Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

Proof. The value of s gives

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=-\Id.

Cayley–Hamilton gives A+A^{-1}=\Id. Multiplication by B and taking traces therefore yield

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 2.1. 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 [18, 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 2.1, \lambda^{24}=1. Consequently

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

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 (2.3).

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 (2.4) has basis

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 (2.4) identifies its kernel with L_i(-D), proving (2.5). 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

because \alpha^\vee\beta=\Id. This is (2.6).□

Proof. Fix m and write

Pullback of (2.5) gives

These sequences remain exact because their original quotients are locally free. Here

and hence

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

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 [12, 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

Thus each pair of steps leaves the residual bundle K\otimes\Sym^{r-2}V, with one line-bundle quotient at each step. For even n, iteration gives a filtration of \Sym^nV with line factors

Indeed, after j pairs the residual bundle is K^j\otimes\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 (3.1), so it remains valid when the two kernel lines have the same fiber image. Concretely, in the pulled-back basis e,f of (2.7), the kernels are generated by k=de-hf and f, with h a unit. For n=2 the filtration is

Since e,k are also a basis of V, the module kV has basis ke,k^2 and is a direct summand of \Sym^2V, with complement generated by e^2. Moreover,

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

By (3.2), the bidegrees of its filtration factors, in the order of (3.5), are

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 \OO_S(c,t). Moreover H^1(\PP^1,\OO(c))=H^1(\PP^1,\OO(t))=0, as computed on the two standard affine charts, so the Künneth formula gives H^1(S,\OO_S(c,t))=0 [24, 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 [24, Tags 01OA, 01N1, and 01WD]. Let \pi:\PP(V)\to S and H=H^0(S,W). Evaluation H\otimes\OO_S\to W induces a closed immersion

whose projective hyperplane line pulls back to \OO_{\PP(W)}(1). The relative Veronese closed immersion \PP(V)\to\PP(\Sym^n V) sends a quotient line \ell to \ell^{\otimes n}. Under the natural identification \PP(\Sym^n V)\simeq\PP(W), its tautological pullback is

Consequently their composition embeds \PP(V) into \PP(H)\times S with this hyperplane pullback from the first factor. Embedding S by P=\OO_S(1,1) and then applying the Segre embedding gives a closed immersion into projective space whose hyperplane line pulls back to

This line is therefore very ample. Finally, \PP(E_m)=\PP(V\otimes P)\simeq\PP(V) identifies \OO_{\PP(E_m)}(1) with \OO_{\PP(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

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=\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,

at the normal point. Thus, for a holomorphic section s along the disk,

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 [26, Remark 1.3 and Lemma 1.7] and [23, Section 2]; the same implication is recorded in [9, 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

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 (4.2) 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

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 (4.2), 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 [4, Chapter 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

Choose smooth background Hermitian norms |\cdot|_0 on F and P^{-1}. By (4.3), the continuous quadratic

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

This rescaling does not change the curvature. Homogeneity gives

A measurable coefficient limit. In a holomorphic trivialization of F|_U, write

On each relatively compact subset of U, the positive matrices H_m are uniformly bounded, by (4.5). 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

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

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 (4.4). 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

On the other hand,

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

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

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 [2, Section 3, Proposition 3.1] and [23, Proposition 1.3(i)]; the use of polarization to recover an everywhere-defined representative is also discussed in [20, 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 \CC^2, supported in the unit ball and of integral one. Put

on the interior domain where this convolution is defined. Convolving H_m in the same way gives

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

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 \RR^4. Polar coordinates give

The spherical mean is nondecreasing in its radius. The integration weight in (4.8) 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

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 [4, Chapter I, Theorem 5.4]. Finiteness is supplied by (4.7). 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

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

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,

Thus the local q_U glue to a single function q on F. Its coefficients represent the original measurable form almost everywhere, so (4.6) implies q\not\equiv0. All the required pointwise and plurisubharmonic properties now hold.□

Proof. Write q(x,v)=H_x(v,v); nonnegativity says that every H_x is positive semidefinite. Pull q back by the line inclusions

For fixed x_i\in\PP^1 and v\in\OO(-b)_{x_i}, the function obtained by holding v fixed while varying the other base coordinate is finite and subharmonic on the complementary \PP^1. It is constant by the maximum principle. Consequently q\circ\iota_i descends to a finite nonnegative squared seminorm a_i on \OO_{\PP^1}(-b) over the i-th factor. Fixing any point in the other factor realizes a_i as a holomorphic pullback of q, so a_i is plurisubharmonic on the total line bundle.

On \Gamma_M, the identity \iota_2\alpha=\iota_1 from Proposition 2.3 and the definition of \alpha give

The case M=\Id identifies a_1 and a_2 as a single function a_\bullet on \OO(-b). This function is not identically zero. Indeed, if it were zero, then for every x\notin D, the form H_x would vanish on each of two spanning lines. The inequality

would then force H_x=0. Hence q would vanish on the full-measure complement of the inverse image of D in F. Nonnegativity and the submean inequality on small total-space coordinate balls would give q=0 everywhere, a contradiction.

The cases M=A,B of (5.1) give the asserted invariances under the holomorphic lifts A^{\otimes b} and B^{\otimes b}.□

Proof. Fix a coordinate disk U and a frame e, and abbreviate u_e to u. The function u is finite, nonnegative, subharmonic, and locally bounded, by restriction to the holomorphic section e. Choose a nonnegative smooth radial function \rho supported in the unit disk, with integral one, and put

These functions are defined on the interior of U for sufficiently small \delta>0. Averaging the holomorphic base translations (z,w)\mapsto(z-\zeta,w) shows that (u*\rho_\delta)(z)|w|^2 is plurisubharmonic. Adding \delta|w|^2 preserves this property, so u_\delta>0 and u_\delta(z)|w|^2 is smooth and plurisubharmonic. Its Levi matrix at w=1 is

Its nonnegative determinant gives

Thus f_\delta:=\log u_\delta is subharmonic.

The radial convolutions are nonnegative weighted averages of circular means of u. These means increase with their radii, and converge to u as the radii tend to zero. Consequently

We establish local integrability before taking distributional derivatives of the limit. Suppose that u(z_0)>0, and choose R>0 with \overline{B(z_0,R)}\subset U. For all sufficiently small \delta, local boundedness supplies a constant C_0\geq0 such that f_\delta\leq C_0 on this disk. Moreover f_\delta(z_0)\geq\log u(z_0). Integrating the circular submean inequality over the radius gives

Writing f^-:=\max\{-f,0\}, we obtain the explicit bound

The negative parts increase as \delta decreases, whereas the positive parts are bounded by C_0. Monotone convergence therefore proves \log u\in L^1(B(z_0,R)). It also gives f_\delta\to\log u in L^1 on this disk.

This conclusion propagates throughout C. Indeed, let \Omega consist of the points near which a local logarithmic weight is integrable. A frame change e'=g e gives

so \Omega is independent of the chosen frame. It is open and nonempty: nontriviality of a_\bullet supplies a point with u>0, to which (5.3) applies. If x is in the boundary of \Omega, choose a coordinate disk containing \overline{B(x,4r)}. There is a small open disk contained in \Omega\cap B(x,r). On that disk integrability of \log u implies u>0 almost everywhere, so it contains a point z_0 with u(z_0)>0. The disk B(z_0,2r) has closure in the original chart and contains a neighborhood of x. Applying (5.3) on this larger disk proves x\in\Omega. Thus \Omega is also closed, and connectedness gives \Omega=C.

On every relatively compact disk, C_0-f_\delta increases to the integrable function C_0-\log u. Hence the local convergence is in L^1, and the decreasing limit \log u is subharmonic. In particular, \ddc\log u is a positive distribution and thus a positive Radon measure. Since g in (5.4) is holomorphic and nowhere zero, \log|g|^2 is harmonic. The local measures therefore glue to \mu.

For the invariance assertion, choose frames e on U and e' on V with T(U)\subset V, and write

where c is holomorphic and nowhere zero. The isometry condition gives

Taking distributional \ddc, which commutes with pullback by the holomorphic diffeomorphism T, gives \mu=T^*\mu. Here pullback is that of a top-degree current: for a test function \varphi,

Equivalently, (T^*\mu)(E)=\mu(T(E)) for Borel sets E. This is ordinary measure invariance under the bijection T; in particular, T^*\delta_p=\delta_{T^{-1}p} with unchanged atom mass.

Finally suppose C=\PP^1 and L=\OO(-b). Let h_0 be the smooth Hermitian metric on \OO(-b) induced by the standard norm on \CC^2. In the tautological affine frame e(z)=(1,z)^{\otimes b} its weight is

The local functions \log u_e-\log h_0(e,e) define a single globally integrable function \Phi on \PP^1, by the common frame law. If \mu_0=\ddc\log h_0(e,e), then

The measure \mu is finite because it is locally finite on the compact sphere. With the convention \ddc f=i f_{z\bar z}\,dz\wedge d\bar z, the smooth reference measure has mass

This proves (5.2). Zeros of u_e and any atoms of \mu are included in the distributional calculation; no points have been excised.□

Proof. Suppose that \mu is such a measure. It is invariant under T=AB. The two eigenvalues of AB have distinct moduli, so after a projective change of coordinate T acts as z\mapsto\lambda z with R:=|\lambda|>1. The Borel sets

are pairwise disjoint, cover \CC^*, and satisfy T(C_n)=C_{n+1}. Each has the same measure. Finiteness and positivity imply that this common measure is zero, and countable additivity gives \mu(\CC^*)=0. The half-open annuli assign every boundary circle to exactly one set, so this argument applies equally to atomic and diffuse measures and to nonreal \lambda.

Thus the nonempty support of \mu consists of one or two of the fixed points of T. Invariance under A and B preserves this support as a set: the inverse image of an open neighborhood of the image of a support point has positive measure. Both projective transformations have order three. Their induced permutations on a set of at most two points are therefore trivial. Each point of the support would be fixed by both A and B, and hence would be a common eigenline of the two matrices. Lemma 2.1 excludes such a line.□

Proof. If such a function were nonzero, Lemma 5.1 would give a nonzero squared seminorm on \OO_{\PP^1}(-b) invariant under the lifts of A and B. Lemma 5.2 would then produce a finite positive measure on \PP^1 of mass 2\pi b>0, invariant under both projective transformations. This contradicts Lemma 5.3.□

Proof of Theorem 1.1. Take the algebraic rank-two bundle G of Proposition 2.3. By Proposition 3.1, 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 4.1 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 5.4 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.□