Proof. The assertion is immediate if \(W\) is a point. Suppose \(\dim W=n>0\) and choose a smooth projective resolution \(\rho:\widetilde W\to W\) on which the metric is bounded. Let \(r_0>0\) make \(r_0P\) Cartier. Riemann–Roch makes \[t\longmapsto\chi(\widetilde W,K_{\widetilde W}+t\rho^*(r_0P))\] a polynomial. Its value at \(t=0\) is \((-1)^n\chi(\widetilde W,\mathcal O_{\widetilde W})\), which is nonzero by hypothesis. It is therefore nonzero for all but finitely many positive integers. For each such integer at least one cohomology group is nonzero. Passing to an infinite subsequence fixes a cohomological degree \(q\). Apply (1) with \(B=t\rho^*(r_0P)\). Its multiplier ideal is trivial. We obtain a fixed \(p=n-q\) and nonzero sections \[\sigma_i\in H^0(\widetilde W,\Omega^p_{\widetilde W}\otimes\mathcal O_{\widetilde W}(n_i\rho^*P)), \qquad n_i\longrightarrow\infty.\] If \(p=0\), projection to \(W\) gives the first alternative because \(\rho_*\mathcal O_{\widetilde W}=\mathcal O_W\).

Assume \(p>0\). Work over the function field \(K=\mathbb C(W)\) and choose a rational trivialization for \(r_0P\). Untwisting the sections defines vectors in \(\bigwedge^p\Omega^1_{K/\mathbb C}\). Let \(F_K\) be their span and \(b=\dim_KF_K\). Its determinant is a one-dimensional subspace of \[\bigwedge^b\bigl(\bigwedge^p\Omega^1_{K/\mathbb C}\bigr) \subset (\Omega^1_{K/\mathbb C})^{\otimes pb}.\] The indicated inclusion uses alternating tensors in characteristic zero. Intersect this line with \((\Omega_W^1)^{[\otimes pb]}\), the reflexive tensor power, and take its saturated reflexive rank-one extension. This is a sheaf \(\mathcal O_W(M)\) for a Weil divisor \(M\). The determinant of any basis selected from the \(\sigma_i\) gives a nonzero section of \(\mathcal O_W(M+\sum n_iP)\) on the smooth locus of \(W\). It is regular at every codimension-one point: the resolution is an isomorphism there, and the original sections are holomorphic. Reflexivity extends it across codimension two. Its zero divisor is an effective Weil divisor \(N\).

The sums of exponents of such bases are unbounded. Given any one of the nonzero vectors at arbitrarily high exponent, extend it to a basis using vectors from the spanning family. All exponents are positive, so the sum is at least the chosen exponent. Selecting a strictly increasing subsequence of these sums gives \(m_i\) and \(N_i\) in (3).

The pair \((W,D)\) is projective and klt, the boundary is effective, and \(-(K_W+D)\) is nef. Thus (Lazić et al. 2023, Theorem 4.1) applies to the saturated rank-one subsheaf \(\mathcal O_W(M)\) of the reflexive cotangent tensor power. It asserts that its negative first Chern class is pseudoeffective. Since \(W\) is \(\mathbb Q\)-factorial, this is the claimed pseudoeffectivity of the \(\mathbb Q\)-Cartier divisor \(-M\). ◻

Proof. Resolve the pencil, remove its fixed part, and combine its morphism to \(\mathbb P^1\) with \(f\). If the pencil is nonconstant on the general fiber, the image of the combined map has dimension greater than \(\dim Y\). A product polarization gives a basepoint-free line bundle \(F\) on the resolved source, dominated by a multiple of \(\pi^*P\) plus an exceptional divisor, using (5).

Choose a general member \(F_0\) of that system. It has no \(\pi\)-exceptional component. Push the class comparison down to \(W\), using the numerical fact in Section 2.1; the exceptional term vanishes. The result says that \(c'P-\pi_*F_0\) is pseudoeffective. Pulling back gives \(c'\pi^*P-\pi^*\pi_*F_0\) pseudoeffective. Since the effective divisor \(\pi_*F_0\) is \(\mathbb Q\)-Cartier, its pullback contains its strict transform \(F_0\) and has an effective exceptional remainder. Hence \(c'\pi^*P-F_0\) is pseudoeffective. This dominates a polarization for the larger projective image. On a smooth model of that image the pulled-back polarization is big, so it dominates a small ample class; scaling and pulling back gives a diagram of the allowed form. This contradicts maximality.

For the last assertion, choose an integer \(b\) such that \(bA\) minus the fixed base divisor is ample. The extra pullback is then dominated by \(bf^*A\), hence by \(bd\pi^*P\) using (5). Increase \(c\) accordingly. ◻

Proof. For \(m_i<m_j<m_k\), set \[\alpha=\frac{m_k-m_j}{m_k-m_i},\qquad \beta=\frac{m_j-m_i}{m_k-m_i}.\] Then \(\alpha,\beta>0\), \(\alpha+\beta=1\), and \[N_j\sim_{\mathbb Q}\alpha N_i+\beta N_k.\] After clearing denominators, the two effective divisors give sections of the same line bundle. That bundle is pseudoeffectively dominated by a multiple of \(P\), because \(-M\) is pseudoeffective in (3). Lemma 4 says that their ratio is constant on the general fiber. The ratio belongs to \(\mathbb C(Y)\), using relative algebraic closure. Its principal divisor therefore has no horizontal component on \(S_0\).

It follows that the horizontal parts of \(\pi_0^*N_i\) are affine functions of \(m_i\) as actual divisors. Fixing the first two indices expresses them as \(A_0+m_iB_0\), where the supports of \(A_0,B_0\) are contained in one finite union of primes. Effectivity for an unbounded sequence of \(m_i\) gives \(B_0\ge0\). Consequently the difference of two members, divided by the difference of their exponents, is \(\mathbb Q\)-linearly equivalent to \(\pi_0^*P\) and has effective horizontal part. Clearing denominators gives a rational section of a positive Cartier multiple of \(\pi_0^*P\) with no horizontal poles. Write its divisor as \(T_0\).

For every base prime \(G\), let \[v_G=\min_{F\mapsto G} \frac{\operatorname{coeff}_F(T_0)}{\operatorname{coeff}_F(f_0^*G)}.\] The denominator is positive, and the minimum is over the finitely many primes dominating \(G\). Only finitely many \(v_G\) are nonzero. Every vertical prime of \(S_0\) occurs above a base prime, by equidimensionality. Thus \[T_0-f_0^*\Bigl(\sum_Gv_GG\Bigr)\ge0.\] The difference is \(\mathbb Q\)-Cartier: \(T_0\) is Cartier up to its chosen multiple, and \(Y\) is smooth. Multiply again to make the base divisor integral and the remainder Cartier. This gives \(V\) and an effective Cartier remainder on \(S_0\). Pull them to \(S\) to obtain (7). The component attaining the minimum has zero remainder coefficient; its strict transform on \(S\) still dominates \(G\) and has coefficient zero in \(H\). ◻

Proof. Over the generic fiber the section of \(E+aH\) gives a nonzero section, by (7). This includes \(a=0\). If the dimension of the generic-fiber section space were at least two, twisting by the pullback of a sufficiently ample line bundle on \(Y\) would extend two independent generic sections to global sections. Their ratio would be nonconstant on the general fiber. Their line bundle is dominated by a multiple of \(\pi^*P\) plus the exceptional divisor \(E\), including the base twist as in Lemma 4. This contradicts that lemma. ◻

Proof. Fix a dense Zariski open \(Y^\circ\) over which \(f\) is smooth with connected fibers, \(\Xi\) is relatively simple normal crossings, and no fiber is contained in \(E\cup H\). In a coordinate neighborhood in \(Y^\circ\), choose frames \(v\) of \(V\) and \(\eta\) of \(-K_Y\). The section \[s=f^*v\,s_H\] of \(r\pi^*P\) has divisor \(H\) over that neighborhood. The section \(s_Es^af^*\eta\) belongs to \(K_{S/Y}+B_a\). By Lemma 6, it generates the direct image at a general point. On its base-change open, this generator locally identifies the adjoint direct image with \(aV-K_Y\). Because the direct image has rank one, its semipositive \(L^2\) metric has a psh weight equal to the negative logarithm of the squared \(L^2\) norm of this generator. Dividing by \(a\) gives the functions below. This is a local identification where the generator is a frame, and asserts no global invertibility at omitted base points. We obtain psh functions \[ u_a(y)=-\frac1a\log\int_{S_y}t(x)^a\,d\mu_y(x), \qquad t=|s|_{h_P^r}^2,\qquad a\ge1. \tag{11}\] The measure \(\mu_y\) is the squared adjoint norm density of \(s_Ef^*\eta\), using the metric on \(\pi^*P+\Xi\). Locally it has a bounded positive factor, the squared modulus of a nonzero holomorphic section, and the factors \(|z_j|^{-2c_j}\) with \(c_j<1\). It is therefore finite on a general compact fiber and has the same null sets as a smooth volume measure. Zeros of \(s_E\) and the divisorial poles occupy sets of measure zero. This description uses boundedness, not continuity, of \(\varphi_P\).

For each \(y_0\in Y^\circ\), choose a point on \(S_{y_0}\) off \(E\cup H\cup\mathop{\mathrm{Supp}}\Xi\). On a small product neighborhood for the smooth map, the holomorphic factors are bounded away from zero and the metric weights are bounded. There are a neighborhood of \(y_0\), constants \(c,b>0\), and open pieces in its fibers on which \(t\ge c\) and whose \(\mu_y\)-mass is at least \(b\). Thus \[\int_{S_y}t^a\,d\mu_y\ge b c^a, \qquad u_a(y)\le-\log c-\frac{\log b}{a}.\] These estimates extend the psh representatives across each exponent-dependent base-change exceptional set and bound them locally uniformly above on \(Y^\circ\).

Take the countable intersection \(Y^*\) of the full-measure sets where these integral formulas agree with the psh representatives for every positive integer \(a\). This is a common full-measure subset of \(Y^\circ\), not a claimed common Zariski open. For \(y\in Y^*\), the elementary \(L^a\)-norm limit on the finite measure space \((S_y,\mu_y)\) gives \[ \lim_{a\to\infty}u_a(y) =-\log\mathop{\mathrm{ess\,sup}}_{S_y}|s|_{h_P^r}^2=:u(y). \tag{12}\] For completeness, if \(M_y=\mathop{\mathrm{ess\,sup}}t\), the upper bound is \((\int t^a\,d\mu_y)^{1/a}\le M_y\mu_y(S_y)^{1/a}\). For any \(c<M_y\), the set \(\{t>c\}\) has positive measure, giving the reverse limiting bound \(\liminf(\int t^a\,d\mu_y)^{1/a}\ge c\). Let \(c\uparrow M_y\).

The psh functions \(u_a\) have locally uniformly bounded upper bounds. They cannot converge to \(-\infty\) locally: on each relatively compact base neighborhood, properness, holomorphicity of \(s\), and boundedness of \(\varphi_P\) bound \(t\) above. The measures \(\mu_y(S_y)\) are bounded above there: in relative simple normal crossings charts their densities are at most \(C\prod_j|z_j|^{-2c_j}\) with \(c_j<1\), and properness gives a finite cover above a compact base neighborhood. Hence (11) also has a uniform lower bound on compact subsets of \(Y^\circ\). Local compactness for psh functions now gives subsequential \(L^1_{\mathrm{loc}}\) convergence, and (12) identifies every subsequential limit almost everywhere. Consequently \(u\) has a psh representative.

Under a frame change \(v'=gv\), the function \(t\) is multiplied by \(|g|^2\) and \(u\) changes to \(u-\log|g|^2\). Changing \(\eta\) contributes a term divided by \(a\), which disappears in the limit. Thus \(u\) defines the weights of a metric on \(V\) over \(Y^\circ\).

The normalized integrals have now produced positivity on the smooth part of the base. It remains to prove boundedness at the boundary; positivity alone would not suffice for the later multiplier-ideal calculation. On the inverse image of a relatively compact base coordinate neighborhood, properness and bounded weights of \(h_P\) give a uniform upper bound for \(|s|^2\). Formula (12) therefore gives a lower bound for \(u\) even near a missing base fiber. For an omitted prime \(G\subset Y\), choose the component \(F\) above \(G\) on which \(H\) has coefficient zero. At a general point of \(F\), the section \(s\) is nonzero, the map \(F\to G\) is submersive, and local transverse coordinates have the form \[z_1\circ f=w_1^e\,a(w),\qquad a(w)\ne0.\] Together with coordinates along \(G\), this shows that every nearby general fiber meets a neighborhood of that point in an open set. On a smaller neighborhood \(|s|_{h_P^r}^2\ge c>0\) because the holomorphic section is nonzero and the metric is bounded. The neighborhoods intersecting such a general fiber have positive \(\mu_y\)-measure, since that measure has the same null sets as smooth volume. No uniform positive lower bound on their measures is needed for an essential supremum. Thus the component may lie in \(E\) or \(\Xi\) without affecting this argument: on nearby general fibers their divisorial supports have measure zero. The essential supremum is at least \(c\). Thus \(u\le-\log c\) near a dense open subset of \(G\).

The psh removable-singularity theorem extends \(u\) across these codimension-one sets, using the upper bound just proved. What remains is an analytic subset of codimension at least two in the smooth base. Psh Hartogs extension extends \(u\) there as well. Its lower bound persists by the upper-semicontinuous extension, and its psh representative has a local upper bound. The local extensions are unique and respect the frame transformations, giving the asserted bounded metric on \(V\) over all of \(Y\). This conclusion is local boundedness. We have used no continuity assertion for the original metric or for its essential supremum. ◻

Proof. By (5), \(d\pi^*P-f^*A\) has a positive-current metric. Tensoring with a positive metric on \(f^*A\) and taking a \(d\)th root gives a metric on \(\pi^*P\) with local weights \(\psi\) and curvature at least \(d^{-1}f^*\omega_A\). Choose \(0<\tau<1\) and mix these weights with the bounded metric: \[\varphi_\tau=(1-\tau)\varphi_P+\tau\psi.\] Its curvature is at least \((\tau/d)f^*\omega_A\). We show that \(\varphi_\tau+\varphi_\Xi\) has trivial multiplier ideal for one sufficiently small \(\tau\).

Choose \(p>1\) so close to one that \(pc_j<1\) for every coefficient \(c_j\) of \(\Xi\), and let \(q=p/(p-1)\). On a log-resolution chart, \[e^{-p\varphi_\Xi}\asymp\prod_j|z_j|^{-2pc_j}\] is integrable. Skoda’s local small-exponent integrability theorem (Skoda 1972) implies that \(e^{-q\tau\psi}\) is integrable for sufficiently small \(\tau\). A finite cover of the compact \(S\) permits one choice of \(\tau\) for all charts. Hölder’s inequality makes \(e^{-(\varphi_\Xi+\tau\psi)}\) integrable; the remaining bounded factor \(e^{-(1-\tau)\varphi_P}\) does not affect integrability. This is exactly the triviality of the multiplier ideal. No statement about analytic singularities of \(\psi\) is needed. ◻

Proof of Theorem 1. The earlier arguments already dispose of \(\dim W=0\), the \(p=0\) alternative of Lemma 3, and a zero-dimensional maximal base. Otherwise take \(V,H,E,\Xi\) as constructed above. Let \(h_V\) be the bounded semipositive metric on \(V\) from Proposition 7. For every integer \(a\ge0\), equip \(B_a\) in (10) with the tensor product of the metric from Lemma 8, the pullback of \(h_V^{\otimes a}\), and the divisor metric of \(aH\). More explicitly, if \(g_H\) is a local equation of \(H\), this metric has local weight \[\varphi_{\mathrm{mix}}+a f^*\varphi_V+a\log|g_H|^2,\] where \(\varphi_{\mathrm{mix}}\) is a weight of the fixed metric from Lemma 8 and \(\varphi_V\) is a weight of \(h_V\). Curvature is at least the same positive multiple of \(f^*\omega_A\). Since \(\varphi_V\) is bounded and \(aH\) is integral, the multiplier ideal is exactly \(\mathcal O_S(-aH)\). Indeed, at the generic point of a component of \(H\), the remaining psh weight is bounded above, so integrability forces divisibility by \(g_H^a\). The quotient extends holomorphically across codimension two by normality. Conversely, after this division, the test is the trivial-ideal test in Lemma 8. Bounded weights do not change that ideal.

The adjoint identity becomes \[K_S+B_a-aH=E+af^*V.\] Apply (2) and the projection formula. Writing \(\mathcal G=f_*\mathcal O_S(E)\), we obtain \[ H^i(Y,\mathcal G\otimes\mathcal O_Y(aV))=0 \quad\text{for every }i>0\text{ and every }a\ge0. \tag{13}\] The sheaf \(\mathcal G\) is coherent and has rank one. Riemann–Roch on the smooth projective base makes \[Q(a)=\chi(Y,\mathcal G\otimes\mathcal O_Y(aV))\] a polynomial in \(a\). The effective divisor \(E\) supplies a nonzero section of \(\mathcal G\). At \(a=0\), (13) therefore gives \(Q(0)=h^0(Y,\mathcal G)>0\). The polynomial is nonzero, so there exists a positive integer \(a\) with \(Q(a)\ne0\). Again by (13), \(Q(a)=h^0(Y,\mathcal G\otimes\mathcal O_Y(aV))\), so this space of sections is nonzero. Thus \[H^0(S,E+af^*V)\ne0.\] Multiplication by \(s_H^a\) gives a nonzero section of \(E+ar\pi^*P\). Normality and the effective exceptional nature of \(E\) imply \(\pi_*\mathcal O_S(E)=\mathcal O_W\): a rational function with possible poles only over codimension at least two is regular on \(W\). The projection formula now gives a nonzero section of \(arP\) on \(W\). This is a positive Cartier multiple, as required. ◻

Proof. First fix a nonzero integral cocharacter \(\lambda\). The Białynicki-Birula decomposition (Białynicki-Birula 1973) has an open attracting stratum; the smooth attracting-piece statement and its tangent-weight description also follow from (Jelisiejew and Sienkiewicz 2019, Theorem 1.5 and Corollary 7.3). At its limiting fixed component, all tangent weights are nonnegative, and some are positive. Otherwise the attracting stratum would have no positive-dimensional directions transverse to a fixed component and openness would make \(\lambda\) act trivially. This contradicts effectiveness. The weight of \(L=\det T_Z\) is the sum of those tangent weights, so it is strictly positive on this open stratum.

The generic pattern for \(J\) and \(J+L\) occurs on one dense open subset, obtained by requiring every nonzero eigensection in chosen bases to be nonvanishing. This open meets the open attracting stratum for each fixed \(\lambda\). Its weight function is therefore positive for every nonzero integral cocharacter. The finitely many rational linear pieces then make it positive on every nonzero real cocharacter: a zero or negative locus on one rational cone would contain a nonzero rational point. On a unit sphere this continuous homogeneous function has a positive minimum. ◻

Proof. The bundle \(L\) is nef, so \(L+\delta J\) is ample for every \(\delta>0\). Subdivide the finitely many weight fans to one finite pointed rational fan. On each cone every weight pattern of \(J\) and \(L\) is linear.

For each pattern failing (15), choose an integral ray on which its weight is negative. There are only finitely many choices. For sufficiently small \(\delta>0\) and sufficiently small rational character shift, all these chosen weights remain negative. Such a pattern cannot occur at a semistable point for the perturbed linearization. On the other hand, Lemma 9 and its uniform positive lower bound on the unit sphere show that the generic pattern stays strictly positive under these perturbations. Thus the stable locus is nonempty.

After fixing such a rational \(\delta\), choose the rational character off the finitely many hyperplanes on which a pattern has zero weight on one of the rays of the common fan. For any semistable point, weights on those rays are now strictly positive. Linearity on pointed cones makes them strictly positive on every nonzero cocharacter. The point is stable by Hilbert–Mumford. Finally clear denominators to obtain an ordinary ample linearized line bundle. This leaves the stated loci unchanged. ◻

Proof. Apply Luna’s slice theorem (Luna 1973) in an invariant affine neighborhood coming from the ample GIT linearization. Near an orbit with finite stabilizer \(G\), there is a smooth slice \(S_{\mathrm{sl}}\) and, after an étale base localization, the local model \[T\times^G S_{\mathrm{sl}}\longrightarrow S_{\mathrm{sl}}/G.\] The slice is smooth because \(U\) is smooth and the orbit has constant dimension. Normality follows from invariants of a normal ring. Finite quotients of smooth varieties are locally \(\mathbb Q\)-factorial: a Weil divisor pulls back to a Cartier divisor locally on the smooth finite cover, and the product of its translates gives a local equation for a positive multiple downstairs. The same assertion descends through the étale localizations. Thus \(W\) is \(\mathbb Q\)-factorial.

Discard an ineffective kernel of the finite slice action when computing ramification. At a general point over a prime \(B\), the finite map has ramification order \(e_B\), and the ramification divisor has coefficient \(e_B-1\). Thus its canonical divisor formula is \[K_{S_{\mathrm{sl}}}=p^*(K_W+D)\] in codimension one. The numbers \(e_B\) are intrinsic: they are the multiplicities along components of the quotient pullback of \(B\). The formulas therefore agree on overlapping charts and define one global effective rational divisor \(D\). Only finitely many primes are ramified.

The usual finite-morphism discrepancy formula shows that this pair is klt. More explicitly, extend any divisorial valuation over \(W\) to the function field of the slice. If the valuation has ramification index \(e\), its log discrepancy upstairs is \(e\) times its log discrepancy for \((W,D)\) downstairs. The upstairs variety is smooth with zero boundary, hence every log discrepancy is positive. The downstairs log discrepancies are therefore positive as well. This is the finite-morphism discrepancy criterion of (Kollár and Mori 1998, Proposition 5.20); the exact finite-group calculation with its ramification boundary is also given in (Braun et al. 2024, Proposition 3.5 and equation (3.4)).

It remains to keep track of the linearization in (17). On \(U\) the infinitesimal action gives an injective subbundle \[\mathop{\mathrm{Lie}}(T)\otimes\mathcal O_U\ \longrightarrow\ T_U.\] Its rank is \(\dim T\) because all stabilizers are finite. Choose a dense smooth open \(W^\circ\subset W\) away from the branch boundary over which \(q\) is a submersion, and put \(U^\circ=q^{-1}(W^\circ)\). There the tangent sequence is \[0\longrightarrow\mathop{\mathrm{Lie}}(T)\otimes\mathcal O_{U^\circ} \longrightarrow T_{U^\circ} \longrightarrow q^*T_{W^\circ}\longrightarrow0.\] The adjoint representation of a torus is trivial, so a nonzero element of \(\det\mathop{\mathrm{Lie}}(T)\) gives an equivariant trivialization of the orbit determinant. Taking determinants of this sequence therefore identifies \(L|_{U^\circ}\) with \(q^*(-K_{W^\circ})\) equivariantly. On each slice the orders of this rational identification are exactly those in the ramification calculation above, which introduces the boundary \(D\). No character twist occurs.

The quotient has constant fiber dimension \(\dim T\). A prime divisor in \(U\) cannot map into codimension at least two in \(W\): the dimension bound for such a preimage would be at most \(\dim U-2\). Thus the preceding slice and codimension-one calculation accounts for every prime divisor upstairs. It gives the equivariant rational-line-bundle identity (17). Equivalently, choose finitely many slice charts covering the projective quotient. Their finite stabilizer groups have bounded exponent, and a common multiple also clears the Cartier indices. At that multiple we obtain an actual isomorphism of line bundles whose action on the pullback from \(W\) is trivial along each fiber. This is an equivariant rational-line-bundle identity, not only a numerical equality. ◻

Proof. We first prove that translated lifts have no poles along algebraic arcs starting in \(U\). A compactification of the action graph will then turn this statement about orders into a locally uniform bound. Let \(s\mapsto x(s)\) tend to \(x_0\in U\), let \(l(s)\) be a nonzero holomorphic lift with nonzero value at \(s=0\), and let \(t(s)\in T\) be meromorphic at zero. Since a torus is a product of multiplicative groups, write \[t(s)=\lambda(s)u(s)\] with an integral cocharacter \(\lambda\) and a holomorphic map \(u\) into \(T\) whose value at zero is invertible. Replace \(x(s),l(s)\) by \(u(s)x(s),u(s)l(s)\). Rename their new initial point \(u(0)x_0\) as \(x_0\). It is still in \(U\), which is invariant, so the weak-weight inequality remains available. It suffices to consider \(t(s)=\lambda(s)\).

To compare the original lift with \(\lambda(s)l(s)\), vary the rate at which its cocharacter parameter tends to zero. For rational \(r\in[0,1]\), let \(b(r)\) be the order of \(\lambda(s^r)l(s)\) relative to a nonvanishing local frame at the limiting base point. Positive order means a zero of the lift and negative order means a pole. Take a common ramified parameter to define this order and divide by its degree. Projectivity gives the limiting base point, and algebraicity gives a finite integral order before normalization. Smoothness and positivity of the metric imply \[ b(r)=\lim_{s\downarrow0} \frac{-\log|\lambda(s^r)l(s)|_h^2}{-\log|s|^2}. \tag{19}\] For each fixed real \(s>0\), the numerator is convex as a function of \(r\). Indeed the pullback metric along the holomorphic orbit map from \(\mathbb C^*\) has psh weight, and invariance under the compact circle makes that weight convex in the real logarithmic coordinate. Composition with \(r\mapsto r\log s\) preserves convexity. The positive denominator in (19) preserves it too. Passing to limits shows that \(b\) is convex on rational points, and \(b(0)=0\).

We claim that for sufficiently small positive rational \(r\), \[ b(r)=r w_L(x_0,\lambda). \tag{20}\] To prove the claim without assuming \(L\) ample, first use the very ample bundles \(J\) and \(J+L\) from Section 3.1. Choose a nonvanishing local lift to \(J\) along the arc and tensor it with \(l(s)\) to obtain the lift to \(J+L\). For either very ample bundle \(C\), choose an eigenbasis of sections, with weights \(\alpha_j\). Let \(\nu_j\ge0\) be their orders relative to the initial lift; set \(\nu_j=+\infty\) for an identically vanishing section. Relative to the transformed lift their orders are \[\nu_j-r\alpha_j.\] Because the sections generate \(C\), the order of the transformed lift is \[-\min_j(\nu_j-r\alpha_j).\] For small enough \(r>0\), every minimizer has \(\nu_j=0\). The order is therefore \(r\max_{\nu_j=0}\alpha_j=r w_C(x_0,\lambda)\). Subtract the formula for \(J\) from that for \(J+L\) to obtain (20).

For such an \(r\), convexity and \(b(0)=0\) give \[b(r)\le r b(1),\qquad b(1)\ge\frac{b(r)}r=w_L(x_0,\lambda)\ge0,\] where the last inequality is (15). Thus a transformed lift has no pole along any such algebraic arc.

We now prove uniformity in \(x\). Suppose (18) were unbounded as \(x\to x_0\). Compactify \(T\) projectively, and compactify the total space of \(L\) as \(\mathbb P(\mathcal O_Z\oplus L)\) with its infinity divisor. Consider the graph of the algebraic map \((t,x)\mapsto t\cdot l(x)\), retaining \(x\) as a separate parameter, and take its closure in the resulting proper space over the parameter neighborhood. Since \(Z\) is projective and \(h\) is smooth, a sequence of unbounded norms gives a point of this closure at fiber infinity above \(x_0\). Algebraic curve selection through that point, with generic point in the graph, gives after normalization an arc \(x(s)\to x_0\) and a meromorphic torus arc \(t(s)\) for which \(t(s)l(x(s))\) has a pole. This contradicts the preceding paragraph. Hence the bound is locally uniform.

A holomorphic frame differs from an algebraic frame by a nowhere vanishing holomorphic function, bounded above and below on a smaller neighborhood. Positive powers preserve the estimate. This proves the remaining assertions. ◻

Proof. Consider a finite slice chart as in Proposition 11. Nearby quotient points have representatives on the slice near a fixed representative. Lemma 12 gives a local upper bound for the norm of the pullback of a frame of \(NP\). The supremum also includes the norm at the representative itself. The latter is bounded away from zero in a small chart, since the original metric is smooth and the frame is nonvanishing. Thus the supremum is finite and bounded above and below in frames. Its negative logarithm is consequently locally bounded.

To prove psh positivity, work over a small smooth slice coordinate ball \(B\) and the universal cover \(\mathbb C^t\to T\). Let \(e\) be the pulled-back local frame of \(NP\) and set \[\Phi(z,\zeta)=-\log\bigl|\exp(\zeta)\cdot e(z)\bigr|_{h^N}^2 \qquad(z\in B,\ \zeta\in\mathbb C^t).\] The action map is holomorphic and the translated frame is holomorphic and nonvanishing, so \(\Phi\) is jointly psh. The metric is invariant under the compact form of \(T\), hence \(\Phi\) is independent of \(\operatorname{Im}\zeta\). The domain \(B\times\mathbb C^t\) is pseudoconvex and has connected tube fibers. Kiselman’s minimum principle (Kiselman 1978), in the form of (Deng et al. 2019, Theorem 1.1), therefore makes \[\inf_{\zeta\in\mathbb C^t}\Phi(z,\zeta)\] psh or identically \(-\infty\). It is the weight of (21), and the bounds already proved exclude the latter possibility. Notice that the minimum principle supplies positivity after boundedness has been established; it does not supply the orbit bound.

The resulting slice weights are invariant under the finite slice group and have the correct transformations for the pullback of \(NP\). They descend to locally bounded upper-semicontinuous functions on the finite quotient. To check positivity, take any holomorphic disc in that quotient. Normalize a component of its finite pullback to the slice that dominates the disc. After a finite ramified disc cover it has a holomorphic lift to the slice. The pulled-back weight is subharmonic there, hence subharmonic downstairs off the branch points; boundedness extends it across those points. This proves the disc test for the descended weight. In particular, after composition with any holomorphic disc in a smooth resolution, the same argument gives a subharmonic function. The pulled-back weights on the resolution are therefore psh. Equivalently, extend from the regular locus using the local upper bounds and (Demailly 2012, I, Theorem 5.24), which is applied on the smooth resolution. Bounds above and below persist. Taking the \(N\)th root gives the claimed metric on \(P\).

Finally let \(C\) be an irreducible curve on a resolution. A local bounded psh weight restricts to a psh function on the normalization of \(C\), rather than being identically \(-\infty\). The resulting positive curvature measure has total mass equal to the degree of the restricted line bundle. This degree is nonnegative. Thus the pullback of \(P\) has nonnegative intersection with every curve, so is nef on the projective resolution. Curves downstairs are dominated by curves upstairs; the projection formula gives nefness of \(P\) on \(W\). ◻

Proof of Theorem 2. Now assume that \(Z\) is rationally connected. If the effective image of \(T\) is trivial, every section is naturally invariant. Theorem 1 applied to \((Z,0)\) gives such a nonzero section, since \(L\) is nef and has a smooth semipositive metric and \(\chi(Z,\mathcal O_Z)=1\).

Otherwise take the projective quotient \((W,D)\) of Proposition 11. It is \(\mathbb Q\)-factorial and klt with effective rational boundary. Proposition 13 proves the nefness and bounded-metric hypotheses for \(P=-(K_W+D)\).

Let \(\widetilde W\to W\) be a smooth projective resolution. The rational map \(Z\dashrightarrow\widetilde W\) is dominant. Resolve its indeterminacy to a morphism \(\widetilde Z\to\widetilde W\). Rational connectedness is preserved by smooth projective birational modification and by a dominant rational map to a smooth projective target; see (Kollár 1996). Alternatively, pullback of differential forms under this dominant morphism is injective and \(H^0(\widetilde Z,\Omega^p_{\widetilde Z})=0\) for \(p>0\). Hence \(H^0(\widetilde W,\Omega^p_{\widetilde W})=0\) for \(p>0\), and Hodge symmetry yields \[\chi(\widetilde W,\mathcal O_{\widetilde W})=1.\] If \(W\) is a point, this identity and the same conclusion hold directly.

Theorem 1 provides a nonzero section \(\sigma\) of a positive Cartier multiple \(mP\). Take a positive power so that \(m\) is also divisible by the integer used in (17). Its pullback is a nonzero invariant section \(\widetilde\sigma\) of \(mL\) on \(U\). The bounded quotient metric makes \(|\sigma|\) bounded on the compact \(W\): in finitely many relatively compact charts this follows from bounded holomorphic coefficients and the bounded frame norms. Definition (21) implies \[ |\widetilde\sigma(x)|_{h^m} \le |\sigma(q(x))|_{h_{mP}}\le C \quad (x\in U). \tag{22}\]

Let \(x\) be any point of \(Z\setminus U\) and choose a holomorphic nonvanishing frame of \(mL\) on a small coordinate neighborhood. Smoothness of \(h\) gives upper and lower positive bounds for the norm of that frame. The coefficient of \(\widetilde\sigma\) is therefore a bounded holomorphic function off the algebraic set \(Z\setminus U\). The removable-singularity theorem extends it holomorphically across that set (Demailly 2012, I, Corollary 5.25). In particular the estimate applies at the generic point of every omitted divisor; no exception for an unstable divisor is made. The same argument, or normality, extends across the remaining codimension-two locus. Unique local extensions glue to a global section of \(mL\).

For any \(t\in T\), the translated global section agrees with the original on the dense open set \(U\), so it agrees everywhere. The section remains nonzero since it was nonzero on \(U\). This proves the invariant nonvanishing assertion. ◻