim A+r\le n-2.\] The image of \(R\) in \(V_0\), hence in \(X\), has dimension at most \(n-2\), since \(V_0\to X\) is a morphism. Thus \(R\) is \(\pi'\)-exceptional. Every other component of \((f')^*D'\) dominates a component of \(D'\) and is covered by its definition. This proves (71), including the divisors introduced by the last resolution.

At a general point of a boundary prime \(Q\), choose a component \(R\subset(f')^*Q\) dominating it, and write \[k=\operatorname{ord}_R(f')^*Q>0,\qquad b=\operatorname{ord}_R\bigl(K_{V'/X}+c(\pi')^*D\bigr)\ge0.\] The psh weight of \(h_M\) is locally bounded above. Hence the pullback measure is bounded below by a positive constant times \(|z_1|^{2b}\) times coordinate volume near a general point of \(R\). Coordinates can be chosen there with \(t_1=z_1^k\), where \(Q=(t_1=0)\), and with the other base coordinates among the remaining source coordinates. Integration in a fixed small fiber polydisc and change of variables give \[ I_0(t)\ge C|t_1|^{2((b+1)/k-1)} \tag{77}\] almost everywhere near a general point of \(Q\). If \(Q\not\subset D'\), choose \(R\) nonexceptional over \(X\) and not mapping into \(D\). Then \(b=0\); the exponent in (77) is nonpositive, so in particular \(I_0\) has a positive constant lower bound there.

There are only finitely many boundary primes. Choose an integer \(c'\ge0\) at least as large as each positive exponent \((b+1)/k-1\) needed for a component of \(D'\). If \(d'\) is an equation of \(D'\), the psh function \[-\log I_0+c'\log|d'|^2\] is locally bounded above at the generic points of all boundary primes. It extends across them, and then across the remaining analytic subset of codimension at least two by the Hartogs extension property for psh functions. These are the weights of a semipositive metric on \(-K_{Y'}+c'D'\). Dividing its metric by the divisor metric of \(c'D'\) recovers the measure with density \(I_0\). It has finite mass because it is the pushforward of the original finite measure.

Finally take a nonzero rational section of \(mB\) with poles on \(D'\), pull it back, and apply the \(m\)-th power of (70). The only possible poles lie in \((f')^*D'\). Equation (71) shows that its pushdown has poles only on \(D\), and it is nonzero at the generic point. In the smooth invariant case this rational section is invariant by (76); since \(D=0\), normality makes it a global regular section on \(X\). ◻

Proof. Its leaves define a rational fibration. Fix a sufficiently ample line bundle \(H\), independent of the integer \(l\ge2\). Nadel vanishing and Castelnuovo–Mumford regularity make \[\mathcal O_X(lM+H)\otimes\mathcal J(l\varphi)\] globally generated. The local Bergman approximation inequality gives, for its local base ideal generators \(g_j\), \[ e^{l\varphi}\le C_l\sum_j|g_j|^2. \tag{79}\] The fixed ample twist can be chosen by applying Nadel vanishing after subtracting \(1,\ldots,\dim X\) times a fixed very ample line bundle. The local inequality follows from point extension in the weighted holomorphic \(L^2\) space; generators of the multiplier ideal represent the singularities of that space. These are the usual Bergman approximation and global generation constructions of Demailly (Demailly 2015, Theorems 0.3 and 1.2).

Let \(\rho:U\to X\) resolve the rational fibration and principalize this base ideal, with its divisor \(R_l\), and let \(E_U=K_{U/X}\). Take a simultaneous log resolution with \(\rho^*D\). Finite mass and (79) imply, along every prime on \(U\), \[ \lfloor R_l/l\rfloor \le E_U+c\rho^*D. \tag{80}\] Indeed the comparison makes \(|s_{E_U+c\rho^*D}|^2|s_{R_l}|^{-2/l}\) locally integrable. At a prime with respective orders \(b\) and \(r\), integrability says \(b-r/l>-1\); since \(b\) is integral this is exactly \(\lfloor r/l\rfloor\le b\).

Pull back the linear subsystem \[H^0\bigl(X,\mathcal O_X(lM+H)\otimes\mathcal J(l\varphi)\bigr) \subseteq H^0(X,lM+H)\] to \(U\), and remove its fixed divisor \(R_l\). Let \(G_l\) be a general member of the resulting basepoint-free system, and put \[\Delta_l=R_l/l-\lfloor R_l/l\rfloor+G_l/l.\] Bertini and the log resolution make \(\Delta_l\) an effective klt boundary, also klt on the general fiber of the resolved fibration. Moreover, \[ K_U+\Delta_l\sim_{\mathbb Q} E_U+c\rho^*D-\lfloor R_l/l\rfloor+\rho^*H/l. \tag{81}\] By (80), a positive divisible multiple of the right side restricts to an effective divisor on the general fiber. One may take a general effective representative of \(H\), which does not contain that fiber.

The relative pseudoeffectivity theorem for an algebraically integrable foliation now gives \(K_{\mathcal F_U}+\Delta_l\) pseudoeffective on \(U\), where \(\mathcal F_U\) is the induced foliation (Ou 2023, Proposition 3.1, arXiv version 4); its relative positivity input is due to Druel (Druel 2017, Proposition 4.1). The hypotheses used are the effective boundary, log canonicity of its general-fiber restriction, and the general-fiber nonvanishing in (81). Push down to \(X\), and add back the effective divisor \(\rho_*\lfloor R_l/l\rfloor\). Since foliation determinants agree away from the exceptional locus, this proves that \(K_{\mathcal F}+M+H/l\) is pseudoeffective. Its limit as \(l\to\infty\) proves (78). ◻

Proof. Induct on \(n=\dim X\); the assertion for a point is immediate. Suppose the conclusion fails. The class \(L+D\) is pseudoeffective and not big, so cone duality gives a nonzero movable class \(\alpha\) with \[L\cdot\alpha=D\cdot\alpha=0\] (Boucksom et al. 2013). Proposition 50 and the following slope argument apply. Rational connectedness gives \(H^i(X,\mathcal O_X)=0\) for \(i>0\), so \(\chi(X,K_X)=(-1)^n\ne0\). Riemann–Roch makes \(\chi(X,K_X+mL)\) a nonzero polynomial. Thus one cohomological degree occurs with nonzero cohomology for an unbounded sequence of positive \(m\). Hard Lefschetz with multiplier ideals supplies twisted holomorphic forms for that sequence; the multiplier ideals are trivial because the metric weights on \(mL\) are bounded (Demailly et al. 2001).

Apply the determinant construction of Lemma 6 to these forms. Its fixed determinant maps nontrivially to a cotangent tensor power, so its \(\alpha\)-degree is nonpositive by the slope consequence of Proposition 50. We obtain a fixed line bundle \(B_0\) and effective divisors \[ N_j\sim jL+B_0,\qquad \alpha\cdot N_j=0 \tag{82}\] for an unbounded set of positive integers \(j\).

Let \(K_\alpha\subset\mathbb C(X)\) be the field of functions whose polar divisors have \(\alpha\)-degree zero. By Lemma 28, it is relatively algebraically closed, its transcendence degree is less than \(n\), and it is represented by a rational fibration with connected generic fiber. Resolve it to \(\pi:V\to X\), \(f:V\to Y\), with \(Y\) smooth projective. For a general sufficiently ample divisor \(H_Y\) on \(Y\), \[ \alpha\cdot\pi_*f^*H_Y=0. \tag{83}\] Indeed a general ratio from \(|H_Y|\) belongs to \(K_\alpha\); the two general pulled-back divisors have no common component, so its pole divisor represents the indicated class.

The affine interpolation of Lemma 13 applied to (82) produces an integer \(a>0\) and a nonzero rational section \(\tau\) of \(a\pi^*L\) that is regular over a dense base open. Explicitly, the linear equivalence \[(k-i)N_j\sim(k-j)N_i+(j-i)N_k \qquad(i<j<k)\] has its defining function in \(K_\alpha\). Consequently the coefficients of \(\pi^*N_j\) at every horizontal prime are affine in \(j\); nonnegativity at unbounded \(j\) makes each slope nonnegative. A difference of two such divisors has no negative horizontal part and gives \(\tau\). If \(Y\) is a point, this already gives a global section of a positive multiple of \(L\).

Suppose therefore that \(\dim Y>0\), and put \(E=K_{V/X}\). For every integer \(p\ge0\), \(s_Es_{c\pi^*D}\tau^p\) is a nonzero generic-fiber section of \(E+c\pi^*D+pa\pi^*L\). Any finite set of other generic-fiber sections becomes global after a common sufficiently ample base twist. The pushdowns of their effective divisors have \(\alpha\)-degree zero, by (83), \(\pi_*E=0\), and \(\alpha L=\alpha D=0\). Ratios of two such sections therefore belong to \(K_\alpha\), proving (69) for every \(p\), including zero. This argument establishes generic ranks and requires no simultaneous base-change assertion for infinitely many sheaves.

Proposition 47 produces \(Y',B,D'\). The variety \(Y'\) is rationally dominated by \(X\), so it is rationally connected, and its dimension is smaller than \(n\). Induction gives a section of \(mB+rD'\) for some \(m>0\), \(r\ge0\). Transfer returns a nonzero rational section of \(md_0aL\) with poles only on \(D\), the required contradiction. ◻

Proof. For the trivial action this is the argument of Section 12.3 with \(D=0\). Otherwise take a rational quotient and an equivariant resolution \(\pi:V\to F\), \(f:V\to Y\). Its invariant field is \(\mathbb C(Y)\), and the generic fiber is an orbit closure. In particular it is connected. On the smooth quotient locus, choose \(\dim V-\dim Y\) generically independent vector fields from the torus action and wedge them. Commutativity makes their wedge an invariant section of \(-K_{V/Y}\). Tensor it with a rational anticanonical frame from \(Y\), then multiply by the Jacobian section \(s_E\), where \(E=K_{V/F}\). This produces an invariant rational section \(\tau\) of \(\pi^*L\), regular over a dense base open.

For every \(p\ge0\), the invariant generic-fiber adjoint space for \(E+p\pi^*L\) contains \(s_E\tau^p\). Two nonzero invariant sections have an invariant ratio, hence a ratio in \(\mathbb C(Y)\). Its invariant rank is exactly one, also at \(p=0\). Apply the smooth invariant part of Proposition 47, using the given smooth metric for both \(L\) and \(-K_F\). If \(Y\) is a point, \(\tau\) is already a global invariant section. Otherwise the resulting original-field base \(Y'\) is rationally connected; apply the argument of Section 12.3 there and transfer its section. The pole-support statement and \(D=0\) make the resulting invariant section regular on \(F\). ◻