Proof. Embed the projective graph over \(Y\) in \(\mathbb P^N\times Y\). Over the flat open set the fibers define a map to the projective Hilbert scheme (Grothendieck 1961, Theorem 3.2 and the Hilbert-scheme specialization after Proposition 3.8). Resolve the closure of its graph, also resolving the base, and pull back the universal family. Flatness over an integral base embeds each affine coordinate ring in its generic localization. The generic fiber is integral, so these coordinate rings have neither nilpotents nor zero divisors. Thus the pulled-back family is integral and is the closure of the original family over the flat open set. Its map to the graph is birational.

Constancy of the Hilbert polynomial bounds the dimension of every fiber above by the generic relative dimension. At any fiber component the height bound for the ideal generated by local parameters of the smooth base gives the opposite bound. This proves purity of fiber dimension. Normalization is finite, so fiber dimensions and dimensions of images of components are preserved. A prime divisor mapping into codimension at least two in \(Y'\) would have dimension at most \(\dim Y'-2+(\dim X-\dim Y')=\dim X-2\), a contradiction.

For connectedness apply Stein factorization to the normal total space. Its finite intermediate base has function field the algebraic closure of \(\mathbb C(Y')\) in \(\mathbb C(X)\); the asserted field condition and normality of \(Y'\) make the finite birational morphism an isomorphism. The same reasoning applies after resolution. In the equivariant case, the family over the original open set is stable under the action. Its closure and normalization are therefore stable, and equivariant resolution in characteristic zero supplies the last assertion. ◻

Proof of Theorem 1. The common ratio field. Put \(n=\dim X\) and write \(N_i=M+iP\). For each \(i\), ratios of invariant sections of the same positive multiple of \(N_i\) form a field \(Q_i\). For \(p<q<b\) in \(I\), the identity \[(b-p)N_q=(b-q)N_p+(q-p)N_b\] holds with linearizations. Multiplication by suitable powers of the fixed \(s_p,s_b\) puts any ratio for \(N_p\) or \(N_b\) into \(Q_q\). This retains the ratio itself: for a ratio in degree \(aN_p\), multiply its numerator and denominator by \(s_p^{a(b-q-1)}s_b^{a(q-p)}\), and use the analogous formula for a ratio in \(aN_b\). Taking a further index beyond any prescribed two interior indices proves that all \(Q_i\) with \(i>\min I\) coincide; denote this field by \(Q\).

Fix the first two indices \(p<q\). For \(b>q\) the ratio \[ \frac{s_q^{b-p}}{s_p^{b-q}s_b^{q-p}} \tag{3}\] belongs to \(Q\). If \(\operatorname{trdeg}_{\mathbb C}Q=0\), it is constant. The section divisors are affine in \(i\), and their slope is effective because the indices are unbounded. Thus \(s_q/s_p\) is a regular invariant section of \((q-p)P\).

Otherwise put \(e=\operatorname{trdeg}Q>0\). Choose \(e\) independent ratios to define a map to \(\mathbb P^e\), resolve its graph and take Stein factorization. The Stein field is the relative algebraic closure in \(\mathbb C(X)\) of the field of the chosen ratios, a finite extension of that field. It contains \(Q\), since \(Q\) is algebraic over the chosen ratio field, and is relatively algebraically closed in \(\mathbb C(X)\). The connected torus acts trivially on the Stein base because its orbits lie in finite fibers over a base with trivial action. As before take a smooth projective base \(Y\), a normal equidimensional \(W\to Y\) with a birational morphism \(W\to X\), and an equivariant smooth resolution \(V\to W\), with maps \(\mu:W\to X\), \(f_W:W\to Y\), and total maps \(\pi:V\to X\) and \(f:V\to Y\). Put \(P_W=\mu^*P\). The map \(f\) has connected fibers by the Stein construction and normality of \(Y\). Write \(L'=\pi^*L\) and \(P'=\pi^*P\).

For each interior index \(i\) and every divisor \(H'\) on \(Y\), \[ H^0(V,a\pi^*N_i-f^*H')^T\ne0 \quad\text{for some integer }a>0. \tag{4}\] Indeed the chosen independent ratios belong to \(Q_i\). Put their numerators and denominator in one common degree by products of invariant sections. On the resolved graph they have a common invariant factor whose line is that multiple of \(N_i\) minus the pullback of \(\mathcal O_{\mathbb P^e}(1)\). The latter line is big on \(Y\), so a sufficiently large multiple dominates \(H'\).

A regular relative section. We next remove the vertical poles while preserving the character. Let \(D_i\) be the section divisor on \(W\). Since (3) comes from \(Y\), its horizontal divisor is zero. The affine relations show that the horizontal part of \((D_q-D_p)/(q-p)\) is effective. Normalize the vertical part by subtracting the minimum coefficient divided by pullback multiplicity above each base prime, and let \(B\) be the resulting rational base divisor. There is an integer \(k>0\), divisible by \(q-p\) and clearing the coefficients of \(B\), for which \[ \gamma=(s_q/s_p)^{k/(q-p)}(f_W^*1_{kB})^{-1} \in H^0(W,kP_W-f_W^*(kB))^T \tag{5}\] is regular. Normality checks regularity in codimension one, and equidimensionality accounts for every vertical prime. Above every base prime \(\gamma\) is nonzero at the general point of at least one component. We also write \(\gamma\) for its pullback to \(V\).

All adjoint ranks, including zero. The ratio field must also control the fiber integrals. Let \(E=K_{V/X}\ge0\). Its Jacobian section \(1_E\) is invariant for the natural linearization on \(L'\), irrespective of the character shift in \(P'\). For all \(j\ge0\), \[ \operatorname{rank} \bigl(f_*\mathcal O_V(E+jkP')\bigr)^T=1. \tag{6}\] The section \(1_E\gamma^j\) gives the lower bound over the generic fiber. If the rank were larger, globalize two invariant sections \(u_1,u_2\) independent over \(\mathbb C(Y)\) after a twist \(f^*(aH)\), where \(H\) is ample. Choose \(i,b\in I\) with \(b>i+jk\). Multiply \(u_1^k\) and \(u_1^{k-1}u_2\) by \(s_i^k\gamma^{b-i-jk}\). Both results are sections of \[kE+k\pi^*N_b+ f^*\bigl(kaH-(b-i-jk)kB\bigr),\] with ratio \(u_2/u_1\). Apply (4) at index \(b\) to cancel this base twist by the same invariant multiplier in the numerator and denominator. Removing the exceptional twist \(kE\) and pushing to \(X\) leaves two invariant sections of an equal multiple of \(N_b\). Normality extends across the image of the exceptional locus. Their ratio must be in \(Q\subset\mathbb C(Y)\), contrary to independence. This proves (6).

The bounded metric and the horizontally positive metric. We now construct the two metrics needed for interpolation. Average a smooth semipositive metric \(h_0\) on \(L\) over the maximal compact torus. It is also invariant for \(P\), because the two linearizations differ by a character, unitary on that compact torus. Equation (4) and pseudoeffectivity of \(-M\) give \(L'-\eta f^*H\) pseudoeffective for some \(\eta>0\) and ample \(H\). Taking a positive current, averaging, and mixing a small positive fraction of its potential with \(h_0\) gives an invariant singular metric \(h\) on \(L'\) with \[ i\Theta_h\ge\delta f^*\omega_Y, \qquad \mathcal I(h)=\mathcal O_V, \qquad\delta>0. \tag{7}\] Skoda integrability and compactness give the last equality. The metric is bounded below by a positive constant times \(h_0\) on compact coordinate patches: its potential relative to the smooth metric is locally bounded above.

Apply Proposition 4 with the normal model \(W\), the chosen linearization \(P\), and the relative section \(\gamma\) of \(kP_W-f_W^*(kB)\). Equivalently use the rational section \((s_q/s_p)^{k/(q-p)}\) before normalization. Equation (6) gives every invariant adjoint rank, the Jacobian has its natural invariant action, and the averaged smooth metric is invariant also for \(P\). The vertical minima defining \(kB\) are exactly those in that proposition. Its smooth coefficient is \(L'+jkP'\), ordinary \((1+jk)L'\), so the orthogonal quotient argument applies without any singular-subbundle claim. We obtain a continuous, in particular locally bounded, semipositive metric on \(kB\).

A fixed ideal on the base. The continuous base metric just obtained controls the varying power. The singular degree-zero metric will supply the positive constant term. For \(j=0\) use the singular coefficient metric \(h\). The singular adjoint direct-image theorem (Păun and Takayama 2018, Setup 3.2.1 and Theorem 3.3.4) applies over the smooth, locally free base-change locus. Its generic multiplier-ideal inclusion is an isomorphism by (7); fiber integrability is an almost-everywhere assertion by Fubini. The invariant summand is an orthogonal holomorphic quotient for the compact-invariant metric. Thus its line metric has semipositive curvature. Subtract a local potential for \(\delta\omega_Y\) from the coefficient weight to see that its curvature is in fact at least \(\delta\omega_Y\).

In a frame \(\xi\) of \(-K_Y\), denote the squared norm of \(1_Ef^*\xi\) by \(F_\xi(y)\). With base volume in the dual frame, this is the pushforward density of the integrable absolute adjoint form \(1_E\) with metric \(h\). Hence \(F_\xi\) is locally integrable on the whole base chart. We next extend its negative logarithm with an explicit exceptional correction.

Boundary extension with exceptional support. For a prime boundary divisor \(P\) choose a prime \(Q\subset V\) mapping onto it, of pullback multiplicity \(l\) and Jacobian order \(a=\operatorname{ord}_Q(E)\). At general points write \(f=(z_1^l,z_2,\ldots,z_e)\). Integrating on one coordinate patch, using \(h\ge c h_0\), gives almost everywhere \[ F_\xi(y)\ge c'|y_1|^{2((a+1)/l-1)}. \tag{8}\] The exponent includes the transverse Jacobian factor \(|z_1|^{-2(l-1)}\). If a nonexceptional component dominates \(P\), choose it for the estimate. Then \(a=0\) and the coefficient of \(P\) in the correction is zero. Otherwise every component dominating \(P\) is exceptional; choose a nonnegative integer coefficient at least \((a+1)/l-1\). Summing over the finitely many boundary primes gives \(D_*\ge0\) with \(f^*D_*\) exceptional over \(X\).

For the last assertion one must also check primes mapping to codimension at least two on \(Y\). The equidimensional normal model \(W\) has relative dimension \(n-e\), so for \(Z\subset Y\) of codimension at least two, \[\dim f_W^{-1}(Z)\le\dim Z+n-e\le n-2.\] Any prime of \(V\) over \(Z\) therefore maps to codimension at least two on \(W\) and on \(X\). Every component dominating a base prime in \(\mathop{\mathrm{Supp}}D_*\) is exceptional by the rule defining that support. These two checks account for every component of \(f^*D_*\).

On \(-K_Y+D_*\) define the metric by \(|\xi1_{D_*}|^2=F_\xi\) on the dense open. Its ordinary-frame weight is \(-\log F_\xi+\log|g_{D_*}|^2\). Estimate (8) bounds this weight above at general boundary points. Subtract a local smooth potential for \(\delta\omega_Y\). The resulting plurisubharmonic weight extends across each boundary divisor away from a codimension-two analytic subset. Hartogs extension for plurisubharmonic functions then fills that subset. Restoring the smooth potential gives a metric on all of \(Y\) with curvature at least \(\delta\omega_Y\). If \(\mathcal J\) is its multiplier ideal, then \[ \mathcal O_Y(-D_*)\subseteq\mathcal J, \tag{9}\] since multiplying the metric coefficient by \(|g_{D_*}|^2\) recovers the locally integrable density \(F_\xi\).

Interpolation and return to \(X\). Tensor by the bounded semipositive metric on \(jkB\). For every \(j\ge0\) the multiplier ideal remains the same \(\mathcal J\), which is not assumed trivial. Nadel vanishing yields \[H^v\bigl(Y,\mathcal O_Y(D_*+jkB)\otimes\mathcal J\bigr)=0 \qquad(v>0,\ j\ge0).\] Put \(\mathcal F=\mathcal O_Y(D_*)\otimes\mathcal J\). Its Euler characteristic after twisting by \(jkB\) is a polynomial in \(j\). The inclusion (9) gives \(\mathcal O_Y\hookrightarrow\mathcal F\), so at zero this polynomial equals \(h^0(Y,\mathcal F)>0\). A nonzero section therefore exists for some \(j>0\). Pull it back and multiply by \(\gamma^j\) to obtain an invariant section of \(jkP'+f^*D_*\). Removing the exceptional twist and pushing down gives the required section of \(jkP\). ◻

Proof. The polynomial in Proposition 5 is not zero. A fixed cohomological degree therefore has nonzero invariant cohomology along an unbounded sequence. Average a Kähler form and the metric weights over the maximal compact torus, and denote the resulting curvature of \(L\) by \(\theta\). The smooth semipositive hard Lefschetz theorem (Demailly et al. 2001, Corollary 2.1.2), applied to \(mL=K_X+(m+1)L\), gives the asserted forms. Concretely, if \(u\) is harmonic of type \((n,q)\) with values in \((m+1)L\), Bochner–Kodaira gives \[0=\|(D')^*u\|^2+\langle[(m+1)\theta,\Lambda]u,u\rangle.\] Both terms are nonnegative. Applying the complex-linear star on form indices to \(u\) therefore gives a holomorphic form of type \((n-q,0)\). The construction is equivariant. For \(q=0\) it already yields invariant sections of \(mL\); for \(q=n\) the holomorphic form itself is a section of \((m+1)L\). If \(r=\max\operatorname{rank}\theta=n\) and \(q>0\), the curvature term makes this form vanish on a nonempty open set, hence everywhere, so this case cannot occur.

The determinant extraction follows the method of (Lazić et al. 2023, Lemma 5.1) and (Lazić and Peternell 2018, Lemma 4.1). We spell out the unbounded-twist step, which here also applies when \(L\) is numerically trivial. Choose a rational frame of \(L\) and divide each form by the corresponding power of that frame. Their span over \(\mathbb C(X)\) is a subspace of the generic fiber of \(\Omega_X^{n-q}\), of some rank \(e\). Changing the frame rescales each vector by a nonzero rational function, so this subspace is intrinsic. Saturate its top exterior power in \(\bigwedge^e\Omega_X^{n-q}\) and call the resulting rank-one subsheaf \(M\). It is reflexive on smooth \(X\), hence a line bundle. The generic span is torus-stable, so its saturation carries the induced linearization.

Choose finitely many of the original forms as a generic basis. Every wedge of \(e\) original forms lies generically in \(M\) tensored by the sum of their powers of \(L\); saturation makes it a regular section of that line. In the fixed basis, one coordinate of the unbounded family is nonzero for unbounded \(m\). Wedge those forms with the other \(e-1\) fixed basis forms. These wedges are nonzero and invariant, and their twist exponents are \(m+1\) plus a fixed sum. They give the required sections with \(a_i\to\infty\). ◻

Proof. Every element of the connected torus is homotopic to the identity, so its action on singular cohomology is trivial. By Hodge decomposition its action on every \(H^q(X,\mathcal O_X)\) is trivial. Consequently the invariant index at zero equals \(\chi(X,\mathcal O_X)\ne0\). Lemma 6 supplies either an invariant section directly or invariant sections of \(M+a_iL\) with \(a_i\to\infty\) and a nonzero map from \(M\) into a positive tensor power of \(\Omega_X^1\). The cotangent-tensor theorem (Lazić et al. 2023, Theorem 4.1), applied to the smooth zero-boundary pair with nef \(-K_X\), says that \(-M\) is pseudoeffective. Apply Theorem 1 with \(P=L\) naturally linearized. ◻

Proof. For every Kähler class \(G\), Yau’s prescribed-Ricci theorem (Yau 1978) gives a Kähler metric in \(G\) with Ricci form \(\theta\). Kähler curvature symmetry identifies the mean curvature of \(T_X\) with Ricci; the induced mean curvature of \(\bigwedge^e\Omega_X^p\) is nonpositive. If \(\deg_G M>0\), solve a scalar Laplace equation to give \(M\) a metric of positive constant mean curvature. Its inverse tensor the cotangent construction then has negative mean curvature, contradicting the Bochner formula for its nonzero section. Thus \(c_1(M)G^{n-1}\le0\). Put \(G=[\theta]+t[\omega]\), divide by \(t^{n-1-r}\), and let \(t\downarrow0\). Since \(\theta^{r+1}=0\), \[c_1(M)[\theta]^r[\omega]^{n-1-r}\le0.\] The same intersection for \(M+aL\) is nonnegative by effectivity and smooth semipositivity, and the contribution from \(aL\) is zero. It is consequently zero. Each component has zero intersection separately; the nonnegative form \(\theta^r\wedge\omega^{n-r-1}\) therefore vanishes on its smooth locus, which gives the last assertion. ◻

Proof. Białynicki-Birula’s theorem (Białynicki-Birula 1973) gives positive normal \(\lambda\)-weights at \(S\). The first nonzero conormal jet of an eigensection subtracts a nonnegative sum of these weights from the line’s fiber weight, proving the section bound. For an ample line, generation of a sufficiently high power gives the fixed-fiber weight bound. Apply this to \(bU+A\), for an ample linearized \(A\), and let \(b\to\infty\) to obtain the bound for nef \(U\).

To verify the full-character identity, evaluate the equivariant fixed-point formula (Atiyah and Segal 1968; Atiyah and Singer 1968) at \(a\lambda(t)\) as \(t\to\infty\), divide by \(t^{m\langle\mu,\lambda\rangle}\), and keep \(a\) generic in \(T\). The inverse normal Euler factors are \[(1-a^{-w}t^{-\langle w,\lambda\rangle}e^{-x})^{-1}.\] They tend to one at \(S\). Every other fixed component has a negative normal weight, hence an inverse factor tending to zero, and its line weight is no larger. Truncating in the ordinary cohomology degree gives the limit \(a^{m\mu}\chi(S,mU|_S)\). Since the index is a finite Laurent polynomial, existence of this limit excludes higher \(\lambda\)-weights. The limit is precisely the sum of its terms on the weight hyperplane with pairing \(m\langle\mu,\lambda\rangle\); varying \(a\) identifies that sum with the single character \(m\mu\) and the stated coefficient. For the trivial action the identity is immediate. ◻

Proof. Take a generic integral cocharacter \(\lambda\) with \(X^\lambda=X^T\). A linearized projective embedding shows that only finitely many walls must be avoided. Białynicki-Birula’s attracting-cell theorem (Białynicki-Birula 1973) gives a unique source component \(S\) with open attracting stratum as \(t\to0\); all its normal \(\lambda\)-weights are positive. Projection of that stratum to \(S\) gives a dominant rational map \(X\dashrightarrow S\). Pullback of forms is injective and extends across the codimension-two indeterminacy locus on smooth \(X\). Thus \(S\) also has no positive-degree holomorphic forms and \(\chi(S,\mathcal O_S)=1\).

Let \(\mu\) be the full \(T\)-character of \(L|_S\). Apply Lemma 10 with \(U=L\).

The right side of (10) is polynomial in \(m\) with value one at zero. For infinitely many \(m>0\), one fixed cohomological degree has a nonzero weight-\(m\mu\) class. Equivariant hard Lefschetz with coefficient \((m+1)L\) produces, for a fixed \(p\), nonzero eigenforms \[\alpha_m\in H^0(X,\Omega_X^p\otimes(m+1)L), \qquad\operatorname{wt}(\alpha_m)=m\mu.\] Use an invariant Kähler form and the maximal compact torus to preserve all character spaces.

Use the determinant construction in Lemma 6: the generic span has rank \(r\), and its saturated determinant is a linearized line \(M_0\subset\bigwedge^r\Omega_X^p\). Wedge forms with unbounded \(m\) against \(r-1\) fixed complementary basis forms. If the indices in one such wedge are \(m_1,\ldots,m_r\), its twist degree and character are \[i=\sum_{j=1}^r(m_j+1),\qquad \sum_{j=1}^r m_j\mu=(i-r)\mu.\] Thus there are sections of \(M_0+iL\) of weight \((i-r)\mu\) for unbounded \(i\). If \(p>0\), the cotangent tensor theorem (Lazić et al. 2023, Theorem 4.1) makes \(-M_0\) pseudoeffective; for \(p=0\), \(M_0=\mathcal O_X\) as an ordinary line bundle. Let \(P\) be \(L\) with its natural linearization shifted by \(-\mu\), and let \(M\) be \(M_0\) with its induced linearization shifted by \(r\mu\). These changes leave the ordinary line bundles unchanged, so \(-M\) is pseudoeffective in both cases above. The eigensections just constructed are invariant sections of \(M+iP\), because \[(i-r)\mu+r\mu-i\mu=0.\] Theorem 1 yields a nonzero section \[v_\mu\in H^0(X,k_\mu L),\qquad \operatorname{wt}(v_\mu)=k_\mu\mu, \qquad k_\mu>0\] in the natural linearization.

There are finitely many source weights as the generic cocharacter varies. Zero lies in the convex hull of the actual section weights \(k_\mu\mu\). Otherwise strict separation gives a generic integral cocharacter pairing negatively with all of them. At its own source, the weight \(\mu\) is a sum of tangent weights of positive pairing, or the empty sum for a trivial action, contradicting strict negativity. The weights are integral, so a convex combination giving zero may be chosen rational. Clear its denominators and multiply the corresponding powers of the nonzero \(v_\mu\). The product is a nonzero invariant section in positive degree. For the trivial torus the same argument starts directly from \(\chi(X,\mathcal O_X)=1\) with \(\mu=0\). ◻

Proof. The input is the injectivity theorem of Fujino–Matsumura (Fujino and Matsumura 2021, Theorem A). On a compact Kähler manifold it applies to a possibly singular coefficient metric and a smooth semipositive multiplying line bundle \(S\), provided the former curvature dominates a positive multiple of the latter. Multiplication by any nonzero section of \(S\) is injective on canonical cohomology with the coefficient multiplier ideal. Here we use \(S=g^*(mA)\); the comparison constant can be \(\epsilon/m\), and the ideal is trivial.

We include the reduction to base cohomology, following the slicing and Leray argument in (Fujino and Matsumura 2021, proof of Proposition 1.9, Step 3). Induct on \(\dim Y\). Choose \(m\) large enough for Serre vanishing for all the finitely many sheaves \(R^jg_*\mathcal F\), where \(\mathcal F=\mathcal O_V(K_V+N)\). A sufficiently general divisor \(C\in|mA|\) is smooth, has smooth inverse image \(V_C\), contains no associated point of these direct images, and contains the image of no component of \(V\). Its restricted coefficient metric has trivial multiplier ideal. To see the last assertion without assuming analytic singularities, pull the locally integrable metric coefficients to the incidence variety of the basepoint-free system on \(V\). That incidence variety is a projective-space bundle over \(V\). Fubini’s theorem gives integrability on almost every smooth member, simultaneously on a finite coordinate cover. This full-measure condition meets the required algebraic open conditions. Restricting the curvature inequality on these members gives the same hypothesis over \(C\).

Adjunction and multiplication by the cutting section give \[0\longrightarrow\mathcal F\longrightarrow \mathcal F\otimes g^*\mathcal O_Y(mA)\longrightarrow \mathcal O_{V_C}(K_{V_C}+N|_{V_C})\longrightarrow0.\] The corresponding long direct-image sequence breaks into short exact sequences: multiplication by the cutting equation is injective on every \(R^jg_*\mathcal F\), by the associated-point choice. The induction hypothesis on \(C\) and Serre vanishing for the middle terms therefore give \(H^i(Y,R^jg_*\mathcal F)=0\) for \(i\ge2\).

The Leray spectral sequence now has only columns zero and one. For each \(j\) it identifies \(H^1(Y,R^jg_*\mathcal F)\) with the first filtration subspace of \(H^{j+1}(V,\mathcal F)\). After the twist by \(g^*(mA)\) this subspace maps to zero, since all positive base cohomology of the twisted direct images vanishes. Functoriality of the Leray filtration and Fujino–Matsumura injectivity imply that the original subspace is zero. This proves the remaining degree \(i=1\). ◻

Proof. Fix \(a<b\) in \(S\). The affine relations \[ (s-a)N_b\sim(s-b)N_a+(b-a)N_s\qquad(s>b) \tag{11}\] will make the fiberwise section direction unique. Put \(e=\kappa(X,N_b)\). If \(e=0\), the two effective divisors in (11) are equal. Letting \(s\) tend to infinity gives \(N_b\ge N_a\), so \(N_b-N_a\sim(b-a)L\) is effective. If \(e=\dim X\), \(bL=N_b+F_0\) is big, which also proves the assertion. We treat \(0<e<\dim X\).

Resolve a multiple system of \(N_b\) attaining its Iitaka dimension and take its Stein factorization. After modifying the base, flattening the main component, normalizing, and resolving, we have \[V\longrightarrow W\longrightarrow X, \qquad f:V\to Y, \qquad f_W:W\to Y.\] Here \(Y,V\) are smooth projective, \(W\) is normal projective and birational to \(X\), \(f_W\) is equidimensional, and \(f\) has connected fibers. The normalization is finite and preserves the fiber-dimension bound from flattening; the resolution \(V\to W\) need not be equidimensional over \(Y\). Equivalently the flat main component is obtained from the closure of the family of generic fibers in the projective Hilbert scheme (Grothendieck 1961, Theorem 3.2). Write \(\pi:V\to X\), \(L'=\pi^*L\), and \(E=K_{V/X}\ge0\). For a very ample \(A\) on \(Y\) and some integer \(t>0\), \[t\pi^*N_b-f^*A\sim G\ge0.\] The initial moving polarization stays big after base modification, so it dominates the chosen \(A\) after increasing \(t\).

For a geometric generic fiber \(F\), \[ \kappa\bigl(F,(\pi^*N_b+E)|_F\bigr)=0. \tag{12}\] Indeed adding an effective exceptional divisor changes no global sections of multiples of a pullback from the smooth \(X\). If two sections on the generic fiber were independent, a sufficiently large ample base twist would globalize two such sections. Absorb that twist using \(t\pi^*N_b\ge f^*A\), up to linear equivalence. Their nonconstant fiberwise ratio, together with the ratios giving the base, would yield image dimension at least \(e+1\) for a multiple of \(\pi^*N_b+E\). The geometric generic fiber is integral because the Stein field is relatively algebraically closed in characteristic zero. Generic base change justifies the globalization over the original function field. This contradiction proves (12). The same assertion holds without \(E\).

Restriction of (11) to \(F\) is therefore equality of effective divisors. Unboundedness of \(S\) shows \((\pi^*(N_b-N_a))|_F\ge0\). If \(d=b-a\), a rational section \(z\) of \(dL\) with divisor \(N_b-N_a\) is consequently regular on \(F\). For every \(k\ge0\), \[ h^0\bigl(F,(E+kdL')|_F\bigr)=1. \tag{13}\] The indicated effective section gives the lower bound. For \(k\ge1\), multiply by the section of \(k\pi^*N_a+(k-1)E\) to inject into the sections of \(k(\pi^*N_b+E)|_F\). For \(k=0\), multiply by the section of \(\pi^*N_b|_F\). Equation (12) gives both upper bounds. This explicitly includes the zero exponent.

On the normal equidimensional model \(W\), take for every prime \(P\subset Y\) the minimum \[c_P=\min_{Q\mapsto P} \frac{\operatorname{ord}_Q(z)}{ \operatorname{ord}_Q(f_W^*P)}.\] Every vertical prime maps to a base prime and every denominator is positive. There are only finitely many nonzero \(c_P\). Subtracting \(f_W^*(\sum c_PP)\) leaves an effective rational Cartier divisor, because the horizontal divisor of \(z\) was already effective. Clear denominators, replacing \(d,z\) by a common power. We obtain \[ dL'\sim H+f^*B, \qquad H\ge0, \tag{14}\] with \(H\) pulled back from an effective Cartier divisor on \(W\) and \(B\) integral on \(Y\). Above each base prime at least one component has coefficient zero in \(H\).

Proposition 4 now applies with the ordinary line \(P=L\), the rational section \(z\), and the normal equidimensional model \(W\). Equation (13) supplies rank one for every \(k\ge0\); the vertical minima giving (14) agree with the proposition’s normalization. Hence \(B\) has a continuous semipositive metric. In a local frame \(b\), with \(\sigma=s_Hf^*b\) and \(h_0=\pi^*h_L\), its weight is \[ \phi_B=-\log\max_{V_y}|\sigma|_{h_0^d}^2. \tag{15}\] Only local boundedness of this metric is needed in the following injectivity argument.

It remains to use the zero exponent rather than merely asymptotic positivity. Since \(F_0\) is pseudoeffective, the ample domination above gives \(nL'-f^*A\) pseudoeffective for some \(n>0\). Its positive current yields a singular metric on \(L'\) with curvature dominating a positive multiple of \(f^*\omega_A\). Mix its weights with the smooth semipositive weights using a sufficiently small positive coefficient. Skoda’s small-exponent integrability (Demailly 2011, Lemma 5.6(a)), in the squared-norm convention \(e^{-\phi}\), and a finite covering of \(V\) make the new multiplier ideal trivial while retaining a strictly positive base curvature coefficient. No analytic-singularity assumption is used. For \(k\ge0\) put \(C_k=L'+kf^*B\) and add \(kf^*\phi_B\) to the coefficient weights. Local boundedness keeps the multiplier ideal trivial and preserves the positive base-curvature lower bound. Since \(K_V+C_k=E+kf^*B\), Lemma 12 and the projection formula give \[H^i\bigl(Y,f_*\mathcal O_V(E)\otimes\mathcal O_Y(kB)\bigr)=0 \quad(i>0,\ k\ge0).\] Riemann–Roch makes the Euler characteristic a polynomial in \(k\). At \(k=0\) it equals \(h^0(V,E)=1\). It is therefore nonzero at some positive integer \(k\). A section of \(E+kf^*B\), multiplied by \(s_H^k\), gives a section of \(E+kdL'\). Since \(\pi_*\mathcal O_V(E)=\mathcal O_X\), it yields a nonzero section of \(kdL\) on \(X\). ◻

Proof. For \(p=0\) the assertion is immediate. Otherwise take the stable generic span of the directions of the twisted forms in \(\Omega_X^p\), of rank \(r>0\), and saturate its determinant in \(\bigwedge^r\Omega_X^p\subset(\Omega_X^1)^{\otimes pr}\). A saturated rank-one subsheaf of a locally free sheaf on smooth \(X\) is reflexive and hence invertible; call it \(M\). Wedge an arbitrarily far-tail member with suitable \(r-1\) members of a fixed generic basis. This gives nonzero sections of \(M+sL\) for unbounded sums \(s\). The cotangent-tensor generic-nefness theorem (Lazić et al. 2023, Theorem 4.1) applies because \(-K_X\) is nef and proves that \(-M\) is pseudoeffective. Apply the proposition with \(F_0=-M\) and the divisors of these sections. ◻

Proof. This is the regularized-maximum construction of Richberg (Richberg 1967); see also (Demailly 2012, Theorem I.5.21). The small negative curvature allowance permits its use for semipositive, rather than strictly positive, weights. Write a weight as a smooth reference weight plus a global continuous function \(u\), with background curvature \(\gamma\). For fixed \(\varepsilon>0\), use background \(\gamma+\varepsilon\eta\), so that the original potential has a strict local margin. On finitely many enlarged coordinate patches, add a smooth local potential of this background, convolve, and subtract that potential. Take inner patches which still cover \(Y\). Add smooth bumps zero on the inner patches and a small negative constant near the outer boundaries, choosing their second derivatives smaller than the strict margin and the convolution errors much smaller than the bump heights. A regularized maximum of the active candidates is smooth: a candidate near its outer boundary is uniformly below a candidate from an inner patch and does not affect the maximum. Convexity, monotonicity, and translation equivariance of regularized maximum preserve the background lower bound and the transition rule for weights. Let the uniform error and then \(\varepsilon\) tend to zero. Equivalently, one may smooth the strictly positive metric on \(NQ+H\), with \(H\) positive, subtract the weight of \(H\), and divide by \(N\). ◻

Proof. Fix \(k\). We will show that every nonzero cohomology class on the source restricts nontrivially to some general fiber. Smooth-coefficient decomposition will then give the asserted vanishing. Choose a Kähler form \(\omega\) on \(V\) and a Dolbeault class \(c\in H^\ell(V,K_V+A)\), where \(A=D_V+kf^*Q\) and \(n=\dim V\). Lemma 15 gives product metrics \(h_j\to h\) uniformly on \(A\), with curvature at least \(\theta-o(1)\omega\). Let \(u_j\) be the harmonic \((n,\ell)\)-representative of \(c\) for \(h_j\). Harmonic norm minimization, comparison with one fixed smooth closed representative, and uniform metric equivalence give a fixed \(L^2\) bound. The Bochner–Kodaira identity gives \[ \|(D'_j)^*u_j\|_{h_j}^2 +\langle[\theta,\Lambda]u_j,u_j\rangle_{h_j} \le o(1)\|u_j\|_{h_j}^2. \tag{18}\] Here curvature commutators in bidegree \((n,\ell)\) are monotone in the curvature form: their eigenvalues are sums of the curvature eigenvalues on the antiholomorphic index set.

Let \(v_j=*u_j\), absorbing a fixed nonzero dimensional constant so that \(u_j=\omega^\ell\wedge v_j\). The complex-linear star acts on form indices, carrying the coefficient line along. Metric compatibility gives \((D'_j)^*=\pm*\bar\partial*\) and \(\bar\partial_j^*=\pm*D'_j*\). Thus \[\bar\partial v_j\longrightarrow0\text{ in }L^2, \qquad D'_jv_j=0.\] These are statements about a fixed holomorphic bundle; the first operator does not differentiate the varying metric. Passing to a weak \(L^2\) subsequence gives a distributionally holomorphic, hence smooth holomorphic, \(v\). Since \(\omega\) is Kähler, the form \(u=\omega^\ell\wedge v\) is \(\bar\partial\)-closed and represents \(c\). Indeed projection to harmonic forms for one fixed smooth reference metric is a bounded finite-rank operator, has the same value on every \(u_j\), and commutes with this weak limit. The nonnegative square root of \([\theta,\Lambda]\) annihilates the limit by (18). Pointwise diagonalization therefore shows that every positive-curvature holomorphic covector is a wedge factor of \(v\).

The approximation also gives a Chern equation before any use of the kernel condition. In any local holomorphic frame of \(A\), let \(h_j\) denote its squared norm. The equation \(D'_jv_j=0\) reads \(\partial(h_jv_j)=0\). Uniform convergence of \(h_j\) and weak \(L^2\) convergence suffice to pass it to distributions: \[ \partial(hv)=0. \tag{19}\] No derivative convergence of metric weights is used.

To detect the class on a fiber, we next show that \(v\) contains all base canonical covectors as wedge factors. On the open set in (16), every pulled-back base covector annihilates \(\ker\theta\), so its wedge with \(v\) vanishes. The holomorphic bundle map \[f^*\Omega_Y^1\longrightarrow\Omega_V^{n-\ell+1}\otimes A, \qquad \alpha\longmapsto (df)^*\alpha\wedge v,\] therefore vanishes everywhere. Over the smooth fibration locus, in base coordinates \(y_1,\ldots,y_b\), this gives \[ v=f^*(dy_1\wedge\cdots\wedge dy_b)\wedge w. \tag{20}\] The relative restriction \(w_y\) is unambiguously defined after choosing the base canonical frame. If \(n-\ell<b\), then \(v=0\) already.

The limiting metric is the smooth metric of \(D_V\) times a positive continuous function pulled back from the base. Because of (20), only vertical differentiations survive in (19). The continuous base function is constant under these differentiations; the remaining expression is smooth in the vertical variables and continuous altogether. It vanishes pointwise and gives \(D'_{D_V|V_y}w_y=0\) on every smooth fiber under consideration.

The restriction in cohomology is restriction of a \((0,\ell)\)-form valued in \(K_V+A\). Use \(K_V|_{V_y}=K_{V_y}\otimes K_Y|_y\) to remove the constant canonical base factor. It is not the ordinary pullback of an \((n,\ell)\)-form to a lower-dimensional fiber. The restricted representative is a nonzero constant times \[\omega_y^\ell\wedge w_y, \qquad \omega_y=\omega|_{V_y}.\] It is harmonic on the compact smooth fiber: it is \(\bar\partial\)-closed, and its star has zero Chern derivative. Thus a nonzero \(v\) gives a nonzero restriction class on some general fiber. The Leray edge map \[ H^\ell(V,K_V+D_V+kf^*Q) \longrightarrow H^0(Y,R^\ell f_*(K_V+D_V)\otimes Q^k) \tag{21}\] is injective, since zero edge image implies zero restriction on the general smooth base-change locus.

Finally apply Fujisawa’s derived decomposition (Fujisawa 2016, Theorem 1), obtained from Takegoshi’s harmonic representatives, to the smooth coefficient \(D_V\): \[Rf_*(K_V+D_V)\simeq\bigoplus_q R^q f_*(K_V+D_V)[-q].\] Tensor the resulting derived isomorphism by the line \(Q^k\) on \(Y\) and use the projection formula. It gives \[h^\ell(V,K_V+D_V+kf^*Q) =\sum_{p+q=\ell}h^p(Y,R^qf_*(K_V+D_V)\otimes Q^k).\] Injection (21) bounds the left side by the summand \(p=0\). All remaining summands vanish. For total degrees exceeding \(n\), the left side is zero directly. This proves every case of (17), including the twist \(k=0\). ◻

Proof. Keep the uniformly convergent metrics and the notation of the proof of Theorem 16. For \(\ell>0\), put \[S=i^{(n-\ell)^2}h\,v\wedge\overline v, \qquad T=S\wedge\omega^{\ell-1},\] with coefficient pairing understood. The distributional Chern equation and holomorphy make \(T\) a continuous closed positive \((n-1,n-1)\)-form. The curvature-energy equality gives \(\theta\wedge T=0\). Choose a positive curvature form \(\eta\) of \(H\). A positive current in the class of \(bD_V-f^*H\) pairs nonnegatively with \(T\), since positive currents have order zero. To compute this pairing cohomologically, apply de Rham regularization, which commutes with \(d\), to approximate \(T\) uniformly by smooth closed forms. For each approximant the pairing agrees with that against the cohomologous smooth form \(b\theta-f^*\eta\); uniform convergence passes this equality to \(T\). It follows that \[0\le\int_V(b\theta-f^*\eta)\wedge T =-\int_V f^*\eta\wedge T\le0.\] Pointwise positivity forces \(f^*\eta\wedge T=0\); diagonalization gives the full base wedge factor in \(v\). The distributional fiber equation, canonical-valued restriction, edge injection, and smooth-coefficient splitting proceed exactly as in that proof. Only positive total degrees are needed to kill a term of positive base degree. ◻

Proof. Let \(\mathcal G=f_*\mathcal O_V(E)\), or its invariant summand in the equivariant case. A torus over a trivially acted-on base has coherent weight summands, so this is a direct summand. Vanishing gives \(h^0(Y,\mathcal G\otimes Q^k)=\chi(Y,\mathcal G\otimes Q^k)\) for every \(k\ge0\). The right side is polynomial in \(k\) by Riemann–Roch for coherent sheaves. At zero it is one: \(\pi_*\mathcal O_V(E)=\mathcal O_X\) and the exceptional canonical section is invariant. The polynomial is therefore nonzero at some positive integer. Multiply the resulting section of \(E+kf^*Q\) by the \(k\)-th power of the injection and push through \(\pi\). The effective exceptional divisor \(E\) changes neither sections nor their characters, by normality of \(X\). ◻

Harmonic proof of Corollary 7. Use Lemma 6, retaining the cases which did not already produce a section, and write \(N_i=M+a_iL\), \(s_i\) for its invariant sections. Write \(n=\dim X\), let \(\theta\) be the averaged smooth semipositive curvature, let \(r\) be its maximal rank, and fix a Kähler form \(\omega\). Choose indices \(a_-<a\) and keep unbounded indices above \(a\). Put \(N=M+aL\). Among rational maps given by all invariant sections of positive powers of \(N\), choose one with maximal image dimension. Resolve its base ideal equivariantly and take Stein factorization. We obtain a smooth projective \(X_0\), a birational \(\pi_0:X_0\to X\), and \(f_0:X_0\to Y_0\) with connected fibers and normal projective base, with \[ d\pi_0^*N=f_0^*H_0+G_0,\qquad G_0\ge0, \tag{22}\] where \(H_0\) is ample. The torus acts trivially on \(Y_0,H_0\), and the defining section of \(G_0\) is invariant.

The invariant sections of \(j\pi_0^*N\) on the generic fiber form a one-dimensional vector space over \(\mathbb C(Y_0)\) for every \(j>0\). Indeed finitely many generic-fiber sections extend rationally with vertical poles. A section of a sufficiently high power of \(H_0\) clears those poles; multiplication by the corresponding power of the section of \(G_0\) gives invariant sections of a higher power of \(\pi_0^*N\), and hence of \(N\) on \(X\). If two original sections were independent, their ratio would not lie in \(\mathbb C(Y_0)\). This field is relatively algebraically closed by Stein factorization, so that ratio would be transcendental over it. Combine it with the original image coordinates, by products to put all ratios in one power. The image dimension increases, contradicting maximality.

For each \(a_i>a\) there are positive integers \(p,q\) such that \(pa_i+qa_-=(p+q)a\). The invariant section \(s_i^p s_-^q\) is thus generically proportional to \(s_a^{p+q}\). Its effective generic-fiber divisor shows that all horizontal components of \(\operatorname{div}(\pi_0^*s_i)\) lie in the fixed finite support of \(\operatorname{div}(\pi_0^*s_a)\). Extract a subsequence of their integer multiplicity vectors that is componentwise nondecreasing. For two indices \(a_j>a_i\) in it, put \[b=a_j-a_i>0,\qquad s=\pi_0^*s_j/\pi_0^*s_i.\] This is an invariant rational section of \(b\pi_0^*L\), regular on the generic fiber. The relative canonical divisor \(E_0=K_{X_0/X}\) has its horizontal support in the same fixed support: \(\pi_0\) can be chosen to be an isomorphism outside the invariant base ideal, which lies in the zero locus of \(s_a\). Fix \(k\ge0\), and work on the generic fiber. The effective divisors of \(s_{E_0}\) and \(s\) have support in the divisor of \(\pi_0^*s_a\). All coefficients are finite, so a sufficiently large integer \(h\) satisfies \[h\operatorname{div}(\pi_0^*s_a) \ge \operatorname{div}(s_{E_0})+k\operatorname{div}(s).\] The quotient \((\pi_0^*s_a)^h/(s_{E_0}s^k)\) is therefore a regular invariant section of the difference line. Multiplication by it injects the entire invariant adjoint section space into the one-dimensional invariant space for \(h\pi_0^*N\). Consequently \[ \dim_{\mathbb C(Y_0)}H^0((X_0)_\eta,E_0+kb\pi_0^*L)^T=1 \qquad(k\ge0). \tag{23}\] If \(\dim Y_0=0\), the section \(s\) is already regular on \(X_0\) and descends to the required section on \(X\).

Otherwise apply Lemma 3 to this fibration and resolve its normal equidimensional model: \[V\longrightarrow W\longrightarrow X_0,\qquad g:W\to Y, \qquad f:V\to Y.\] Write \(\mu:W\to X\) and \(\pi:V\to X\) for the composed birational maps. We also use \(f_0\) for its pullback to \(V\). Here \(Y\) is smooth projective, \(W\) is normal equidimensional over \(Y\), and \(V\) is smooth. All constructions are equivariant, with trivial base action. The ranks (23) persist: a further effective birational canonical divisor has trivial pushforward also on the generic smooth fiber.

It remains to relate this algebraic fibration to the curvature. The kernel condition for \(D_V=\pi^*L\) follows from Lemma 8 and (22). If \(r<n\), \[\int_V (\pi^*\theta)^r\wedge f_0^*\eta_{H_0} \wedge(\pi^*\omega)^{n-r-1}=0.\] Indeed the whole class \(dN\) has zero intersection, and both the moving and effective fixed parts have nonnegative intersection. On an ordinary open where \(\pi\) is an isomorphism, \(\theta\) has rank \(r\), and the base-to-image map has injective differential, positivity forces the horizontal form to annihilate \(\ker\pi^*\theta\). Its kernel there is \(\ker df\). Thus \(\ker\pi^*\theta\subset\ker df\). These algebraic generic conditions meet the nonempty ordinary maximal-rank open. For full curvature rank the kernel inclusion holds directly.

Proposition 4 normalizes \(s\) to \(g^*Q\to b\mu^*L\), with effective zero divisor \(G\), and gives a continuous semipositive metric on \(Q\). Theorem 16 and Proposition 19 now produce an invariant section of a positive power of \(L\). ◻

Proof. An ample line has a linearized positive power for a projective \(\mathbb C^*\)-action. Indeed the action’s map to the Picard variety is constant, since a torus has no nonconstant homomorphism to an abelian variety. The pairs consisting of an element and an isomorphism from the pulled-back line to the line form an algebraic extension of \(\mathbb C^*\) by the scalar \(\mathbb C^*\); this extension is a torus and splits. This supplies the needed linearization. The section-weight bound, the comparison of fixed weights, and the index formula are Lemma 10 for \(T=\mathbb C^*\) and \(U\) as given. The source retraction itself is the attracting-cell map of (Białynicki-Birula 1973). The additional assertion needed by the harmonic construction is its bundle map, which we now prove.

For the bundle map, identify the generic fibers of \(U\) and its restriction at the source by transporting a vector along an orbit, dividing by \(t^{w_{S_0}(U)}\), and taking the limit. The orbit extension over \(\mathbb A^1\), over the relevant function field, pulls the line back to an equivariantly trivial line of that weight. This constructs a multiplicative rational identification, hence the claimed rational map. If \(U\) is ample, a generated power has enough source-weight sections to generate its restriction to \(S_0\). Their pulled-back restrictions map to the original sections by the limit construction, proving absence of poles. For nef \(U\), a negative order in the rational map would give a negative order for \(kU+H\) for large \(k\), contrary to ampleness. This proves regularity along every prime divisor and hence everywhere by normality. The trivial action has \(S_0=S\) and the identity map. ◻

Proof. Determinant sections with a prescribed weight. A point is immediate. Otherwise \(S_0\) is rationally connected, being a smooth rational image of \(S\). Thus \(\chi(S_0,\mathcal O_{S_0})=1\). The highest-weight index in Lemma 21 is a nonzero polynomial in \(m\), so for unbounded \(m\) a fixed group \(H^q(S,mL)\) contains weight \(mw\). The smooth coefficient Lefschetz map is equivariant for a circle-invariant Kähler form. It gives twisted sections \[v_m\in H^0(S,\Omega_S^{\dim S-q}\otimes(m+1)L) =H^0(S,\bigwedge^q T_S\otimes mL)\] of weight \(mw\). For \(q=0\) they are the desired sections. Otherwise saturate their generic span in \(\bigwedge^qT_S\), of rank \(e>0\), and call it \(\mathcal U\). The determinant extraction used earlier gives nonzero eigensections \[ d_i\in H^0(S,\det\mathcal U+k_iL),\qquad \operatorname{wt}(d_i)=k_iw,\qquad k_i\to\infty. \tag{25}\] If \(D_i\) is the zero divisor, we claim \[eL-c_1(\mathcal U)\text{ pseudoeffective},\qquad (e+k_i)L-D_i\text{ pseudoeffective}.\] For \(q=\dim S\), saturation gives \(\mathcal U=\bigwedge^{\dim S}T_S=L\) and \(e=1\), so \(eL-c_1(\mathcal U)=0\) directly. For \(0<q<\dim S\), the cotangent-subsheaf theorem (Lazić et al. 2023, Theorem 4.1) applies: \(\mathcal U\otimes(-L)\subset\Omega_S^{\dim S-q}\), and exterior powers are direct summands of tensor powers in characteristic zero; saturating can only add an effective determinant correction in the required direction.

A base on which dominated systems become rigid. Call a divisor class dominated if a positive multiple of \(L\) minus it is pseudoeffective. Maximize the image dimension of linear systems of dominated effective integral divisors, and choose a full system \(|P_0|\) realizing this maximum. Rational connectedness gives \(\operatorname{Pic}^0(S)=0\), so the system may be linearized and taken equivariantly. Resolve its graph, take Stein factorization, flatten by the Hilbert-family construction, resolve the base and normalize the main component. We obtain \[V\longrightarrow W\longrightarrow S,\qquad g:W\to Y,\] with smooth projective \(V,Y\), normal equidimensional \(W\), and connected fibers. Write \(f:V\to Y\), \(\mu:V\to S\), \(A=\mu^*L\), \(E=K_{V/S}\), and let \(L_W\) be the pullback of \(L\) to \(W\). An ample pullback from \(Y\) is dominated: the original moving polarization is dominated, and its pullback to a birational base model is big, hence dominates a small ample class. Also \(E\) is dominated. Horizontal exceptional primes have centers in the original base locus and are bounded by pullbacks of divisors in \(|P_0|\); vertical primes are bounded by sufficiently ample base pullbacks. These statements prove domination of the whole finite effective canonical discrepancy.

A line \(B\) on \(V\) is dominated when \(cA-B\) is pseudoeffective for some \(c>0\). No such line can have two independent sections on the generic fiber. If it did, tensor by a sufficiently ample base pullback to globalize them. This new line remains dominated because the base polarization is dominated. Combine their ratio with the original base coordinates, using products to put all coordinates in a single linear system. Its line is still dominated. Push the effective system and its pseudoeffective domination relation down to \(S\). The new ratio is nonconstant along the generic fiber, so the image dimension exceeds the chosen maximum, a contradiction. In particular generic-fiber sections of both \(jaA\) and \(E+jaA\) have dimension at most one for every \(a,j>0\), and the same adjoint assertion for \(j=0\) follows from domination of \(E\).

There are only finitely many dominated horizontal primes on \(V\). Here \(\operatorname{Pic}^0(V)=0\), since \(V\) is birational to the rationally connected \(S\). Thus its Picard group has finite rank. Infinitely many dominated horizontal primes would give, after clearing torsion, a nontrivial integral linear equivalence among them. Splitting positive and negative coefficients gives two effective divisors in the same class and a pencil varying along the generic fiber, contrary to the preceding paragraph. The pullbacks of \(D_i\) therefore have horizontal supports in a fixed finite set. Order their multiplicities along a subsequence. A ratio from (25) gives a rational eigensection \(s\) of \(aL_W\), weight \(aw\), with no horizontal poles.

The metrized base line and its source weight. Divisorial normalization, as in Proposition 4, gives \[ g^*Q\longrightarrow aL_W, \tag{26}\] with effective zero divisor omitting one lift over every base prime. Shift the linearization of \(Q\) to make this map equivariant. Since the canonical exceptional section times \(s^j\) is nonzero generically, all required adjoint ranks are exactly one. The maximum construction gives a continuous semipositive metric on \(Q\). Here the base action may be nontrivial, but full rank one is available and the ordinary direct-image line theorem applies without taking an invariant summand.

We must still control the character produced by interpolation on the base. Let \(Y_0\) be the source of its induced action. The original unbounded eigensections imply \[ w_{Y_0}(Q)\ge aw. \tag{27}\] After fixing \(a\), take an infinite subsequence of \(k_i\) in one residue class modulo \(a\), still ordered horizontally. Divide by its first section. The resulting eigensections of \(jaL_W\), for unbounded \(j\), have weight \(jaw\), no horizontal poles and vertical poles bounded by one fixed divisor. Clear these poles by an invariant effective Cartier divisor \(C\) on \(Y\), with its canonical section of weight zero. Generic rank one and the map (26) show that each resulting section comes from \(jQ+C\). It has no residual poles: at a base prime test on a lift omitted by the zero divisor of (26). The source upper bound gives \(jaw\le j w_{Y_0}(Q)+w_{Y_0}(C)\). Divide by \(j\) and pass to the unbounded subsequence to obtain (27).

Euler interpolation on the source of the base. The continuous semipositive metric makes \(Q\) nef by Lemma 15. Resolve the source retraction of \(Y\) and dominate this resolution by a smooth model \(V'\) of \(V\). Let \(p:V'\to Y_0\), \(\mu':V'\to S\), \(A'=\mu'^*L\), \(E'=K_{V'/S}\), and \(Q_0=Q|_{Y_0}\). Lemma 21 and (26) give \(p^*Q_0\to aA'\). An ample pullback by \(p\) is dominated by \(A'\): the pullback of the original ample base polarization is big on the resolution of \(Y\), and hence dominates an ample class there, which in turn dominates an ample pullback from \(Y_0\). Proposition 18 applies to every twist \(jQ_0\), including zero. The action on \(Y_0\) is trivial, and \(Q_0\) has the constant fiber weight \(w_{Y_0}(Q)\). Let \(\mathcal B\) be the weight-zero summand of \(p_*\mathcal O(E')\). Thus sections of \(\mathcal B\otimes Q_0^j\) have precisely weight \(j w_{Y_0}(Q)\). Vanishing identifies their dimension with the Euler polynomial for every \(j\ge0\). At zero this polynomial is positive, because the canonical exceptional section is invariant. For some \(j>0\) we therefore obtain a nonzero section of \(E'+jp^*Q_0\) of that weight. Map it into \(E'+jaA'\) and push down to \(S\). Its weight is at least \(jaw\) by (27), and at most \(jaw\) by the source bound on \(S\). Equality gives the required extremal section. ◻

Proof. For a generic integral one-parameter subgroup \(\lambda\), its source \(C\) is a torus-fixed component with all normal weights positive on \(\lambda\). Denote by \(\gamma_C\) the fiber character of \(L\). Choose a full torus eigenvector in the extremal section space of Proposition 22. It cannot vanish on \(C\), since a nonzero leading conormal jet would strictly lower its \(\lambda\)-weight. Its character is therefore \(\ell\gamma_C\). The finitely many characters of such sources have convex hull containing zero. Otherwise a separating functional, approximated by a generic rational one, would be negative on all of them, whereas the source for that one-parameter subgroup has \(\gamma_C(\lambda)>0\), the sum of its positive normal weights. Use the effective torus quotient if the action has a kernel; a trivial action is already covered. A rational convex combination of the source characters is zero. Clear its denominators and the finitely many section exponents, then multiply the corresponding powers of their nonzero eigensections. On the integral variety this product is nonzero and invariant. ◻

Proof. We construct a globally strictly positive metric with exactly the same multiplier ideal.

Producing an integrable divisor metric. Choose a very ample \(A\) with curvature \(\omega_A\), and a smooth metric \(\psi_0\) on \(R\). For sufficiently small fixed \(\delta>0\) and arbitrarily large integers \(l\), we first claim that \[ H^0\bigl(Y,\mathcal O_Y(lR-A)\otimes \mathcal I(l(1+\delta)\psi)\bigr)\ne0. \tag{32}\] On \(lR\) use weights \(l(1+\delta)\psi-l\delta\psi_0\), and add the smoothly metrized line \(j_lA-K_Y\), with \(j_l=\lceil C\delta l+C'\rceil\). Fixed sufficiently large constants \(C,C'\) absorb the negative smooth curvatures and leave a positive lower bound. The multiplier ideal is \(\mathcal I_l=\mathcal I(l(1+\delta)\psi)\). Within a smaller ball of strict positivity choose a point and a cutoff logarithmic pole equal to \(\log|z|^2\) near that point. Add \(\eta l\) times this pole, with \(\eta>0\) fixed sufficiently small independently of \(\delta,l\). Its negative curvature costs at most \(O(\eta l)\) on the ball, where the original positive lower bound has order \(l\). Nadel vanishing (Nadel 1990), in the strict-current form of (Demailly 2011, Theorem 5.11), therefore gives surjectivity to jets of order \(\lfloor\eta l\rfloor-d\) at the point. Continuity of \(\psi\) there makes \(\mathcal I_l\) trivial near that point, and polar-coordinate integration gives the ideal \(\mathfrak m^{\lfloor\eta l\rfloor-d+1}\) for the added pole. Consequently \[h^0\bigl(Y,(lR+j_lA)\otimes\mathcal I_l\bigr)\ge c l^d\] for some \(c>0\) independent of sufficiently small \(\delta\).

Remove \(j_l+1\) copies of \(A\). At each step choose a smooth member of \(|A|\) containing no associated center of \(\mathcal O_Y/\mathcal I_l\). If a section vanishes on that member, division by its equation preserves membership in \(\mathcal I_l\), because that equation is a nonzerodivisor on the quotient. The number of conditions lost at a step is at most \(C''l^{d-1}\) uniformly for \(0<\delta\le1\). Indeed choose \(p\) with \(pA-R\) globally generated, restrict a general nonzero section to the hyperplane, and bound the restriction space by that of \((pl+h)A\) there, where \(0\le h\le j_l\). The Hilbert polynomial and the hyperplane exact sequence give this bound. The total loss is \(O(\delta l^d)+O(l^{d-1})\). Choosing \(\delta\) small and then \(l\) large proves (32).

Let \(\sigma\) be a nonzero section in (32). In compatible local frames, the weight \(\phi_A\) of the positive metric on \(A\) defines a divisor metric on \(R\) by \[\psi'=\frac{1}{l}\bigl(\log|\sigma|^2+\phi_A\bigr).\] Its curvature dominates \(\omega_A/l\). If \(V=\psi'-(1+\delta)\psi\), membership of \(\sigma\) in the multiplier ideal in (32) says exactly that \(\exp(lV)\) is locally integrable.

Preserving the ideal. Take \(l\ge1/\delta\) and put \(\rho=(1-\varepsilon)\psi+\varepsilon\psi'\). The identity \[-\psi=-\frac{\rho}{1+\varepsilon\delta} +\frac{\varepsilon V}{1+\varepsilon\delta}\] and Hölder’s inequality, with the integrability of \(\exp(V/\delta)\), prove \(\mathcal I(\rho)\subseteq\mathcal I(\psi)\). For the converse, strong openness (Guan and Zhou 2015, Theorem 1.1), coherence, and compactness give \(\gamma>0\) with \(\mathcal I((1+\gamma)\psi)=\mathcal I(\psi)\) on a finite cover. Small-exponent integrability of the divisor metric \(\psi'\) gives \(\gamma'>0\) with \(\exp(-\gamma'\psi')\) locally integrable. Apply Hölder with \(s=(1-\varepsilon)/(1+\gamma)\) and \(1-s\). For small \(\varepsilon\) one has \(\varepsilon/(1-s)<\gamma'\), proving \(\mathcal I(\psi)\subseteq\mathcal I(\rho)\). The metric \(\rho\) has a global positive curvature lower bound and the same ideal. Nadel vanishing proves the assertion. ◻

Proof. We spell out the consequence of Berndtsson’s curvature formula (Berndtsson 2009, sec. 4, Proposition 4.2, Lemmas 4.3–4.4, and equation (4.8)), since strict vertical positivity is not required. Restrict to a disk with coordinate \(z\) in a chosen base direction, and let the fiber dimension be \(a\). If \(a=0\), the direct image is locally the orthogonal sum of the coefficient lines on the finitely many sheets. Its curvature pairing is a sum of nonnegative terms, with a positive term on the patch where \(u_0\) is nonzero. Assume now \(a>0\). A holomorphic direct-image section extending \(u_0\) with vanishing covariant derivative at zero has a smooth absolute \((a,0)\) representative \(u\) satisfying \[D'u=dz\wedge\lambda,\qquad \bar\partial u=dz\wedge\eta, \qquad D'=\partial-\partial\phi\wedge.\] On the central fiber, \(\lambda\) is orthogonal to holomorphic adjoint forms and \(\eta\wedge\omega\) is \(\bar\partial\)-exact. Replace \(u\) by \(u-dz\wedge\xi\) so that \(\lambda=0\) and \(\eta\) is primitive there. These two requirements prescribe, respectively, \(\bar\partial^*(\omega\wedge\xi)\) and \(\bar\partial(\omega\wedge\xi)\), up to the Kähler-identity constants. Orthogonality and exactness place the two prescriptions in their required images; Hodge decomposition solves them simultaneously. Lefschetz linear algebra identifies the relevant \((a,1)\)-forms with \(\omega\wedge\xi\), for \(\xi\) of type \((a-1,0)\).

With \(c_a=i^{a^2}\), the resulting curvature pairing is \[c_a p_*\bigl(u\wedge\bar u\,e^{-\phi} \wedge i\partial\bar\partial\phi\bigr) -i\,dz\wedge d\bar z\,c_a \int_{V_0}\eta\wedge\bar\eta\,e^{-\phi}.\] The second term is nonnegative by the primitive middle-degree Hodge–Riemann sign. The first integrand is nonnegative everywhere. On the prescribed patch the pullback lower bound contributes the positive base form times the squared vertical norm of \(u_0\). This is positive on a nonempty open subset, so its integral is strictly positive. For a torus action, averaging is a holomorphic orthogonal projection. The Chern connection preserves the orthogonal character splitting, so the same calculation applies on the invariant summand. ◻

Proof. The sheaf \(\mathcal B\) is torsion-free. Its hull is determined by regularity over the generic points of base primes. More precisely, for a prime \(P\subset Y\) its coefficient in \(B_0\) is \[b_P=\min_{Q\mapsto P} \left\lfloor\frac{\operatorname{ord}_Q(t)}{ \operatorname{ord}_Q(f^*P)}\right\rfloor.\] These are exactly the regularity tests for the rational section \((f^*a)t\). On the smooth base-change open, the entire invariant adjoint summand is a line. Its fiber-integral metric is semipositive by the compact-torus orthogonal quotient argument of Proposition 4.

Put \(k=\dim Y\). This metric extends plurisubharmonically across every base prime. At a general point of a component attaining \(b_P\), write the map as \((z_1^e,z_2,\ldots,z_k)\), where \(e>0\) is the pullback multiplicity, and let \(a\) be the order of \(t\). The relative canonical coefficient for the local frame of \(R\) is a nonvanishing smooth factor times \[z_1^{a-b_Pe-(e-1)}.\] Since \(0\le a-b_Pe<e\), this exponent is nonpositive. Integration on one fixed small vertical polydisk bounds the squared norm below by a positive constant near the base prime. The weight is therefore locally bounded above there. Extend first along these divisorial neighborhoods and then across codimension at least two; write the extended weight as \(\psi_R\).

Choose local frames \(g\) of \(\mathcal O_Y(B_0)\) and \(\eta\) of \(K_Y\). The frame \(g/\eta\) of \(R\) corresponds on the smooth locus to the relative adjoint section \((f^*g)t/f^*\eta\). For a holomorphic function \(a\), Fubini identifies integrability of \(a\) with this frame’s squared norm, using the base volume from \(\eta\), with square integrability of the absolute top form \((f^*(ag))t\) with the smooth \(L'\) metric upstairs. The excluded analytic sets have measure zero. Such an integrable meromorphic top form has no divisorial pole; smoothness upstairs then extends it across codimension two. Conversely a holomorphic lift is integrable over inverse images of relatively compact base charts by properness. The lift is invariant and the identification agrees on the generic fiber. This proves both inclusions in (33), so it retains any codimension-two difference between \(\mathcal B\) and its hull.

For the final assertion apply Lemma 26 to the invariant section determined by \(t\) on the stated patch. The fiber-integral metric is smooth on the base-change open, so its strictness at one point gives a positive lower curvature bound on a smaller coordinate ball. ◻

Proof. The null-pole field and the relative section. Average a metric on \(L\) and a Kähler form \(\omega\) over the maximal compact torus, and write \(\alpha\ge0\) for the resulting curvature form, in first-Chern normalization. Proposition 5, with its actual value at zero, followed by the equivariant hard Lefschetz construction in Lemma 6, has the following consequence: if the asserted invariant sections do not exist, there is a fixed \(q>0\) and unbounded positive \(j\) with \[0\ne u_j\in H^0(X,\Omega_X^{n-q}\otimes jL)^T.\] The coefficient in hard Lefschetz is \(j=m+1\), since \(K_X+jL=mL\) equivariantly. In the Bochner identity for the associated harmonic \((n,q)\)-forms the curvature term is a sum of \(q\) eigenvalues of \(j\alpha\). Full rank on an open set would force vanishing there, hence vanishing of the holomorphic Lefschetz preimage everywhere. Thus \(r=\max\operatorname{rank}\alpha<n\).

Define the degree of a divisor by \[\ell(D)=[D]\,[\alpha]^r[\omega]^{n-r-1}.\] It is nonnegative on effective divisors and vanishes on \(L\). Lemma 6 gives a linearized determinant line \(S\) and, for unbounded positive integers \(p\), nonzero invariant sections \[s_p\in H^0(X,S+pL)^T.\] Put \(N_p=\operatorname{div}(s_p)\). Lemma 8 gives \(\ell(N_p)=0\): prescribed Ricci curvature first gives \(\ell(S)\le0\), and effectivity supplies the reverse inequality. The degree-zero divisors are exactly the curvature-null divisors of Section 9.1. We use their invariant pole field again, now to construct the exact direct-image ideal.

Let \(K\) be the field of invariant rational functions whose polar divisors have degree zero. Sums and products preserve this property; inverses do also, because polar and zero divisors are linearly equivalent and effective degrees are nonnegative. The field is relatively algebraically closed in \(\mathbb C(X)\). Indeed a root of a monic equation over \(K\) can have poles only on the polar divisors of its coefficients, by the discrete-valuation inequality. The connected torus fixes each member of the finite root set, so the root is invariant as well. The field is finitely generated: choose a transcendence basis over \(\mathbb C\); algebraic subextensions over its rational field inside the finitely generated ambient field have bounded degrees, after adjoining an ambient separating transcendence basis. Their compositum is therefore finite.

Take a smooth projective model \(Y\) of \(K\). Flatten the graph of the induced rational map, and normalize its main component, to obtain a normal equidimensional \(G\to Y\) birational to \(X\); then take an equivariant resolution \(V\to G\). The action on \(Y\) is trivial. Write \(f_G:G\to Y\) for the normal-model morphism and \(L_G\) for the pullback of \(L\) to \(G\). Write \(\pi:V\to X\), \(f:V\to Y\), \(L'=\pi^*L\), and let \(t\) be the invariant Jacobian section of \(K_V+L'=\mathcal O_V(J)\), with \(J=K_{V/X}\ge0\). The generic fiber is geometrically integral because \(K\) is relatively algebraically closed. These constructions use only equivariant resolution of the graph and the Hilbert-family flattening; they do not assert equidimensionality of the smooth resolution.

Ratios of products of the \(s_p\) having the same bidegree in \((S,L)\) belong to \(K\). Their horizontal divisor on \(G\) is zero, so for \(p>p_1>p_0\), \[(p-p_0)N_{p_1}^{\rm hor} =(p-p_1)N_{p_0}^{\rm hor}+(p_1-p_0)N_p^{\rm hor}.\] Equivalently, \[N_p^{\rm hor}=N_{p_0}^{\rm hor} +\frac{p-p_0}{p_1-p_0} (N_{p_1}^{\rm hor}-N_{p_0}^{\rm hor}).\] Effectivity for unbounded \(p\) forces \(N_{p_1}^{\rm hor}-N_{p_0}^{\rm hor}\ge0\). Put \(d_0=p_1-p_0>0\) and \(\sigma_0=s_{p_1}/s_{p_0}\), a rational invariant section of \(d_0L\) with effective horizontal divisor. We use the same symbol for its pullback to \(G\). If \(Y\) is a point, it has no poles and proves the assertion. Assume \(\dim Y=k>0\).

For each prime \(P\subset Y\), put \[c_P=\min_{Q\mapsto P} \frac{\operatorname{ord}_Q(\sigma_0)} {\operatorname{ord}_Q(f_G^*P)}.\] Only finitely many \(c_P\) are nonzero. Choose a positive integer \(h\) clearing their denominators, and set \[d=hd_0,\qquad \sigma=\sigma_0^h,\qquad B=\sum_P h c_P P.\] Thus \(\sigma\) is a rational invariant section of \(dL_G\), and \(B\) is an integral divisor with trivial action. Divisorial normalization gives \[ s_D=\sigma(f_G^*1_B)^{-1} \in H^0(G,dL_G-f_G^*B)^T, \qquad D=\operatorname{div}(s_D)\ge0. \tag{34}\] Here \(1_B\) is the rational canonical section of \(\mathcal O_Y(B)\). At least one component above every base prime has coefficient zero in \(D\). We also write \(\sigma\) for its pullback to \(V\).

Adjoint rigidity and a horizontal curvature bound. For every \(m\ge0\), \[ \operatorname{rank} \bigl(f_*\mathcal O_V(J+mdL')\bigr)^T=1. \tag{35}\] There is a generic section \(t\sigma^m\). If the invariant rank were larger, a large very ample base twist would give two invariant global sections independent over \(K\). Their pushed divisors on \(X\) have degree zero. For the base twist this follows by taking the ratio of two general base hyperplanes: their pullbacks to the equidimensional \(G\) have no common prime component, so their pushed divisors are the actual polar and zero divisors of an element of \(K\). The remaining class is \(mdL\), and the exceptional Jacobian pushes to zero. Thus the ratio of the two sections would again belong to \(K\), contradicting their independence. This proves (35), including \(m=0\).

At a rank-\(r\) point of \(\alpha\), the differential of every rational function in \(K\) kills \(\ker\alpha\) wherever it is regular. For a regular value, the closure of its level hypersurface has degree zero. The nonnegative form \(\alpha^r\wedge\omega^{n-r-1}\) therefore vanishes on that hypersurface. A tangent hyperplane failing to contain \(\ker\alpha\) would still carry rank \(r\), a contradiction. Apply this to \(k\) independent base coordinates. It gives \(k\le r\). Intersect the maximal-rank open with the dense smooth base-change locus and the complement of the exceptional locus. On a small patch there, \[ \pi^*\alpha\ge c f^*\omega_Y\qquad(c>0). \tag{36}\] This is semipositive linear algebra on a complement of the kernel, not an inference from numerical degree alone. If \(r=0\), a nonzero effective divisor cannot have degree zero, so \(k>0\) cannot occur.

Apply Proposition 4 to the equivariant normal equidimensional model \(G\to Y\), with \(P=L\) naturally linearized and the rational section \(\sigma\). Equation (35) verifies the invariant adjoint ranks, and (34) is its vertical normalization. The result is a positive continuous semipositive metric on \(B\).

The exact ideal and interpolation from zero. The varying twist is now metrized. The degree-zero metric must retain the entire direct image, including its codimension-two defect.

Apply Proposition 27 to \(\mathcal B=(f_*\mathcal O_V(J))^T\). Equation (35) at \(m=0\) gives its rank, and the patch in (36) is disjoint from the exceptional locus, so \(t\) is nonzero there. The proposition supplies the metric \(\psi_R\) on \(R=B_0-K_Y\), strictly positive on a base ball, with the exact identity (33).

Adding \(m\) times the continuous bounded metric on \(B\) preserves the ideal and the strictness on the ball. Lemma 25 therefore gives \[H^q(Y,\mathcal B\otimes\mathcal O_Y(mB))=0 \qquad(q>0,\ m\ge0).\] At zero, canonical pushforward gives \(h^0(Y,\mathcal B)=h^0(V,\mathcal O_V(J))^T=1\). Thus the Euler polynomial has value one at zero. For some \(m>0\) lift a nonzero section and multiply by \(s_D^m\). The result is an invariant section of \(J+mdL'\). Remove the exceptional Jacobian divisor by pushforward to the smooth \(X\). It gives the required natural invariant section of \(mdL\). ◻

Proof. Generic adjoint rigidity. Write \(D(j)=M+jL\) and fix \(j_0<j_1\) in \(J\). Choose \(k>0\) so that the invariant system \(H^0(X,kD(j_1))^T\) has maximal image dimension among these systems for all positive degrees. No finite-generation assertion is needed to attain this maximum, an integer between zero and \(\dim X\). An equivariant principalization of its base ideal gives a smooth projective \(\pi_0:W_0\to X\), isomorphic off the base locus, and a connected-fiber Stein map \(f_0:W_0\to Y_0\) such that \[\pi_0^*kD(j_1)=f_0^*H+G,\] where \(H\) is ample on the normal projective \(Y_0\) and \(G\) is the effective fixed divisor. The base action and the fiber action on \(H\) are trivial, while the defining section of \(G\) is invariant. These identifications follow by dividing the invariant generating sections by the common fixed factor. The action on the finite Stein factor is trivial by connectedness of \(T\).

Let \(K=\mathbb C(Y_0)\) and let \(F\) be the generic fiber. It is normal and integral, with \(H^0(F,\mathcal O_F)=K\). The field \(K\) is relatively algebraically closed in \(K(F)\): an algebraic element has no poles on proper normal \(F\), as follows from a monic equation over \(K\). For every \(h>0\), \[ \dim_K H^0(F,h\pi_0^*D(j_1)|_F)^T=1. \tag{37}\] There is the section \(\pi_0^*\sigma_{j_1}^h\). If there were two independent invariants, their ratio would be transcendental over \(K\). They extend as invariant rational sections on \(W_0\) with only vertical poles. A common pullback of a section of a sufficiently large multiple \(lH\) clears these poles: prescribe vanishing on their finitely many proper base images. Multiplying also by \(s_G^l\) produces invariant sections of a multiple of \(D(j_1)\) on \(X\). Products with the original system retain its image field and add their new ratio, contradicting maximal image dimension. This proves (37).

Let \(D_i^F\) be the effective divisor of \(\sigma_i|_F\). The invariant products of the same linearized degree are proportional, so for \(c>j_1\) in \(J\), \[(c-j_0)D_{j_1}^F =(c-j_1)D_{j_0}^F+(j_1-j_0)D_c^F.\] Thus \(D_{j_1}^F-D_{j_0}^F\ge0\). Put \(d=j_1-j_0>0\) and \(z=\sigma_{j_1}/\sigma_{j_0}\), a rational invariant section of \(dL\). It is regular on \(F\), with zero support in \(D_{j_1}^F\). If the base is a point we already have a regular invariant section on \(W_0\), hence on \(X\).

Otherwise let \(e_0\) be the Jacobian section of \(K_{W_0}+\pi_0^*L\). Its zeros lie over the principalized base locus and, on \(F\), in the support of \(D_{j_1}^F\). For every fixed \(s\ge0\), choose \(h\) large enough that, on \(F\), \[hD_{j_1}^F\ge \operatorname{div}(e_0|_F) +s\operatorname{div}(z|_F).\] Such an \(h\) exists because both effective divisors on the right have support in \(D_{j_1}^F\). The invariant quotient \((\sigma_{j_1}|_F)^h/(e_0z^s|_F)\) is regular. Multiplication by this nonzero quotient injects the entire invariant adjoint section space into the one-dimensional space in (37). Consequently \[ H^0\bigl(F,(K_{W_0}+(1+sd)\pi_0^*L)|_F\bigr)^T =K\cdot(e_0z^s)|_F\qquad(s\ge0). \tag{38}\]

Actual lines and affine boundary orders. The generic rank computation will now be upgraded to line bundles whose local frames lift to adjoint sections. After further birational modifications, the same adjoint ranks are preserved by canonical pushforward on a common smooth resolution. No support claim about new exceptional primes is needed. Now apply birational weak toroidalization (Abramovich et al. 2013, Theorem 1.1), followed by equidimensionalization by toroidal subdivisions (Abramovich and Karu 2000, Proposition 4.4). This gives a birational modification \(Y\to Y_0\) with \(Y\) smooth, a normal toroidal \(Z\) birational over \(W_0\), and an equidimensional \(g:Z\to Y\). These are birational modifications; no alteration or assertion of reduced fibers is being used. Take a smooth resolution \(W\to Z\) and write \(f:W\to Y\), \(\pi:W\to X\), and \(\pi_Z:Z\to X\).

Toroidal singularities are rational. Therefore, for \[\mathscr E_s=f_*\mathcal O_W(K_{W/Y}+(1+sd)\pi^*L),\] the sheaf \(\mathscr E_s\otimes\mathcal O_Y(K_Y)\) equals \[g_*\bigl(\omega_Z\otimes \pi_Z^*\mathcal O_X((1+sd)L)\bigr).\] It is reflexive: deleting a codimension-two subset of \(Y\) deletes only codimension at least two from \(Z\), by equidimensionality, and the reflexive canonical sheaf on \(Z\) extends its sections. The pushforward is torsion-free as well, since the source is integral and the map is dominant.

The toroidal modifications may fail to be equivariant. Nevertheless \(\mathscr E_s\) has a \(T\)-linearization. Compute the same direct image on an equivariant smooth resolution \(W'\) of the main component of \(W_0\times_{Y_0}Y\). A common smooth resolution of \(W\) and \(W'\) identifies their canonical direct images, since all coefficient twists are pulled back from \(X\) and a birational morphism of smooth varieties pushes its canonical sheaf to the canonical sheaf. The identifications are the identifications of rational forms, so preserve the generic torus action. The invariant part \[A_s=(\mathscr E_s)^T\] is now an actual reflexive direct summand of rank one by (38); on smooth \(Y\) it is a line bundle. Birational canonical pushforward gives \[ H^0(Y,K_Y+A_s)=H^0(X,sdL)^T, \qquad h^0(Y,K_Y+A_0)=1. \tag{39}\] The last equality uses the natural invariant trivialization of \(K_X+L\).

We compute these actual lines. On the regular locus of \(Z\) let \(e_Z\) be the Jacobian section with effective divisor \(E_Z\). Use the rational generator \(e_Zz^s\) for \(K_Y+A_s\). For each prime \(B\subset Y\), let \(Q_i\) run over primes of \(Z\) mapping onto \(B\), and let \[b_i=\operatorname{ord}_{Q_i}(g^*B)>0, \quad e_i=\operatorname{ord}_{Q_i}(E_Z), \quad c_i=\operatorname{ord}_{Q_i}(z).\] A rational base multiplier \(a\) is regular above the generic point of \(B\) precisely when \(b_i\operatorname{ord}_B(a)+e_i+sc_i\ge0\) for all \(i\). There are no horizontal poles by (38). The exact allowed pole order is \[u_s(B)=\left\lfloor\min_i\frac{e_i+sc_i}{b_i}\right\rfloor, \qquad K_Y+A_s=\mathcal O_Y(C_s), \quad C_s=\sum_Bu_s(B)B.\] Only finitely many primes occur with a nonzero slope or intercept. Replace \(d,z\) by a common positive multiple and power so that all \(v_B=\min_i(c_i/b_i)\) are integers, and write \(V=\sum_Bv_BB\). For all sufficiently large integer \(s\), \[ C_s=C_\infty+sV, \qquad C_\infty\ge C_0. \tag{40}\] Indeed only the indices of minimal slope can minimize for large \(s\). Among them the least intercept, after taking its floor, is at least the unrestricted least intercept at zero. This proves the stated inequality coefficient by coefficient. Set \(A_\infty=\mathcal O_Y(C_\infty)-K_Y\). Then \(A_s=A_\infty+sV\) eventually, and \[ A_0+sV\hookrightarrow A_s \tag{41}\] for those \(s\). The eventual family has intercept \(A_\infty\), but this inclusion compares it with the actual zero-degree line \(A_0\). It is this inclusion that will transfer sections produced by interpolation from exponent zero.

Integrable metrics and interpolation from zero. The original moving part, together with \(-M\) pseudoeffective, shows that \(\pi^*L-\epsilon f^*H_1\) is pseudoeffective for some ample \(H_1\) on \(Y\) and \(\epsilon>0\). The pullback of \(H\) to \(Y\) is big, so it still dominates a small ample class. Mix a metric representing this pseudoeffectivity with the smooth semipositive metric on \(\pi^*L\), using a small positive coefficient. This gives at \(s=0\) a singular metric with trivial multiplier ideal and curvature at least \(\delta f^*\omega_Y\), with \(\delta>0\). For \(s>0\) use the smooth semipositive metrics on \((1+sd)\pi^*L\).

The singular adjoint metric and its extension (Păun and Takayama 2018, Theorems 3.3.4–3.3.5) give semipositive metrics on \(\mathscr E_s\) on its locally free locus. The generic multiplier-ideal inclusion is an isomorphism since the upstairs ideal is trivial. Give \(A_s\) the quotient metric for the holomorphic projection onto this summand; quotient positivity is (Păun and Takayama 2018, Lemma 2.3.4(2)). Write its weights as \(\psi_s\). Their description is the logarithm of the squared norm of the dual line frame in \(\mathscr E_s^\vee\). They extend plurisubharmonically across the remaining codimension-two subset of smooth \(Y\), since \(A_s\) is a line bundle. They are not identically \(-\infty\), by the nondegenerate finite fiber metric at almost every smooth base-change parameter. An invariant choice of upstairs metric is unnecessary for this quotient argument.

Crucially, \[ e^{-\psi_s}\in L^1_{\rm loc}(Y)\qquad(s\ge0). \tag{42}\] A local frame of the actual summand lifts to a local section of \(\mathscr E_s\). Its fiberwise squared norm bounds the quotient squared norm from above. The lift is a holomorphic adjoint section upstairs, after adding a base canonical frame. Its absolute density is integrable over inverse images of relatively compact charts by the trivial upstairs ideal and properness. Fubini on the smooth locus identifies this integral with the fiber norms integrated against the base canonical volume, proving (42). A frame of a reflexive hull without such a lift would not suffice.

The weight \(\psi_0\) has a strictly positive lower curvature bound on all of \(Y\). To check this across the boundary, choose a smooth function \(p\) on \(Y\) whose Hessian dominates \(\omega_Y\) near a given point and is bounded above by \(C\omega_Y\) globally. Multiply the upstairs squared metric by \(e^{t f^*p}\) for small \(t>0\). It remains semipositive and has the same ideal. Its direct-image quotient weight is \(\psi_0-tp\) almost everywhere, so it is plurisubharmonic everywhere by the extension theorem. A finite covering gives \(i\partial\bar\partial\psi_0 \ge\delta'\omega_Y\) for some \(\delta'>0\).

We now put strictly positive metrics with trivial multiplier ideal on \(A_0+sV\) for every \(s\ge0\). These are the lines on the left of (41), so vanishing for them will retain the known section count at zero. Fix an integer \(s>0\). Take a sufficiently large integer \(u\) in the range of (40), and choose a smooth metric \(\psi_\infty\) on \(A_\infty\). In compatible frames the formula \[ \rho_s=\psi_0+\frac{s}{u}(\psi_u-\psi_\infty) \tag{43}\] defines a metric on \(A_0+sV\). Its curvature is strictly positive once \(s/u\) is small enough. Strong openness (Guan and Zhou 2015, Theorem 1.1) and compactness give \(\eta>0\) such that \(e^{-(1+\eta)\psi_0}\) is locally integrable on a finite cover. Hölder, together with (42) for this \(u\), proves \(e^{-\rho_s}\) integrable if \[\frac{s}{u}\frac{1+\eta}{\eta}\le1.\] The smooth \(\psi_\infty\) contributes only bounded factors. No exponent improvement uniform in \(u\) is needed. At \(s=0\) the original \(\psi_0\) already has these properties.

Nadel vanishing consequently gives \[H^i(Y,K_Y+A_0+sV)=0\qquad(i>0,\ s\ge0).\] The Euler polynomial in \(s\) has value one at zero by (39). Thus it is nonzero for all sufficiently large integers \(s\). Using (41) and then (39) gives a nonzero invariant section of \(sdL\) on \(X\). ◻