Fourfold nonvanishing by minimal metrics and moving jets

Proof. Start with a smooth counterexample H. Run the canonical minimal model program. Existence of the necessary flips follows from [3, Corollary 1.4.1]. Termination in the pseudo-effective fourfold case is [7, Theorem 1.1], applied with zero nef data to the ordinary pair with zero boundary. Divisorial contractions lower the Picard number, so together these statements give termination of the program. Pseudo-effectivity is preserved and excludes a Mori fiber space as its endpoint. We obtain a projective \Q-factorial terminal minimal model Y.

Canonical section spaces in sufficiently divisible degrees are birationally invariant here. On a common resolution the differences from the respective pulled-back canonical divisors are effective exceptional divisors, and pushing their sheaves forward does not add sections. Thus \kappa(Y,K)=-\infty. In particular K is not big; as it is nef, this says K^4=0.

Terminal singularities are rational. Hence q(Y) agrees with the irregularity of a resolution. If it were positive, [11, Corollary 4.7] would make K semiample, a contradiction. Likewise, if the nef dimension were at most three, [1, Theorem 4.3], with the established log minimal model program and log abundance in those dimensions, would make K semiample. This includes nef dimension zero. Thus the nef dimension is four.

The curve property of nef reduction [2, Theorem 2.1] says in this case that K\cdot C>0 for every integral curve through a very general point. Choose \iota with \iota K Cartier. Its degree on an integral complete curve is an integer, so positivity gives Equation (2.1). This uses only the curve property of nef reduction, not an Iitaka fibration on Y.□

Proof. We first work with one family of subvarieties, and at the end make the very general locus independent of the family. Hilbert schemes give countably many finite-type parameter spaces for integral subvarieties of Y. Stratification, resolution of the universal families, and generic smoothness give countably many smooth families

whose fibers are smooth projective resolutions of their images and whose fiber maps are birational onto those images. To obtain this description, resolve the total universal family on an integral parameter stratum, shrink until its fibers are smooth and the fiber maps birational, and repeat on the complement by noetherian induction. Non-dominant evaluation families can be discarded by excluding their image closures in Y.

Consider a dominant family with fibers of dimension d, where 0<d<4. We may slice the parameter space to dimension 4-d so that e becomes generically finite and remains dominant. Indeed, the parameters incident to a general point of Y have a component of dimension \dim B-(4-d) on the locus where the fiber maps are generically embeddings. General ample cuts in a compactification of B meet this locus in finitely many points. We take the cuts very generally, so a very general point of the slice has the generic value of every plurigenus of the original family. There are only countably many such conditions.

On the resulting smooth total space, ramification and terminality give

For completeness, the rational lift of e to a resolution of Y is defined at codimension-one points by properness. The usual ramification formula there, followed by the effective discrepancies of the terminal target, proves this divisor inequality. Its restriction to a very general fiber is

In particular K_V is pseudo-effective.

The dimension of V is at most three. The minimal model program and abundance in these dimensions give a good terminal minimal model V_{\min}. Suppose that V is not of general type. The semiample canonical divisor of V_{\min} defines a morphism with connected positive-dimensional general fibers, on each of which a multiple of K_{V_{\min}} is trivial. On a common resolution \widetilde V of V and V_{\min}, there is a complete curve through a very general point with K_{\widetilde V}-degree zero. Here is the avoidance needed for this assertion. The singular locus of the terminal model and the centers of the exceptional divisors have codimension at least two. A general fiber either misses a given vertical center or meets a dominant center in codimension at least two. General sufficiently ample complete intersections in that fiber through a prescribed very general point give a curve avoiding all such centers; for a one-dimensional fiber the fiber itself does so. Its strict transform has canonical degree zero by the exceptional discrepancy formula.

Pulling back Equation (2.2) to \widetilde V and adding the effective discrepancies of \widetilde V\to V gives

Choose the preceding curve through a point outside \Supp\widetilde E and outside the exceptional locus of the birational map onto the image subvariety. It is not contained in \widetilde E, and its image in Y is a curve. Thus its pulled-back K-degree is at most zero. Dominance of the total evaluation lets us choose the point in the very general locus of Equation (2.1), which gives a contradiction.

We have proved that the very general fiber of the sliced family is of general type. By the choice of slice its plurigenera are the generic plurigenera of the original smooth family. Upper semicontinuity shows that all fibers of that family have at least these plurigenera, hence are of general type as well. Finally exclude the image closures of all non-dominant families. Countability of the parameter strata gives the asserted very general locus.□

Proof. We recall the argument of [38, Lemmas 7.3–7.4] to explain why the estimate concerns the entire kernel and ordinary ideal powers. On the open where the jet maps have constant rank, their kernels are vector bundles with fibers \mathcal H_i(x). Their base loci increase with i. Starting with a positive-dimensional component through the mark, follow a containing component at each level. The d+1 components all have dimensions in [1,d-1], so two successive ones coincide. After a finite parameter extension and shrinking, their geometric generic components and inclusions spread to one family \Gamma\subset T\times Z. Its evaluation dominates Z, since each member contains its moving mark x(t).

Take a local frame of the higher kernel and trivialize kP near z with a frame independent of t. In a fixed basis of the global section space, each frame section has the form

Differentiation in the parameter with z fixed therefore gives another global section of kP. A derivative of order r lowers the vanishing order at x(t) by at most r. It belongs to the entire lower kernel whenever

The right side is at least k\delta-1, so every derivative of order less than h=\ceil{k\delta/2} belongs to that kernel and vanishes on \Gamma. This is where using the full kernels matters: an arbitrarily chosen subseries need not contain these derivatives.

To turn these vanishings into ordinary ideal membership, work at a general smooth incidence point. Since \Gamma\to Z is submersive, parameter directions span its normal space in T\times Z. Choose parameter coordinates t=(t',t'') so that the implicit function theorem writes \Gamma as t'=F(t'',z). With t'' and z fixed, derivatives in t' are exactly derivatives in the normal coordinates u=t'-F(t'',z). All normal Taylor coefficients of degree below h therefore vanish, proving s\in\cI_\Gamma^h. Faithful flatness of analytic local rings over algebraic local rings gives the same membership algebraically. There are finitely many frame sections, so this containment holds for the entire incidence base ideal on a dense algebraic open.

Generic smoothness of \Gamma\to T identifies its scheme fiber there with the reduced member V_t. Restriction gives \cI_\Gamma^h\cO_{\{t\}\times Z}=\cI_{V_t}^h, and the frame specializes to a basis of the entire higher kernel. Isolation holds because V_t is a base-locus component. Finally remove all other base components and choose the incidence point where V_t is smooth over its actual image in B. Its scheme fiber is then reduced locally, so restriction to the ambient fiber sends \cI_{V_t}^h to \cI_F^h and the base support to F. If F is proper in the chosen integral ambient fiber component, that support is proper there; hence the restricted series is nonzero and has F as an isolated base component at its generic point.□

Proof. Uniform effective birationality for projective lc pairs of dimension f with big adjoint and coefficients in the fixed DCC set \{0,1\} supplies an integer b=b(f) such that |b(K_{V_t^*}+G_t)| is birational [18, Theorem 4.0.1]. Resolve its base ideal and denote the free moving divisor by H_b. We keep the same notation for the pullbacks of P_t and L_t. The fixed divisor is effective, so

Both H_b and P_t are nef and big; P_t is big because P_t^f>0. The Khovanskii–Teissier log-concavity inequalities for their mixed intersections give, when f>1,

Indeed the successive ratios of the positive numbers P_t^{f-i}H_b^i decrease, which gives the first inequality.

Choose a point where the birational morphism defined by H_b is an isomorphism onto an open of its image, where e_t is quasi-finite, and whose image lies in the very general locus of Y. These conditions are compatible. To spell out the last one, write the excluded subset of Y as a countable union of proper closed subsets. The image of e_t is not contained in any of them because it meets their complement. Their inverse images are therefore proper closed subsets of V_t^*. They can be avoided together with the finitely many dense-open conditions. For a sweeping family one first chooses a very general member not contained in any excluded subset, and then such a point on that member.

For f>1, cut by f-1 general members of |H_b| passing through the chosen point. These give a proper effective curve cycle: on the birational image they are general hyperplanes through a point of the isomorphism open, and their pullbacks have no positive-dimensional base component. A component C through the selected point maps nonconstantly to Y, since e_t is quasi-finite there. Nefness and Equation (3.7) give

In the first inequality the positive degree of the finite map of the normalization of C onto its image has merely been discarded. If f=1, use V_t^* itself as C; the same argument gives L_t\cdot e_t(C)\leq P_t\cdot C\leq C_0.

The image is an integral curve through the very general locus, so Equation (3.2) bounds its degree below by r_t/\iota. This tends to infinity, whereas the upper bound depends only on f,C_0,C_1. The contradiction proves the lemma. If several dimensions 1\leq f\leq n occur, take the maximum of the finitely many resulting constants.□

Proof. Fix t for the moment. For each Cartier multiple kP_{\mathrm s} and each integer a, the condition

is a rank condition on the jet evaluation map over Y_{\mathrm{sm}}. We choose x outside all proper rank-jumping loci of these countably many maps. Thus if the order bound fails at such a point, the relevant high-order kernel is nonzero for general marks as well. Taking powers of the offending sections gives the same strict inequality in arbitrarily large divisible degrees. This permits all subsequent choices of a sufficiently large divisible k.

Choose n+1 levels equally spaced from 2\rho_t to 3\rho_t; their gap is

If \ord_x(s)>4k\rho_t in a sufficiently large degree, every one of these jet kernels is nonzero. The base locus of the lowest kernel cannot have x as an isolated component. Indeed its base ideal at x is contained in \mathfrak m_x^{\ceil{2k\rho_t}}, so the zero-dimensional case of Lemma 3.2 would give

a contradiction.

Apply Lemma 3.3. It produces a dominant family of proper positive-dimensional base components V, of some dimension 1\le f<n, such that the entire higher jet kernel’s generic base ideal along V is primary and contained in \cI_V^h, where

For a general member, Lemma 3.2 gives

where V^* is a smooth projective resolution. The second inequality uses K\le P_{\mathrm s}/(2r_t).

The constants on the right are bounded independently of small t, of k, and of the member V, by Equation (3.9). We choose a very general incidence point before choosing the member V. Its image belongs to Y^{\mathrm{vg}}, so V^* is of general type by Proposition 2.2. Moreover P_{\mathrm s} dominates L in nef order. The inclusion of V into Y, composed with its resolution, satisfies all the hypotheses of Lemma 3.4. Equations (3.11) to (3.12) would therefore give a curve through Y^{\mathrm{vg}} of bounded L-degree, contradicting Equation (3.2) as t becomes small.

The bounds used in this contradiction do not depend on the initial degree in which excessive vanishing occurred. Consequently, after one restriction to sufficiently small t, all Cartier multiples satisfy Equation (3.10) at a very general point.□

Proof. Put P_W=K_W+D_W. Terminality gives

with E_W exceptional. In particular, both K_W and P_W are pseudo-effective. We assume that P_W is not big and derive a contradiction. We first reduce its Kodaira dimension to at most zero, then construct a nef boundary with a nonempty reduced floor. We will lift a nonzero section from that entire floor before using any semiampleness theorem that requires nonnegative Kodaira dimension.

Positive Kodaira dimension would already imply bigness

Suppose first that \kappa(P_W)>0. Resolve the rational map defined by a moving subsystem of a sufficiently divisible multiple bP_W:

Here \widetilde W is smooth, U is the projective image, and, for a very ample line bundle \cO_U(1),

If f is generically finite, then P_W is big, contrary to the assumption. Otherwise 0<\dim U<4. A smooth fiber component through a very general point of \widetilde W is birational to a proper positive-dimensional subvariety of Y meeting the very general locus. It is therefore of general type. We work over a smooth open of U where generic smoothness applies. On its smooth fibers the restriction of K_{\widetilde W} is the canonical bundle of the fiber, up to the constant one-dimensional factor coming from the base.

Fix an ample Cartier divisor H_0 on \widetilde W. For some integer j>0, the restriction of jK_{\widetilde W}-H_0 has a nonzero section on such a fiber component. Distinct components of a smooth fiber are disjoint, so the section can be taken to be zero on the other components. Choose the fiber over a point where the fiber dimension of the space of sections is the generic one for every integer j; this requires only countably many exclusions. Consequently

is generically nonzero for one j. After tensoring by a sufficiently large power \cO_U(c), this coherent sheaf has a nonzero global section. Thus

The divisor on the left before subtracting H_0 is big. By Equation (5.1), adding an effective divisor gives bigness of jK_{\widetilde W}+cb\pi^*P_W. Birational pushforward gives bigness of jK_W+cbP_W. For completeness, the ample divisor H_0 has big pushforward: for a fixed ample divisor H_W on W and small positive rational \eta, H_0-\eta\pi^*H_W is ample, and pushing an effective rational representative proves this assertion. Finally

is big. This contradiction proves

A minimal model and a crepant klt perturbation

Run a log minimal model program for the rational dlt pair (W,D_W). We use flip existence for klt pairs, the usual dlt contraction properties, and termination of flips for pseudo-effective lc fourfolds, applied with zero nef part [3, 7]. The dlt flips needed here are obtained by lowering the finitely many boundary coefficients slightly while retaining negativity of the given extremal ray. The resulting pair is klt. On the relative Picard-number-one contraction, the original and perturbed negative adjoints are relatively proportional with positive rational ratio; the contraction theorem descends a Cartier multiple of their numerically trivial difference. The klt flip therefore has the required positivity for the original adjoint as well. This supplies the flips used in this dlt program.

There are only finitely many divisorial contractions. Pseudo-effectivity persists under each pushforward and strict transform and excludes a negative fiber space, by intersection with covering curves in a general fiber. These pushforwards on numerical divisor classes are well-defined on the \Q-factorial models: on a common resolution they are pullback followed by pushforward, and exceptional divisor classes give no ambiguity. We obtain a projective \Q-factorial dlt minimal model (Z,D_Z) without extracting divisors. Write

The divisor D_Z is reduced, M is nef, and the standard exceptional comparison on a common resolution preserves section spaces in sufficiently divisible degrees. Hence

Moreover K_Z is pseudo-effective, since K_W is pseudo-effective.

The signed representative supplies an additional linear relation. Indeed

and the divisor on the right is supported on D_W. Its strict pushforward to Z is a signed rational divisor G_Z supported on D_Z, with

We next make a small klt perturbation while keeping a fixed Cartier multiple of M. Choose an integer \ell>0 for which \ell M is Cartier, and fix a rational number

The klt adjoint

is pseudo-effective because K_Z is pseudo-effective and (1-\delta)D_Z is effective. Run its minimal model program. We show inductively that every step is crepant for M, preserves nefness of M, and preserves the Cartier property of \ell M.

At a given step let R be a negative extremal ray for the driving adjoint. Since M is nef and \delta<1/2, the ray is also negative for the klt adjoint M-D_Z/2. The length bound gives a rational curve C spanning R such that

Set a_C=M\cdot C and b_C=D_Z\cdot C. Then

It follows that

Because \ell M is Cartier, a_C\in\ell^{-1}\Z, and therefore a_C=0.

The line-bundle clause of the contraction theorem now descends \cO_Z(\ell M) to an actual line bundle on the contraction base [12, Theorem 1.1(4)(iii)]; the same theorem’s part (5) gives the length bound just used. For a divisorial contraction its pullback is the original bundle. For a flip it pulls back on both sides. Consequently M remains nef, the same multiple \ell M remains Cartier, and its pullbacks to a common resolution agree. This is crepancy for the full adjoint, rather than numerical triviality alone. More explicitly, choose a rational section of the descended line bundle. The divisor \ell M differs from its pullback divisor by a principal divisor; transform the same rational function across the flip. The two exact divisor pullbacks then agree on a common resolution, with compatible canonical divisors. The driving pair stays klt. The boundary with coefficient 1/2 is smaller than the driving boundary, so it is again klt at the next step. The preceding argument therefore repeats with the same \ell and \delta.

Let Z' be the resulting model, and denote strict transforms by D', M', and G'. Put K'=K_{Z'}. We have

Here the last nefness assertion follows by interpolation between M' and M'-\delta D'. The klt assertion follows by interpolation between the klt driving pair and the lc full-boundary pair. The underlying variety Z' is klt.

We have reduced the problem to a fixed nef interval of actual klt adjoints and a signed representative supported on its boundary. We next use that representative to locate a nonempty boundary floor.

A nonempty floor and the reduced non-klt ideal

Some coefficient of G' is positive. Suppose otherwise. Intersect K'+D'\sim_{\Q}G'\leq0 with the third power of an ample divisor. Pseudo-effectivity of K' and effectivity of D' show that all inequalities are equalities; positivity for nonzero effective divisors then gives D'=0 and G'=0.

To see why this is impossible, take a common smooth resolution V of W and Z'. The pullback to V of the signed divisor J is supported over D_W. Since the programs extract no divisors and D'=0, this pullback is exceptional over Z'. It is rationally linearly equivalent to the pullback of K_Y, and hence is nef. The negativity lemma makes this exceptional divisor nonpositive. Its intersection with an ample divisor to the third power is nonnegative by nefness and nonpositive by its sign; it must vanish. Thus the pullback of K_Y is rationally linearly trivial, contrary to \kappa(Y,K_Y)=-\infty.

Write D'=\sum_j D'_j and G'=\sum_j g_jD'_j, allowing g_j=0. Let a=\max_j g_j>0. Choose a sufficiently small positive rational s and define

Choose s so that every coefficient 1-s+(s/a)g_j is nonnegative and s/\alpha\in(0,\delta). All these coefficients are at most one, with equality precisely when g_j=a. Therefore

The divisor N is nef by Equation (5.5). The pair (Z',B) is lc and klt outside S. Indeed, outside S its finitely many boundary coefficients are bounded above by a number less than one; interpolate the lc full-boundary pair with the underlying klt variety. The same reasoning shows that (Z',bB) is klt whenever 0<b<1.

The multiplier ideals are actual ideals

Here S carries its reduced induced structure. To verify the first identity, use the discrepancy description on a log resolution. Since the pair is lc, a regular function satisfies every positive log-discrepancy condition automatically. At a log-discrepancy-zero divisor it must vanish to positive order. All such centers lie in S, and each component of S itself gives such a condition. A function vanishing on reduced S has positive order at every valuation centered in S; conversely the component valuations force vanishing on S. This proves the identity. These are multiplier ideals for an lc pair, not a substitution of a non-lc ideal for the reduced ideal.

Lifting sections from the floor

We claim that, for every sufficiently large and divisible integer m, the following restriction map to the whole reduced floor S is surjective:

The interior injectivity theorem will compare the boundary B with b_0B for b_0<1, where the pair is klt. It also requires an endpoint b_2B beyond B. The corresponding residual bundle is (m-1)N-(b_2-1)B; its positivity is supplied by the interval Equation (5.5). We now verify this for both endpoints.

Fix rational numbers 0<b_0<1=b_1<b_2 and a projective log resolution h:R\to Z' of (Z',D'), hence also of (Z',B). Choose a fixed effective integral h-exceptional divisor A_R, with coefficients large enough that

Their supports lie in one simple normal crossings divisor, and C_0\leq C_1\leq C_2. Moreover C_1 is the strict interior convex combination of C_0 and C_2 with parameter (1-b_0)/(b_2-b_0). For mN Cartier put

This is an integral divisor. Its endpoint residuals are

For i=0,2, t_i(m)>0 once m is large, and

Consequently each endpoint residual is rationally linearly equivalent to a positive rational multiple of h^*(K'+(1-u_i(m))D'), a nef klt adjoint pulled back to its log resolution. By Theorem 4.1, it admits a semipositive singular metric with zero Lelong numbers at every point. Taking positive rational powers and transporting by the actual rational line-bundle isomorphisms preserves this property. The endpoint hypotheses of Theorem 4.2 are therefore satisfied.

That theorem gives injectivity from the first ordinary cohomology group of K_R+L_R-\floor{C_1} to that of K_R+L_R-\floor{C_0}. The precise floor calculation is

Multiplier-ideal local vanishing and the projection formula [12, Theorem 3.3] identify these groups with

The comparison map is the map induced by inclusion of the multiplier ideals. By Equation (5.7), we have proved injectivity of

The exact sequence of S proves Equation (5.8).

The restriction map is now surjective. To obtain a nonzero global section, we must produce one on the whole reduced scheme S. We do this on a dlt floor using adjunction and threefold abundance. The next step therefore includes both gluing on that floor and actual line-bundle descent to S.

Whole-floor abundance and descent

Take a projective \Q-factorial dlt blowup

extracting only lc places; this is the lc case of the dlt blowup theorem [14, Theorem 2.10]. Set T=\floor{B_Q}. It is nonempty. Whole-floor dlt adjunction equips T with an effective rational different \operatorname{Diff}_T such that

as the actual rational adjoint line bundle, including the conductor and residue identifications on the normalization. The pair on T is semi-dlt; see [10, Remark 1.2(3)] and [13, Remark 2.7].

We record the depth condition needed for this whole-floor statement. The underlying Q is klt and \Q-factorial. Both \cO_Q and \cO_Q(-T) are Cohen–Macaulay. For the latter assertion, locally trivialize a Cartier multiple of T and take the normal cyclic index cover. It is finite and quasi-étale, hence klt and Cohen–Macaulay. Its finite pushforward decomposes into the corresponding rank-one reflexive sheaves, including \cO_Q(-T); in characteristic zero these are direct summands. They are therefore Cohen–Macaulay. The sequence

then gives S_2 for T (in fact the required Cohen–Macaulay depth along its support). Its codimension-one crossings and the conductor description come from the dlt structure. Thus the semi-log-canonical adjunction in Equation (5.11) is on the whole projective threefold, not on a disjoint collection of its components.

The adjoint in Equation (5.11) is nef. Projective semi-dlt threefold abundance makes it semiample [10, Corollary 4.10]. Hence

for all sufficiently divisible positive m.

These sections descend to S. Indeed \cJ(Q,B_Q)=\cO_Q(-T), and crepancy together with multiplier-ideal local vanishing gives

One can compute both assertions on a common log resolution of (Q,B_Q) and (Z',B). The equality of their pulled-back adjoints identifies the discrepancy round-up sheaves, and local vanishing for each log resolution followed by Leray gives Equation (5.12). Since Z' is normal, g_*\cO_Q=\cO_{Z'}. Pushing forward the divisor sequence therefore gives

The projection formula identifies the nonzero section spaces above with H^0(S,\cO_S(mN)). Choose m both sufficiently divisible for this semiampleness and sufficiently large for Equation (5.8). A nonzero section on T then descends to S and lifts to Z'. We have proved \kappa(N)\geq0.

The final klt adjoint

Because B\leq D', section inclusion in divisible degrees gives \kappa(N)\leq\kappa(M')\leq0. Thus \kappa(N)=0. Now consider the klt pair

Its nef adjoint is rationally linearly equivalent to N/\alpha, so its Kodaira dimension is zero. By Theorem 4.1, on a smooth projective log resolution a positive Cartier multiple of its pullback has a semipositive singular metric with zero point Lelong numbers. All hypotheses of the fourfold semiampleness result of Gongyo–Matsumura are now satisfied: the pair is projective klt with rational boundary, the adjoint has nonnegative Kodaira dimension, and the specified pullback metric exists [17, Corollary 5.3].

Semiampleness and Kodaira dimension zero make this adjoint numerically trivial. On the other hand, K' is pseudo-effective, D'\ne0 because G' has a positive coefficient, and 1-s/\alpha>0. Intersecting K'+(1-s/\alpha)D' with the third power of an ample divisor gives a strictly positive number. This contradiction proves the proposition.□

Proof. Let p:\widetilde Y\to Y be a smooth projective resolution. Rational singularities and the irregularity conclusion in Proposition 2.1 give q(\widetilde Y)=0. The product description of the Picard group, or the seesaw principle and the vanishing of \Hom(\Alb(\widetilde Y),\Pic^0(\widetilde Y)), gives

For a prime Weil divisor on Y\times Y, take its strict transform on this smooth product. Its Cartier class is a sum of pullbacks from the two factors. Pushing down the corresponding linear equivalence expresses the original Weil divisor class as a sum of pullbacks of Weil divisor classes on Y. These are \Q-Cartier because Y is \Q-factorial. Thus Y\times Y is \Q-factorial. Exceptional components have images of codimension at least two and do not affect this divisor-class calculation.

The bundle over \widetilde Y\times\widetilde Y is a smooth resolution of Z. Its Picard group is the Picard group of the base plus the integral tautological class. Pushing down as above proves \Q-factoriality of Z. The identical argument over \widetilde Y proves it for the slice. A product of canonical varieties is canonical, as follows directly from the product of resolutions and the canonical discrepancy formula. Projective bundles are smooth over their bases, so the bundles are klt as well.

The direct sum of nef line bundles is nef; hence its tautological class is nef. That class is also relatively ample. Adding a positive pullback of an ample class therefore makes it ample: for example, write the sum as a positive multiple of a relatively ample class made ample by a sufficiently large base twist, plus nef classes. This proves the assertions about P and P_{\mathrm{sl}}. Finally, the canonical bundle formula for a rank-two projective bundle gives the equality in Equation (6.5). Since \xi is nef and K_\Sigma\le L_\Sigma/r_t, the stated nef inequality follows.□

Proof. For a rank-two split bundle with first Chern classes E_1,E_2, the projective-bundle formula is

All classes in the expansion of P^d are nef. Put L_\Sigma=L_1+L_2. Each E_j=S_{0,j} is bounded above in nef order by (\iota/r_t)L_\Sigma. Consequently

Here L_\Sigma^{2n}=\binom{2n}{n}(L^n)^2, and L^n\to\epsilon^n>0. This proves Equation (6.6).

On a slice one summand is trivial, so

For small t, we may use L^n\le2\epsilon^n, giving Equation (6.7). Positivity follows either from ampleness or from the i=1 term.□

Proof. We first rule out branch divisors that sweep Y, using the intersection bounds uniformly in t. We then fix t and the family: a common branch complement gives finitely many covers, whose birational graph families lead to an abelian variety dominating Y.

Since P-g_i^*L is nef, the projection formula gives

The degrees are uniformly bounded because L^n\to\epsilon^n>0. This degree bound alone does not give a finite list of covers; we must first control their branch loci.

Numerical bounds for branch divisors. Resolve the general members of the incidence family in family. After shrinking the irreducible parameter space, there is a smooth projective family \mathcal V^*\to T with integral fibers birational to the corresponding V, together with the two generically finite morphisms g_i:V^*\to Y on each fiber. For either projection, let T_i\to Y denote the finite normal Stein factor. Its function field is \C(V).

If J is a branch prime of T_i\to Y, a ramification prime on T_i has a strict transform R on V^*. A proper birational morphism to a normal variety is an isomorphism at the generic point of each target divisor, so this transform is present. Over the smooth locus of Y, its ramification coefficient is at least one. There is an effective canonical comparison

For completeness, resolve the rational lift to a resolution of the terminal target. Ramification between smooth varieties and the effective terminal discrepancies give this formula upstairs; pushing down to V^* gives Equation (6.12).

Set d_J=L^{n-1}\cdot J. Projection and nef order imply

where C_3=B_1B_0 is independent of t and of the family. The same estimate applies to every prime divisorial image of ramification that meets Y_{\mathrm{sm}}.

We also need a canonical bound on J, which need not be normal. On Y_{\mathrm{sm}}, the divisor J is a Gorenstein hypersurface. If \nu:J^\nu\to J is its normalization, finite duality gives, in codimension one,

The conductor therefore has the sign that decreases the normalized canonical intersection. Terminal singularities are smooth in codimension two, so J\cap Y_{\mathrm{sing}} has dimension at most n-3. Its inverse image on the normalization has codimension at least two. It introduces no omitted codimension-one term in this comparison. On a resolution J^*, the exceptional canonical cycles push to zero against n-2 pullbacks of L. Consequently

Here K\le L/r_t. The square term is the Hodge-index inequality

It applies to the \Q-Cartier divisor J; one can pull back to a resolution and approximate the nef class L by ample classes. The bound in Equation (6.13) and the positive lower bound for L^n make C_4 uniform.

If a family of these prime images dominates Y, choose a very general member and then a very general point on its resolution mapping to Y^{\mathrm{vg}}. Such a choice is legitimate despite the countable exceptional set. For each excluded proper closed subset of Y, dominance says that the generic image divisor is not contained in it; omit the corresponding proper parameter locus. On the remaining very general member, omit the inverse images of the countably many excluded subsets, together with the finite birational-system and quasi-finiteness exceptions. The chosen image divisor is of general type by Proposition 2.2. Its top L-degree is d_J, and Equation (6.14) is the required canonical bound. Lemma 3.4 excludes this sweeping family for sufficiently small t.

A fixed branch complement and finitely many covers. The preceding exclusion holds uniformly over all families in the statement. Now fix one such sufficiently small t and its incidence family. No further decrease of t will be made in this proof. The smooth projective family \mathcal V^*\to T has integral fibers, and both projections to the normal variety Y are dominant and generically finite. Their degrees are bounded by Equation (6.11). After a finite parameter extension and shrinking, follow the finitely many geometric generic components of the relative Jacobian over Y_{\mathrm{sm}}. Every followed component whose fiber image is a divisor has proper total image closure: the preceding argument excludes a dominant such family using Equations (6.13) and (6.14).

These are exactly the hypotheses of the simultaneous finite-cover lemma [38, Lemma 7.5]. It gives one smooth nonempty open Y^\circ\subset Y over which every general finite Stein cover in either slot is finite étale, and finite lists of the resulting normal covers of Y. The lemma accounts for every branch prime by its ramification strict transform, then applies purity and finiteness of bounded-degree étale covers of the fixed open. In particular, this is a finite-type family argument; a numerical branch bound alone would not suffice. The open and lists may depend on the fixed t and incidence family. No uniformity in those choices is needed.

Countably many birational graph families. Each V gives a birational map between one cover from each list: both finite Stein factors have function field \C(V). Its image in Y\times Y is the finite image of the closure of that birational graph. Fix one pair of covers that are birational. A smooth projective resolution of their common birational class has pseudo-effective canonical class, by the effective canonical comparison with the nef K on Y. It is non-uniruled by [5].

The birational-group structure theorem of Hanamura [19], in the form stated in [4, Proposition 3.7 and Theorem 3.8], permits a smooth projective model U of this class for which the reduced flat-graph birational scheme is a group scheme locally of finite type and

is its identity component, an abelian variety. The flat graphs are represented in graph Hilbert schemes. There are countably many Hilbert polynomials and finitely many components in each finite-type Hilbert scheme, so the birational maps occur in countably many translates A_U b.

For such a translate, its graphs are parametrized on a dense open by the rational evaluation map

If this map is not dominant, its image closure is a proper closed subset of U\times U. Fix birational identifications of U with the two covers. Every graph meets the product of their fixed birational opens densely, because it dominates both factors. Transporting the image closure through those opens, then taking its closure in the product of covers, preserves its dimension. The subsequent finite map to Y\times Y also preserves dimension. Thus all correspondences belonging to this translate are contained in a proper closed subset of Y\times Y. The possible boundary of an individual birational map introduces no new component: the generic graph has already been included.

There are finitely many cover pairs and countably many translates for each pair. If every map in Equation (6.15) were nondominant, the image of the original incidence would be contained in a countable union of proper closed subsets of Y\times Y. But \Gamma\to Z is dominant, so its composite to Y\times Y is dominant and its image contains a nonempty open subset. Over the uncountable field \C, such an open cannot be covered by countably many proper closed subsets. Therefore the map in Equation (6.15) is dominant for at least one translate.

The abelian curve contradiction. For a general x in this last evaluation, A_U\cdot b(x) is a dense orbit in U. A stabilizer element fixes every point of that orbit by commutativity; it fixes U by density. The action of \Aut^0(U) is faithful, so the stabilizer is zero-dimensional. Hence the orbit map from A_U is generically finite dominant. Composing with the rational map through either fixed cover gives a generically finite dominant rational map

In particular \dim A_U=\dim Y=n.

Resolve this map as \pi_A:T_A\to A_U and g:T_A\to Y, with T_A smooth and projective. Since the canonical bundle of A_U is trivial, and since Y is terminal, the canonical comparisons give

where E_{\pi_A} is exceptional over A_U. The images of the exceptional divisors of \pi_A have codimension at least two on the smooth abelian variety.

Choose a point in the isomorphism open of \pi_A where g is quasi-finite, outside \Supp(R), and mapping to Y^{\mathrm{vg}}. Dominance and the countable-exclusion description ensure that such a point exists. A general sufficiently ample complete-intersection curve in A_U through its image avoids all the exceptional centers: they have codimension at least two and do not contain the prescribed point. Its strict transform C is disjoint from E_{\pi_A} and is not contained in R. The map g is nonconstant on C, since it is quasi-finite at the chosen point. Intersecting Equation (6.16) with C gives

The final strict inequality uses K\cdot g(C)\ge1/\iota, the defining curve property of Y^{\mathrm{vg}}, multiplied by the positive degree of C\to g(C), together with R\cdot C\ge0. This contradiction excludes the remaining correspondence family.□

Proof of Proposition 6.3. All constants used to exclude moving centers below are independent of the section degree and of small t, once \epsilon,\delta,q have been fixed. We first produce such a center from failure of Equation (6.10); we then exclude each possible dimension of its fibers over the two factors.

A moving component from failure of the estimate. Suppose that Equation (6.10) fails at very general torus points. At such a point there is a curve whose ratio is strictly less than 2n+3. This conclusion does not require the infimum defining the Seshadri constant to be attained.

Take d+1 levels

By Equation (6.9), ample Hilbert asymptotics and the dimension of the jet space at a smooth point show that, for large divisible k, every kernel of jets of order \ceil{k\tau_j} is nonzero. Explicitly, the leading coefficients of the two dimensions are P^d/d! and \tau_j^d/d!. If a section in the lowest kernel does not contain the curve just chosen, its local intersection with that curve at the marked point is at least \ceil{k\tau_0}\mult_z C. This exceeds kP\cdot C. Thus every section in that kernel contains the curve, and the lowest base locus has a positive-dimensional component through the mark.

These conditions produce a dominant marked-point family in the sense of Lemma 3.3. For generality, choose the marked point outside the proper rank-jumping and base-dimension loci for all divisible degrees; there are only countably many such conditions. For the degree k now chosen, restrict to the open where the finitely many jet kernels have constant rank. The dimension of the base scheme at its marked point is upper semicontinuous. The existence of the witnessing curve at very general marks therefore forces the positive-dimensional-base condition on a dominant locus; no family of the witnessing curves is assumed. The finitely many generic base components and their inclusions can then be followed after a finite parameter extension and shrinking.

We may increase the divisible integer k to arrange, in addition, that all divisors

are Cartier and globally generated. Here is one simultaneous verification. For 1\le a\le n, subtract aA and then K. The resulting class is

which is ample once kr_tt>n and kr_t\ge1. Klt Kawamata–Viehweg vanishing followed by regularity with respect to A proves global generation for every j in Equation (6.17).

The moving-multiplicity lemma supplies an irreducible parameter space T, a dominant incidence \Gamma\subset T\times Z, and integral fiber components V=V_u of a fixed dimension 1\le f<d. The high-kernel subseries has V as an isolated base component generically, with ordinary generic order at least

For a smooth resolution V^*, the base-component estimates and Equation (6.5) give

The right sides are bounded uniformly in the specified sense.

We first exclude positive-dimensional slot fibers. The chosen incidence point will ensure that all image subvarieties below meet the prescribed very general locus of Y. For an intermediate fiber F in a slice, let W\subseteq Y be its image and write s=\dim F, w=\dim W. The three possibilities are summarized below; the subsequent paragraphs prove all the indicated exclusions.

A full slot fiber is excluded by exchanging slots. This leaves generically finite projections. Proper images are again excluded by general type, and dominant images are excluded by Proposition 6.4.

Restriction to a slot fiber. Consider V over either factor of Y\times Y. Choose a very general incidence point first. We require that its two base coordinates belong to Y^{\mathrm{vg}}, that it lies in the torus, and that the ordinary ideal-power and isolated-component conclusions above hold near it. We also impose the smoothness conditions for the incidence over T and for V over the smooth open of its actual slot image. These are available on dense opens.

Now take the slot value of this point, and let F be the reduced fiber component through it. Write s=\dim F, the general fiber dimension. Suppose first that

The restricted high-kernel subseries on the opposite slice is nonzero and has F as an isolated base component generically. Indeed, near the selected point the original base support is just V. No other base component can contain the fiber component through that point. The scheme fiber over the ambient slot agrees with the scheme fiber over the actual image, and smoothness over that image makes it reduced at the selected generic component. Thus the image of \cI_V^h in the slice is contained in \cI_F^h, as an ordinary ideal power. A restriction that vanished identically on the whole slice would contradict this local proper base support.

Applying Lemma 3.2 in the slice gives

Let W\subseteq Y be the image of F in the other base, and put w=\dim W. It meets the prescribed very general locus. Since the bundle fiber has dimension one, either w=s or w=s-1. We treat every possibility.

A generically finite proper image. If w=s<n, the resolution of W is of general type by Proposition 2.2. The same holds for F^*: after resolving the generically finite map to a resolution of W, ramification adds an effective divisor to the pulled-back canonical divisor. The polarization P_{\mathrm{sl}} dominates the pullback of L. Equations (6.20) and (6.21) therefore contradict Lemma 3.4 for sufficiently small t.

The whole bundle over a proper image. Suppose s=w+1, where necessarily w<n. A positive-dimensional closed subset of the generic \PP^1 fiber is the entire fiber. Consequently

as reduced integral subvarieties. Nef intersection and the projective-bundle formula give

For w=0, the left side is q. The right side is at most

by Equation (6.8), a contradiction.

Assume w>0. We will apply the degree obstruction to W, rather than to the ruled variety F. Expand the restricted high sections using the two homogeneous fiber coordinates. Their coefficients of weight j, for 0\le j\le kq, are actual sections of

Let \mathfrak a be the ideal generated by all these coefficients in the regular local ring

This ring is regular because the generic point of W lies in the smooth locus of Y.

We need both the order and the support of this coefficient ideal. After choosing frames, the local ring of the slice at the generic point of F is

The residue of u is transcendental over the residue field of R. If a coefficient had order less than h, the least \mathfrak m-adic homogeneous part of its polynomial would remain nonzero after adjoining that residue variable. Therefore membership of every high section in \mathfrak m^hR[u]_{\mathfrak m R[u]} implies \mathfrak a\subseteq\mathfrak m^h. Moreover, if a prime \mathfrak p\subsetneq\mathfrak m contained \mathfrak a, its extension to this local polynomial ring would be a proper prime strictly below the generic ideal of F containing the whole high base ideal. This contradicts the isolated support of F. Hence

Multiply every weight-j coefficient by the globally generated space in Equation (6.17). The products all belong to one linear subseries of

At \eta_W, each filler system generates a unit after choosing a frame. Thus this new subseries has precisely the joint ideal \mathfrak a, not merely an ideal with the same support. Its threshold construction in Lemma 3.2 gives an effective rational divisor

such that the pair is lc at the generic point of W and has an lc place centered there. Apply Lemma 3.1 with the testing class L. Since qS_0\le L in nef order, we obtain

The top degree L^w\cdot W is bounded by Equation (6.22) and Equation (6.20). The variety W^* is of general type. Equation (6.24) and the top-degree bound now contradict Lemma 3.4 on W^*.

A non-axis multisection over all of Y. The only remaining intermediate-fiber case is s=w=n. Here F is a multisection of positive degree e over Y. It is not an axis, since it contains the selected torus point. Let \Sigma_0,\Sigma_\infty be the axes on the slice, with classes

The intersections with F are effective integral cycles. Their pushforwards define effective integral Weil divisors D_0,D_\infty on Y, allowing a divisor to be zero. Projection and rational equivalence of cycles give

This is linear equivalence of Weil divisors: codimension-one rational equivalence on the normal variety Y is precisely linear equivalence. In particular,

is an actual signed rational representative.

For either axis, P_{\mathrm{sl}}-q\Sigma is nef: it is L or L+qS_0. As F\cap\Sigma is effective and P_{\mathrm{sl}}\ge\pi_{\mathrm{sl}}^*L in nef order,

Take a log resolution p:Y^\dagger\to Y of D_0+D_\infty, and let G be the reduced snc divisor consisting of their strict supports and all exceptional divisors. By Proposition 5.1, K_{Y^\dagger}+G is big. The exceptional terms push to zero against L^{n-1}, while the reduced strict support is bounded by the integral divisor D_0+D_\infty. Hence

The second term is uniformly bounded by Equation (6.26) and Equation (6.20). Also L^n is bounded above and bounded away from zero. Thus Equation (6.27) is the canonical-to-volume bound in the log version of Lemma 3.4, with polarization p^*L and map p. That lemma gives the contradiction in this final intermediate-fiber case.

Reduction to finite correspondences. We have excluded all slot-fiber dimensions 1,\ldots,n. A fiber dimension n+1 over one slot image W would force V to be the entire bundle over W\times Y. Since V\ne Z, one has \dim W<n; its fiber over the other slot then has dimension \dim W+1, which has just been excluded. It follows that both projections of V to Y are generically finite onto their images.

If one image were proper, it would have dimension f<n and be of general type. Its generically finite cover V^* would also be of general type, and the total bounds Equations (6.18) to (6.19) would contradict Lemma 3.4. Therefore

are generically finite dominant morphisms. The bounds Equations (6.18) to (6.19) supply constants B_0,B_1 independent of small t, of the section degree k, and of the family. We are therefore in the situation excluded by Proposition 6.4.

Every possibility for a moving component has now been excluded. The slice degree obstructions and Proposition 6.4 impose restrictions on t independent of k and of the moving family. The assumed failure of Equation (6.10) is therefore impossible for all sufficiently small t, as required.□

Proof. Choose a very general smooth torus point z of Z above the isomorphism open of p in both slots and satisfying Equation (6.10), so that \varepsilon(P;z)>10. If b:\widehat Z\to Z is its blowup and E_z is the exceptional divisor, the Seshadri criterion shows that the following rational divisor is ample:

Indeed one may choose a rational number strictly between 10 and \varepsilon(P;z), and then use the usual ample interval below the Seshadri threshold. Choose a>0 so that aP and a(b^*P-10E_z) are Cartier. Serre vanishing, applied to the fixed sheaf \cO_{\widehat Z}(-E_z), gives

for all sufficiently large multiples k of a. The standard local calculation for the blowup of a smooth point identifies the pushforward of \cO_{\widehat Z}(-(10k+1)E_z) with \mathfrak m_z^{10k+1}, with vanishing higher direct images. The restriction sequence therefore gives surjectivity onto jets through order 10k at z.

The natural morphism Z_W\to Z is an isomorphism near the lift z_* of z. Pullback preserves the sections in this jet surjection and identifies their jets. Increase the divisible integer k if necessary so that kL is Cartier, and call it k_0. Choosing finitely many inverse images of a basis of the finite-dimensional jet target proves the assertion.□

Proof. If \pi_x^*D^+-40\epsilon J_x were pseudo-effective, then for any rational c with 32\epsilon<c<40\epsilon the divisor \pi_x^*D^+-cJ_x would be big: it is a positive convex combination of the preceding pseudo-effective divisor and the big divisor \pi_x^*D^+. A sufficiently divisible effective multiple would contradict Equation (7.5). Thus \pi_x^*D^+-40\epsilon J_x is not pseudo-effective.

Movable-curve duality [5, Theorem 2.2] gives a strongly movable class with negative pairing against that divisor. Since the inequality is strict, we may take an actual pushforward of an ample complete intersection on one smooth birational projective model, approximating the ample classes rationally and clearing their denominators if needed. The big divisor \pi_x^*D^+ has positive pairing with this nonzero class: write it as an ample class plus an effective class and pull back to the chosen model. The strict inequality therefore implies J_x\cdot\gamma>0.□

Proof. Fix the equations y_i=0 at all coordinates vanishing in the given real solution. Together with the displayed equations they define a nonempty affine space over \Q. Gaussian elimination over \Q gives a rational particular solution and a basis over \Q for its direction space. Thus its rational points are dense in its real points. Approximate the given point closely enough to keep all of the finitely many remaining coordinates y_i strictly positive. This proves the first assertion, including the case where all y_i vanish or the affine space has dimension zero.

For the divisor assertion, choose a finite expression

List the prime divisors in the supports of D,G, and these finitely many principal divisors as P_1,\ldots,P_w. Write D=\sum d_iP_i and let m_{ij}\in\Z be the coefficient of P_i in \Div(f_j). The coefficients of G and the a_j give a real solution of (8.1). A rational solution gives G_{\Q}=\sum y_iP_i\ge0 with the same support and D-G_{\Q}=\sum b_j\Div(f_j).

Choose a positive integer N divisible by r and by the denominators of all y_i,b_j. Then

The effective integral divisor NG_{\Q} is Cartier, since ND is Cartier and their difference is principal. It defines a nonzero section of \cO_X(ND). Write N=mr. No \Q-factoriality is needed: all equalities were equalities of Weil divisors on X.□

Proof of Corollary 1.2. Hashizume’s reduction [20, Theorem 1.4] takes smooth canonical nonvanishing in dimension four, supplied by Theorem 1.1, and gives nonvanishing for projective lc pairs with effective real boundaries and pseudo-effective real Cartier adjoints in dimensions at most four. Applied to the given rational lc pair, it produces D\sim_{\R}G\ge0 on the original normal variety; nefness implies pseudo-effectivity. Lemma 8.1 gives a nonzero section of mrD for some positive integer m, with the prescribed index r.□