Why finite generation attains the limit

The proof first places the principal corrections to the given divisors in one function field, and then uses finite generation to attain the limiting class. Let \(k\subset\mathbb C(X)\) be the field of rational functions whose polar divisors have \(\ell\)-value zero. It is relatively algebraically closed. Two fixed input divisors give representatives of \(L+M/m_j\) on a common finite support, and the principal corrections between those representatives and \(N_j/m_j\) belong to \(k\).

The geometric task is to find a model \(Y\) of \(k\) on which these corrections lie in a finitely generated divisor section ring. On a smooth resolution \(\pi:Z\to X\) of the rational map to \(Y\), the null-polar field makes the section space of the effective divisor \(K_{Z/X}\) one-dimensional on the generic fiber over \(Y\). The null intersection also forces the curvature kernel to be vertical on a nonempty open set. Section 3 verifies these two inputs to the volume-metric theorem of Metric descent and rank-preserving contractions (OpenAI 2026d, Theorem (Toroidal volume metric)). It gives an effective rational correction \(A_0\) on \(Y\) whose pullback is exceptional over \(X\). The semipositive metric it supplies on \(-K_Y+A_0\) has locally integrable squared frame norms and is smooth and strictly positive on an ordinary open set. The correction-ring theorem of Integrable metrics and effectivity with controlled boundary (OpenAI 2026b, Theorem (Finite generation near a correction divisor)) then gives finite generation in the two rational divisor directions selected by the input sequence.

The remaining argument uses the actual rational functions representing sections. In a finitely generated bigraded ring, a monomial whose normalized degree approaches a boundary ray uses off-ray generators in negligible degree. For each input section, the valuation inequality selects a monomial with no larger valuation. One boundary-ray generator therefore has sufficiently small normalized valuation. Its divisor is effective up to the correction \(A_0\). Birational pushforward removes that correction; centeredness then turns the resulting upper bound on the valuation into exact equality with zero. This order of operations allows the exceptional correction itself to have positive valuation.

The endpoint argument belongs to the study of asymptotic vanishing and multigraded section rings. Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa explain how finite generation controls asymptotic orders and base loci (Ein et al. 2006, sec. 4), with the corrected polyhedral argument (Ein et al. 2023, Proposition 1.1); Corti and Lazić give a multigraded formulation (Corti and Lazić 2013, Theorem 3.5). Lemma 3 gives the direct one-ray, one-valuation argument used here. The geometric finite-generation input rests on the adjoint-ring theory of Birkar, Cascini, Hacon and McKernan (Birkar et al. 2010) and Cascini–Lazić (Cascini and Lazić 2012), in the form (Corti and Lazić 2013, Theorem 3.2(1)).

From orders to invariant sections

For a torus action, the differential of the action gives \(-K_X\) its natural linearization. A nonzero section can transform by a nontrivial character, so scalar nonvanishing does not by itself give invariance. Along the generic orbit of a one-parameter subgroup, an eigensection of weight \(w\) has order \(\mu-w\), where \(\mu\) is the fiber weight at the limit. Exact order zero therefore attains that fiber weight.

Müller’s equivariant nonvanishing theorem gives invariant anti-log-canonical sections for projective sub-log-canonical pairs with semiample anti-log-canonical divisor, under commutative linear algebraic group actions (Müller 2025, Theorem C). For smooth varieties with a torus action, we obtain the following result under smooth semipositivity and a cohomological hypothesis.

The proof starts with the source component, which attracts the generic orbit as the one-parameter subgroup tends to zero. The attracting decomposition of Białynicki-Birula identifies this component (Białynicki-Birula 1973). The holomorphic fixed-point formula of Atiyah–Bott, Atiyah–Segal and Atiyah–Singer (Atiyah and Bott 1968; Atiyah and Segal 1968; Atiyah and Singer 1968) supplies cohomology classes of the corresponding extremal weight. Semipositive hard Lefschetz (Demailly et al. 2001, Theorem 0.1) turns them into twisted differential forms. Saturated determinants, following the method of Lazić–Peternell (Lazić and Peternell 2018, Lemma 4.1) and its anticanonical adaptation (Lazić et al. 2023, Lemma 5.1), give effective divisors with bounded order along the generic orbit. Their character calculation retains the determinant-rank shift. Theorem 1 then supplies actual normalized section weights whose convex hull contains zero. A rational convex combination becomes an invariant product after clearing denominators. The comparison of fixed-fiber and section-weight polytopes in the semiample setting is developed in (Müller 2025, Proposition 3.13 and Corollary 3.14); here exact attainment supplies the supporting section weights.

Together with the structural inputs described below, the invariant sections yield the following scalar consequence: for every smooth connected projective complex variety \(X\) with smoothly semipositive \(-K_X\), there is an integer \(m>0\) such that \[H^0(X,-mK_X)\ne0.\] Sections 13 and 14 give two descents. The first uses the compact-monodromy structure theorem of Anticanonical nonvanishing from smooth semipositivity (OpenAI 2026a, Theorem (Finite cover with compact torus monodromy)), based on the universal-cover structure of Demailly–Peternell–Schneider and its rational-connectedness refinement by Campana–Demailly–Peternell (Demailly et al. 1996; Campana et al. 2015). The second uses a locally constant maximal rationally connected fibration, supplied by Matsumura–Wang (Matsumura and Wang 2023), following the earlier smooth structure theorem of Cao–Höring (Cao and Höring 2019). It follows Müller’s invariant-section reduction (Müller 2025, Theorem 2.2) and includes the unipotent part of the commutative algebraic monodromy. The compact-monodromy argument uses only the structural and finite-cover norm results of the nonvanishing companion; its nonvanishing theorem is not an input.

Independent methods and their additional outputs

The exact-order theorem and its invariant application are followed by two independent constructions. Their purpose is to retain different information during the passage from twisted sections to anticanonical sections.

The first uses nef reduction and extension. Given bounded orders in \(|m_jL+M|\), with \(q(X)=0\) and \(-M\) pseudoeffective, it separates a fixed divisor of zero arc order from a line bundle on a base. A fiber-maximum metric makes that line nef; the nef reduction of Bauer and collaborators (Bauer et al. 2002) removes its numerical null directions. Extension of first jets by the theorem of Cao–Demailly–Matsumura (Cao et al. 2017) proves that the remaining base line is big and yields arbitrarily small normalized arc orders. Theorem 14 records this approximate transfer, which the exact theorem already strengthens under the same hypotheses. The independent proof also gives the bigness and jet construction. Its character application, Proposition 15, uses \(q(X)=0\) and \(\chi(X,\mathcal O_X)\ne0\) instead of vanishing of all higher structure-sheaf cohomology, together with the stated equivariant very ample linearization.

There is a second ending to this nef-reduction construction. Starting with invariant sections of \(jL-G\) for unbounded \(j\), where \(G\) is a fixed pseudoeffective linearized line bundle, it forms a base from invariant rational functions with pseudoeffectively controlled poles. It then extends one nonzero invariant class in a multiplier-ideal quotient. Theorem 16 thus removes the fixed error while keeping invariance throughout. This ending needs \(q(X)=0\) but does not require the first-jet or weight-polytope steps. The common tools occupy Section 8, the arc and character applications Sections 9 and 10, and the invariant conversion Section 11.

The other finite-generation construction assumes rational connectedness and controls an entire collection of divisor classes. A prime divisor \(D\) is controlled when \(aL-D\) is pseudoeffective for some positive integer \(a\). Proposition 19 places the naturally linearized classes of all invariant controlled primes in one rational polyhedral cone generated by invariant effective rational divisors. Its generators need not themselves be controlled. Here the effectivity companion’s weighted-approximation theorem produces an adjoint boundary on a base; its correction disappears under birational pushforward. Finite generation of the full multigraded ring and then of its invariant subring gives the cone. Character separation supplies another proof of invariant nonvanishing in the rationally connected case, as well as the fiberwise input used in the locally constant MRC descent. Section 12 keeps this weighted metric construction distinct from the unweighted integrability used for exact attainment.

The proof of exact attainment is organized as follows. Section 2 proves the finite-generator lemma and its birational application. Sections 3 and 4 construct the null-polar base and prove Theorem 1; Section 5 gives the formal-arc consequence. Section 6 establishes the order/weight identity, and Section 7 proves Theorem 2. The independent methods then supply the additional conclusions just described, before the paired monodromy descents finish the scalar nonvanishing consequence.

Removing a birational correction

We now specify the geometric data to which the lemma applies. The main point is that the correction on the base need not have valuation zero. It will disappear under birational pushforward before the last valuation comparison.

All varieties below are integral projective varieties over \(\mathbb C\). For a Cartier divisor \(B\) on a normal variety \(Y\), we identify sections with rational functions: \[H^0(Y,\mathcal O_Y(B)) =\{h\in\mathbb C(Y)^\times:\mathop{\mathrm{div}}_Y(h)+B\geq0\}\cup\{0\}.\] For two Cartier divisors \(B_1,B_0\), their bigraded section ring is \[ R(Y;B_1,B_0) =\bigoplus_{a,b\in\mathbb N} H^0\bigl(Y,\mathcal O_Y(aB_1+bB_0)\bigr)x^ay^b \subset\mathbb C(Y)[x,y]. \tag{3}\] Projectivity and integrality give \(H^0(Y,\mathcal O_Y)=\mathbb C\), as required in Lemma 3.

We also use rational powers of rational functions, interpreted as elements of \(\mathbb C(Y)^\times\otimes_{\mathbb Z}\mathbb Q\), written multiplicatively. The divisor map and every valuation extend \(\mathbb Q\)-linearly to this group. Each element becomes an ordinary rational function after a positive integral power is taken. No extension of the function field by actual roots is intended.

A valuation of \(\mathbb C(X)\) is centered on \(X\) if its valuation ring dominates \(\mathcal O_{X,\xi}\) for some point \(\xi\in X\). For a Cartier divisor \(D\), define \(v(D)\) by applying \(v\) to a local defining rational function at \(\xi\). This is independent of the equation, because units have valuation zero, and extends \(\mathbb Q\)-linearly to rational Cartier divisors. In particular, \(v(D)\geq0\) for every effective rational Cartier divisor \(D\). The zero valuation is centered at the generic point.

The minimum in (4) is over a finite nonempty set, since \(f\) is surjective and equidimensional. Only finitely many base primes occur: the pullbacks of \(D_0,D_1\) have fixed finite support. Each coefficient is a minimum of finitely many rational affine functions, so \(P(t)\) is piecewise affine. In applications one restricts to a sufficiently small rational segment ending at \(t=0\) and rescales the parameter to obtain the stated affine hypothesis. If \(Y\) is a point, the sum is empty and means \(P(t)=0\).

The proof never evaluates the correction \(f^*A\) at \(v\). Its removal is a divisor identity under pushforward. This permits an exceptional correction with positive valuation on a higher model.

An equidimensional model without a field extension

Assume \(d=\dim Y>0\). We need a normal projective variety \(W\) and a diagram \[ X\xleftarrow{\ p\ }W\xrightarrow{\ f\ }Y \tag{9}\] with \(p\) birational, \(Y\) smooth projective, and \(f\) equidimensional and toroidal for strict toroidal embeddings. All modifications here are birational; in particular the field of the base remains \(k\). A toroidal embedding is locally modeled on a toric variety with its dense torus; the complement is its boundary. Strictness means that the irreducible boundary components are normal. Toroidal morphisms are locally toric morphisms, so boundary strata determine rational cones and the morphism determines their lattice-linear maps.

Weak toroidalization (Abramovich et al. 2013, Theorem 1.1) first gives a birational toroidal diagram with smooth projective target. For a toroidal morphism, equidimensionality is equivalent to the condition that every source cone maps onto a target cone (Abramovich and Karu 2000, Lemma 4.1 and Remark 4.2). Refine the target cone complex so that the image of every source cone is a union of cones. Projective refinements exist by (Abramovich and Karu 2000, Lemma 4.3); further projective toroidal resolution makes the target nonsingular. Intersecting the source cones with inverse images of target cones gives projective source subdivisions and a compatible toroidal morphism. In a target cone of the new subdivision, an old cone image meets it in a union of faces which is convex, hence in a face. Thus the new source cones map onto target cones. This is the equidimensionalization of (Abramovich and Karu 2000, Proposition 4.4).

In local toric models, a source stratum maps submersively onto its target stratum. Cone surjectivity bounds the dimension of the fiber in each stratum above by \(n-d\); the opposite bound for full fibers follows because the target is smooth. This gives equidimensionality. The normal source may remain singular, and no covering to reduce multiplicities is needed. Resolve it by \(Z\to W\) when using smooth differential forms; that resolution need not itself be equidimensional. We continue to write \(\pi:Z\to X\) and \(g:Z\to Y\) for the induced maps.

The horizontal curvature patch

The next calculation verifies the remaining analytic hypothesis of the toroidal-volume theorem. Let \(\omega_0\) be a positive form representing \(c_1(G_0)\), and let \(\omega_H\) represent a very ample class on \(Y\). When \(\nu<n\), equation (7) implies \[ (\pi^*\alpha)^\nu\wedge g^*\omega_H \wedge\pi^*\omega_0^{n-\nu-1}=0. \tag{10}\] Indeed the left side is a semipositive top-degree form whose integral is the displayed intersection number. It therefore vanishes pointwise.

At a maximum-rank point away from the Jacobian divisor, diagonalize \(\pi^*\alpha\) in coordinates orthonormal for \(\pi^*\omega_0\). Its first \(\nu\) eigenvalues are positive and the rest are zero. The left side of (10) is a positive multiple of the sum of the diagonal entries of \(g^*\omega_H\) in the remaining \(n-\nu\) directions. These entries vanish. A semipositive Hermitian form vanishing on a vector has zero mixed terms with that vector, so \[\ker\pi^*\alpha\subset\ker dg.\] For \(\nu=n\) this inclusion is automatic. The maximum-rank locus is a nonempty ordinary open and intersects the dense smooth base-change locus, away from the Jacobian divisor. Near a point in their intersection the rank is constant. On a smaller relatively compact neighborhood the positive eigenvalues are bounded below, and the kernel inclusion gives \[ \pi^*\alpha\ge c\,g^*\omega_H\qquad(c>0). \tag{11}\] The Jacobian section is nonzero on this patch. If \(\nu=0\), the same calculation would force \(dg=0\) on a nonempty open; thus a positive- dimensional base cannot occur in that case.

We have now proved both analytic inputs from the null-polar field: full rank one in (8) and the patch (11). Apply the toroidal-volume theorem (OpenAI 2026d, Theorem (Toroidal volume metric)) to (9). For a prime \(E\) of \(W\) put \(a_E=\mathop{\mathrm{ord}}_E K_{W/X}\). Its conclusion uses \[ A_0=\sum_Q\max\left\{0, \min_{E\mapsto Q}\left(\frac{1+a_E}{\mathop{\mathrm{ord}}_E f^*Q}-1\right) \right\}Q. \tag{12}\] It supplies \(A_0\ge0\), \(p_*f^*A_0=0\), and a semipositive metric on \(-K_Y+A_0\) with locally integrable squared frame norms, smooth and strictly positive on a nonempty ordinary open. The cited theorem proves the all-valuation discrepancy calculation and extension across boundary divisors and codimension two. Its strictness requirement is exactly (11); no global strict curvature bound is being inferred directly from that patch.

The correction-ring theorem (OpenAI 2026b, Theorem (Finite generation near a correction divisor)) now applies. For every finite list of rational divisors \(P_i\) on \(Y\), there exist \(\epsilon\in\mathbb Q_{>0}\) and \(k_0\in\mathbb Z_{>0}\) such that all \(k_0(A_0+\epsilon P_i)\) are Cartier and their multigraded section ring is finitely generated. This statement retains the actual rational linear equivalences and the common value of \(\epsilon\). We next choose the two directions dictated by the divisors in Theorem 1.

Source weights and cohomology

Fix a cocharacter \(\lambda:\mathbb G_m\to T\). Its unique source component \(S\) attracts a dense open subset as \(t\to0\), by the Białynicki-Birula decomposition (Białynicki-Birula 1973); see also the smooth attracting decomposition in (Jelisiejew and Sienkiewicz 2019, Theorem 1.5 and Corollary 7.3). Let \(c\) be the weight of \(L\) on \(S\). This is the sum of the positive normal tangent weights, so \(c>0\) for a nonzero cocharacter. For the zero cocharacter take \(S=X\) and \(c=0\). The rational attracting map to \(S\) is dominant. Resolve it and pull back holomorphic forms; birational invariance and Hodge symmetry give \(H^i(S,\mathcal O_S)=0\) for \(i>0\).

Average the logarithmic weights of the metric over the circle in \(\lambda(\mathbb G_m)\). For a nonzero vector \(u\in L_x\), put \(q_u(s)=\log|\lambda(e^s)u|\). The logarithmic norm of this nonvanishing holomorphic orbit section is superharmonic by semipositive curvature. Circle invariance therefore makes \(q_u\) concave in \(s\). There is a smooth function \(\theta:X\to\mathbb R\) such that \(q_u'(s)=\theta(\lambda(e^s)x)\); it is independent of the scaling of \(u\) and equals the fiber weight on each fixed component. If \(x\) is in the dense attracting set, continuity gives \(q_u'(s)\to c\) as \(s\to-\infty\). Concavity then gives \(\theta(x)\le c\). Density and continuity extend this inequality to all of \(X\), so every fixed-component fiber weight is at most \(c\).

For all sufficiently large integers \(l\), the weight \((l-1)c\) occurs in some \(H^q(X,(l-1)L)\). Indeed, expand the equivariant holomorphic Lefschetz formula at infinity in its weight variable \(z\). For a normal tangent weight \(w\), the inverse normal factor is \((1-z^{-w}e^{-x})^{-1}\). If \(w>0\), it has constant term one and lower powers; if \(w<0\), all its powers are strictly negative. Every component other than the source has a negative normal weight. Maximality of \(c\) therefore makes the coefficient of \(z^{(l-1)c}\) precisely \(\chi(S,(l-1)L|_S)\). This polynomial in \(l-1\) has value \(\chi(S,\mathcal O_S)=1\) at zero, and is nonzero for all large \(l\). See (Atiyah and Segal 1968; Atiyah and Singer 1968) for the fixed-component formula.

Equivariant hard Lefschetz with the smooth semipositive coefficient \(lL\) (Demailly et al. 2001, Theorem 0.1) gives a surjection \[H^0(X,\Omega_X^{n-q}\otimes lL)\longrightarrow H^q(X,K_X+lL)=H^q(X,(l-1)L).\] A circle-invariant Kähler form makes this map equivariant. On an infinite subsequence we obtain nonzero eigensections \(u_l\) of \(\Omega_X^a\otimes lL\), with one fixed \(a\), of weight \((l-1)c\) for \(\lambda\). Decompose each into \(T\)-weight components and select a nonzero component; all the components have the same \(\lambda\)-weight. Thus the \(u_l\) can be chosen to be \(T\)-eigensections.

Determinants with bounded orbit order

The use of saturated determinants follows the method of Lazić–Peternell (Lazić and Peternell 2018, Lemma 4.1) and its anticanonical adaptation in (Lazić et al. 2023, Lemma 5.1). Those statements include a numerical-nontriviality hypothesis on the twisting line. Here the positive unbounded twists are already available, and the generic-span and wedge construction below proves exactly the assertion needed, also for a numerically trivial \(L\).

Each section \(u_l\) defines a map \(L^{-l}\to\Omega_X^a\). Take the span of their image lines over \(\mathbb C(X)\), of rank \(e>0\), and saturate its determinant line in \(\bigwedge^e\Omega_X^a\). This construction is independent of a choice of rational frame for \(L\). Denote the resulting line bundle by \(M\). Rank-one saturation in a locally free sheaf is reflexive and hence a line bundle on the smooth variety; its natural linearization extends from the subbundle locus across codimension two. The needed curvature-intersection sign is \(\ell(M)\leq0\); it is automatic when \(\nu=n\). Suppose \(\nu<n\). The smooth metric on \(L=-K_X\) determines a positive volume form whose Ricci form is \(\alpha\). For every \(b>0\), scale this volume to the volume of the Kähler class \(G_0+bL\) and apply the Calabi–Yau theorem (Yau 1978). The resulting Kähler metric has Ricci form \(\alpha\ge0\). Kähler curvature symmetries identify the mean curvature of its tangent bundle with the Ricci endomorphism. Duals and exterior powers therefore make the mean curvature of \(\bigwedge^e\Omega_X^a\) nonpositive.

Apply the subbundle curvature inequality to the inclusion of \(M\) where it is nonzero. Its logarithmic norm extends as a locally quasiplurisubharmonic function across the zero set. Its Hessian has a smooth lower bound there, so the singular trace measure is nonnegative and the extension preserves the degree inequality. Hence \[M\cdot(G_0+bL)^{n-1}\leq0.\] Since \(L^{\nu+1}=0\), all powers above \(b^\nu\) vanish in this intersection polynomial. The coefficient of \(b^\nu\) is a positive multiple of \(\ell(M)\), so letting \(b\to\infty\) proves the required sign.

Choose a fixed finite collection of the \(u_l\) whose image lines form a generic basis. Every further nonzero \(u_l\) can be completed to a basis by \(e-1\) members of this collection. There are finitely many choices; on an infinite subsequence one choice is fixed. Taking these wedges gives nonzero eigensections of \(M+m_jL\), where \(m_j\) is the sum of the \(e\) twists and tends to infinity. Their weights are \((m_j-e)c\), since each of the \(e\) constituent forms contributes its twist minus one. Their zero divisors \(N_j\) satisfy \(\ell(N_j)=0\), by effectivity and \(\ell(M)\leq0\). Let \(v\) be the order obtained by substituting \(\lambda(t)x\) for the generic point \(x\) of \(X\). Properness extends this orbit to a formal arc at zero, with center at the generic point of \(S\); its order is a discrete valuation up to rescaling, or is zero. If \(c_M\) is the fiber weight of \(M\) at \(S\), then \[v(N_j)=m_jc+c_M-(m_j-e)c=c_M+ec.\] This is Lemma 8 for the declared left action. The constant \(c_M+ec\) retains the determinant-rank shift.

Exact source weights and an invariant product

Theorem 1 now gives an effective rational representative of \(L\) of order zero at the generic source. After clearing denominators, some section of \(mL\) restricts nontrivially to \(S\). Its weight decomposition consequently contains a normalized weight whose value on \(\lambda\) is exactly \(c\). If \(T\) is trivial, this is already a nonzero invariant section. Assume from now on that \(T\) has positive dimension.

The function \(c\) is the maximum of the finitely many \(T\)-fixed fiber weights evaluated on \(\lambda\): the source is \(T\)-stable and contains a \(T\)-fixed point by the Borel fixed-point theorem. It thus extends to a continuous rational polyhedral homogeneous function on real cocharacters. It is strictly positive away from zero. Indeed a nontrivial zero or negative locus would, by rational polyhedrality, contain a nonzero rational cocharacter, contrary to the source-weight calculation. Every normalized section weight is bounded above by \(c\) on each cocharacter, by the same orbit-order test, and the preceding argument attains this upper bound. The comparison between fixed-fiber and section-weight polytopes for semiample linearized line bundles, and the resulting invariant-product criterion, are established in Müller (Müller 2025, Proposition 3.13 and Corollary 3.14). Here exact order attainment supplies the supporting section weights.

Testing a lattice basis of cocharacters and its negatives bounds all normalized section weights. Their closed convex hull is therefore compact; its support function equals \(c\) on rational cocharacters by exact attainment, and then on all real cocharacters by continuity. Strict positivity of \(c\) away from zero puts zero in its interior. Choose a small simplex surrounding zero in that convex hull and approximate its vertices by finite convex combinations of actual normalized weights. Sufficiently close approximations still surround zero, so the finitely many actual weights in these combinations have zero in their convex hull. Lemma 9 now multiplies finitely many corresponding eigensections to produce a nonzero invariant section.

This proves Theorem 2. For an arbitrary torus action, replace \(T\) by its effective quotient \(T/\ker(T\to\operatorname{Aut}(X))\). The entire kernel, including its finite part, acts trivially on the natural anticanonical line, so a section invariant under the quotient is invariant under \(T\).

Models and connected fibers

The relative section must be normalized over every base divisor. For that purpose we use a normal equidimensional model. Its smooth resolution is used for integration, and is not required to remain equidimensional.

A bounded metric from all relative moments

The two nef-reduction arguments below construct a section of a relative line bundle and normalize its orders over base divisors. The purpose of this normalization is analytic: it prevents the maximum of the section’s norm on nearby fibers from tending to zero. Together with rank one for every relative adjoint twist, it produces a nef line bundle on the base. Ordinary and invariant rank one are separate hypotheses in the statement.

The conclusion over the entire base is local boundedness. Continuity on the smooth locus is used for the power-mean limit; the extension argument does not require continuity at every boundary point.

Descending through the numerical null directions

A nef divisor has degree zero on the general fibers of its nef reduction. The next argument upgrades this to every contracted curve, which is the hypothesis needed for numerical pullback. Irregularity zero is then used to turn that numerical identity into a rational linear identity of line bundles.

The reduced base now has positive degree on every curve through a very general point. We next turn this qualitative condition into an arbitrarily large Seshadri bound after adding a fixed small ample term. This will give an isolated logarithmic pole strong enough to detect either a first jet or a nonzero quotient class.

We use the following metric conventions in this extension argument. If a singular Hermitian metric has squared frame norm \(h=e^{-\varphi}\), its multiplier ideal \(\mathcal I(h)\), also denoted \(\mathcal J(h)\), consists of holomorphic germs \(u\) for which \(|u|^2e^{-\varphi}\) is locally integrable. The metric \(he^{-\psi}\) has weight \(\varphi+\psi\) and curvature \(i\Theta_h+i\partial\bar\partial\psi\). A quasi-plurisubharmonic function has neat analytic singularities if locally it has the form \(c\log\sum_j|f_j|^2+u\), where \(c>0\), the \(f_j\) are holomorphic and \(u\) is smooth.

The field of functions with controlled poles

Let \(P\) consist of zero and the rational functions whose poles are supported on controlled primes of \(V\). Sums and products stay in \(P\). If \(u\ne0\) belongs to \(P\), its zero divisor is linearly equivalent to its polar divisor, hence is dominated by a multiple of \(L\). Thus \(u^{-1}\in P\) as well. This proves that \(P\) is a \(T\)-stable subfield of \(\mathbb C(V)\).

The field is relatively algebraically closed. In fact, if \(u\) is algebraic over \(P\) and had a pole at an uncontrolled prime, all coefficients of a monic equation for \(u\) would have nonnegative order there. Its leading power would then have strictly smaller order than every other term, which is impossible. Also \(P\) is finitely generated over \(\mathbb C\). For completeness, choose a transcendence basis \(t_1,\ldots,t_d\) for \(P/\mathbb C\) and extend it to a transcendence basis of \(\mathbb C(V)\). The algebraic closure of \(\mathbb C(t_1,\ldots,t_d)\) inside this finitely generated field is finite over \(\mathbb C(t_1,\ldots,t_d)\), so its intermediate algebraic extension \(P\) is finite too.

Choose controlled primes \(D_1,\ldots,D_s\) spanning the classes of all controlled primes in \(\operatorname{Pic}(V)_{\mathbb Q}\). If a controlled prime \(D\) is not among them, clearing denominators and torsion gives a principal divisor supported on \(D,D_1,\ldots,D_s\) whose coefficient at \(D\) is nonzero. Its defining function belongs to \(P\). Thus the valuation of \(D\) is nontrivial on \(P\), and \(D\) is vertical for any rational map with target field \(P\). In particular, only finitely many controlled primes can be horizontal. If \(P=\mathbb C\), this proves Proposition 19 directly: there are only finitely many controlled primes, and we take the cone of the invariant ones.

Suppose \(\operatorname{trdeg}_{\mathbb C}P>0\). Choose a smooth projective equivariant model \(Y\) of \(P\) and an equivariant diagram \[ V\xleftarrow{\pi}U\xrightarrow{f}Y, \qquad \mathbb C(Y)=P, \tag{29}\] with \(U\) smooth projective and \(\pi\) birational. We require that the strict transform of each prime of \(V\) either dominate \(Y\) or map onto a prime divisor of \(Y\). Here is a construction with this property.

Finitely many generators of \(P\) have a common controlled polar bound \(N\ge0\). A sufficiently large full linear system of a multiple of \(N\) contains these generators as ratios, and every ratio of its sections still has controlled poles. After linearizing a multiple, its image is an equivariant projective model with function field \(P\). Over a dense open, the closures in \(V\) of the fibers form a flat family. Resolve equivariantly the graph of its rational map to the Hilbert scheme and pull back the universal family. This uses the Hilbert-family and flattening constructions of Grothendieck and Raynaud–Gruson (Grothendieck 1961; Raynaud and Gruson 1971). Flatness with fixed Hilbert polynomial bounds fiber dimension by \(\dim V-\dim Y\). The main component is birational to \(V\). A divisor coming from \(V\) therefore cannot map to codimension at least two: that would force its generic fiber dimension to exceed this bound. Resolve the total space equivariantly; strict transforms retain the asserted images. Relative algebraic closedness of \(P\) makes \(f\) have connected fibers by Stein factorization. Pullback of holomorphic one-forms, followed by birational invariance, gives \(\operatorname{Pic}^0(Y)=0\).

Define rational pullback of divisors and linearized line bundles by \[p^\diamond=\pi_*f^*.\] For line bundles, one can equivalently pull back on the big open set where \(V\dashrightarrow Y\) is defined and extend reflexively across codimension two. Since \(V\) is smooth, this extension is a line bundle; the equivariant structure extends uniquely. If \(\sigma:Y'\to Y\) is a smooth birational modification, resolve the diagram above it and use the same notation. Then \(p^\diamond\) commutes with \(\sigma^*\) and kills \(\sigma\)-exceptional divisors, including their divisor linearizations. Indeed, a prime of \(V\) above such a divisor would have image of codimension at least two on \(Y\), contrary to the model property. That property persists over \(Y'\): a horizontal prime remains horizontal, and a prime mapping onto a divisor of \(Y\) maps onto its strict transform, because \(\sigma\) is an isomorphism at its generic point.

We can now reduce the proposition to a statement about divisors on \(Y\). On a nonempty Zariski open subset of \(Y\), the morphism \(f\) is smooth with geometrically connected, hence geometrically integral, fibers. If the generic point of a prime \(Q\) lies in this open, exactly one prime of \(U\) dominates \(Q\), and its multiplicity in \(f^*Q\) is one. Every other component of \(f^*Q\) maps into a smaller subset of \(Y\); by the model property it is exceptional over \(V\) and pushes to zero. Consequently, if a prime \(D\) of \(V\) maps onto such a \(Q\), then \[p^\diamond Q=D.\] The controlled primes not covered by this statement form a finite set: they are the finitely many horizontal controlled primes already identified and the finitely many primes over the excluded base divisors. If \(D\) is invariant, its image \(Q\) is invariant, and the equality also holds for their divisor linearizations.

It therefore remains to find a rational polyhedral cone of invariant effective classes containing \(p^\diamond[R]^T\) for every invariant effective integral divisor \(R\) on \(Y\). Adding the classes of the finitely many invariant controlled exceptions will finish the proof. We may construct this cone on any smooth equivariant modification \(\sigma:Y'\to Y\): the effective total transform \(\sigma^*R\) has its canonical divisor linearization, and \(p^\diamond[\sigma^*R]^T=p^\diamond[R]^T\).

The field construction has now supplied a rank-one adjoint image and an exceptional correction \(J\). We next use smooth semipositivity on \(V\) to produce a strictly positive metric on \(-K_Y+J\). Its integrable object will be the canonical section of \(J\), which is the precise input needed for weighted approximation.

The weighted metric and its disappearing boundary

By (30), \(\pi^*L\) has a singular metric whose curvature dominates a positive multiple of \(f^*\omega_H\). Mix its weight with the pulled-back smooth semipositive weight, using a sufficiently small positive coefficient for the singular term. Skoda’s local integrability theorem (Skoda 1972) and compactness give a metric \(h\) such that \[ \Theta_h(\pi^*L)\ge\epsilon f^*\omega_H, \qquad \mathcal J(h)=\mathcal O_U, \qquad \epsilon>0. \tag{33}\]

The identity \[K_{U/Y}+\pi^*L=E-f^*K_Y\] identifies the reflexive hull of the adjoint direct image with \(-K_Y+J\). Apply the singular direct-image metric theorem of Păun–Takayama (Păun and Takayama 2018, Theorem 3.3.5). Its multiplier-ideal inclusion is an isomorphism here by (33). The rank-one fiber integral therefore gives a singular metric with weights \(\varphi\) on \(-K_Y+J\), extending across divisors and then across codimension two. It has curvature at least \(\epsilon\omega_H\): locally subtract a potential of this form from the coefficient weight before applying positivity, and restore it afterward.

For clarity, its required integrability follows directly from the same integral. The section induced by the Jacobian section of \(E\) is represented by \(s_J\) in a local frame of the reflexive hull. Integrating its squared fiber norm against base coordinate volume gives the integral upstairs of the corresponding canonical-valued form with coefficient metric \(h\). This is finite by \(\mathcal J(h)=\mathcal O_U\) and properness. Fiber integration on the smooth locus, which has full measure, proves \[ |s_J|^2e^{-\varphi}\in L^1_{\mathrm{loc}}(Y), \qquad \mathcal J(\varphi)\supseteq\mathcal O_Y(-J). \tag{34}\] This is the weighted input of (OpenAI 2026b, Theorem (Weighted anticanonical approximation)). It does not require an unweighted integrability assertion for \(e^{-\varphi}\).

We record the complete output of that theorem at this use. A power of an ample line on \(Y\) is \(T\)-linearizable, since \(\operatorname{Pic}^0(Y)=0\); the zero divisor of an eigensection of a very ample power is invariant and ample. Choose a small positive rational multiple \(A\) of this divisor so that \(B=-K_Y+J-A\) still has a strict curvature bound. Resolve the full system of a sufficiently divisible \(rB\) together with \(J\) by an equivariant sequence of smooth blowups \(\sigma:Y'\to Y\) and write \[\sigma^*|rB|=G+|\mathrm{Mov}|,\] with the mobile system base-point free. Weighted approximation gives \[ \Psi=G/r-\sigma^*J-K_{Y'/Y},\qquad \operatorname{coeff}\Psi<1. \tag{35}\] Writing \(\Psi=\Psi^+-\Psi^-\) with disjoint positive and negative parts, put \(D_0=\Psi^-\). Then \[ D_0\sim_{\mathbb Q}K_{Y'}+\Psi^+ +\mathrm{Mov}/r+\sigma^*A. \tag{36}\] Here \(\Psi^+\) is an effective simple-normal-crossings klt boundary, and the last two summands are nef and big together. The support of \(D_0\) is contained in that of \(\sigma^*J\) and the exceptional divisor of \(\sigma\). The divisors \(G,K_{Y'/Y},\Psi^+,D_0\) are invariant and the free system \(|\mathrm{Mov}|\) is torus-stable. Identity (36) is an ordinary rational linear equivalence; we construct the needed linearized classes separately below.

The support statement has the crucial additional consequence \[ p^\diamond[D_0]^T=0. \tag{37}\] Indeed \(p^\diamond\) kills the effective divisor \(\sigma^*J\) by (32), and it kills every \(\sigma\)-exceptional prime by the model property. An effective divisor supported on this union has zero pushdown. Its canonical invariant section also pushes to the constant rational section, so the equality includes the linearization. No character shift is lost in (37).

The multigraded ring and the finite cone

We now build the cone of base pullbacks required above. Let \(R_0\) be any invariant effective integral divisor on \(Y'\). Since \([D_0]^T\) disappears under \(p^\diamond\), the divisor \(D_0+tR_0\) for a small rational \(t>0\) has the same rational pullback class as \(tR_0\). We will place these divisors in a finite multigraded ring and express their invariant sections through its generators. This is why we need a whole neighborhood of \([D_0]^T\) in the linearized Picard space, rather than just the single adjoint ring of \(D_0\).

Because \(\sigma\) is a sequence of smooth blowups, there is an effective exceptional rational divisor \(F\) with \(-F\) relatively ample. One constructs it successively by giving sufficiently small positive coefficients to the later exceptional divisors. For small rational \(\delta>0\), \(\sigma^*A-\delta F\) is ample and \(\Psi^++\delta F\) is still a klt boundary with simple normal crossings. If \(\sigma\) is the identity, take \(F=0\). Thus (36) expresses \(D_0\) as \(K_{Y'}\) plus that boundary and an ample rational class.

Choose a fixed small ample rational part \(A_*\). Every rational class in a sufficiently small neighborhood of the underlying class of \(D_0\) can be written \[K_{Y'}+A_*+\Delta_i,\] with \(\Delta_i\ge0\) and klt. For a finite set of classes, choose general members of sufficiently divisible very ample systems for the remaining ample parts, divided by their multiples. They can be chosen transverse to one another and to the fixed boundary, with coefficients less than one. Consequently their union has simple normal crossings. Small character changes only change linearizations, so the same construction gives a neighborhood in \(\operatorname{Pic}^T(Y')_{\mathbb R}\).

Choose rational vertices \(U_1^T,\ldots,U_s^T\) of a polytope with \([D_0]^T\) in its interior. After a common multiple \(k\), let \(\mathcal E_i\) be integral linearized representatives of \(kU_i^T\). The ordinary ring \[ R=\bigoplus_{\boldsymbol\ell\in\mathbb N^s} H^0\!\left(Y',\bigotimes_{i=1}^s \mathcal E_i^{\otimes\ell_i}\right) \tag{38}\] is finitely generated by the multigraded adjoint-ring theorem of Cascini–Lazić (Cascini and Lazić 2012, Theorem A); compare (Corti and Lazić 2013, Theorem 3.2(1)). We take \(k\) sufficiently divisible to turn the rational linear equivalences above into isomorphisms of line bundles. This identifies (38) with a divisible subring of the stated adjoint ring. The linearizations give an algebraic \(T\)-action on \(R\), independently of the general boundary representatives used to prove finite generation.

Here is the invariant-ring step explicitly. Enlarge finitely many multihomogeneous generators of \(R\) to their finitely many character components. They define an equivariant multigraded surjection from a polynomial ring to \(R\). Projection to weight zero is exact for a torus, so the invariant polynomial ring surjects onto \(R^T\). If the variable weights are \(\eta_1,\ldots,\eta_N\), its invariant monomials have exponents \[(a_1,\ldots,a_N)\in\mathbb N^N, \qquad \sum a_i\eta_i=0.\] These are the lattice points of a rational polyhedral cone; Gordan’s lemma makes their semigroup finitely generated. Thus \(R^T\) has finitely many nonzero multihomogeneous generators \(u_1,\ldots,u_b\).

Let \(\mathcal C_0\) be the cone generated by the rational pullbacks \(p^\diamond[\operatorname{div}(u_j)]^T\). These are classes of invariant effective divisors on \(V\). We claim that \(\mathcal C_0\) contains \(p^\diamond[R_0]^T\) for every invariant effective integral divisor \(R_0\) on \(Y'\). Choose a small rational \(t>0\) such that \([D_0]^T+t[R_0]^T\) belongs to the vertex polytope. Express it as a rational convex combination of the \(U_i^T\). Clearing denominators and torsion gives an equivariant isomorphism between a positive multiple of \(D_0+tR_0\) and one multidegree in (38). Its canonical section gives a nonzero element of \(R^T\) in that multidegree. Express it as a polynomial in the \(u_j\) and retain the terms of that multidegree. At least one nonzero monomial occurs, and its degree is a nonnegative integral sum of generator degrees. Thus the corresponding linearized class lies in their cone. Only equality of homogeneous degrees is used here; the selected monomial need not have the same zero divisor as the original section. Apply \(p^\diamond\), use (37), and divide by the positive factor containing \(t\). This proves the claim.

Apply this claim to the total transforms of invariant effective divisors on the original base \(Y\). Compatibility of \(p^\diamond\) with \(\sigma^*\) gives the required cone of their rational pullbacks. Add the invariant effective classes of the finitely many controlled primes excluded in the initial comparison with base divisors. The resulting rational polyhedral cone proves Proposition 19.

A weight bound and character separation

The cone gives finitely many effective generators. To obtain an invariant anticanonical section, we must still put the specified natural linearization of \(L\) in their cone. The following fixed-point calculation supplies the sign needed for separation. It is a bound on some cohomology weight, rather than the weight-zero index theorem.

We now obtain an independent proof of the rationally connected special case of Theorem 2. Its additional output is the controlled-divisor cone just constructed. The proof uses that cone and character separation, and will supply the fiberwise section in the locally constant MRC application of Section 14.

Making the monodromy group connected and commutative

The structure theorem of Demailly–Peternell–Schneider (Demailly et al. 1996, Structure Theorem (iii), p. 218) shows that \(\pi_1(X_0)\) is virtually abelian. Since \(f_0\) is a fiber bundle with connected fiber, its homotopy sequence makes \(\pi_1(X_0)\to\pi_1(B_0)\) surjective. The image of an abelian finite-index subgroup is again abelian of finite index. Consequently \(\pi_1(B_0)\) is virtually abelian.

To place monodromy in a fixed algebraic group, choose an ample line bundle \(A\) on \(X_0\) and put \(A_F=A|_F\). The restrictions of \(c_1(A)\) form a flat section of \(R^2(f_0)_*\mathbb Z\), so monodromy fixes the integral class \(c_1(A_F)\). Since \(H^1(F,\mathcal O_F)=0\), the exponential sequence makes the Chern-class map \(\operatorname{Pic}(F)\to H^2(F,\mathbb Z)\) injective. It follows that \(\rho_0(\gamma)^*A_F\simeq A_F\) for every \(\gamma\in\pi_1(B_0)\). Choose \(a>0\) such that \(A_F^a\) is very ample. The projective embedding by \(A_F^a\) identifies \(\operatorname{Aut}(F,A_F^a)\) with the closed stabilizer of the embedded \(F\) in \(\operatorname{PGL}(H^0(F,A_F^a)^*)\). Monodromy therefore lies in a linear algebraic group. This argument uses a projective action and requires no choice of linearization on \(A\) or \(A_F\).

Pass to the connected finite étale cover of \(B_0\) corresponding to an abelian subgroup of finite index in \(\pi_1(B_0)\), and pull back \(X_0\). The resulting monodromy has commutative Zariski closure \(H\): the commutator morphism is trivial on the dense product of the monodromy image with itself. Pass next to the cover corresponding to the inverse image of the identity component \(H^\circ\). The new monodromy is dense in \(H^\circ\), since each component of an algebraic group is open and closed. Write \[f:Y\longrightarrow B,\qquad p:Y\longrightarrow X, \qquad G=\overline{\rho(\pi_1(B))}^{\mathrm{Zar}}=H^\circ\] for the resulting fibration, composite finite étale cover, and monodromy closure. These finite covers of projective varieties are projective, and \(K_B\) is still torsion. In particular \(G\) is a connected commutative linear algebraic group acting on the same fiber \(F\).

From a torus-fixed section to a section on \(X\)

We use the natural action on anticanonical bundles: an automorphism acts on \(K_F^{-1}=\det T_F\) by the determinant of its differential. It induces an algebraic representation on every finite-dimensional space \(H^0(F,K_F^{-m})\). This specified action is also the action in the relative canonical transition functions of the locally constant fibration.

Let \(T\subset G\) be a maximal algebraic torus. Theorem 22 gives an integer \(m>0\) for which \[W:=H^0(F,K_F^{-m})^T\ne0.\] If \(F\) is a point, take \(m=1\) and the constant section; the rest of the argument is unchanged. The trivial torus is allowed. The inclusion \(G\subset\operatorname{Aut}(F)\) makes the action faithful, so there is no ineffective subgroup that could introduce a scalar character on the natural anticanonical line.

For completeness, the following elementary argument explains why \(W\) contains a vector fixed by all of \(G\). Jordan decomposition in a connected commutative linear algebraic group over \(\mathbb C\) gives \[G=T\times U,\] where \(U\) is connected and unipotent. Commutativity makes \(W\) stable under \(U\). Its Lie algebra acts on \(W\) by commuting nilpotent endomorphisms \(N_1,\ldots,N_e\), after choosing a basis of the Lie algebra. Their kernels have nonzero intersection: successively restrict \(N_j\) to the intersection of the preceding kernels, which it preserves by commutativity; a nilpotent endomorphism of a nonzero finite-dimensional space has nonzero kernel. A nonzero vector in this intersection is fixed by \(U\), because \(U\) is connected and is generated by the exponentials of its Lie algebra. We obtain \[0\ne s\in W^U=H^0(F,K_F^{-m})^G.\] This is the connected-group step in (Müller 2025, Lemma 2.3 and the proof of Theorem 2.2); we have retained the unipotent argument because \(G\) need not be a torus. The input here is a fixed vector for the natural linearization, not a section of an unspecified character.

Since \(s\) is fixed by monodromy, its constant extension to \(\widetilde B\times F\) descends to a nonzero section \[\sigma\in H^0(Y,K_{Y/B}^{-m}).\] The canonical bundle formula reads \(K_Y=K_{Y/B}\otimes f^*K_B\). Choose \(r>0\) with \(K_B^r\simeq\mathcal O_B\) and a nowhere-zero section \(\tau\in H^0(B,K_B^{-r})\). Then \[\sigma^r\otimes f^*\tau^m \in H^0(Y,K_Y^{-mr})\setminus\{0\}.\] All these holomorphic sections are algebraic by projectivity. Finally, let \(d=\deg p\). Since \(p\) is finite étale, \(K_Y^{-mr}\simeq p^*K_X^{-mr}\), and the norm of this section is a section of \(K_X^{-mrd}\). It is nonzero: over the generic point the norm is a product of nonzero conjugates in a finite field extension. Thus \(X\) has a nonzero section of a positive anticanonical power. The construction uses the locally constant product only after passing to the universal cover of the base; no product decomposition of a finite cover of \(X\) is required.