The boundary construction and its antecedents

An effective rational boundary \(\Delta\) is klt (Kawamata log terminal) if, on a log resolution \(\sigma:Y'\to Y\), every coefficient of \(\sigma^*(K_Y+\Delta)-K_{Y'}\) is less than one. For a simple normal crossing boundary on a smooth variety, this amounts to each boundary coefficient being less than one.

The proof connects multiplier-ideal approximation with big-boundary nonvanishing. Nadel vanishing (Nadel 1990), followed by very ample regularity, makes the relevant multiplier-ideal systems globally generated. Point extension of Ohsawa–Takegoshi (Ohsawa and Takegoshi 1987), in the weighted Bergman approximation developed by Demailly (Demailly 2015), compares the fixed divisor of a complete system with the metric’s singularities. On a resolution \(\sigma:Y'\to Y\), weighted integrability produces a signed divisor \(\Psi=\Psi^+-\Psi^-\) whose coefficients are strictly below one. Its negative part lies only over \(C\) and the exceptional locus.

The approximation leaves an ample rational class \(A\) in reserve. For a prescribed pseudoeffective \(M\), choose a small rational \(t>0\) so that \(A+tM\) remains ample. General free and ample representatives then turn the positive part of the signed divisor into an effective big klt boundary \(\Delta\), with \[K_{Y'}+\Delta\sim_{\mathbb Q}t\sigma^*M+\Psi^-.\] This is the pair to which the nonvanishing theorem of Birkar–Cascini–Hacon–McKernan (Birkar et al. 2010, Theorem D) applies. A finite rational linear system upgrades its real-linear conclusion to rational linear equivalence. Pushforward gives the correction \(\sigma_*\Psi^-/t\), supported on \(C\). Sections 2 and 3 give this construction. The latter also retains an analytic perturbation argument: strong openness of Guan–Zhou (Guan and Zhou 2015) and small-exponent integrability of Skoda (Skoda 1972) permit a small pseudoeffective perturbation of the metric while preserving the integrability of the distinguished section.

A different consequence begins with an unweighted integrable semipositive metric that is smooth and strictly positive on one ordinary open set. Boucksom’s volume criterion (Boucksom 2002) first gives bigness; a small mixture with a strict metric then gives a global curvature lower bound while preserving integrability. Section 4 makes this preparation explicit and permits perturbation in finitely many arbitrary rational directions. The resulting klt boundaries fit the multigraded adjoint finite-generation results of Cascini–Lazić (Cascini and Lazić 2012) and Corti–Lazić (Corti and Lazić 2013, Theorem 3.2(1)). More precisely, if \(-K_Y+A_0\) has this unweighted metric, then for every finite list of rational divisors \(P_1,\ldots,P_s\) there are a common rational \(\epsilon>0\) and Cartier multiples \(B_i=k_0(A_0+\epsilon P_i)\) whose multigraded section ring is finitely generated (Theorem 9). This controls the sections of specified Cartier divisors, not merely a numerical cone.

Anticanonical nonvanishing and fixed-error conversion

For a smooth projective variety \(X\), anticanonical nonvanishing asks whether some \(H^0(X,-mK_X)\), \(m>0\), is nonzero when \(-K_X\) is nef. Numerical effectivity is weaker: an effective divisor in the numerical class need not supply a section of the actual line bundle. For pairs \((X,\Delta)\), the corresponding divisor is \(-(K_X+\Delta)\), the anti-log-canonical divisor. Lazić–Matsumura–Peternell–Tsakanikas–Xie (Lazić et al. 2023, Theorem A) established numerical nonvanishing for projective log canonical threefold pairs with nef anti-log-canonical divisor, assuming either that the variety is \(\mathbb Q\)-factorial or that it has rational singularities. Müller (Müller 2025, Corollary B) proved nonvanishing for projective klt threefold pairs with nef anti-log-canonical divisor. His higher-dimensional theorem assumes, in addition to nefness, that the anti-log-canonical divisor of the induced pair on the general fiber of the maximal rationally connected fibration is semiample (Müller 2025, Theorem A).

Our principal conversion result uses smooth metric semipositivity and works in every dimension. Let \(X\) be smooth connected projective, let \(L=-K_X\) carry a smooth semipositive Hermitian metric, and let \(P\) be any fixed pseudoeffective Cartier divisor. Then Theorem 10 gives the implication \[H^0(X,m_jL-P)\ne0\ \text{for unbounded positive }m_j \quad\Longrightarrow\quad H^0(X,mL)\ne0\ \text{for some }m>0.\] The fixed error \(P\) may itself have no effective representative. A single Iitaka base converts the effective divisors into a pseudoeffective rational divisor on that base. The controlled-boundary theorem makes this divisor rationally effective with correction on base primes whose pullbacks disappear over \(X\). A rational function from the base then corrects the original rational section without introducing a pole on \(X\).

Twisted differential forms supply the sequence through the determinant method of Lazić–Peternell (Lazić and Peternell 2018, Lemma 4.1), adapted in (Lazić et al. 2023, Lemma 5.1). Ou’s generic tangent-nefness theorem (Ou 2023, Theorem 1.4), in the cotangent-subsheaf form (Lazić et al. 2023, Theorem 4.1), makes the negative determinant pseudoeffective. Our extraction argument starts with positive unbounded exponents and also includes numerically trivial \(L\). Thus nonzero sections of \((\Omega_X^1)^{\otimes p}\otimes m_jL\) for one fixed \(p\ge0\) and unbounded positive \(m_j\) give an anticanonical section by Corollary 15. The same holds for exterior forms. This is an ordinary conversion; invariant sections require further control of characters.

Volume metrics and the distinct geometric refinements

Let \(X\) be smooth projective and let \(L=-K_X\) carry a smooth semipositive metric. In a resolved rational fibration \[X\xleftarrow{\ \pi\ } Z\xrightarrow{\ g\ }Y,\] assume \(Z,Y\) are smooth projective, \(\pi\) is birational, and \(g\) is surjective. Write \(K_{Z/X}=K_Z-\pi^*K_X\) for the relative canonical (Jacobian) divisor. Its canonical section belongs to \(K_Z+\pi^*L\). When \(\operatorname{rank}g_*\mathcal O_Z(K_{Z/X})=1\), its squared fiberwise \(L^2\) norm is a volume density \(\rho\) on the base. On the smooth base-change open where this section generates the adjoint line, \(-\log\rho\) is a metric on \(-K_Y\). Berndtsson’s direct-image theory supplies its positivity (Berndtsson 2009, 2011). Extending it over the boundary with enough integrability is the geometric task preceding effectivity.

The main conversion proof uses a singular coefficient metric whose curvature dominates a positive base form and whose total-space multiplier ideal is trivial. Singular adjoint direct-image positivity of Păun–Takayama (Păun and Takayama 2018) then gives a metric with a global strict lower bound on \(-K_Y+C\). Its integral boundary has exceptional pullback, and the distinguished section is locally square integrable.

Keeping the coefficient metric smooth gives a different output. The coarea formula and a primitive-representative curvature calculation produce an integral correction \(C\) with \(\pi_*g^*C=0\) and \[|s_C|^2e^{-\phi}\in L^{1+\eta}_{\mathrm{loc}} \quad\text{for some }\eta>0.\] The metric is smooth and strictly positive on one ordinary open set. The higher integrability exponent permits a mixture with a globally strict metric while keeping \(s_C\) integrable. Thus the global bound needed for Theorem 1 follows from an explicit additional step.

The companion Metric descent and rank-preserving contractions supplies the toroidal-volume construction used here (OpenAI 2026c, Theorem 2.1). For a normal projective model \(p:W\to X\) and a surjective equidimensional toroidal map \(f:W\to Y\), its theorem assumes adjoint rank one at degree zero on a smooth resolution and a patch, away from the Jacobian divisor, where the pulled-back curvature dominates a positive base form. It produces an effective rational correction \(A_0\) with \[p_*f^*A_0=0\] and a semipositive metric on \(-K_Y+A_0\) with unweighted locally integrable squared frame norms, smooth and strictly positive on a nonempty ordinary open set. That companion proves the toroidal Jacobian calculation, the all-valuation integrability and the boundary extension. We use its metric after checking the stated model, rank and curvature premises in each application. The rational correction is used directly in Corollary 13; no rounding or replacement of its unweighted metric by an integral-boundary hypothesis is needed.

For the invariant problem, Müller (Müller 2025, Theorem C) obtains naturally invariant sections for projective sub-log-canonical pairs with semiample anti-log-canonical divisor and a commutative linear algebraic group action. Theorem 24 here assumes \(X\) smooth connected projective, \(-K_X\) smoothly semipositive and \(\chi(X,\mathcal O_X)\ne0\). For any algebraic torus acting on \(X\), it gives \[H^0(X,-mK_X)^T\ne0\quad\text{for some }m>0\] with the natural tangent-determinant linearization. The companion invariant-index theorem, including its actual value at zero, supplies invariant cohomology (OpenAI 2026b, Theorem 1.1); compact-averaged hard Lefschetz then gives forms with the coefficient shift \(m+1\). The local determinant and character arguments supply the sections needed here. The finite-cover structure and norm used in the isometric application are those of the companion headline paper, (OpenAI 2026a, Theorem 2.1 and Lemma 2.2), which develops the geometric work of Demailly–Peternell–Schneider and Campana–Demailly–Peternell (Demailly et al. 1996; Campana et al. 2015). Our algebraic boundary criterion, smooth coarea proof and correction ring are independent of that application.

Finally, a suitable auxiliary base has an effective big klt boundary \(\Delta\) with \(K_Y+\Delta\sim_{\mathbb Q}C\) for an effective rational \(C\) supported on divisors disappearing over \(X\). When \(\kappa(Y,C)=0\), a log terminal model of the base is of Fano type: it admits an effective rational boundary \(\Theta\) with klt singularities such that \(-(K+\Theta)\) is ample. This is a statement about the auxiliary model. Its proof retains the precise contracted-prime support needed to return the resulting divisors to \(X\).

The main route is the boundary and ring results followed by the single-Iitaka conversion in Section 5. Sections 6 and 7 retain the smooth coarea estimate and direct jet-separation proof of bigness; their applications reuse the common return of sections. Section 8 then develops the supported Fano-type model. The invariant base constructions in Sections 9 and 10 lead to compact-monodromy descent in Section 11. Finally, Section 12 constructs invariant tensors from extremal characters of the full equivariant Euler characteristic. That construction is independent of the weight-zero index input and descends tensors before applying ordinary conversion.

Preserving integrability under a pseudoeffective perturbation

One can also insert \(M\) at the level of metrics, before approximation. This gives analytic information in addition to the effectivity proved above: a small pseudoeffective perturbation preserves the integrability of the distinguished section and the global strict curvature bound.

Subtract a sufficiently small ample rational \(A\) from the perturbed metric, retaining its strict lower bound. Applying Lemma 2 to \(-K_Y+C+\epsilon M-A\) with \(r>1\) then gives a second boundary construction. For its signed divisor \(\Psi\) and a general free member \(D_{\mathrm{mov}}\), add a general small-coefficient representative \(U\sim_{\mathbb Q}A\). Then \(\Delta=\Psi^++D_{\mathrm{mov}}/r+\sigma^*U\) is effective, big and klt, and \(K_{Y'}+\Delta\sim_{\mathbb Q}\epsilon\sigma^*M+\Psi^-\). Nonvanishing, rational recovery and pushforward apply as before. Here the approximation itself depends on the metric on \(M\); the support conclusion is unchanged.

Direct selection of an integrable divisor

There is a second way to perform the last step of Lemma 7. It chooses a section by averaging negative powers of its norm, instead of using a general member on a log resolution. This gives a direct analytic description of the effective klt representative.

Let \(D\) be a rational line bundle with a globally strictly positive metric of weights \(\phi_D\), and suppose \(e^{-\phi_D}\) is locally integrable. For a large divisible integer \(m>1\), take an orthonormal basis \(s_1,\ldots,s_N\) of the global sections of \(mD\) that are square integrable with weight \(m\phi_D\) and a fixed smooth volume form. The global Bergman estimate is \[ \frac1m\log\sum_{j=1}^N|s_j|^2\ge\phi_D-C_m \tag{7}\] on each relatively compact coordinate patch. We recall the construction because this version uses global sections of \(mD\) without a fixed auxiliary twist.

Fix a point \(y\) where \(\phi_D(y)\) is finite. Point extension on a coordinate ball gives a local holomorphic section taking squared value \(e^{m\phi_D(y)}\) at \(y\), with uniformly bounded weighted norm. Multiply it by a cutoff equal to one on a smaller ball and solve the resulting \(\bar\partial\) equation. In top holomorphic degree the coefficient line is \(mD-K_Y\). Add a cutoff logarithmic pole \(d\log|z-y|^2\) to the estimating weight, where \(d=\dim Y\). For sufficiently large \(m\), the strict curvature of \(mD\) absorbs both the fixed canonical curvature and the negative curvature of the cutoff pole, uniformly for \(y\) in a smaller ball. The correction is holomorphic near \(y\); its integrability against \(|z-y|^{-2d}\) forces its value there to vanish. The corrected global section retains the specified value and has uniformly bounded \(m\phi_D\)-weighted norm.

For the singular-weight solution, first choose rational trivializing sections of an integral multiple of \(D\) and of \(K_Y\). Enlarge the support of their zeros and poles by an effective divisor so that the resulting divisor is ample. Its complement is affine and trivializes the required integral line bundles. Use Stein exhaustions of that complement, approximate the psh weights on relatively compact exhaustion sets and pass to weak limits. In bidegree \((d,1)\) the \(L^2\) estimate uses contraction with the inverse positive line-curvature matrix; its bound is independent of an auxiliary complete Kähler metric chosen to dominate a fixed reference metric. The estimates therefore persist through the exhaustion. Since the original psh weights are locally bounded above, weighted square integrability implies ordinary square integrability near the removed divisor. The resulting holomorphic section extends across it. The local weighted estimate is Demailly’s Bergman approximation (Demailly 2015, sec. 1, Theorem 1.2), with the point-extension input of Ohsawa–Takegoshi (Ohsawa and Takegoshi 1987). The strict global curvature is what permits the untwisted global construction just given. Expanding the constructed section in the orthonormal basis proves (7); the estimate is automatic on the polar set of the weight.

It follows that \((\sum_j|s_j|^2)^{-1/m}\) is locally integrable. Average over the unit sphere in \(\mathbb C^N\). Unitary invariance gives \[\int_{\|a\|=1}\left|\sum_{j=1}^Na_js_j(y)\right|^{-2/m}\,da =c_{m,N}\left(\sum_{j=1}^N|s_j(y)|^2\right)^{-1/m}.\] For \(N\ge2\), the possible singularity is an inverse power of one complex coordinate and \(c_{m,N}<\infty\) because \(m>1\). For \(N=1\) the identity is immediate. Fubini on a finite covering gives a nonzero section \(s\) of \(mD\) with \(|s|^{-2/m}\) locally integrable. On a log resolution, change of variables identifies this condition with all discrepancy coefficients of \(\operatorname{div}(s)/m\) being less than one. Thus \(\operatorname{div}(s)/m\) is the required effective klt rational representative of \(D\).

Vertical minima and return of sections

We record the return step once, including the exceptional divisors that can appear when a smooth model ceases to be equidimensional.

The required model condition follows from flattening, rather than from an assertion that a final smooth resolution is flat. Starting with a resolved rational map and its Stein factorization, flatten over a modification of the base (Raynaud and Gruson 1971, Theorem 5.2.2). Normalize the main component, resolve the base, flatten again if necessary, and then resolve the source. On the intervening flat model a prime cannot map into base codimension two: its inverse image has dimension at most \(\dim X-2\). Any prime on the final resolution with that property is consequently exceptional over the intervening model, and hence over \(X\).

The Iitaka base and its strict metric

The smooth toroidal metric for the same base

A toroidal embedding is locally modeled on a toric variety with its dense torus; its boundary corresponds to the toric boundary. We use strict toroidal embeddings, whose boundary components are normal. A toroidal morphism is locally a toric morphism, described on the dense tori by monomials. This local description keeps track of the exact ramification exponents in the metric correction.

There is another way to supply the metric on the Iitaka base. It keeps the original smooth anticanonical metric throughout fiber integration and gives unweighted integrability. We give the geometric verification because the two metric constructions have different conclusions.

We also isolate the algebraic consequence of unweighted integrability.

Prepare the rational Iitaka map birationally as an equidimensional strict toroidal morphism \(f:W\to Y\), with \(W\) normal, \(Y\) smooth, and \(p:W\to X\) birational. Such a preparation follows from weak toroidalization (Abramovich et al. 2013, Theorem 1.1) and equidimensional toroidal subdivision (Abramovich and Karu 2000, Proposition 4.4). Mark the nonisomorphism locus in the source boundary before these operations. No finite base alteration or reduced-fiber assertion is used here. Take a smooth resolution \(Z\to W\), with maps \(\pi:Z\to X\) and \(g:Z\to Y\). The resolution itself need not be equidimensional. The generic rank-one argument (10) is unchanged. Likewise the horizontal interpolation and the vertical minima may be computed on \(W\), where every vertical prime dominates a base prime.

Equation (8), pulled to \(Z\), says that a multiple of \([\Theta_{\pi^*L}]\) dominates the pullback of a positive base class pseudoeffectively. Lemma 12 gives a horizontal strictness patch above the smooth base-change locus, away from the Jacobian divisor. The Jacobian section is nonzero there. The model, rank-one, and strict-patch hypotheses have now been verified for the original smooth coefficient metric. These are the hypotheses of the toroidal-volume theorem in (OpenAI 2026c, Theorem 2.1). That theorem supplies a rational \(A_0\ge0\) with \(p_*f^*A_0=0\) and the unweighted integrable metric of Corollary 13. Its coefficients are \[\operatorname{ord}_Q A_0= \max\left\{0,\min_{D\mapsto Q} \left(\frac{1+\operatorname{ord}_D K_{W/X}} {\operatorname{ord}_D f^*Q}-1\right)\right\}.\] The all-valuation integrability and extension assertions here are part of that theorem, rather than consequences of generic rank one alone. Corollary 13 and Lemma 11, on a common resolution, complete the same conversion. The output here is unweighted integrability for a rational correction: a nonvanishing frame has finite norm. The smooth coarea construction in Section 6 instead gives an integral correction and an \(L^{1+\eta}\) bound for the distinguished section. Lemma 20 converts that output to the same supported effectivity premise in Lemma 11.

The field of all pseudoeffectively dominated systems

A second base construction uses every effective system dominated by a multiple of \(L\). It is useful when no distinguished divisor should be chosen in advance. We retain the different maximality argument and the explicit integral correction it produces.

Call a subsystem of an integral divisor \(D\) admissible if it has a nonzero section and \(bL-D\) is pseudoeffective for some integer \(b>0\). Choose finitely many admissible systems whose joint image has maximum dimension \(d\). Every further admissible ratio is algebraic over this image field. Take its relative algebraic closure and a smooth projective connected-fiber model \(\pi:Z\to X\), \(f:Z\to Y\), prepared by flattening as before. For every ample \(H\) on \(Y\) there is a \(b_0>0\) with \[ b_0\pi^*L-f^*H\quad\hbox{pseudoeffective}. \tag{12}\] Indeed the sum of the chosen divisors dominates the pullback of the product polarization; its restriction to the joint image is big. The sum itself is pseudoeffectively dominated by a multiple of \(L\), and a large multiple of the product polarization dominates \(H\). If \(d=\dim X\), this makes \(L\) big.

Take sections \(u_j\) defining \(N_j\sim m_jL-P\), put \(r=m_2-m_1\), and let \(t=\pi^*(u_2/u_1)\), a rational section of \(r\pi^*L\). For \(j>2\) the expressions \(u_j^r u_1^{m_j-m_2}\) and \(u_2^{m_j-m_1}\) belong to the same admissible system, namely a multiple of \(m_2L-P\). Writing \(D_t=\operatorname{div}(t)\) gives \[r\pi^*N_j=r\pi^*N_1+(m_j-m_1)D_t+ f^*\operatorname{div}(g_j), \qquad g_j\in\mathbb C(Y)^*.\] Thus \(D_t\) has nonnegative horizontal part, and its divisor of vertical minima \(M\) is pseudoeffective by the same fixed-error inequality used in (9). The case \(d=0\) is already settled.

For \(d>0\), the full adjoint sheaf \(f_*\mathcal O_Z(E)\), \(E=K_{Z/X}\), has rank one. Otherwise two generic-fiber independent sections can be made global in \(E+kf^*H\). After pushforward their effective divisors give a subsystem of \(k\pi_*f^*H\) on \(X\) whose ratio is not in \(\mathbb C(Y)\). By (12) this system is admissible, contradicting maximality. Mixing a positive-current metric furnished by (12) with the smooth pullback metric again gives a globally defined coefficient \(h\) with \(\Theta_h\ge\delta f^*\omega_Y\) and \(\mathcal J(h)=\mathcal O_Z\). The singular direct-image theorem therefore applies to the full actual sheaf as in the first proof. Let \(w\) be the pushed-forward Jacobian volume density on its smooth base-change locus. Then \(-\log w\) has curvature at least \(\delta\omega_Y\) there, and \(w\) has finite local mass.

Here the boundary can be written directly. Set \[c_Q=\min_{D\mapsto Q} \left\lfloor\frac{\operatorname{ord}_D E} {\operatorname{ord}_D f^*Q}\right\rfloor, \qquad C=\sum_Qc_QQ.\] It is effective and integral. If a prime over \(Q\) is nonexceptional then its coefficient in \(E\) is zero, so \(c_Q=0\). Hence \(f^*C\) is exceptional, including the primes over codimension two by the model condition. Near a general point of a prime \(D\) attaining the minimum, write \(f=(z_1^e,z_2,\ldots,z_d)\) and \(a=\operatorname{ord}_D E\). A plurisubharmonic coefficient weight is bounded above on a relatively compact chart. Its metric norm is therefore bounded below by a positive constant times a smooth norm. Integration of this chart gives \[w(y)\ge c|y_1|^{2(a+1-e)/e}.\] If \(a=qe+s\), \(0\le s<e\), then \((a+1-e)/e=q+(s+1-e)/e\le q=\lfloor a/e\rfloor\). It follows that \(\psi=-\log w+\log|s_C|^2\) is locally bounded above at general points of every missing divisor. Extend plurisubharmonically across these divisors and then across codimension two, subtracting local potentials of \(\delta\omega_Y\) to preserve the lower bound. This gives a metric on \(-K_Y+C\) with global strict curvature and \[|s_C|^2e^{-\psi}=w\in L^1_{\mathrm{loc}}.\] Theorem 1 applies to \(M\). Lemma 11, now with the rational section \(t\), completes the alternative construction. Thus its distinct ingredients are the field of all admissible systems, full rank one from maximality, and the explicit floor correction; the algebraic return is unchanged.

From twisted tensors to a fixed error

The determinant method of Lazić–Peternell (Lazić and Peternell 2018, Lemma 4.1), and its formulation in (Lazić et al. 2023, Lemma 5.1), provides the link to twisted differentials. Those cited extraction statements include a numerical-nontriviality hypothesis. The following direct construction also covers a numerically trivial \(L\), because the starting exponents here are positive and unbounded.

A movable-slope proof of the determinant input

For completeness we give a second proof of the pseudoeffectivity used in Lemma 14, under its smooth-metric application hypotheses. This proof explains the ramification correction in the relative-volume argument. It uses movable slope theory (Greb et al. 2016), duality between pseudoeffective divisors and movable curves (Boucksom et al. 2013), and the positive-minimal-slope algebraicity theorem of Campana–Păun (Campana and Păun 2019, Theorem 1.1).

For a torsion-free sheaf \(\mathcal E\) and movable curve class \(\alpha\), its slope is \(\mu_\alpha(\mathcal E)=c_1(\mathcal E)\cdot\alpha/ \operatorname{rank}\mathcal E\). The symbols \(\mu_{\max,\alpha}\) and \(\mu_{\min,\alpha}\) denote the greatest and least slopes in its Harder–Narasimhan filtration, whose successive quotients are semistable with decreasing slopes.

Suppose a saturated subsheaf of a cotangent tensor power on smooth projective \(X\) had determinant of positive degree against a movable curve class \(\alpha\). Tensor-product slope inequalities imply \(\mu_{\max,\alpha}(\Omega_X^1)>0\), equivalently \(\mu_{\min,\alpha}(T_X)<0\). Since \(c_1(T_X)\cdot\alpha\ge0\), the positive-slope part \(\mathcal F\) of the Harder–Narasimhan filtration of \(T_X\) satisfies \[\mu_{\min,\alpha}(\mathcal F)>0, \qquad\mu_{\max,\alpha}(T_X/\mathcal F)\le0, \qquad c_1(T_X/\mathcal F)\cdot\alpha<0.\] The bracket obstruction from \(\bigwedge^2\mathcal F\) to \(T_X/\mathcal F\) vanishes by the slope inequality. Thus \(\mathcal F\) is a foliation, and the cited algebraicity theorem realizes it as the relative tangent foliation of a rational map with connected general fibers. Prepare an equidimensional model of that map, then take a smooth resolution with maps \(\mu:W\to X\) and \(f:W\to Y\). Put \(L=-K_X\) and retain its smooth semipositive metric.

The relative adjoint bundle \(K_{W/Y}+\mu^*L\) has a nonzero section on a general fiber: there it is represented by the effective Jacobian \(K_{W/X}\). On the smooth bundle and base-change locus, evaluation with the \(L^2\) direct-image metric supplies its Bergman weight \[\Phi(x)=\log\sup_{\|u\|_{L^2}\le1}|u(x)|^2\] in a local relative-adjoint frame. Semipositivity follows from the dual-norm form of direct-image positivity (Berndtsson 2009, 2011); zeros of the evaluation map give allowed singularities. On the isomorphism locus of \(\mu\), the submersive differential identifies this line with \(\det(T_X/\mathcal F)\). The only extra codimension-one issue is ramification.

A prime of \(X\) vertical for the map dominates a base prime on the normal equidimensional model. Near its general point, and hence on its strict transform in \(W\), choose coordinates \[f=(z_1^e,z_2,\ldots,z_{\dim Y}).\] The differential contributes \(z_1^{e-1}\). Therefore \(\det(T_X/\mathcal F)\) corresponds to the relative adjoint line twisted down by \((e-1)\{z_1=0\}\). A relative form written in total-space/base canonical frames acquires the factor \((ez_1^{e-1})^{-1}\) when restricted to a fiber. Integration on a fixed local fiber chart and the holomorphic submean estimate consequently bound every unit-norm section coefficient by \(C|z_1|^{e-1}\). It follows that \[\Phi\le(e-1)\log|z_1|^2+O(1).\] After the indicated twist, the induced weight is locally bounded above at the general point of every such prime. At horizontal primes the ordinary submean estimate gives the same upper boundedness without a ramification term. These weights extend plurisubharmonically across codimension one, and then across the remaining codimension-two locus on \(X\). They define a semipositive singular metric on \(\det(T_X/\mathcal F)\), making that line pseudoeffective. This contradicts its negative degree against \(\alpha\).

By movable-cone duality, the negative determinant of every cotangent tensor subsheaf is therefore pseudoeffective. The argument retains the factor \(e-1\) because omitting it would construct a metric on the wrong line at ramified divisors.

The primitive representative and horizontal strictness

For a higher-dimensional base we apply Lemma 16 to small embedded disks in its tangent directions. In rank one a local holomorphic frame may be chosen Chern-normal at a specified point. Dividing (13) by its squared norm then gives the curvature of minus the logarithm of that norm. A lower bound by the pullback of a positive base form on one patch yields strict positivity in every nonzero tangent direction at the image point. If the relative dimension is zero and the generic rank is one, the smooth map is locally an isomorphism and this assertion follows directly from the coefficient metric.

A coarea estimate

Extension across base divisors

From local strictness to a global strict metric

The metric just constructed is strictly positive on one open set. The next step uses the extra integrability exponent to introduce a global lower curvature bound while preserving the distinguished integrable section.

The smooth metric on the maximal dominated base

We now apply the smooth coarea construction to the maximal base of Section 5.4. That base already supplies the interpolation and the pseudoeffective divisor of vertical minima. The additional points here are a valuation proof of the model condition and a direct intersection argument for horizontal strictness. These allow us to use the original smooth metric, and hence retain the exponent \(1+\eta\) in Proposition 18.

Retain the hypotheses of Theorem 10, with effective integral divisors \(N_j\sim a_jL-D\), where \(D\) is pseudoeffective and \(a_j\) is a strictly increasing unbounded sequence of positive integers. A rational map is dominated by \(L\) if, on a smooth resolution \(X\xleftarrow{\ g\ }Z\xrightarrow{\ f\ }Y\), one has \[g^*L-\epsilon f^*H\quad\hbox{pseudoeffective}\] for an ample \(H\) on the projective base and a rational \(\epsilon>0\). Include the point map. This is the map formulation of the admissible systems used in Section 5.4. Indeed, an admissible system gives this inequality after resolving its base ideal. Conversely, after multiplying \(H\) so that it defines a projective embedding, its pulled-back sections push down to a subsystem of \(|g_*f^*H|\); pushing down the domination inequality makes that system admissible.

The condition persists on higher source resolutions and under birational base modifications. It also persists under a generically finite change of base to which the rational map lifts dominantly: the pullback of the old ample class is big and dominates a small ample class. Finally, the joint image of two dominated maps is dominated. On a common source resolution, add suitable positive multiples of their inequalities and use the sum of the pulled-back polarizations, whose restriction to the joint image is ample. Thus the maximality construction in Section 5.4 provides a base field \(\mathbb C(Y)\), relatively algebraically closed in \(\mathbb C(X)\), containing the functions supplied by every dominated map.

For the resulting smooth projective diagram put \[P=\frac{g^*N_2-g^*N_1}{a_2-a_1}\sim_{\mathbb Q}g^*L, \qquad B=a_1P-g^*N_1.\] The interpolation argument of Section 5.4 gives \(\mathbb Q\)-principal divisors \(U_j\) on \(Y\) with \[ g^*N_j=a_jP-B+f^*U_j,\qquad U_1=U_2=0. \tag{20}\] For \(j>2\), the two effective divisors in \((a_j-a_1)N_2\sim(a_j-a_2)N_1+(a_2-a_1)N_j\) lie in a multiple of the admissible system \(N_2\), so their ratio belongs to \(\mathbb C(Y)\). In particular \(P\) has nonnegative horizontal part. If \(Y\) is a point, pushing \(P\) down already proves the theorem. Assume henceforth that \(b=\dim Y>0\).

A base lemma proved by jets and a nef-and-big decomposition

The Iitaka base revisited

We apply the jet construction to the Iitaka base already used in Section 5.2. The algebraic return remains Lemma 11; the new point is a fiber-intersection proof that the original smooth metric is strictly positive in base directions. We also record the stronger generic-fiber rank statement behind this version of the argument.

Retain \(X\) and \(L=-K_X\) with its smooth semipositive metric from Theorem 10. Write the given effective integral divisors as \(N_m\sim mL-D\) for an unbounded set of positive integers \(m\), with \(D\) pseudoeffective Cartier, and fix two exponents \(j<q\). Use the Iitaka fibration of \(N_q\) constructed in Section 5.2. That construction settles the cases of zero-dimensional and generically finite base. In the remaining case it gives \[X\xleftarrow{\ \pi\ }S\xrightarrow{\ f\ }Y, \qquad 0<b=\dim Y<n=\dim X, \qquad u\pi^*N_q\sim f^*A_0+I,\] where \(A_0\) is ample Cartier, \(u\) is a positive integer, and \(I\ge0\). The varieties are smooth projective, \(f\) has connected fibers, and every source prime mapping into base codimension at least two is \(\pi\)-exceptional. Every ratio from a multiple of \(N_q\) belongs to \(\mathbb C(Y)\), which is relatively algebraically closed in \(\mathbb C(X)\). These are precisely the model and field properties established there; no flatness of the final smooth resolution is needed.

Put \(E=K_{S/X}\) and \(D_S=\pi^*N_q+E\). Adding positive multiples of \(E\) does not change the sections of multiples of \(\pi^*N_q\), because \(E\) is exceptional and \(X\) is normal. Thus \(\kappa(D_S)=b\) and \(uD_S-f^*A_0\) has an effective representative. In fact \[ \operatorname{rank}f_*\mathcal O_S(\ell D_S)=1 \qquad(\ell\in\mathbb Z_{>0}). \tag{31}\] For if this rank exceeded one, an ample base twist would yield two sections of \(\ell D_S+af^*A_0\) independent over \(\mathbb C(Y)\). Multiplication by the section of \(a(uD_S-f^*A_0)\) places them in \((\ell+au)D_S\). Their ratio is transcendental over \(\mathbb C(Y)\). Other sections of a sufficiently divisible multiple recover \(b\) algebraically independent base functions through \(A_0\). Products put all these ratios into one multiple of \(D_S\), contradicting \(\kappa(D_S)=b\). The canonical sections show that the rank cannot be zero. At the generic point \(\eta\) of \(Y\), this gives \(h^0(S_\eta,E|_{S_\eta})=1\), and \(\pi^*N_q|_{S_\eta}\) has Iitaka dimension zero.

The interpolation and vertical-minimum construction of Section 5.2, with these indices, gives \[R=\frac{\pi^*N_q-\pi^*N_j}{q-j}\sim_{\mathbb Q}\pi^*L, \qquad W=\sum_T\min_{V\mapsto T} \frac{\operatorname{ord}_V R}{\operatorname{ord}_V f^*T}\,T.\] Here \(R\) has nonnegative horizontal part and \(W\) is pseudoeffective. The rank statement above also gives the interpolation directly: for each given \(m>q\), restrict the two effective divisors \((m-j)N_q\sim(m-q)N_j+(q-j)N_m\) to the generic fiber. Its one-dimensional space of sections makes their horizontal divisors equal. The horizontal slope is therefore nonnegative as \(m\) tends to infinity. The principal difference has vertical divisor, so its restriction to the projective normal generic fiber is constant; hence it is pulled back from \(Y\). Taking vertical minima in this equality recovers the pseudoeffectivity of \(W\) by the fixed-error limit used above.

Call a base prime missed when every source prime above its generic point is \(\pi\)-exceptional. The model property also makes all other components of its pullback exceptional. By Lemma 11, it is enough to construct a weighted metric satisfying Theorem 21 on \(-K_Y+A\), with \(A\) supported on missed primes. We now verify the only new ingredient, horizontal strictness, using intersections on the fibers.

Horizontal positivity from fiber intersections

Let \(\widetilde L=\pi^*L\) and let \(\alpha\) be its smooth semipositive curvature form. Since \(D\) is pseudoeffective and \(u\pi^*N_q-f^*A_0\) has an effective representative, \(k\widetilde L-f^*A_0\) is pseudoeffective for some \(k>0\). Let \(J\) be ample on \(S\), put \(d_0=n-b\), and choose the largest \(t\in\{0,\ldots,d_0\}\) such that \[(\widetilde L^tJ^{d_0-t}\cdot S_y)>0\] on a smooth general fiber. The choice exists because \(t=0\) works. These intersections are constant on the smooth base locus. Semipositivity then makes the restriction of \(\alpha\) have rank at most \(t\) at every point of every such fiber.

The pseudoeffective comparison and nefness give \[((k\widetilde L)^b\cdot\widetilde L^tJ^{d_0-t}) \ge ((f^*A_0)^b\cdot\widetilde L^tJ^{d_0-t})>0.\] Indeed the difference of the \(b\)-th powers factors as the pseudoeffective class \(k\widetilde L-f^*A_0\) times a sum of products of nef classes. The final strict inequality is the projection formula and the definition of \(t\). The smooth representative shows that \(\alpha\) has rank at least \(b+t\) on a nonempty open set.

A vertical null vector for a nonnegative Hermitian form is null for the full form. The vertical null space has dimension at least \(d_0-t\), so the total rank is at most \(b+t\) on the smooth locus. On the preceding open set equality holds, and the full kernel is vertical. A smaller constant-rank patch, disjoint from \(E\), therefore satisfies \[\alpha\ge c f^*\omega_Y\] for some \(c>0\) and a positive base form \(\omega_Y\).

The adjoint rank is one by (31). Proposition 18 now gives an integral effective \(A\) supported on missed primes and a semipositive metric on \(-K_Y+A\), smooth and strictly positive on an ordinary open set, with weighted density in \(L^{1+\gamma}_{\rm loc}\) for some \(\gamma>0\). Theorem 21 makes \(W+C_A\) rationally effective with \(C_A\ge0\) supported on \(A\). Lemma 11 applied to \(R\) gives a section of a positive multiple of \(L\). This completes the jet-based proof of Theorem 10.

A fibration tangent to the null directions

The cases \(\dim X=0\) and \(p=0\) are immediate. Suppose otherwise. Write \(\mathcal E=(\Omega_X^1)^{\otimes p}\) and apply the determinant extraction in Lemma 14. Retain its determinant line \(M\), rather than its negative: there is a nonzero map \[M\longrightarrow\bigwedge^s\mathcal E\] for some \(s>0\), and effective integral divisors \[ N_m\sim M+mL \tag{32}\] for an unbounded set of positive integers \(m\). The generic wedges used in that lemma are regular because they take values in the saturated subsheaf and its reflexive determinant. This is the determinant method of (Lazić and Peternell 2018, Lemma 4.1) and (Lazić et al. 2023, Lemma 5.1), with the unbounded exponents supplied here as part of the hypothesis.

Write \(n=\dim X\), let \(\alpha\) be the normalized curvature of \(L\), and let \(\nu\) be its maximum rank. If \(\nu=n\), then \(L\) is nef and \(L^n=\int_X\alpha^n>0\), so \(L\) is big and conversion follows. Fix an ample class \(A\) and assume \(\nu<n\). We will use \[ N_m\cdot[\alpha]^\nu A^{n-1-\nu}=0. \tag{33}\] Here is a differential-geometric proof of this equality. For each \(\epsilon>0\), Yau’s prescribed-Ricci theorem (Yau 1978) gives a Kähler metric in \([\alpha]+\epsilon A\) whose Ricci form equals the prescribed semipositive form \(\alpha\), up to the fixed positive normalization. Kähler symmetry identifies the mean curvature of the tangent bundle with its nonnegative Ricci endomorphism. Thus the mean curvature of every cotangent tensor power, and of its exterior powers, is nonpositive. Consequently \[c_1(M)([\alpha]+\epsilon A)^{n-1}\leq0.\] Indeed, a positive degree would allow a metric on \(M\) with constant positive mean curvature, by the scalar Poisson equation. The induced mean curvature on \(\bigwedge^s\mathcal E\otimes M^{-1}\) would then be negative definite, and the integrated Bochner formula would forbid its nonzero section. Since \(\alpha^{\nu+1}=0\), the coefficient of the lowest possible power \(\epsilon^{n-1-\nu}\) is a positive constant times \(c_1(M)[\alpha]^\nu A^{n-1-\nu}\) and is nonpositive. The same intersection for \(N_m\) equals it by (32), and is nonnegative by effectivity. This proves (33).

Choose a finite sum \(D\) of the \(N_m\), allowing repetitions, whose Iitaka dimension \(d\) is maximal among such sums. These dimensions are integers in \(\{0,\ldots,n\}\), so a maximum exists. Choose \(q>0\) for which \(|qD|\) has image dimension \(d\). On the open set where \(\alpha\) has rank \(\nu\), the differential of this rational map annihilates \(\ker\alpha\). To see this, each member of \(|qD|\) has zero intersection with \([\alpha]^\nu A^{n-1-\nu}\). A smooth local component whose tangent hyperplane failed to contain \(\ker\alpha\) would retain rank \(\nu\) for the restricted form, and hence contribute positive mass to that intersection. Away from the base locus, a projective hyperplane detecting a nonzero differential on \(\ker\alpha\) would produce exactly such a smooth local component. Hyperplanes detect the differential of the map, so the assertion follows. In particular \(d\leq\nu<n\).

Resolve the base ideal by \(p_1:X_1\to X\), with \(X_1\) smooth and the resolution an isomorphism off that ideal, and take the Stein factorization \(f_1:X_1\to B\). The normal projective base \(B\) has relatively algebraically closed function field in \(\mathbb C(X)\). There are an ample divisor \(A_B\) and an effective fixed divisor \(I\) with \[ p_1^*(qD)\sim f_1^*A_B+I. \tag{34}\] Every \(p_1\)-exceptional prime lies in \(\operatorname{Supp}I\), since its center lies in the base ideal and that ideal has positive order on the prime.

Fix \(m_1<m_2\) among the determinant exponents and put \[G=\frac{N_{m_2}-N_{m_1}}{m_2-m_1}\sim_{\mathbb Q}L, \qquad J=N_{m_1}-m_1G.\] On any compatible higher model \(\mu:W\to X\), \(f:W\to Y\), with \(Y\) birational over \(B\), there are \(\mathbb Q\)-principal divisors \(T_m\) on \(Y\) such that \[ \mu^*N_m=m\mu^*G+\mu^*J+f^*T_m\qquad(m>m_2). \tag{35}\] For the principal difference of \[(m_2-m_1)N_m+(m-m_2)N_{m_1} \quad\hbox{and}\quad(m-m_1)N_{m_2}\] is a rational function on \(B\). Otherwise it is transcendental over \(\mathbb C(B)\), by relative algebraic closedness, and adjoining it to the map for \(|qD|\) increases the image dimension. The joint map is given by products of sections in the system for the permitted finite sum \(qD+(m-m_1)N_{m_2}\), contradicting maximality. Dividing its principal divisor by \(m_2-m_1\) proves (35). If \(d=0\), these functions are constant: \(N_m=mG+J\), so every coefficient of \(G\) is nonnegative by letting \(m\to\infty\). This already proves conversion. We henceforth have \(0<d<n\).

We next establish the adjoint rank required for the volume metric. For each integer \(\ell>0\), the general fiber of \(f_1\) satisfies \[ h^0(X_{1,b},\mathcal O_{X_1}(\ell I)|_{X_{1,b}})=1. \tag{36}\] If the dimension were larger, generic base change and a sufficiently ample base twist would give sections of \(\ell I+t f_1^*A_B\) whose ratio varies on the general fiber. For \(t\gg\ell\), multiply them by the section of \((t-\ell)I\). They become sections of \(t p_1^*(qD)\). The same complete system recovers the base using the sections from \(|tA_B|\) multiplied by \(s_I^t\), so its image dimension exceeds \(d\), a contradiction. The Jacobian divisor \(K_{X_1/X}\) is effective and bounded by some \(\ell I\). Its nonzero canonical section and (36) therefore give rank one for its full direct image. Further smooth modifications do not change this rank: their extra relative canonical divisors are effective and exceptional, and their section pushforwards equal \(\mathcal O\) by normality. Thus on any smooth resolution \(\pi:Z\to X\), \(g:Z\to Y\) dominating these models, \[ \operatorname{rank}g_*\mathcal O_Z(K_{Z/X})=1. \tag{37}\] The Jacobian section identifies \(g_*(K_{Z/Y}+\pi^*L)\) with \(-K_Y\) on the smooth base-change locus. This is an ordinary rank assertion, involving the entire adjoint space.

The integrable correction and the model it permits

Apply birational weak toroidalization (Abramovich et al. 2013, Theorem 1.1), marking the nonisomorphism locus over \(X\), and equidimensional toroidal subdivision (Abramovich and Karu 2000, Proposition 4.4). We obtain \(\mu:W\to X\), \(f:W\to Y\) with \(W\) normal projective, \(Y\) smooth projective, and \(f\) surjective, equidimensional and toroidal for strict toroidal embeddings. All modifications are birational; no reduced-fiber alteration is required. The source can be singular, and its resolution is used only for integration. The effective exceptional Weil divisor \(K_{W/X}\) is supported in the toroidal boundary.

Choose a global Kähler form \(\omega_Y\) on \(Y\). By generic smoothness and cohomology-and-base-change, there is a nonempty Zariski open \(U\subset Y\) on which the resolved map \(g\) is smooth and its adjoint direct image is the line identified above. The kernel calculation persists off the exceptional loci, since the map from the base to the original image is generically finite. On a small patch above \(U\), in the maximum-rank region and away from the Jacobian divisor, it gives \[\ker\pi^*\alpha\subset\ker dg, \qquad \pi^*\alpha\geq c\,g^*\omega_Y\quad(c>0).\] The second inequality follows from the first by semipositive linear algebra and a lower bound for the nonzero eigenvalues on a smaller patch. Together with (37), these verify all hypotheses of the toroidal volume theorem in (OpenAI 2026c, Theorem 2.1).

Explicitly, for a base prime \(P\) and primes \(S\subset W\) dominating it, put \(a_S=\operatorname{ord}_S K_{W/X}\) and \(e_S=\operatorname{ord}_S f^*P\). The theorem supplies \[ C=\sum_P\max\left\{0,\min_{S\mapsto P} \left(\frac{1+a_S}{e_S}-1\right)\right\}P \tag{38}\] and a semipositive metric on \(\Lambda=-K_Y+C\) whose local squared frame norms are integrable and whose curvature is smooth and strictly positive on a nonempty ordinary open set. Moreover \(C\) is effective rational with finite support and \(f^*C\) is entirely exceptional over \(X\). The companion theorem proves the all-valuation integrability and the extensions across divisors and codimension two; these conclusions use the precise ordinary rank and patch verified here.

In particular \(\kappa(Y,C)=0\). Indeed, a rational function with poles bounded by a positive divisible multiple of \(C\) pulls back to a function with only \(\mu\)-exceptional poles. It extends across \(X\) by normality and is constant because \(X\) is connected projective. Also there is an effective big rational divisor \[ \Delta\sim_{\mathbb Q}\Lambda, \qquad (Y,\Delta)\ \hbox{klt}. \tag{39}\] Apply Corollary 8 to the unweighted integrable metric on \(\Lambda\). Its proof first subtracts a small ample rational class while keeping a klt representative, then adds a general representative of that ample class. Thus the boundary here is effective, big and klt, exactly as required by the birational theorem below.

Returning the correction to the original variety

Apply Theorem 22 to (39). We now finish this proof of Corollary 15. For each base prime \(P\) set \[g_P=\min_{S\mapsto P}\frac{\operatorname{coeff}_S\mu^*G}{e_S}, \qquad j_P=\max_{S\mapsto P}\frac{\operatorname{coeff}_S\mu^*J}{e_S}, \qquad B_g=\sum_Pg_PP,\quad B_j=\sum_Pj_PP.\] These are finite rational sums. The horizontal coefficients of \(\mu^*G\) are nonnegative by (35) and \(m\to\infty\). At a component attaining the minimum \(g_P\), that same equation and effectivity of \(\mu^*N_m\) give \[m g_P+j_P+\operatorname{coeff}_P T_m\geq0.\] Thus \(mB_g+B_j+T_m\geq0\). Push this inequality by strict transform to \(Y_0\). Principal terms remain principal because \(\chi\) extracts no divisors. Dividing numerical classes by \(m\) proves that \((B_g)_0\) is pseudoeffective. Theorem 22 supplies a \(\mathbb Q\)-principal adjustment making \((B_g)_0\) effective. Use its same rational functions and coefficients on \(Y\), and call the resulting principal divisor \(T_*\). Then \(B_g+T_*\) is effective outside the primes contracted by \(\chi\).

The divisor \(\mu^*G+f^*T_*\) has nonnegative coefficient at every prime of \(W\) that is not exceptional over \(X\). Horizontals were already treated. Above an uncontracted prime this follows from the defining minimum \(g_P\). Above a contracted prime, every component is exceptional over \(X\), since that prime lies in \(\operatorname{Supp}C\) and \(f^*C\) is exceptional. Equidimensionality leaves no other vertical primes. Pushing forward to \(X\) therefore gives an effective rational divisor. Its class is \(\mathbb Q\)-linearly equivalent to \(G\sim_{\mathbb Q}L\), because the base principal adjustment pulls back and pushes forward as a principal divisor. Clearing denominators completes the proof.

The Fano-type conclusion belongs to \(Y_0\), the model of an auxiliary linear-system base. The argument on \(X\) uses exactly the supported exceptional correction; it does not assert that \(X\) itself is of Fano type.

The invariant ratio field

Put \(B_j=A+m_jL\). Form the subfield generated by ratios of invariant sections of the same line bundle \(\sum_j k_jB_j\), where \(k_j\ge0\) are integers of finite support. Let \(F\) be its relative algebraic closure in \(\mathbb C(X)\). Both fields are finitely generated: choose a transcendence basis for the smaller field; the algebraic closure of the associated purely transcendental field in the finitely generated extension \(\mathbb C(X)\) is finite, and the intervening algebraic fields are finite as well. The field \(F\) is fixed by \(T\). Indeed an element algebraic over the invariant subfield has finite orbit among the roots of a polynomial, and connectedness makes this orbit a point.

Finitely many ratios can be put over a common denominator by multiplication of sections. Thus one bundle \(B=\sum k_jB_j\) and invariant sections of \(B\) define an image \(Y_0\) with the smaller function field. Normalize in \(F\), resolve the base, and obtain the invariant rational map \(X\dashrightarrow Y\), with \(\mathbb C(Y)=F\) and geometrically integral generic fiber. Let \(H_0\) be the pullback of the image polarization; it is big and base point free. On a smooth equivariant graph resolution \[X\xleftarrow{\ \pi\ }Z\xrightarrow{\ g\ }Y\] there is a nonzero invariant section of \(\pi^*B-g^*H_0\): divide the pulled-back defining sections by the homogeneous coordinates on the image. Hence for a chosen ample \(H\) on \(Y\) there is an integer \(b>0\) and an invariant section of \(b\pi^*B-g^*H\).

If \(F=\mathbb C\), consider \[ r_j=\frac{\sigma_j^{m_2-m_1}\sigma_1^{m_j-m_2}} {\sigma_2^{m_j-m_1}},\qquad j>2. \tag{42}\] Its numerator and denominator are invariant sections of the same multiple of \(B_2\), so \(r_j\) is constant. Divisor comparison and \(m_j\to\infty\) show that \(\operatorname{div}(\sigma_2/\sigma_1)\ge0\). This is the desired invariant section of \((m_2-m_1)L\). We may therefore suppose \(\dim Y>0\).

There is a horizontal strictness patch on \(Z\). If \(\nu=n\), use any maximum-rank point away from the exceptional and critical loci. If \(\nu<n\), Lemma 25 gives \(c_1(bB)\cdot[\theta]^\nu[\omega_0]^{n-1-\nu}\le0\). Since \(b\pi^*B-g^*H\) is effective, and all the forms involved are semipositive, it follows that \[\int_Z g^*\omega_H\wedge \pi^*(\theta^\nu\wedge\omega_0^{n-1-\nu})=0.\] At a maximum-rank point where \(\pi\) is an isomorphism, nonnegative linear algebra implies \(\ker\pi^*\theta\subset\ker dg\). Choose such a point over the smooth base-change locus. Shrinking a constant-rank neighborhood gives \[ \pi^*\theta\ge c g^*\omega_H,\qquad c>0, \tag{43}\] on a nonempty ordinary patch, disjoint from the Jacobian divisor. Rank zero is impossible here when \(\dim Y>0\).

On this equivariant model, with the trivial action on \(Y\), the invariant part of \(g_*\mathcal O_Z(K_{Z/X})\) has rank one. The natural Jacobian section gives rank at least one. If it were larger, global generation after a sufficiently ample base twist would give two invariant sections independent on the generic fiber. Multiplying them by a power of the invariant section of \(b\pi^*B-g^*H\) puts them in \(K_{Z/X}+k b\pi^*B\). Birational pushforward, using \(\pi_*\mathcal O_Z(K_{Z/X})=\mathcal O_X\), gives a ratio of invariant sections of \(kbB\) not in \(F\), a contradiction. This is invariant rank one; no assertion about the rank of the full direct image has been made.

We have obtained the analytic data: smooth invariant coefficients, invariant adjoint rank one, the natural nonvanishing Jacobian generator, and the horizontal patch (43). To obtain unweighted integrability with exceptional support, we now use a separate normal toroidal model. Equivariance of that model is not required because the meromorphic data on \(X\) are already invariant.

A klt model and its unweighted volume

By weak toroidalization and equidimensional subdivision (Abramovich et al. 2013; Abramovich and Karu 2000), prepare \[p:W\to X,\qquad f:W\to Y\] with \(W\) normal and simplicial toroidal, \(Y\) smooth, \(f\) equidimensional and toroidal, and the \(p\)-exceptional locus in the boundary. The base may be modified birationally, and the smooth equivariant graph resolution above is then replaced accordingly. For clarity, the simplicial condition means that the primitive rays of each local toric cone are linearly independent. Such toric charts are finite abelian quotients of smooth toric charts. The simplicial refinement can be made projectively using generic heights on the existing rays. Since the base cones are simplicial and source rays map to base rays or zero, each subdivided cone still maps onto a face of a base cone. Equidimensionality is preserved. Simplicial toroidal charts have quotient singularities; in particular \(W\) is \(\mathbb Q\)-factorial and klt.

Put \(D_j=\operatorname{div}(p^*\sigma_j)\) and \(V=(D_2-D_1)/(m_2-m_1)\sim_{\mathbb Q}p^*L\). Equation (42) gives \[D_j=D_1+(m_j-m_1)V+ \frac{f^*\operatorname{div}_Y(r_j)}{m_2-m_1}.\] Thus \(V\) has nonnegative horizontal part. Every vertical prime of \(W\) dominates a base prime. Its minimum divisor \[M=\sum_Q\left(\min_{D\mapsto Q} \frac{\operatorname{ord}_D V}{\operatorname{ord}_D f^*Q} \right)Q\] is pseudoeffective by the coefficientwise finite-error argument following (9). On \(W\) one has the actual inequality \(V\ge f^*M\).

The smooth metric of \(L\) defines a positive volume \(\mu\) on \(X\). On a smooth base-change open, let \(\rho\) be the coordinate density of \(g_*\pi^*\mu\). This measure can equally be computed on \(W_{\rm reg}\); birational changes affect only sets of measure zero. Its density is the squared norm of the Jacobian adjoint generator divided by the base canonical frame. Compact averaging on the smooth direct image is a holomorphic orthogonal projection onto the invariant line. Its norm is therefore the quotient norm. For positive-dimensional fibers, Lemma 16 gives \[i\partial\bar\partial(-\log\rho)\ge0,\] with strictness on a nonempty ordinary open by (43). If the relative dimension is zero, connected general fibers make \(g\) birational. Over its isomorphism open the integral metric is simply the transported coefficient metric; its curvature gives the same assertion directly, including strictness on the patch. This use of a smooth coefficient and an orthogonal quotient does not assert positivity of an arbitrary subbundle of a singular direct image.

Write \(E_W=K_{W/X}\ge0\). For a base prime \(Q\), set \[a_D=\operatorname{ord}_D E_W,\qquad e_D=\operatorname{ord}_D f^*Q,\qquad b_Q=\max\left\{0,\min_{D\mapsto Q} \frac{1+a_D}{e_D}-1\right\}, \quad F_0=\sum_Q b_QQ.\] This effective rational divisor has finite support, and \(f^*F_0\le E_W\). In fact, if \(b_Q>0\), then \(e_Db_Q\le1+a_D-e_D\le a_D\) for every \(D\mapsto Q\). If a prime above \(Q\) is nonexceptional over \(X\), its \(a_D\) is zero and \(b_Q=0\). Hence \(f^*F_0\) is exceptional over \(X\).

On \(-K_Y+F_0\) consider the weight \[\Phi=-\log\rho+\sum_Qb_Q\log|q_Q|^2,\] initially off the indicated divisors, with \(q_Q\) a local equation of \(Q\). At a general point of a prime \(D\) attaining the minimum, choose coordinates with \(f=(z_1^{e_D},z_2,\ldots,z_{\dim Y})\). The pulled-back volume density is comparable to \(|z_1|^{2a_D}\) times a smooth positive density. Integration of this chart gives \[\rho(y)\ge c|q_Q(y)|^{2((1+a_D)/e_D-1)}.\] Consequently \(\Phi\) is locally bounded above at a general point of every omitted divisor. Plurisubharmonic extension, first across these divisors and then across codimension two, gives a semipositive metric on the rational line \(-K_Y+F_0\), still smooth and strictly positive on an ordinary open.

Its squared frame norm is unweighted locally integrable. To check all valuations, take a log resolution \(h:W'\to W\) of the relevant data. Integrating \(e^{-\Phi}\) on the base is the same as integrating the pulled-back volume divided by the local divisor factors of \(f^*F_0\). The exponent divisor upstairs is \[K_{W'}-(ph)^*K_X-h^*f^*F_0 =K_{W'/W}+h^*(E_W-f^*F_0).\] The first summand has coefficients greater than \(-1\) because \(W\) is klt, and the second is effective because \(E_W-f^*F_0\) is effective and rationally Cartier. On a normal-crossings resolution all coefficients are therefore greater than \(-1\), which is exactly the local Jacobian integrability test. This includes exceptional divisors above higher-codimension centers.

Apply Corollary 13 to the pseudoeffective \(M\). It gives \(F_0+\epsilon M\sim_{\mathbb Q}G\ge0\) for some rational \(\epsilon>0\). Pullback and the inequality \(V\ge f^*M\) make \(V\) effective after a principal correction from \(Y\) and a divisor exceptional over \(X\). Equivalently apply Lemma 11 on a common resolution. The underlying rational section is a power of \(\sigma_2/\sigma_1\) times a base function, so it is invariant. Its extension on \(X\) proves Theorem 24.

Maximality and all positive adjoint ranks

Consider invariant dominant rational maps to projective bases with trivial action, subject on a resolution \(\pi:Z\to X\), \(g:Z\to Y\) to \[ \pi^*L-\delta g^*H\quad\hbox{pseudoeffective} \tag{44}\] for some ample \(H\) and rational \(\delta>0\). Include the point map. This condition is unchanged by replacing the resolution. It persists under birational modification of the base or its finite Stein refinement: the pullback of an ample divisor under a generically finite map is big and dominates a small ample class. It also persists for the joint image of two such maps, by addition and a product polarization. Choose a map of maximal base dimension, take the relative algebraic closure of its field in \(\mathbb C(X)\), and resolve the base. The resulting field is still invariant, as in the finite-orbit argument above. Every other admissible invariant ratio belongs to it: its adjoining cannot increase dimension, and the field is already relatively algebraically closed.

Fix two exponents \(a<b\) in the determinant sequence, set \(m=b-a\), and put \(t=\sigma_b/\sigma_a\), an invariant rational section of \(mL\). For every larger exponent \(i\), the ratio \[\frac{\sigma_b^{i-a}}{\sigma_i^{b-a}\sigma_a^{i-b}}\] belongs to \(\mathbb C(Y)\). It is a pencil in a bundle \(jL+dA\) with \(j,d>0\), and \(-A\) pseudoeffective makes that pencil satisfy (44). Thus the horizontal orders of the \(\sigma_i\) vary affinely in \(i\). Their nonnegativity at unbounded \(i\) makes \(D_t=\operatorname{div}(\pi^*t)\) effective horizontally. If \(Y\) is a point, this gives the desired section directly.

On a smooth equivariant graph resolution set \(E=K_{Z/X}\). For every integer \(k\ge1\) the invariant generic adjoint rank is \[ \operatorname{rank}\bigl(g_*\mathcal O_Z(E+km\pi^*L)\bigr)^{T_c}=1. \tag{45}\] The rational section \(s_Et^k\) is regular on the generic fiber, hence supplies a nonzero invariant element. If the rank exceeded one, global generation after twisting by \(jg^*H\) would give two invariant sections in \(E+km\pi^*L+jg^*H\) whose ratio is not a base function. Their effective divisors push to a pencil in \(D_0=kmL+j\pi_*g^*H\) on \(X\); exceptionality removes \(E\). Pushing (44) forward gives an integer \(b>0\) for which \(bL-D_0\) is pseudoeffective. Resolve the pushed-down pencil by \(\sigma:U\to X\) and \(h:U\to\mathbb P^1\). Its fixed part \(F\) is effective and \(\sigma^*D_0\sim h^*\mathcal O_{\mathbb P^1}(1)+F\). Consequently \[b\sigma^*L-h^*\mathcal O_{\mathbb P^1}(1) \sim_{\mathbb Q}\sigma^*(bL-D_0)+F \quad\hbox{is pseudoeffective}.\] This is exactly the domination premise for the new invariant pencil. Its ratio is not a base function, contradicting maximality.

We use a normal \(\mathbb Q\)-factorial simplicial toroidal equidimensional model \(p:W\to X\), \(f:W\to Y\), as above, for the divisorial calculations. It need not carry the action. All positivity on invariant summands will instead be proved on the smooth equivariant model, with base change and the ranks (45).

The signed boundaries and their integrals

For a prime \(D\mapsto Q\) on \(W\), put \[d_D=\operatorname{ord}_D\operatorname{div}(p^*t),\qquad a_D=1+\operatorname{ord}_D K_{W/X},\qquad e_D=\operatorname{ord}_D f^*Q.\] Here \(a_D\ge1\), and \(a_D=1\) for nonexceptional primes. Define \[\gamma_Q=\min_{D\mapsto Q}\frac{d_D}{e_D},\qquad \lambda_Q(k)=\min_{D\mapsto Q}\frac{kd_D+a_D}{e_D}.\] The first minimum records the vertical slope of \(t\); the second includes the Jacobian order relevant to integration. We choose the coefficient \(\beta_Q(k)\) of the base correction to satisfy \[\lambda_Q(k)-1\le\beta_Q(k)<\lambda_Q(k),\qquad k\gamma_Q\le\beta_Q(k),\qquad \beta_Q(k)\le\frac{kd_D}{e_D} \quad(D\text{ nonexceptional}).\] These inequalities have different roles. Near general points of \(D\) and \(Q\), the local map \(y_1=z_1^{e_D}\) turns a volume density of order \(2(kd_D+a_D-1)\) into a contribution to the base density with exponent \(2((kd_D+a_D)/e_D-1)\). A prime attaining the minimum therefore gives a lower bound with exponent \(2(\lambda_Q(k)-1)\). Dividing the base density by \(|y_1|^{2\beta_Q(k)}\) adds \(\beta_Q(k)\log|y_1|^2\) to its negative logarithm. The first lower bound makes this corrected weight locally bounded above at \(Q\). The divided volume upstairs has exponent \(2(kd_D+a_D-1-e_D\beta_Q(k))\); the strict upper bound makes it integrable at each prime over \(Q\). The toroidal calculation below will check the remaining valuations. The final upper bound ensures that the section returned to \(X\) has no pole at a nonexceptional prime. The comparison with \(k\gamma_Q\) will be used after taking the limit of the correction divisors.

All four requirements are met by \[\beta_Q(k)=\max\{k\gamma_Q,\lambda_Q(k)-1\},\qquad B_k=\sum_Q\beta_Q(k)Q.\] The supports lie in one finite set. Since every \(a_D/e_D>0\), both terms in the maximum are strictly below \(\lambda_Q(k)\). For a nonexceptional \(D\), both \(k\gamma_Q\) and \(\lambda_Q(k)-1\le kd_D/e_D+1/e_D-1\) are at most \(kd_D/e_D\). Finally, \(\lambda_Q(k)-k\gamma_Q\) is bounded independently of \(k\), because the minimum is over a fixed finite set. Thus \[ \beta_Q(k)<\lambda_Q(k),\qquad \beta_Q(k)\le\frac{kd_D}{e_D}\quad(D\text{ nonexceptional}), \qquad \frac{B_k}{k}\longrightarrow B_\infty:=\sum_Q\gamma_QQ. \tag{46}\] The chosen lower bound by \(k\gamma_Q\) also gives \(B_1\ge B_\infty\). After the metrics show that \(-K_Y+B_k\) is pseudoeffective for every \(k\), the displayed limit will imply pseudoeffectivity of \(B_\infty\) and hence of \(B_1\).

The pair \[ (W,f^*B_k-k\operatorname{div}(p^*t)-K_{W/X}) \tag{47}\] is sub-klt. Horizontal coefficients are nonpositive. A vertical coefficient equals \(e_D\beta_Q(k)-kd_D-(a_D-1)<1\). If \(e_D=1\), both terms in the maximum defining \(\beta_Q(k)\) give this coefficient at most zero. Thus the positive part is supported on the toroidal boundary. On a simplicial toroidal chart the discrepancy function has positive values \(1-b_D\) on the boundary rays, and every new ray has a positive linear combination of those values. Hence an effective boundary with coefficients below one is klt there. Subtracting an effective rational divisor only improves discrepancies. This proves (47), including all divisorial valuations.

Let \(d\mu\) be the smooth anticanonical volume on \(X\). On the smooth base-change locus the invariant rank-one generator of \(K_{Z/Y}+(km+1)\pi^*L\) is the relative form associated to \(s_Et^k\). The squared norm in its coordinate frame is \[\rho_k=\frac{g_*\pi^*(\|t\|^{2k}d\mu)}{dV_Y}.\] Smooth direct-image positivity applies to the smooth invariant coefficient \((km+1)\pi^*L\). Compact averaging is a holomorphic orthogonal projection, so the invariant line has the quotient metric. The squared frame norm on \(G_k=-K_Y+B_k\) is \[H_k=\rho_k\prod_Q|q_Q|^{-2\beta_Q(k)}.\] Near a prime attaining the minimum in \(\lambda_Q(k)\), the local map has form \(y_1=z_1^{e_D}\) and the numerator volume is comparable to \(|z_1|^{2(kd_D+a_D-1)}\) times smooth volume. Therefore \[\rho_k(y)\ge c|q_Q(y)|^{2(\lambda_Q(k)-1)}.\] Since \(\beta_Q(k)\ge\lambda_Q(k)-1\), the weight \(-\log H_k\) is locally bounded above at general points of all missing divisors. It extends plurisubharmonically there and across codimension two. Thus every \(G_k\) is pseudoeffective.

Moreover \(H_k\) is unweighted locally integrable. On a log resolution \(h:W'\to W\), the divisor of poles in the integrand is \[h^*(f^*B_k-k\operatorname{div}(p^*t))-K_{W'/X} =h^*(f^*B_k-k\operatorname{div}(p^*t)-K_{W/X})-K_{W'/W}.\] Its coefficients are below one by (47). The normal-crossings Jacobian test and properness prove integrability on the base. This signed boundary calculation is separate from the effective exceptional correction in Section 9.2.

A global strict metric and the final section

We give the strict-metric construction with its actual sheaf and ideal conditions. From (44) choose a singular metric on \(\pi^*L\) with curvature at least \(\delta g^*\omega_H\). Average its global weight difference \(u\) from the smooth invariant metric over \(T_c\), in the sense of distributions. This preserves the quasiplurisubharmonic lower bound. On \((m+1)\pi^*L\), add \(\epsilon u\) to the smooth coefficient weight. For small \(\epsilon>0\) the resulting metric \(h_\epsilon\) satisfies \[\Theta_{h_\epsilon}\ge\epsilon\delta g^*\omega_H, \qquad\mathcal J(h_\epsilon)=\mathcal O_Z.\] Small-exponent integrability on compact \(Z\) proves the ideal assertion. Consequently the singular adjoint direct-image theorem (Păun and Takayama 2018, Theorem 3.3.5) acts on the full ordinary direct-image sheaf. Let \(U\subset Y\) be a smooth base-change open on which this sheaf is locally free. Fubini makes the fiber integral norm finite almost everywhere on \(U\).

On \(U\), averaging over the compact torus defines a holomorphic projection \(P\) onto the invariant summand. Invariance of \(h_\epsilon\) makes \(P\) orthogonal almost everywhere, where the integral norm is finite. The dual of the quotient embeds holomorphically in the dual full bundle, and its dual norm is the restricted norm. Logarithms of these dual norms are plurisubharmonic by direct-image positivity. They therefore define the semipositive quotient metric; its rank is one by (45). Subtracting a base potential before applying this argument retains the lower curvature bound \(\epsilon\delta\omega_H\). This proves positivity for this specified quotient, without a claim about arbitrary singular invariant subbundles.

For \(k=1\) the integral norm is obtained by inserting \(e^{-\epsilon u}\) in the density \(\rho_1\). Since \(u\) is locally bounded above, the lower density estimates remain valid up to constants. The same extension across divisors, and then codimension two, gives a metric on \(G_1=-K_Y+B_1\) with a global strict curvature lower bound. Together with the already constructed integrable semipositive metric, strong openness and Hölder’s inequality give a globally strict metric on \(G_1\) whose squared frame norms remain locally integrable. Indeed take a sufficiently small positive fraction of the strict weight; the integrable metric has some \(L^{1+\eta}\) room by strong openness, and the new singular weight has small-exponent integrability.

It remains to verify the algebraic premise for the adjoint class. Every \(G_k/k\) is pseudoeffective, and \(G_k/k=-K_Y/k+B_k/k\to B_\infty\). The closedness of the pseudoeffective cone gives \(B_\infty\) pseudoeffective. Since \(B_1\ge B_\infty\), \(B_1\) is pseudoeffective as well. Choose a small ample rational \(A'\) whose smooth curvature can be subtracted from the global strict lower bound on \(G_1\). The resulting metric on \(G_1-A'\) remains globally strict and unweighted integrable. Apply Lemma 2 with \(J=0\) and a sufficiently large divisible \(r>1\). On its log resolution, the fixed part divided by \(r\), minus the relative canonical divisor, has coefficients below one. A general member of the free part, divided by \(r\), meets this support transversely and has coefficient \(1/r<1\). It therefore gives an effective klt \(\Delta_0\sim_{\mathbb Q}G_1-A'\). Add a general effective rational representative of \(A'\) with small coefficients, transverse on a common log resolution. The resulting \(\Delta\sim_{\mathbb Q}G_1\) is effective, big, and klt. Only the global strict bound and unweighted integrability are used here; the mixed metric need not be smooth on an open set. Now \(K_Y+\Delta\sim_{\mathbb Q}B_1\) is pseudoeffective, so big-boundary nonvanishing (Birkar et al. 2010, Theorem D) gives \[\operatorname{div}(h)+NB_1\ge0\] for some \(N>0\) and \(h\in\mathbb C(Y)^*\), after applying Lemma 4. The rational section \(t^Nf^*h\) has no pole at horizontal primes. At a vertical prime nonexceptional over \(X\) this follows from \(\beta_Q(1)\le d_D/e_D\) in (46). Equidimensionality handles every such prime on \(W\). It therefore extends to a nonzero section of \(NmL\) on \(X\). Both \(t\) and the base function are invariant, proving invariant nonvanishing under the stated cohomology-vanishing hypothesis by this second construction.

Recovering an invariant factor section

There is also a useful converse transport after ordinary conversion. It identifies precisely the analytic role of isometric monodromy.

Descent before ordinary conversion

We finish this method by specifying the tensor descent. Use the finite-cover structure statement as in Section 11: on the universal cover of a connected finite étale cover \(\widehat X\to X\) there is an isometric holomorphic product \[\widetilde X=\mathbb C^a\times R\times F.\] Here \(R\) is the compact Ricci-flat product, \(F\) is projective with no positive-degree holomorphic forms, and \(-K_F\) is smoothly semipositive. The chosen deck subgroup acts by translations on the flat factor, fixes the canonical frame on \(R\), and acts on \(F\) through a connected compact torus of holomorphic isometries. Point factors are allowed. These are the geometric structure and frame assertions, independent of the anticanonical nonvanishing conclusion.

For completeness, the compact action has the algebraic realization needed by Proposition 28. Since \(H^1(F,\mathcal O_F)=0\), it preserves a very ample line up to isomorphism and hence acts projectively in its complete linear-system embedding. The representation is continuous, as one sees by tracking a projective frame of points on the embedded variety. A connected compact abelian subgroup of the projective linear group is conjugate into a diagonal torus. Its Zariski closure is an algebraic torus preserving \(F\); quotient by its ineffective kernel if necessary. A diagonal projective torus has a linear lift after fixing one diagonal coordinate, so an ample linearization is available. The action on \(-K_F\) remains the intrinsic tangent-determinant action.

Pull each invariant tensor \(u_k\) from the proposition to \(\widetilde X\) and use the cotangent inclusion from the product projection to \(F\). If \(\omega_{\mathrm{flat}}\) and \(\omega_R\) are the invariant canonical frames of the other factors, then \[u_k\otimes(\omega_{\mathrm{flat}}\otimes\omega_R)^{-k} \in H^0\bigl(\widetilde X, (\Omega_{\widetilde X}^1)^{\otimes p} \otimes(-kK_{\widetilde X})\bigr)\] is nonzero and deck-invariant. It descends to \(\widehat X\) and is algebraic by projectivity. Corollary 15 now gives a nonzero anticanonical plurisection on this actual projective finite cover. Finally the finite étale norm recalled in Section 11 gives a nonzero positive anticanonical multiple on \(X\). Thus the only invariant objects needed for this method are the tensors before descent; the conversion and its Fano-type auxiliary model are ordinary.