Proof. The value of \(s\) gives \[s(1-s)=1+\frac{\sqrt2}{3},\qquad a^{-1}s+a(1-s)=-\sqrt2.\] These identities prove the determinant and trace assertions except for the trace of \(AB\). The characteristic polynomial of each of \(A,B\) is \(t^2-t+1\), with roots \(e^{\pm i\pi/3}\), while that of \(A^{-1}B\) is \(t^2+\sqrt2t+1\), with roots \(e^{\pm3i\pi/4}\). The roots in each pair are distinct, proving the claimed orders; in particular \(A^3=B^3=-\mathop{\mathrm{Id}}\).

Cayley–Hamilton gives \(A+A^{-1}=\mathop{\mathrm{Id}}\). Multiplication by \(B\) and taking traces therefore yield \[\mathop{\mathrm{tr}}(AB)=\mathop{\mathrm{tr}}B-\mathop{\mathrm{tr}}(A^{-1}B)=1+\sqrt2>2.\] Since \(\det(AB)=1\), its eigenvalues are positive real reciprocals, one greater than one. Finally, the eigenlines of \(A\) are the two coordinate lines, and neither is preserved by \(B\), whose two off-diagonal entries are nonzero. ◻

Proof. Each graph is a smooth divisor of bidegree \((1,1)\). An intersection of \(\Gamma_M\) and \(\Gamma_N\) corresponds to a fixed point of \(M^{-1}N\), hence to an eigenline of that matrix. The relative matrices are \(A,B,A^{-1}B\) or their inverses. Each has two distinct eigenvalues, and the multiplier at either projective fixed point is the ratio of those eigenvalues in one order or the other. This ratio is not one. The fixed points are therefore simple, so each pair of graphs meets transversely at two points. A point on all three graphs would give a common eigenline of \(A\) and \(B\), contrary to Lemma 2. Thus the six pairwise intersections are distinct. Their local equations are products of two regular parameters, and the asserted description follows. ◻

Proof. The bundle will be an elementary transformation of \(L_1\oplus L_2\) along the effective Cartier divisor \(D\), as in (Maruyama 1986, Introduction, (0.1)). We verify the gluing and the local basis directly, since the resulting quotient maps are needed in both subsequent arguments.

At a point of \(\Gamma_M\cap\Gamma_N\), let \(v\) span the first-coordinate line. For some eigenvalue \(\lambda\) of \(M^{-1}N\) we have \(Nv=\lambda Mv\). By Lemma 2, \(\lambda^{24}=1\). Consequently \[(Nv)^{\otimes b}=(Mv)^{\otimes b}.\] Thus the two maps agree as maps of the actual line fibers. This also covers the reversed pair, for which the eigenvalue is \(\lambda^{-1}\).

We check descent at a node. If \(R\) is the local ring of \(S\) there and \((x),(y)\) are the branch ideals, transversality gives \((x)\cap(y)=(xy)\) and the exact sequence \[0\longrightarrow R/(xy)\longrightarrow R/(x)\oplus R/(y) \longrightarrow R/(x,y)\longrightarrow0,\] whose last map is the difference of the residue values. Tensoring this sequence by the locally free rank-one sheaf \(\operatorname{Hom}(L_1^{-1},L_2^{-1})|_D\) shows that the equal fiber values are precisely the compatibility needed to descend the two branch sections. Their common value is nonzero, so the descended scalar is a unit in the nodal local ring. The component maps therefore glue to the isomorphism (3).

Near a point of \(D\), choose frames of \(L_1,L_2\) and a local equation \(d\) of \(D\). The scalar defining \(\beta\) modulo \(d\) lifts, after shrinking, to a regular unit \(h\). In these frames the module (4) has basis \[ e=(1,h),\qquad f=(0,d). \tag{7}\] Indeed, every element has a unique expression \(u_1e+kf\), since \(u_2-hu_1\) is divisible by the non-zero-divisor \(d\). Away from \(D\) the sheaf is simply \(L_1\oplus L_2\), proving local freeness everywhere.

The first projection maps \(e\) to \(1\), and the second maps \(e\) to the unit \(h\), so both are surjective. Setting the \(j\)-th coordinate equal to zero in (4) identifies its kernel with \(L_i(-D)\), proving (5). Their duals are line subbundles of \(F\), and away from \(D\) they span. On \(D\), if \(v\in L_1^{-1}\) and \((u_1,u_2)\in G\), then \[\bigl(\iota_2(\alpha v)\bigr)(u_1,u_2) = (\alpha v)(\beta u_1)=v(u_1) =\bigl(\iota_1(v)\bigr)(u_1,u_2),\] because \(\alpha^\vee\beta=\mathop{\mathrm{Id}}\). This is (6). ◻

Proof. Fix \(m\) and write \[V=f_m^*G,\qquad \widetilde L_i=f_m^*L_i,\qquad \widetilde D=f_m^*D,\qquad K_i=\widetilde L_i(-\widetilde D),\qquad K=K_1\otimes K_2.\] Pullback of (5) gives \[ 0\longrightarrow K_i\longrightarrow V\longrightarrow \widetilde L_j\longrightarrow0, \qquad\{i,j\}=\{1,2\}. \tag{10}\] These sequences remain exact because their original quotients are locally free. Here \[K_1=\mathcal O_S(21m,-3m),\qquad K_2=\mathcal O_S(-3m,21m),\] and hence \[ K=\mathcal O_S(18m,18m),\qquad \mathcal O_S(\widetilde D)=\mathcal O_S(3m,3m). \tag{11}\] We write powers of line bundles for tensor powers throughout this proof.

Alternating the quotient sequences. For an extension \(0\to K'\to V'\to L'\to0\) with \(K',L'\) line bundles, multiplication gives \[0\longrightarrow K'\otimes\mathop{\mathrm{Sym}}^{r-1}V' \longrightarrow\mathop{\mathrm{Sym}}^r V'\longrightarrow(L')^r \longrightarrow0\qquad(r\geq1).\] Indeed, a local splitting gives a kernel generator \(k'\) and a lift \(e'\) of a quotient generator. The monomials containing \(k'\) form the kernel of the symmetric quotient, and the remaining monomial \((e')^r\) maps to a quotient generator. This is the usual symmetric-power filtration of an extension; compare the proof of (Hartshorne 1966, Corollary 3.4).

Apply this construction first to \(V\to\widetilde L_2\) and then to \(V\to\widetilde L_1\). For \(r\geq2\) it gives \[\begin{align*} 0&\longrightarrow K_1\otimes\mathop{\mathrm{Sym}}^{r-1}V \longrightarrow\mathop{\mathrm{Sym}}^r V\longrightarrow\widetilde L_2^{\,r} \longrightarrow0,\tag{12}\\ 0&\longrightarrow K\otimes\mathop{\mathrm{Sym}}^{r-2}V \longrightarrow K_1\otimes\mathop{\mathrm{Sym}}^{r-1}V \longrightarrow K_1\otimes\widetilde L_1^{\,r-1} \longrightarrow0. \tag{13}\end{align*}\] Thus each pair of steps leaves the residual bundle \(K\otimes\mathop{\mathrm{Sym}}^{r-2}V\), with one line-bundle quotient at each step. For even \(n\), iteration gives a filtration of \(\mathop{\mathrm{Sym}}^nV\) with line factors \[ \begin{gathered} K^j\otimes\widetilde L_2^{\,n-2j},\qquad K^j\otimes\widetilde L_1^{\,n-2j}(-\widetilde D), \qquad 0\leq j<n/2,\\ K^{n/2}. \end{gathered} \tag{14}\] Indeed, after \(j\) pairs the residual bundle is \(K^j\otimes\mathop{\mathrm{Sym}}^{n-2j}V\), whose second quotient is \(K^jK_1\widetilde L_1^{\,n-2j-1} =K^j\widetilde L_1^{\,n-2j}(-\widetilde D)\). Only the second quotient contains an additional twist by \(-\widetilde D\) beyond the factors absorbed into \(K^j\).

Each step uses just one of the locally split sequences (10), so it remains valid when the two kernel lines have the same fiber image. Concretely, in the pulled-back basis \(e,f\) of (7), the kernels are generated by \(k=de-hf\) and \(f\), with \(h\) a unit. For \(n=2\) the filtration is \[\mathcal O(kf)\subset kV\subset\mathop{\mathrm{Sym}}^2V.\] Since \(e,k\) are also a basis of \(V\), the module \(kV\) has basis \(ke,k^2\) and is a direct summand of \(\mathop{\mathrm{Sym}}^2V\), with complement generated by \(e^2\). Moreover, \[kf=\frac{d}{h}ke-\frac1h k^2.\] The coefficient \(-1/h\) is a unit, so \(kf\) extends to a basis of \(kV\). Both inclusions therefore have locally free quotients. This verification uses no smoothness of \(d=0\) and applies at nodes and to nonreduced pullbacks of the divisor.

Global generation after twisting. Choose \(n\) even with \(n-1\geq3m\), and set \[ W=\mathop{\mathrm{Sym}}^n V\otimes P^{n-1}. \tag{15}\] By (11), the bidegrees of its filtration factors, in the order of (14), are \[\begin{align*} &(18mj+n-1,\;18mj+24m(n-2j)+n-1),\\ &(18mj+24m(n-2j)-3m+n-1,\;18mj-3m+n-1),\\ &(9mn+n-1,\;9mn+n-1). \end{align*}\] Every entry is nonnegative: the only possibly negative contribution is \(-3m\), and it is offset by \(n-1\).

For \(c,t\geq0\), products of homogeneous monomials generate \(\mathcal O_S(c,t)\). Moreover \(H^1(\mathbb P^1,\mathcal O(c))=H^1(\mathbb P^1,\mathcal O(t))=0\), as computed on the two standard affine charts, so the Künneth formula gives \(H^1(S,\mathcal O_S(c,t))=0\) (The Stacks Project Authors 2026, Tags 01XD and 0BED). If \(0\to U\to Z\to Q\to0\) is an extension of globally generated bundles with \(H^1(S,U)=H^1(S,Q)=0\), the cohomology sequence gives \(H^1(S,Z)=0\) and lifts every global section of \(Q\) to \(Z\). The lifted generators of \(Q\), together with generators of \(U\), span each fiber of \(Z\). Starting with the innermost line factor and applying this observation successively proves that \(W\) is globally generated.

The projective embedding and its line bundle. We record the tautological twists in the quotient-projectivization and Segre constructions (The Stacks Project Authors 2026, Tags 01OA, 01N1, and 01WD). Let \(\pi:\mathbb P(V)\to S\) and \(H=H^0(S,W)\). Evaluation \(H\otimes\mathcal O_S\to W\) induces a closed immersion \[\mathbb P(W)\hookrightarrow\mathbb P(H)\times S\] whose projective hyperplane line pulls back to \(\mathcal O_{\mathbb P(W)}(1)\). The relative Veronese closed immersion \(\mathbb P(V)\to\mathbb P(\mathop{\mathrm{Sym}}^n V)\) sends a quotient line \(\ell\) to \(\ell^{\otimes n}\). Under the natural identification \(\mathbb P(\mathop{\mathrm{Sym}}^n V)\simeq\mathbb P(W)\), its tautological pullback is \[\mathcal O_{\mathbb P(V)}(n)\otimes\pi^*P^{n-1}.\] Consequently their composition embeds \(\mathbb P(V)\) into \(\mathbb P(H)\times S\) with this hyperplane pullback from the first factor. Embedding \(S\) by \(P=\mathcal O_S(1,1)\) and then applying the Segre embedding gives a closed immersion into projective space whose hyperplane line pulls back to \[\bigl(\mathcal O_{\mathbb P(V)}(n)\otimes\pi^*P^{n-1}\bigr) \otimes\pi^*P =\bigl(\mathcal O_{\mathbb P(V)}(1)\otimes\pi^*P\bigr)^n.\] This line is therefore very ample. Finally, \(\mathbb P(E_m)=\mathbb P(V\otimes P)\simeq\mathbb P(V)\) identifies \(\mathcal O_{\mathbb P(E_m)}(1)\) with \(\mathcal O_{\mathbb P(V)}(1)\otimes\pi^*P\), proving the proposition. ◻

Proof. Put \(r=\operatorname{rank}G\). Choose an unbounded sequence of the indicated integers and let \(\|\cdot\|_m\) denote the smooth dual metric on \[(f_m^*G\otimes P)^\vee=f_m^*F\otimes P^{-1}.\] Its squared norm is plurisubharmonic on the total space. We recall the sign in this standard implication. For a smooth Hermitian metric with matrix \(h\), choose a holomorphic frame normal at the point in question, so that \(h=\mathop{\mathrm{Id}}\) and its first derivatives vanish there. Along a holomorphic disk with coordinate \(\zeta\), write the Hermitian curvature as \(iC\,d\zeta\wedge d\bar\zeta\). In the convention used here, \[C=-\frac{\partial^2h}{\partial\zeta\,\partial\bar\zeta}\] at the normal point. Thus, for a holomorphic section \(s\) along the disk, \[ \frac{\partial^2}{\partial\zeta\,\partial\bar\zeta}\|s\|^2 =\|D'_{\partial/\partial\zeta}s\|^2-\langle Cs,s\rangle. \tag{16}\] Duality reverses Griffiths positivity, so the second term on the right is nonnegative for the dual metric. Applying this calculation to the section of the pulled-back bundle defined by any disk in its total space proves the assertion. For this classical norm criterion and duality, see (Umemura 1973, Remark 1.3 and Lemma 1.7) and (Raufi 2015, sec. 2); the same implication is recorded in (Forstnerič and Kusakabe 2025, Proposition 2.6).

Trace quadratics and normalization. Fix, for each \(m\), the holomorphic isomorphism \(f_m^*P^{-1}\simeq P^{-m}\). If \(t\) is a nonvanishing holomorphic frame of \(P^{-1}\) on an open set \(U\subset S\), define on \(F|_U\) \[ q_{m,t}(x,v) =\frac{1}{m^2}\sum_{y\in f_m^{-1}(x)} \bigl\|v\otimes(f_m^*t)^{1/m}(y)\bigr\|_{m,y}^2, \tag{17}\] where the sum counts multiplicities. The root in this expression is local on the source. Indeed, in a local frame \(s\) of \(P^{-1}\) the section \(f_m^*t\) has the form \(\eta(y)s^m\), with \(\eta\) holomorphic and nowhere zero. A local holomorphic logarithm of \(\eta\) gives the root \(\exp((\log\eta)/m)s\). Two choices differ by an \(m\)-th root of unity, which does not change the squared norm. The expression (17) is consequently well defined and is a positive definite Hermitian quadratic in \(v\).

We verify its regularity through the branch locus. Off that locus, every local inverse branch of \(f_m\) gives a holomorphic map \[(x,v)\longmapsto \bigl(y(x),v\otimes(f_m^*t)^{1/m}(y(x))\bigr)\] into \(f_m^*F\otimes P^{-1}\). Hence each summand, and their sum, is plurisubharmonic. The unordered multiset of roots of \(y^m=x\) varies continuously with \(x\), counting coalescing roots with their multiplicities. Applied in the two factors, and using local source frames for the norm in (17), this proves continuity of the sum across the branch locus, including a crossing of its two components.

The continuous trace is plurisubharmonic across the branch locus as well. In a local total-space chart, the inverse image of the branch locus under the bundle projection \(F|_U\to U\) is an analytic set, contained in \(\{z_1z_2=0\}\) after choosing affine coordinates on the two base factors. It is therefore pluripolar. The trace is locally bounded above there by continuity, so the removable-singularities theorem for plurisubharmonic functions (Demailly 2012, I, Theorem 5.24) gives a unique plurisubharmonic extension. Its upper-limit values agree with the continuous trace. This proves joint plurisubharmonicity of \(q_{m,t}\) on all of \(F|_U\).

If \(t'=gt\) is another nonvanishing frame, then \[ q_{m,t'}(x,v)=|g(x)|^{2/m}q_{m,t}(x,v). \tag{18}\] Choose smooth background Hermitian norms \(|\cdot|_0\) on \(F\) and \(P^{-1}\). By (18), the continuous quadratic \[N_m(x,v)=q_{m,t}(x,v)|t(x)|_0^{-2/m}\] is globally defined. We use \(N_m\) to normalize the traces globally; the local functions \(q_{m,t}\) retain the plurisubharmonicity used below. The background unit sphere bundle is compact, and \(N_m\) is positive there. Multiplying the dual metric’s squared norm by a positive constant, we may therefore arrange \[ \max_{x\in S,\ |v|_0=1}N_m(x,v)=1. \tag{19}\] This rescaling does not change the curvature. Homogeneity gives \[ 0\le q_{m,t}(x,v)\le |t(x)|_0^{2/m}|v|_0^2. \tag{20}\]

A measurable coefficient limit. In a holomorphic trivialization of \(F|_U\), write \[q_{m,t}(x,v)=v^*H_m(x)v.\] On each relatively compact subset of \(U\), the positive matrices \(H_m\) are uniformly bounded, by (20). Weak compactness in \(L^2\) and a diagonal subsequence on a countable collection of charts and compact exhaustions give distributional limits \(H\) for all the coefficient matrices. These limits belong to \(L^\infty_{\mathrm{loc}}\), are Hermitian, and are positive semidefinite almost everywhere. For example, positivity and the same local upper bound pass to \(c^*Hc\) against every nonnegative test function for each \(c\in(\mathbb Q+i\mathbb Q)^r\); density then gives the matrix inequalities on a single set of full measure.

The coefficient limits have the untwisted transition law of a form on \(F\). More explicitly, let \(v=T(x)w\) be a holomorphic change of bundle coordinates and let \(t'=gt\). The original matrices satisfy \[H'_m=|g|^{2/m}T^*H_mT.\] On a compact subset \(K\) of the overlap, the error obtained by deleting the scalar factor has, against a smooth compactly supported matrix test function \(\Phi\), the bound \[\left|\left\langle \bigl(|g|^{2/m}-1\bigr)T^*H_mT,\Phi \right\rangle\right| \le C_K\|\Phi\|_{L^1} \sup_K\bigl||g|^{2/m}-1\bigr|\longrightarrow0.\] Here the local matrix bound and the boundedness of \(T\) are absorbed into \(C_K\). Multiplication by the fixed smooth matrices \(T,T^*\) preserves distributional convergence, as does a change of base coordinates. Consequently \(H'=T^*HT\) almost everywhere.

Nonvanishing of the measurable limit. The limiting measurable form is not zero. Choose maximizing unit vectors \((x_m,v_m)\) in (19). After a further subsequence they converge in the compact unit sphere bundle to \((x_*,v_*)\). This further subsequence retains all the distributional limits already obtained. In one fixed coordinate and frame chart around \(x_*\), let \(c_m\to c\) be their fiber coordinates. Then \(c\ne0\), and a local operator bound \(\|H_m\|\le C\) gives \[\bigl|c^*H_m(x_m)c-c_m^*H_m(x_m)c_m\bigr| \le C|c-c_m|(|c|+|c_m|)\longrightarrow0.\] On the other hand, \[c_m^*H_m(x_m)c_m=|t(x_m)|_0^{2/m}\longrightarrow1,\] since the norm of the nonvanishing frame is bounded above and away from zero on a fixed compact neighborhood. Thus the nonnegative plurisubharmonic function \(u_m(x)=c^*H_m(x)c\) satisfies \(u_m(x_m)\ge\tfrac12\) for large \(m\). Choose \(\varrho>0\) so that \(B(x_*,4\varrho)\) lies in the coordinate chart, and choose a fixed smooth nonnegative function \(\chi\) supported in \(B(x_*,3\varrho)\) and equal to one on \(B(x_*,2\varrho)\). For all sufficiently large \(m\), the ball \(B(x_m,\varrho)\) lies in that latter ball. The submean inequality yields \[ \int\chi(x)c^*H_m(x)c\,dx \ge\int_{B(x_m,\varrho)}u_m(x)\,dx \ge\frac12\operatorname{Vol}(B(0,\varrho)). \tag{21}\] The left side is a pairing against the fixed smooth matrix test function \(\chi cc^*\), so the inequality passes to \(H\). It proves that \(H\) is nonzero as a measurable form.

A quadratic psh representative on every fiber. The measurable form is defined only up to a set of measure zero in the base. Even a pointwise limit of the traces may fail to be plurisubharmonic: in one base variable, tracing \((1+|y|^2)|v|^2\) through \(x=y^m\) gives \[(1+|x|^{2/m})|v|^2.\] Its pointwise limit has coefficient \(1\) at \(x=0\) and \(2\) elsewhere, whereas its plurisubharmonic representative has coefficient \(2\) everywhere. We now recover the intrinsic plurisubharmonic representative of our measurable form and show that it remains a Hermitian quadratic on every fiber.

Convolution regularization of negative singular Hermitian metrics appears in (Berndtsson and Păun 2008, sec. 3, Proposition 3.1) and (Raufi 2015, Proposition 1.3(i)); the use of polarization to recover an everywhere-defined representative is also discussed in (Păun and Takayama 2018, Remark 2.2.3). We supply the argument for the present possibly rank-deficient limit, including the compatibility of the representatives on overlaps. Work in a coordinate and bundle chart \(U\), and let \(\rho\) be a nonnegative smooth radial function on \(\mathbb C^2\), supported in the unit ball and of integral one. Put \[\rho_\delta(y)=\delta^{-4}\rho(y/\delta),\qquad H_\delta=H*\rho_\delta,\qquad q_\delta(x,v)=v^*H_\delta(x)v,\] on the interior domain where this convolution is defined. Convolving \(H_m\) in the same way gives \[q_{m,\delta}(x,v) =\int q_{m,t}(x-y,v)\rho_\delta(y)\,dy.\] It is plurisubharmonic jointly in \((x,v)\), since each base translation \((x,v)\mapsto(x-y,v)\) is holomorphic and the average has nonnegative weights. For fixed \(\delta\), distributional convergence of \(H_m\) implies convergence of the convolved coefficients and all their derivatives at every interior point: these are pairings against translates of derivatives of the fixed smooth kernel. The Levi matrices of \(q_{m,\delta}\) therefore converge pointwise to that of \(q_\delta\). Hence \(q_\delta\) is smooth and plurisubharmonic, and on each smaller chart \[ 0\le q_\delta(x,v)\le C|v|^2 \tag{22}\] for all sufficiently small \(\delta\).

These quadratics decrease as \(\delta\) decreases. To check this for any fixed nonnegative radial kernel, write \(\rho(y)=\phi(|y|)\) and let \(M_u(x,s)\) denote the spherical mean of a subharmonic function \(u\) on \(\mathbb R^4\). Polar coordinates give \[ (u*\rho_\delta)(x) =|\mathbb S^3|\int_0^1 M_u(x,\delta s)\phi(s)s^3\,ds. \tag{23}\] The spherical mean is nondecreasing in its radius. The integration weight in (23) is nonnegative and does not depend on \(\delta\), so the convolution is nondecreasing in \(\delta\). Apply this to \(u(x)=q_{m,t}(x,v)\) for each constant \(v\), then pass to the limit in \(m\) at two fixed radii. It follows that \[H_{\delta_1}(x)\le H_{\delta_2}(x) \qquad(0<\delta_1<\delta_2)\] in the order of Hermitian forms.

Thus \(q_\delta\) decreases, as \(\delta\downarrow0\), to a finite nonnegative plurisubharmonic function \(q_U\) on the total-space chart. The decreasing-limit assertion follows also by passing to the submean inequalities; see (Demailly 2012, I, Theorem 5.4). Finiteness is supplied by (22). At every fixed base point \(x\), this limit is a Hermitian quadratic. Indeed, all entries of \(H_\delta(x)\) are determined by the finite set of values at \(e_j\), \(e_j+e_k\), and \(e_j+i e_k\), through \[\begin{align*} 2\operatorname{Re}(H_\delta)_{jk} &=q_\delta(x,e_j+e_k)-q_\delta(x,e_j)-q_\delta(x,e_k),\\ 2\operatorname{Im}(H_\delta)_{jk} &=q_\delta(x,e_j)+q_\delta(x,e_k)-q_\delta(x,e_j+i e_k). \end{align*}\] Each value on the right has a finite limit. Hence the entries converge to one Hermitian matrix \(\widehat H_U(x)\), and passage to the finite quadratic polynomial gives \[q_U(x,v)=v^*\widehat H_U(x)v\qquad\text{for every }v\in\mathbb C^r.\] This matrix is positive semidefinite at every \(x\), by nonnegativity. At almost every \(x\), approximate-identity convergence also gives \(\widehat H_U(x)=H(x)\).

Finally these local representatives agree on overlaps. After the holomorphic change of bundle coordinates, their measurable matrices agree outside a base-null set, by the transition law already proved. The quadratic functions therefore agree for every fiber vector outside that set, hence almost everywhere in the total-space chart by Fubini’s theorem. Both functions are locally bounded and plurisubharmonic. Such a function is determined everywhere by its almost-everywhere values: the submean inequality and upper semicontinuity give, in local Euclidean coordinates, \[u(p)=\lim_{\varepsilon\downarrow0} \frac{1}{\operatorname{Vol}(B(p,\varepsilon))} \int_{B(p,\varepsilon)}u.\] Thus the local \(q_U\) glue to a single function \(q\) on \(F\). Its coefficients represent the original measurable form almost everywhere, so (21) implies \(q\not\equiv0\). All the required pointwise and plurisubharmonic properties now hold. ◻

Proof of Theorem 1. Take the algebraic rank-two bundle \(G\) of Proposition 4. By Proposition 5, every \(E_m=f_m^*G\otimes P\) is ample in the quotient convention.

If the set of positive integers \(m\) for which \(E_m\) has a smooth strictly Griffiths-positive metric were infinite, it would contain a sequence tending to infinity. Proposition 6 would then give a finite, nonnegative, nonzero psh function on the total space of \(G^\vee\) that is a Hermitian positive semidefinite quadratic form on every fiber. Proposition 11 excludes precisely such a function. Consequently that set of integers is finite. Choose \(m_0\) larger than all its elements, taking \(m_0=1\) if the set is empty. Every \(m\ge m_0\) gives the required counterexample. ◻