Metric descent and rank-preserving contractions

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 (2.2) 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 (2.2).

If E dominates Q with a_Q>0, its coefficient in \Theta satisfies

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

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

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

Its coefficients are less than one by the sub-klt assertion, giving (2.4).□

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

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

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

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

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

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 (3.1) makes

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

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

For a local canonical frame \eta_Y on the base, the section

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 [22, Theorem 21.1], based on [36, 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 [9, Theorem 0.1]. In the invariant case, use smooth direct-image positivity [8, 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

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 [18, 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 [27, Lemma 4.1], used for anticanonical nonvanishing in [26, Lemma 5.1].□

Proof. The divisor

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

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

In (4.10), 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

Choose analytic units V_i' having the same first jets as V_i. We seek analytic units U_j' satisfying

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

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

If some b_{ij}>0, strict positivity follows from (4.9) 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 (4.6) almost everywhere, and hence on the general locus where the density is continuous.

For (4.7), take a general point of a component which realizes the minimum in (4.5). At that point take coordinates

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

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 3.2 and 3.1. 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 3.2 are therefore satisfied.

The zeroth density is \rho from (4.4); hence -\log\rho is psh on the smooth base-change locus. The normalized base divisor in Lemma 3.1 is rV, by the full-component minimum in (4.2). 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

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 3.2; no singular invariant-subbundle positivity is invoked.

We now use Lemma 4.2 to construct the signed log-anticanonical metric. Put M=-K_Z-B. On the general open its local metric weight is

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

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 [30], applied after adding the smooth positive metric on (d+1-i)A, gives

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 (4.14) 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

is pseudoeffective.

Apply the orbifold cotangent theorem [13, 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

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

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 (4.17) gives

Letting t decrease to zero proves the assertion.□

Proof. Choose b>0 with bV Cartier. Lemma 4.3 gives bV bounded semipositive weights, and the base has no positive-degree forms by the argument preceding (4.1). Apply Lemma 3.3 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 4.4 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 (4.19). For j<k<l, linear equivalence gives

Both sides are effective divisors in the same zero-degree system. By Lemma 4.6, their ratio is a rational function from W. Its pullback has no horizontal divisor over W. Thus (4.20), 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

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

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 [30, 31, 16].

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

We claim that

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

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 (5.1) 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

This globalization by a fixed ample twist is the point-extension construction in [15, Section 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 [16, Corollary 13.9, Remark 13.10, and Theorem 14.2]. For independent standard complex Gaussians c_i and m>1,

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 (5.1).□

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

The bracket map vanishes by the slope inequality, and the Campana–Păun algebraicity theorem makes \mathcal F algebraically integrable [35, 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 [21, 13]. Movable-cone duality proves the assertion [11]. 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 5.1 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

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 [20, Proposition 4.1], [35, 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 5.2 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 5.1 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

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} [13, 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

Let m increase to obtain (5.2).□

Proof. In a local frame of N, write \sigma for the regular normalized section. On the smooth open set, the moments have limit

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, (5.3) 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 (5.4). The generator has squared norm

where d\mu is the smooth volume defined by the metric of L. Lemma 3.2 applies: Berndtsson positivity is used on the smooth base-change locus and the invariant line is a holomorphic orthogonal summand. Lemma 5.5 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

Outside B choose a nonexceptional component, so a_E=0. On the finitely many components of B choose c large enough that

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

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 3.2 supplies the psh moment weights, and Lemma 5.5 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

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 2.2. 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,

On a simultaneous normal-crossing resolution, these are precisely the exponents of the corrected source measure in the projection formula

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

For all sufficiently large m, (5.7) 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

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

The first extends with locally bounded weights by regularity of the normalized section and a minimizing component above every divisor, by Lemma 5.5.

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

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

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

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

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

It is big on that fibre and hence has a nonzero section after taking a positive divisible multiple. Apply Druel’s relative pseudoeffectivity theorem [20, Proposition 4.1], in the form of [35, Proposition 3.1], to conclude that K_{\mathcal F'}+C^++A_\delta is pseudoeffective. Its pushforward is numerically

Letting \delta decrease to zero proves K_{\mathcal F}-K_Y+B^- pseudoeffective. Lemma 5.2 now proves the assertion for every movable class, and therefore in the pseudoeffective cone.□

First proof of Theorem 6.1. 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

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 3.3 gives N_i\sim M+m_iL, while Proposition 5.3 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

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 3.2 and 3.1 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 3.1 carries it to a section with poles only on D. An integral multiple of D clears those finitely many poles.□

Second proof of Theorem 6.1. 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

The zero-degree part of Proposition 5.3 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

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 (6.1), 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

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 3.2 and 3.1 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

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

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 3.2 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 6.1 applies with D=0 and A=L=-K_V. Otherwise take the rational quotient with field \mathbb C(V)^T [38, 5], 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

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 3.2 and Lemma 3.1 gives a bounded semipositive base bundle and a finite-volume donor metric. The base has no holomorphic forms by dominant pullback, and Theorem 6.1 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

For a smooth real weight b whose Hessian b_{xx} is positive definite on \bar D, put

The Brascamp–Lieb bound is

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 [25, 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

The mean-zero condition is precisely its solvability condition on the ball. Weighted integration by parts and the differentiated equation give

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

Combining the two identities gives (7.2), including the case of zero variance by continuity.

Apply (7.2) 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

The factors 1/2 in complex vertical derivatives cancel against the factor 1/4 in the vertical Hessian. Equations (7.1) and (7.2) 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 [7, p. 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

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 7.1. If the torus acts trivially, apply Theorem 6.1 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 [38, 5] for the quotient and [28, 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

is smooth and psh in (y,x+i\theta), and is independent of \theta by compact invariance. In one coordinate,

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

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 7.2. 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 7.2 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 3.1 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 6.1 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 4.6 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 3.2 and 3.1 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 8.1 verifies the smooth invariant adjoint rank condition for every j\geq0. Thus the two required psh functions are supplied by Lemma 3.2.

The full-component normalization in Lemma 3.1 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

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 [11], 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 8.2. 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 [4, 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 (8.1), choose g_0\in K^* satisfying

Set

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

Let \Gamma be a non-null prime of Y. It is nonomitted, and d^*_\Gamma=d_\Gamma by Lemma 8.4. 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

Taking the minimum over these nonexceptional components proves

The rational divisor \operatorname{div}_Y(g)+kbD has total \alpha-degree zero. Its coefficients at every non-null prime are nonnegative by (8.3), 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

This includes k=0, so fixed-section uniqueness is preserved in its full stated range.

Strict transcendence-degree decrease follows from Lemma 8.2. 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 [38]. 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 [14, 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 [10]. 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 [29, Section 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 [37]. 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

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,

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

Write e_\pi for the invariant Jacobian section of K_Z+\pi^*L. On a dense open of U, the sheaf

is locally free, commutes with base change, and has its adjoint L^2 metric. Berndtsson’s theorem gives Griffiths semipositivity [8]. 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

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 (9.2) 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

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

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

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

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

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

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

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

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 [21, 13, 35, 26]. 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 (9.6) 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 (9.6). 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

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

The assumed cotangent estimate and the determinant proof in Lemma 4.5 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 4.6 therefore shows that ratios in any one of these systems lie in \mathbb C(Y'). Moreover \dim Y'<\dim Y. The interpolation of Lemma 4.7 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

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

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 8.1 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

Rational coefficients cause no ambiguity: this is an inequality of rational divisors. These spaces need not be finite-dimensional. Their multiplication law is

By (9.9) 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 (9.11), 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

The closure in the first condition of (9.9) 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

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)^*,

Away from the omitted primes this is precisely the hypothesis of (9.8), 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

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

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 (9.14) implies

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

with pullbacks to the common resolution understood. Its horizontal coefficients are nonnegative by (9.7) 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

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

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

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

The parameter is still greater than a, also when r=0. It follows that

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 5.6. 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

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 3.3, 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 (4.20), 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 10.1 makes the resulting representatives of (j-i)Q_S equal, and cancellation gives

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

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

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 10.1.

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

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 (10.1) 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 5.7. 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 5.7 is supported on B. Proposition 5.4 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 10.1 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

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 10.1, 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 5.8 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,

The unweighted metric of J=-K_Y+kN+R and Proposition 5.4 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 8.2 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 8.5 therefore gives j>i, v=m_j-m_i>0 and c\in K^* with

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 (5.6).

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

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:

The hypothesis is the fixed-section uniqueness condition of Lemma 8.1. It gives invariant adjoint rank one in every nonnegative multiple of a: divide by the Jacobian and test the nonexceptional horizontal primes. Hence Lemma 4.3 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

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:

In particular \alpha\cdot B_{\mathrm{sgn}}^-=0, so Proposition 5.9 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 8.2; it has smaller transcendence degree. On a common resolution, the affine interpolation of Lemma 4.7 gives j>i and

with effective horizontal part over K'. One may instead use the finite-support extraction of Proposition 8.5 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

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 8.2 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 8.6 supplies the invariant section. Ratios of invariant rational tensors of the same power of L lie in \mathbb C(X)^T=K, which proves (10.2).□

Proof. Suppose first that N+tB is big for some sufficiently small rational t>0. Choose a nonzero section of a divisible multiple. Pull it back, use the corresponding power of s, and push its divisor to X. Condition (ii) leaves an effective divisor. All rational coefficients can be cleared by the power, so this is a regular nonzero invariant section of a positive multiple of L. We may therefore assume that N+tB is not big for all small positive rational t.

For every line bundle M admitting a nonzero map M\to(\Omega_Y^1)^{\otimes p}, Proposition 5.9 gives

The case p=0 follows directly from M\to\mathcal O_Y. Choose an integer b>0 clearing all divisor denominators, divisible by \ell, and compatible with the generic-rank hypothesis. If s_{bN} denotes the canonical rational section of bN, the normalized section

It is regular and invariant. Consider the line bundles

where the last map is nonzero. This collection is closed under tensor product: tensoring two nonzero line maps gives a nonzero map at the generic point and hence a nonzero global map into the corresponding tensor power.

No member of (11.2) is big. Indeed, (11.1) would make kN+aB+p\Delta^- big if Q were big. Choose a'\geq a with a'B-aB-p\Delta^-\geq0. Then kN+a'B would be big. For any prescribed small t>0, choose \delta>0 so that \delta k<1 and \delta a'\leq t. Since N and B are pseudoeffective,

would be big, contrary to the assumption.

Choose a nonempty system in a member Q_0 of (11.2) whose image has maximal dimension. The constant system shows that such systems exist. Resolve its map and pass to the connected-fiber base to obtain

We can choose Y' normal projective; its smooth birational model may be used later. Its dimension is strictly smaller than \dim Y, since Q_0 is not big. A suitable power of the chosen system supplies an ample line A on the normal Stein base with \eta^*Q_0-q^*A effective, after replacing Q_0 by that power. Every ratio from every system in (11.2) belongs to \mathbb C(Y'). Otherwise the product with the chosen system would increase the image dimension; relative algebraic closure removes the remaining algebraic extension.

For the generic fiber F of q this implies

To check the assertion for generic sections, spread two putative independent sections after a sufficiently ample base twist q^*(nA). The effective difference \eta^*(nQ_0)-q^*(nA) puts them in \eta^*(Q+nQ_0). Their pushdowns are sections on the smooth Y, and their ratio contradicts the preceding field statement.

Because Y is a rational image of the rationally connected X, it has no positive-degree holomorphic forms. Apply Lemma 3.3 to bN. We obtain one cotangent line M_0 and effective divisors D_i\sim M_0+m_iN for an unbounded increasing sequence m_i\in b\mathbb Z_{>0}. Their restrictions to F are nonzero section divisors. By (11.3), the products realizing any three-index linear relation are proportional on F. Therefore each horizontal multiplicity is affine in m_i. All these multiplicities are nonnegative at unbounded exponents, so the slope is nonnegative. A difference D_j-D_i, j>i, consequently gives a nonzero section of (m_j-m_i)\eta^*N on F.

Write \delta=m_j-m_i, and choose \gamma\in\mathbb C(Y)^* with

The integer \delta is divisible by b and hence by \ell. Resolve the composite map from X to Y' by a smooth model Z' dominating Z and \widetilde Y; write j:Z'\to Y' and continue to write g:Z'\to Y, \pi:Z'\to X, with D pulled back from Z. The rational section

The first summand is effective, and the second is effective on the complete generic fiber over Y', as just proved. Thus \tau is a nonzero invariant generic section. The composite generic fiber is geometrically connected, since both successive function-field inclusions are relatively algebraically closed in characteristic zero.

It remains to restore the adjoint ranks on this larger fiber. Suppose two invariant sections of K_{Z'/X}+k\pi^*L are independent over \mathbb C(Y'), for a sufficiently divisible k\geq0. Clear their poles over Y' by j^*(nA), and pull back the effective difference \eta^*(nQ_0)-q^*(nA) to regard them as sections of

Over \mathbb C(Y) the old rank is one. Divide the two sections by the Jacobian section and \sigma^{k/b}, using the pullback of the normalized section defined above. The quotients descend to rational sections v_1,v_2 of kN+nQ_0 on Y. For a prime P outside B, condition (iii) supplies a prime R above it whose strict transform has order zero in both D and K_{Z'/X}. If e_R=\operatorname{ord}_R g^*P, regularity of the two sections upstairs gives

Thus neither v_i has a pole outside B. A sufficiently large a therefore makes both sections regular in kN+nQ_0+aB, which again belongs to (11.2). Their ratio lies in \mathbb C(Y'), a contradiction.

The new generic section, multiplied by the Jacobian section, supplies a nonzero adjoint section at every positive multiple of its exponent; the Jacobian supplies one at zero. Combining these lower bounds with the upper bound proves all required ranks after enlarging the fixed divisibility integer. Finally replace the normal Stein base Y' by a smooth projective model and resolve the composite map. Its function field is unchanged, and generic sections and Jacobian-twisted ranks are unchanged by this birational replacement, as in Section 4.1. This completes the contraction.□

Proof. Choose a sufficiently high power A of a relatively ample line bundle so that

are vector bundles, their formation commutes with arbitrary base change, and g^*E_A\to A is surjective. Relative Serre vanishing and flatness supply these assertions simultaneously. Multiplication by \tau defines a nonzero element of \operatorname{Hom}(E_A,F_A)_{\eta}. Take the closure of the associated rational section of the projective bundle of lines in \operatorname{Hom}(E_A,F_A), and resolve this closure. The resulting base modification carries a line subbundle

In particular its map to each fiber of this Hom bundle is nonzero.

The induced map g^*J\otimes g^*E_A\to A\otimes P factors through g^*J\otimes A. Indeed, the kernel of g^*E_A\to A is locally free. Its composite with this map vanishes over the generic base point, where the map is multiplication by \tau. It therefore vanishes everywhere: flatness over the integral base makes the generic part schematically dense, and the target is locally free. Dividing by A gives a:g^*J\to P. If its restriction to some fiber were zero, the corresponding map (E_A)_t\to(F_A)_t would be zero, contrary to the line subbundle construction. The prescribed generic equality defines \eta_J.□

Proof. Properness gives the upper bound. On a reduced fiber, a nonzero section of a line bundle is nonzero at a point in its smooth locus. Choose such a point over t. Flatness and smoothness of the fiber at that point make g smooth there. After shrinking, a local analytic section of g, together with continuity of the metric, supplies a positive lower bound on a neighborhood of t. Finitely many such neighborhoods cover the compact base. Reducedness is used precisely here: a nonzero section supported only in the nilpotent structure of a fiber would not supply a point at which its norm is positive.□

Proof. On a dense smooth base open, use coordinates y and write I_p(y) for the density of

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 3.2 applies to (12.1), including p=0. It gives psh representatives of -\log I_0 and

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 [3, 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

Let P be the pullback to W of a\pi^*L. Lemma 12.1 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

by Lemma 12.2. 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

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 [37]. The second map is finite: it is projective, flat and has zero-dimensional fibers. Its degree

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

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

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 (12.6) 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 [39, Tags 0BUT, 0BCX and 0BD2].

Over the generically unramified locus the norm is the product over the d_0 sheets. Applying (12.5) on those sheets gives

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

It is psh on that open. Equation (12.7) 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

defines a rational homomorphism in (12.2) 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, (12.7) and the definition of the essential supremum give

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 (12.8) 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 (12.3) 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

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 (12.3), 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

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

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 (12.9) 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

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 (12.2). The only possible poles lie in (f')^*D'. Equation (12.3) 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 (12.8); 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

globally generated. The local Bergman approximation inequality gives, for its local base ideal generators g_j,

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 [17, 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 (12.11) imply, along every prime on U,

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

to U, and remove its fixed divisor R_l. Let G_l be a general member of the resulting basepoint-free system, and put

Bertini and the log resolution make \Delta_l an effective klt boundary, also klt on the general fiber of the resolved fibration. Moreover,

By (12.12), 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 [35, Proposition 3.1, arXiv version 4]; its relative positivity input is due to Druel [20, Proposition 4.1]. The hypotheses used are the effective boundary, log canonicity of its general-fiber restriction, and the general-fiber nonvanishing in (12.13). 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 (12.10).□

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

[11]. Proposition 12.7 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 [18].

Apply the determinant construction of Lemma 3.3 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 12.7. We obtain a fixed line bundle B_0 and effective divisors

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 8.2, 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,

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 4.7 applied to (12.14) 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

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 (12.15), \pi_*E=0, and \alpha L=\alpha D=0. Ratios of two such sections therefore belong to K_\alpha, proving (12.1) for every p, including zero. This argument establishes generic ranks and requires no simultaneous base-change assertion for infinitely many sheaves.

Proposition 12.4 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 12.4, 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.□