Projective Hodge lines and ordinary Iitaka subadditivity

Proof. In a polarizable rational Hodge structure, intersect all rational Hodge substructures containing the specified line. A finite intersection already attains the minimum dimension, and gives the unique smallest one, say W. Semisimplicity shows that W is simple: in a decomposition into simple summands exactly one summand can have a nonzero deepest Hodge piece, since that piece has dimension one, and that summand already contains the line.

Here is also the generic-fibre justification, which is useful when passing between variations and individual Hodge structures. Trivialize the rational local system on the connected universal cover of U. For each rational endomorphism e, the condition that e preserve every F^p is closed analytic. Choose a lifted point outside all such loci which are proper. A rational Hodge projector onto W at this point therefore preserves the filtration everywhere on the cover. Its action on the Hodge line is the identity everywhere: it is an idempotent on a line bundle, and equals the identity at the chosen point. Its constant image is thus a subvariation containing the line. At every other point chosen in the same way, applying the argument in both directions shows that its image is again the smallest Hodge substructure containing the line. Uniqueness and the deck action now imply monodromy invariance, so this subvariation descends to U. Any rational subvariation containing the line contains it, by inspection at the chosen point.

Restriction of Q to a Hodge substructure is nondegenerate, and V^{\min}\cap V_{\Z} supplies the lattice. The polarization gives a complementary subvariation whose deepest Hodge piece is zero. Canonical extensions preserve this direct sum, proving the last claim. This is the pure case of the transcendental-part construction of [5, Definition 2.9].□

Proof. The monodromy and fixed-part theorems give that G is semisimple and normal in the connected reductive group P; see [3, §5, Theorem 1 and Corollary 1]. Over an algebraic closure, a connected normal subgroup of a semisimple group is the product of a collection of its almost simple factors. Conjugation by P preserves G. Since P is connected, it acts trivially on the finite set of these factors, and therefore preserves H.

Consequently V^H is a rational P-subrepresentation. It is a rational Hodge substructure at a Hodge-generic point, and it contains the deepest Hodge line. Minimality gives V^H=V. The inclusion G\subset\mathrm{GL}(V) is faithful, so H=1.□

Proof. Write \rho for monodromy based at e. Flatness and b(e)=e give

Thus a normalizes both algebraic monodromy and its identity component G. Let \lambda be the eigenvalue in the hypothesis. The condition that the deepest line lies in \ker(a-\lambda) is a holomorphic linear condition on the universal cover, so it holds throughout that connected cover by analytic continuation.

Fix a Hodge-generic lifted point and let W\subset V_{\C} be the complex span of the monodromy orbit of its deepest line. This space is monodromy invariant, hence G-invariant. Equivariance places every line in this orbit among the deepest lines on the universal cover, so a|_W=\lambda\,\mathrm{id}_W. Every element aga^{-1}g^{-1}, with g\in G, therefore acts trivially on W, as do all its G-conjugates. Let H be their normal algebraic closure in G. It is defined over \Q, since a and G are rational. It is connected: the commutator map from the connected group G has connected image containing the identity, and the algebraic subgroup generated by its conjugates is the closure of products of such connected sets. Although W need not be rational, V^H is rational and contains the chosen deepest line. Lemma 2.2 gives H=1. Hence a centralizes G.

Pass to the connected finite étale cover U_1\to U corresponding to the inverse image of G under monodromy, and choose e_1 over e. Since a centralizes all monodromy on U_1, it extends to a global flat rational section of \End(\mathbb V|_{U_1}). The latter is a polarizable pure variation of weight zero. The flat section a has Hodge type (0,0) at e_1, since \Phi_e is a Hodge isomorphism. The base U_1 is a smooth algebraic variety and hence a Zariski open subset of a compact complex variety, so the fixed-part theorem applies. By [25, Theorem 7.22 and Corollary 7.23], a has type (0,0) everywhere. Thus a preserves the full filtration on the universal cover.

For completeness, take the lift of b to the universal cover fixing a chosen lift of e. Flatness of \Phi gives F^\bullet(b\widetilde y)=aF^\bullet(\widetilde y) in the reference space. The assertion just proved makes the right-hand side equal to F^\bullet(\widetilde y).□

Proof. Put t=\dim T, and fix an ample divisor H on T. For 1\le a\le t, choose a sufficiently general smooth complete intersection C_a\subset T of dimension a, using a sufficiently large multiple of H. It meets the boundary transversely. At general points of C_a\cap U, the restriction of d\ell to T_{C_a} has rank \min(a,r). To see the required genericity, at a point where d\ell has rank r, the a-planes with this property form a nonempty open subset of the tangent Grassmannian. A sufficiently ample linear system prescribes the required first-order tangent conditions; Bertini then gives the assertion for general complete intersections. By Theorem 3.1, J|_{C_a} is big exactly when a\le r. Since it is nef, this is equivalent to (J|_{C_a})^a>0, and hence to c_1(J)^aH^{t-a}>0. This proves the formula, also when r=0; dimension zero is immediate.□

Proof. Put a_0=a(0,v). In the Deligne bigrading of the limiting mixed Hodge structure, F^m is a single one-dimensional summand I^{m,q}: all first indices are at most m, and \dim F^m=1. Set \ell=m+q-w. The primitive decomposition on weight w+\ell expresses each summand as a sum of terms

where P_{w+b} denotes the primitive part for b\ge0. A contribution to Hodge type (m,q) with j>0 would come from type (m+j,q+j), impossible because its first index exceeds m. Thus j=0, \ell\ge0, and the top line is primitive. Hard Lefschetz for the monodromy weight filtration gives nonvanishing of its image under N^\ell and vanishing under N^{\ell+1} on the associated graded. Since N has bidegree (-1,-1) in the Deligne bigrading, these statements hold for a_0 itself: a vector in a single Deligne summand maps injectively to its weight-graded piece. The integer \ell is constant on a sufficiently small open stratum, so the identities hold throughout the chosen v-chart.

Let J_E^{\mathrm{gr}} be the extended top line of the pure variation on \Gr^{W(N)}_{w+\ell} along E^\circ. The boundary extension input identifies it with J|_E. Projecting N^\ell a_0 to \Gr^{W(N)}_{w-\ell} yields the image of this graded top line under the flat isomorphism

Consequently the projective map of the graded line has rank no greater than v\mapsto[N^\ell a_0(v)]: projection is a fixed linear map on the flat chart, defined near the line in question. Apply Proposition 3.2 to the graded variation on E^\circ and its smooth projective compactification E. This proves (3.4).□

Proof. Use a common coordinate chart in M, shrinking B and the sector. Cauchy estimates give convergence of all derivatives in the B variables on smaller polydiscs. The (r+1)-minors of df vanish identically, so those of df_0 vanish as well. Thus f_0 has constant rank r near v_0. The holomorphic constant rank theorem supplies source coordinates v=(x,y)\in\C^r\times\C^{n-r} and target coordinates in which f_0(x,y)=(x,0).

Write f=(F,G) in these coordinates, with F(u,x,y)=x+E(u,x,y). On fixed smaller polydiscs, both E and \partial_xE tend uniformly to zero. Choose nested x-balls, a smaller ball W for w, and a ball Y for y. For sufficiently small \epsilon', the maps

send a fixed closed x-ball into itself and have Lipschitz constant less than 1/2, uniformly for (u,w,y)\in S_{\epsilon'}\times W\times Y. The contraction theorem gives a unique solution x=\chi(u,w,y) of F(u,x,y)=w on this common product. The holomorphic inverse function theorem and uniqueness make \chi holomorphic in all interior parameters; moreover \chi\to w uniformly on smaller products.

In the new source coordinates the map is

The w-derivatives already have rank r. The rank bound therefore forces \partial_uH=0 and \partial_yH=0: either derivative, if nonzero, would give an additional image vector with zero first r coordinates. Since the sector and Y are connected, H depends only on w. Its limit as u\to0 is zero, hence H=0. Shrinking the original source neighborhood so that its F-values belong to W proves the assertion. If r=0, the rank bound directly makes f constant on the connected source, and the same conclusion follows.□

Proof. Choose k with N^{k+1}=0, and write a(u,v)=a_0(v)+u q(u,v). Set b(v)=N^\ell a_0(v)/\ell!. Direct expansion gives

For every fixed tangential derivative, uniformly on compact subsets of B, the remainder is

the first term is absent when \ell=0. On the sector |z| is comparable to |\log|u||, so both terms tend to zero. Choose a linear functional nonzero on b(v_0) and shrink the v-chart. Dividing by that functional gives convergence with all tangential derivatives in a fixed affine projective chart, proving (3.5). Finally Nb=0, so the limiting image belongs to the fixed projective linear subspace \mathbb P(\ker N), and the last assertion follows from Lemma 3.4.□

Proof. Proposition 3.2 identifies r with the generic rank of the highest-line map. We first show that this rank is positive, then consider the nilpotent and finite cases.

If r=0, the highest line is constant on the connected universal cover of U_N. Monodromy preserves it. Its connected algebraic monodromy group G is semisimple and hence acts trivially on that line. Lemma 2.2, applied with the normal subgroup equal to G, gives G=1. After a finite étale cover the monodromy is trivial, and the fixed-part theorem makes the entire variation constant. Its full period map then has rank zero, contrary to generic immersivity on the positive-dimensional variety T. Thus r>0.

A component with nonzero monodromy logarithm. Let E\subset D_N and write N\ne0 for its rational monodromy logarithm. In coordinates (u,v) near a general point of E, with E=(u=0), the lifted highest-line map has the form

Proposition 3.3 provides an integer \ell\geq0 with

Lemma 3.5 gives its normal tangential convergence on a sector from the interior line maps. All interior (r+1)-minors vanish. Cauchy convergence of tangential derivatives therefore bounds the rank of the limiting map by r.

Suppose that \nu(J|_E)\geq r. The limiting map has rank exactly r at a suitable general point, and its image lies in \mathbb P(\ker N). Lemma 3.4 forces all nearby interior lines to lie in this same linear subspace. The condition that N annihilates the highest line is holomorphic, so it continues over the connected universal cover of U_N.

The one-parameter group \exp(tN) belongs to G: it is the connected Zariski closure of the cyclic unipotent local monodromy. Every monodromy conjugate of this group fixes the highest line pointwise. Let G_N be the algebraic subgroup they generate. It is connected, defined over \Q, and normalized by monodromy; Zariski density makes it normal in G. Its invariant space contains the highest line at a Hodge-generic point. By Lemma 2.2, G_N=1, contradicting N\ne0. This proves the required inequality for components of D_N.

We have controlled all divisors at which the pure variation fails to extend. The remaining obstruction is a divisor of finite ramification where the variation does extend. Its inertia element can act on the rational fibre even when it fixes every point of the divisor, so we must use the full-period conclusion of Lemma 2.3.

A divisor fixed by finite inertia. Let b\in G_f be nonidentity and fix E pointwise. If E is contained in D_N, the preceding case applies. Otherwise a general point e\in E lies in U_N. Choose a lift h\in H of b. The retained H-action supplies a flat Hodge isomorphism

on U_N. Use V_e as a flat reference space on a small simply connected b-invariant neighborhood, and set a=\Phi_e. If x is the full lifted period map, flatness gives

The operator a is a rational Hodge isometry at e. Because H acts on the variation and b(e)=e, the group-action law gives a^{\operatorname{ord}(h)}=1. Since b fixes E, every highest line on the nearby part of E is an eigenline of a. Its eigenvalue is constant on a connected small patch, so these lines lie in one eigenspace \ker(a-\lambda).

Proposition 3.3 with N=0 bounds \nu(J|_E) by the rank of this extending line map on E.

Suppose again that \nu(J|_E)\geq r. The rank on E is at most r, since all (r+1)-minors of the ordinary extending line map vanish by continuation from the interior. It is therefore r. Apply Lemma 3.4 in a transverse coordinate with N=0. A nonempty open set of lifted highest lines lies in \ker(a-\lambda). Lemma 2.3, applied on the smooth algebraic variety U_N with fixed point e, now shows that a preserves every full period. Consequently x(by)=x(y).

Projecting to \Gamma'\backslash\mathcal D gives \varphi(by)=\varphi(y) on a nonempty open set. Since B is the normalization of the actual image, its normalization map is birational and hence injective on a dense open subset of that image. It follows that b acts trivially on \C(B). The chosen model has \C(T)=\C(B), so b acts trivially on T, contrary to its nonidentity in G_f. This proves the strict inequality.

Finally, in a finite quotient in characteristic zero, ramification at a divisorial valuation is detected by a nontrivial inertia group, which acts identically on that divisor’s function field. Its elements therefore fix the divisor pointwise. The last assertion follows from the case just proved.□

Proof. Choose an ample rational divisor H_1 such that A+D-H_1 is effective. In this proof m tends to infinity through sufficiently divisible positive integers. For each fixed j\geq0, asymptotic Riemann–Roch for the ample divisor H_1+jJ gives

for a constant c>0 and all j\geq1. The last inequality follows from J^rH_1^{t-r}>0.

Remove mD one prime divisor at a time. Each quotient in a divisor exact sequence is a line bundle on some E_i, of the form

There is a single ample rational divisor H_2 on T for which every such quotient has at most h^0(E_i,m(H_2+jJ)|_{E_i}) sections, in divisible degrees and for every j\geq0. To see this, choose an ample Cartier divisor B_0 with a globally generated multiple clearing the denominator of B_0-(A+D), and ample Cartier divisors B_h such that both B_h and B_h+E_h are globally generated. Enlarge B_0 if necessary and take H_2=B_0+\sum_h d_hB_h. The difference, restricted to E_i, is

It has a section nonzero on E_i, by global generation and multiplication, giving the claimed bound.

There are m\sum_i d_i quotient terms. For fixed j, asymptotic Riemann–Roch on each E_i bounds their total contribution on the scale m^t/t! by

This is O(j^{r-1}), since every intersection containing at least r factors of J|_{E_i} vanishes. Consequently

for all sufficiently large j, proving bigness.□

Proof. Factor \pi as T\xrightarrow{\rho}\bar S\xrightarrow{\tau}S, where \bar S is the normalization of S in \C(T). Then \rho is birational and \tau is finite. Every component of E is \rho-exceptional, since a finite map preserves dimensions. Normality gives \rho_*\mathcal O_T(mE)=\mathcal O_{\bar S} in divisible degrees: a rational function with poles only on exceptional divisors has no codimension-one pole downstairs. Projection formula identifies the sections of m(\pi^*D+E) with those of m\tau^*D. Finally, bigness is invariant under surjective finite pullback.□

Proof of Theorem 1.2. We may first make the birational modification allowed in the hypothesis. Replace \mathbb V by the minimal rational subvariation from Lemma 2.1, and equip it with the intersection lattice and the restricted polarization. Its deepest Hodge line and the line J_{\widehat Y} are unchanged. We continue to denote this variation by \mathbb V.

The actual period image and its finite quotient. Choose a rational reference space with lattice and polarization, and let \mathcal D be the connected period-domain component containing a chosen lift of the periods. Let \Gamma be the arithmetic group of integral polarizing isometries preserving this component; it contains the monodromy of \mathbb V. Let \Gamma'\triangleleft\Gamma be a normal neat subgroup of finite index. The kernel of

defines a connected finite étale Galois cover U'\to U. Denote its deck group by H; thus H is the image of this homomorphism, not necessarily all of \Gamma/\Gamma'.

The period map of \mathbb V|_{U'} factors algebraically through the normalization B of the closure of its image in \Gamma'\backslash\mathcal D. The space B is a normal quasiprojective variety, and on a smooth dense open subset its period realization has generically immersive differential; this is the period-image theorem [4, Theorem 1.1]. Here and below the function field of B is the function field of this normalized actual image. We do not replace it by its relative algebraic closure in \C(U').

Neatness makes the arithmetic action free. The universal local system on the quotient is

with its tautological filtration. On the smooth open of B this filtration is horizontal: the dominant map U'\to B is generically submersive and its pullback is the original variation, so Griffiths transversality holds on a dense open and hence everywhere. Thus the restricted local system and filtration form a polarizable integral variation. Moreover, the action of \Gamma/\Gamma' on this local system is an honest action:

Normality of \Gamma' makes the formula independent of the representative \gamma. Restricting gives an action of H on the variation over B, covering its action on B. The action on B is algebraic: apply uniqueness in [4, Theorem 1.1] to the period map and its expression obtained by a deck transformation on U' and the corresponding arithmetic translation on the target, and then lift to the normalization. Let

be the effective group on the image. The full group H will still be retained for its action on the variation.

Choose a G_f-equivariant smooth projective compactification S' of a smooth dense invariant open of B, with snc boundary, by equivariant compactification and resolution. All boundary monodromies of its variation are unipotent. Indeed they are quasi-unipotent by the monodromy theorem, and they belong to the neat group \Gamma'; their root-of-unity eigenvalues must therefore all be one. Let J' denote the extended deepest Hodge line and put L'=cJ'. This is nef by Theorem 3.1. The H-action extends to J' by functoriality of the unipotent Deligne and Hodge extensions.

Let q:S'\to Q_0=S'/G_f be the finite quotient. Although an element of \ker(H\to G_f) may act nontrivially on J', it acts on its fibres through finite-order characters. Taking a tensor power divisible by |H| kills the fibre action of every stabilizer in H, including this kernel. That tensor power therefore descends to an actual line bundle on Q_0 by finite-group descent. Consequently there is a rational line bundle \overline L on Q_0 with

It is nef, since its pullback by the finite surjective map q is nef. Choose a smooth projective resolution s:S_0\to Q_0 and put L_0=s^*\overline L.

The equivariant period map on U' descends to a dominant rational map Y\dashrightarrow Q_0, and hence to S_0. Resolve its graph by a smooth projective birational modification of Y, obtaining

We claim that

To verify this claim, choose a common smooth projective alteration W dominating the alteration in the hypothesis and the level cover, and resolve the graph of its rational map to S'. Write \mu:W\to Y and t:W\to S' for the resulting morphisms. Arrange snc boundaries by further resolution, containing the inverse images of the boundaries in question. The pulled-back variations agree on a dense open subset. Unipotent Deligne extensions and their extended Hodge filtrations commute with these pullbacks between smooth snc pairs; see [5, Lemma 5.6], or the unipotent extension theory [25, 11]. The extended top line J_W is consequently the pullback of the top line on either model. The hypothesis and the commutative maps to Q_0 give

Pullback by a proper generically finite map is injective on rational line-bundle classes: applying divisor pushforward, or the norm of a principal rational function, multiplies the original class by the degree. This proves (5.1). In particular no boundary twist has been inferred from an equality merely on the open locus.

If \dim S_0=0, equation (5.1) gives M\sim_{\Q}0, and the theorem follows. Assume henceforth that \dim S_0>0.

A common equivariant model and log general type. Normalize S_0 in \C(S'), and take a G_f-equivariant smooth projective resolution of the resulting model and its birational map to S'. Denote the resulting maps by

Thus \eta is birational, \pi is generically finite with Galois function-field group G_f, and G_f acts on T over S_0. Choose the resolution so that there is a G_f-invariant snc boundary D_T containing the complement of the open period locus and the support of the Jacobian divisor of \pi. The pulled-back variation has unipotent boundary monodromy. If J_T is its extended top line, then

Let D_N be the reduced sum of those components of D_T having nonzero monodromy logarithm. The variation extends as a pure polarized variation over

Indeed, near a point where all the remaining logarithms are zero, the nilpotent-orbit extension is a pure polarized Hodge filtration; the several-variable extension gives the same statement at crossings [25, 11]. These extensions are unique and preserve the H-equivariance of the variation. The divisor D_N is snc, since it is a union of components of D_T.

The full period map on U_N has generically immersive differential, because T is birational to B. The log-general-type theorem for a polarized variation with generically immersive period map therefore gives

Here the applicable theorem is [7, Theorem 1.1]. Its boundary is precisely D_N, because the pure variation has just been extended over all the other boundary components.

We now have precisely the geometric data of Theorem 4.1: T is birational to the actual normalized period image, the full period map is generically immersive, and the retained finite group H acts on its integral polarized variation. The effective group on T is G_f. Minimality is preserved here: the variation on T and the original variation have the same Hodge structures at Hodge-generic points, since the original source dominates the period image. Consequently

for every component of D_N and every divisorial ramification component of \pi mapping onto a divisor of S_0. For the latter, the inertia group of the Galois extension \C(T)/\C(S_0) contains a nonidentity element of G_f fixing the divisor pointwise. These fixed loci were included in D_T. Positive rational rescaling from J_T to L_T leaves all these numerical dimensions unchanged. The use of the actual image, before Stein factorization, is exactly the function-field hypothesis needed in the finite-inertia part of Theorem 4.1.

Removal of the boundary and ramification. Write the effective Jacobian divisor as

where R_{\mathrm{div}} is supported on components mapping onto divisors of S_0, and each component of R_{\mathrm{exc}} has image of codimension at least two. In the volume Lemma 5.1, take

All components of D satisfy the rank inequality above, and A+D=K_T+D_N is big by (5.3). The volume lemma gives

Lemma 5.2 removes the effective divisors exceptional over S_0 and descends bigness through the generically finite map. Therefore

Connected fibres. Finally take the Stein factorization of p_0, and resolve its finite normal intermediate variety to a smooth projective variety S. Resolving the induced rational map from Y gives, by further smooth projective birational modification, a morphism p:Y\to S with connected fibres. Indeed, its Stein factor is finite birational over the normal variety S, because \C(S) is already algebraically closed in \C(Y), and is therefore S itself. If \sigma:S\to S_0 is the resulting generically finite morphism, set L=\sigma^*L_0. It is nef, and (5.1) pulls back to M\sim_{\Q}p^*L. Since both S and S_0 are smooth, the Jacobian divisor of \sigma is effective. Thus

is big for j\gg0 by (5.5) and preservation of bigness under generically finite pullback. This proves the theorem.□

Proof. Put n=\dim C and D_C=\operatorname{div}(s). The case n=0 is immediate. For n>0, first note that every nonzero pluricanonical space on C has dimension one: two independent sections in the same degree would give a nonconstant rational map and hence positive Kodaira dimension.

The field \C(C) contains all roots of unity. If s/\omega^b=u^p for a prime p\mid b, the rational (b/p)-canonical form u\omega^{b/p} has p-th power s. At every prime divisor its order is p^{-1} times the nonnegative order of s, so this form has no poles. A rational section of a line bundle on the smooth variety C with no divisorial poles is regular. This contradicts the minimality of b. The Kummer irreducibility criterion now gives a field extension of degree b. Denote the normalization map by \pi:T\to C, and a smooth projective resolution by \rho:W\to T, so h=\pi\rho.

The differential of h gives a nonzero morphism h^*K_C\to K_W at the generic point, and a regular morphism of line bundles everywhere. Use this morphism and its tensor powers when pulling back canonical forms. The rational canonical form \eta=\xi h^*\omega then satisfies

The right side is a regular section of bK_W. Thus b\,\operatorname{ord}_E(\eta)\geq0 for every prime divisor E\subset W. Normality of W again implies that \eta is regular, and therefore h^0(W,K_W)\geq1.

We next bound the canonical divisor from above. Choose K_C=\operatorname{div}(\omega), and on T use the canonical Weil divisor determined by the differential pullback of \omega. At a prime divisor P\subset C outside \Supp D_C, the order of s/\omega^b is divisible by b. Locally over the discrete valuation ring at P, after dividing the root by a power of a uniformizer, the defining equation is a b-th root of a unit. It is unramified, since the characteristic is zero. Consequently all ramification divisors lie over \Supp D_C.

If Q\subset T lies over P\subset\Supp D_C, write e=e(Q/P) and d=\operatorname{ord}_P(D_C)\geq1. Tameness gives the ramification coefficient e-1, and

The canonical divisor formula for finite maps, interpreted at codimension-one points, therefore gives

On W the same inequality holds away from the \rho-exceptional divisors. Hence there is an effective integral \rho-exceptional divisor E such that

This argument does not require a pullback of the Weil divisor K_T.

Set A=K_C+D_C. Normality of T implies \rho_*\mathcal O_W(mE)=\mathcal O_T for every m\geq0: a rational function allowed poles only on exceptional divisors is regular in codimension one on T, hence regular on T. Projection formula and (6.3) imply

The last equality is invariance of Iitaka dimension under a finite surjective pullback. In this situation it can be seen directly from section rings: a homogeneous section upstairs satisfies its monic characteristic polynomial over \C(C); its coefficients, in degree jm, are sections of jmA, by the valuation bounds for sections of m\pi^*A. Thus the upstairs graded ring is integral over the downstairs ring, and their fraction fields have the same transcendence degree. Finally D_C\sim bK_C, so

Together with the nonzero form \eta, this proves \kappa(W)=0, and then h^0(W,K_W)=1.□

Proof. Over \C(Z), take a sufficiently divisible pluricanonical map of the generic fibre and its Iitaka fibration, including the relative algebraic closure of the image field. The Iitaka fibration theorem gives image dimension d and Kodaira dimension zero for the geometric generic fibre of this fibration. Formation of its pluricanonical spaces commutes with extension to an algebraic closure of \C(Z); thus these statements can be checked geometrically. Extend the rational map to projective models over Z, resolve source and target, and take the connected-fibre factorization. This gives X_0\to Y_0\to Z with the required dimensions. The second map has connected fibres as well: pushing \mathcal O_{X_0} first to Y_0 and then to Z gives \mathcal O_Z, because the original f and the birational map to its smooth source have this property.

Apply Theorem 6.1 to X_0\to Y_0. All its modifications can be performed over Z. Exceptionality over X_0 implies exceptionality over X. Lemma 6.3 and the subsequent family construction give the required rational variation and its top line. The two comparisons are proved in Lemma 6.5 below.□

Proof. The formula gives

Write K_{\widetilde X}=\mu^*K_X+E_\mu, using compatible canonical divisors. Since X and \widetilde X are smooth, E_\mu is effective and \mu-exceptional. The same is true of E_\mu+R^-. For sufficiently divisible m, normality and projection formula therefore give

Multiplication by the canonical section of mR^+ injects H^0(Y,mD) into the space on the left; here we also use g_*\mathcal O_{\widetilde X}=\mathcal O_Y. This proves \kappa(Y,D)\leq\kappa(X).

Now make the flat base change \bar\eta\to Z. The varieties \widetilde X_{\bar\eta} and Y_{\bar\eta} are smooth; the former is birational to the geometric generic fibre F of f. Restrictions of canonical divisors agree, up to linear equivalence, with the corresponding canonical divisors of these fibres. The restricted formula consequently implies

Here \leq_{\Q} means that the difference is \Q-linearly equivalent to an effective divisor. Formation of a proper direct image commutes with this flat base change, so (6.2) yields, for sufficiently divisible m,

Projection formula now gives

It follows that

Thus D|_{Y_{\bar\eta}} is big, including the zero-dimensional case.

For completeness, the bigness assertion spreads to a dense open subset after restricting to the smooth locus of q. Fix a relatively ample divisor H. Bigness on the geometric generic fibre gives, for some sufficiently divisible integer m, a nonzero section of mD-H there. Flat scalar extension of cohomology first gives such a section over \C(Z). Shrinking the base, the section extends to the family and remains nonzero on its fibres. Hence mD|_{Y_z} is the sum of the ample divisor H|_{Y_z} and an effective divisor for general z, proving the stated very-general-fibre assertion.□

Proof. The nonexceptional coefficients of B^{\mathrm{cr}} are those of the effective divisor B, so its negative part is exceptional. All coefficients of B^{\mathrm{cr}} are less than one by the klt hypothesis. Its positive part has simple normal-crossing support and coefficients in [0,1), proving that (Y',B') is klt. The displayed identity follows directly from the definitions. Exceptional effective twists do not change section spaces of a pullback to a normal base, so they do not change the Iitaka dimension. On the geometric generic fibre over Z, the induced map is birational and the restricted identity expresses D' as the pullback of a big divisor plus an effective divisor. Nefness is preserved by pullback. For the Hodge assertion, use a common smooth alteration of Y' and the alteration in (6.4); the unipotent Deligne and Schmid extensions commute with the resulting morphisms of smooth normal-crossing pairs.□

Proof. Take the Stein factorization of the map from Y to its image in S\times Z. Resolve its intermediate normal variety and resolve the induced rational map from Y, choosing the modification of Y also to be a log resolution of (Y,B). We obtain morphisms

where Y,W are smooth projective, r has connected fibres, and W\to S\times Z is generically finite onto its image. We continue to write Y,B,M,D for the modified data. This replacement is harmless: for such a log resolution \mu:Y'\to Y, take the positive part B' of the crepant boundary. Then

with E exceptional, and (Y',B') is klt. The same identity on the general p-fibres preserves their log Kodaira dimensions. Pullback of M and the effective correction preserve the required bigness on the geometric generic q-fibre.

Since p_*\mathcal O_Y=\mathcal O_S and r_*\mathcal O_Y=\mathcal O_W, the fibres of t are connected. For general s\in S, the fibre W_s is smooth and connected, and h_s:W_s\to Z is generically finite onto its image. Write

The generically finite map onto its image gives a\leq m. We claim that \kappa(W_s)\geq0. The assertion is immediate for a=0; assume a>0. Along a smooth fibre of t, the cotangent sequence is

Since the last term has rank a, the entire m-th exterior power lies in the filtration term with at least m-a base factors. Its quotient by the next term gives, after pulling back top forms on Z, a morphism

At a general point of a general W_s, the differential of h has rank m, and its restriction to the tangent space of W_s has rank a. Thus (7.1) is nonzero. A suitable linear functional on the constant exterior-power factor yields a nonzero morphism h_s^*\omega_Z\to\omega_{W_s}.

Choose n>0 and 0\ne\tau\in H^0(Z,nK_Z). For general s, the image h_s(W_s) is not contained in the zero divisor of \tau: otherwise dominance of h would fail. The n-th tensor power of the preceding morphism therefore sends h_s^*\tau to a nonzero section of nK_{W_s}. This proves the claim.

Let G be a general fibre of r. For general z\in Z, such fibres form a moving family covering Y_z. A big divisor restricts to a big divisor on a general member of this family: restrict an ample-plus-effective decomposition, choosing a member not contained in the support of the effective part. Consequently D|_G is big. Since G maps to a point of S, M|_G\sim_{\Q}0, and adjunction gives

For general s, the morphism r_s:Y_s\to W_s has connected fibres, (Y_s,B|_{Y_s}) is klt, and its general fibre has big log canonical divisor. Theorem 7.1 gives

Finally choose s outside the countably many proper jumping loci for the spaces of sections of all divisible log pluricanonical bundles on the smooth p-fibres. Generic base change identifies their dimensions with those on the geometric generic fibre. Thus the last nonvanishing holds on that fibre as well.□

Proof. Choose a dense open over which p is flat. Flatten p by a projective birational modification of the base and then resolve the base, retaining flatness of the main transform by base change; see [26, Lemma 38.31.1, Tag 081R]. Resolve the resulting main transform V\to S_1, choosing the projective resolution so that its composite \mu:Y_1\to Y is also a log resolution of (Y,B). Thus

are compatible morphisms, S_1,Y_1 are smooth projective, \sigma,\mu are birational, and Y_1\to V is a resolution with V\to S_1 flat. Any prime divisor of Y_1 whose image in S_1 has codimension at least two is \mu-exceptional. Indeed, flatness implies that the inverse image of such a base subset has codimension at least two in V; hence the divisor is exceptional over V, and its image in Y also has codimension at least two.

Take the positive part B_1 of the crepant boundary, so that (Y_1,B_1) is effective klt and

Generic-fibre nonvanishing and field-extension base change allow us to choose a sufficiently divisible k>0 for which

has positive rank. Theorem 7.2 applies to \mathcal E.

Fix an ample Cartier divisor A on S_1, and choose a positive integer \alpha so large that

is big. For some \beta>0, weak positivity makes

generically generated. Set \ell=\alpha\beta. Let U\subset S_1 be the locally free locus of the torsion-free sheaf \mathcal E; its complement has codimension at least two. On all of U, multiplication of sections gives

This map is nonzero at the generic point: a nonzero section on the integral generic fibre has nonzero powers. Generic generation therefore supplies a global section of the twisted reflexive symmetric power whose product is nonzero. Its image is a regular section over p_1^{-1}(U) of

Here U is the locally free locus, not merely a smaller open where the chosen sections generate.

The big divisor

has an effective sufficiently divisible multiple. Taking the same power of the preceding section and multiplying by the pullback of a nonzero section of that multiple gives a nonzero section over p_1^{-1}(U) of a multiple of

As a rational section on Y_1, it has poles only above S_1\setminus U. Thus there are n>0, a rational function \varphi\ne0, and an effective \mu-exceptional divisor T such that

Choose compatible canonical divisors and put

Equation (7.2) gives the exact identity

Push (7.3) forward to the smooth variety Y. Since \mu_*E=\mu_*T=0, we obtain

This proves rational linear effectivity of D_0.□

Proof of Theorem 1.1 over \C. If either \kappa(F) or \kappa(Z) is -\infty, the assertion follows from the convention in the statement. Assume therefore that

Proposition 6.4 and Lemma 6.5 provide algebraic fibre spaces

and an effective klt pair (Y,B), together with a nef \Q-divisor M having the required Hodge realization. Writing D=K_Y+B+M, their conclusions include

where Y_{\bar\eta} is the geometric generic q-fibre.

Apply adjoint positivity, Theorem 1.2, and the compatible modifications of Lemma 6.6. We obtain an algebraic fibre space p:Y\to S with S smooth projective and

If \dim S=0, then M\sim_{\Q}0. Theorem 7.1, applied directly to q, and (7.4) give the result. We may consequently suppose \dim S>0. Adjoint positivity also gives

Openness of the big cone on Y_{\bar\eta} permits a rational number v with 0<v<1 such that

Lemma 7.3 shows that the geometric generic p-fibre has nonnegative log Kodaira dimension. Choose a sufficiently large integer j>1 and an ample \Q-divisor H on S such that

is big. Lemma 7.4 gives

Choose

This balances the coefficients of M; to cancel the corresponding term -\lambda p^*H, put

Then A_S is ample and

Choose a sufficiently divisible integer N>1 such that |Np^*A_S| is basepoint-free. A general member T_N\in|Np^*A_S| gives

with (Y,B+\Gamma) klt. This follows from Bertini, or after pulling the basepoint-free system to a log resolution of (Y,B) and choosing N sufficiently large. Thus E_2\sim_{\Q}K_Y+B+\Gamma is the log canonical class of an effective klt pair. Its restriction to Y_{\bar\eta} is the big divisor in (7.5) plus the nef divisor \lambda p^*H/(1-\lambda), and hence is big. Theorem 7.1 now yields

In sufficiently divisible degrees, multiplication by powers of a nonzero section in (7.6) embeds the corresponding linear systems of (1-\lambda)E_2 into those of D. Positive rational scaling leaves Iitaka dimension unchanged, so

as required.□

Proof. Flat base change for proper coherent cohomology gives, in every positive degree n,

The canonical bundle commutes with this smooth scalar extension. Nonvanishing is therefore unchanged. When the space is nonzero, choose a K-basis. The rational map defined by that basis and its image base-change to the complete pluricanonical map over K'; dimensions of finite-type schemes are unchanged by extension of the ground field. Taking the supremum of these image dimensions proves the first assertion, including the case -\infty.

For the fibre assertion, put E=K(Z) and E'=K'(Z_{K'}), and write F_K for the generic fibre over E. The generic fibre after ground-field extension is F_K\times_E E'. Choose an algebraic closure \Omega of E' and an algebraic closure \overline E\subset\Omega of E. The two geometric generic fibres are F_K\times_E\overline E and F_K\times_E\Omega; the latter is a scalar extension of the former. Their Kodaira dimensions agree by the first assertion.□

Proof. In the unipotent case the canonical extension and relative monodromy weight filtration are supplied by nilpotent-orbit theory [25, 11]; precise formulations for the top line are given in [5, §2.5.3, Lemma 2.16, Lemma 5.6]. The graded pieces are polarized by the primitive decomposition.

For a boundary stratum, take the weight filtration associated with an interior point of its monodromy cone; it is independent of that choice. Strictness and \rank F^p=1 select one nonzero highest graded piece. The global object in (ii) is this pure graded variation. To check its gluing when the normal bundle is nontrivial, choose a local normal coordinate t and write N for its monodromy logarithm. Replacing t by t'=u(s)t, with u invertible along the stratum, changes the local limiting filtration by

Since N lowers the monodromy weight filtration by two, this transition acts identically on each weight graded. For several normal coordinates the same calculation uses the sum of the commuting monodromy logarithms, which also acts trivially on their common weight graded. The pure graded data therefore glue. Strictness identifies the rank-one top filtration with its unique nonzero top graded piece. At the remaining corners, [5, §2.5.3 and Lemma 2.16(3)] identifies the canonical extensions on the entire smooth stratum closure, so the identification of lines holds there as well.

For completeness, the rational convention is compatible with these operations as follows. In a crossing chart, take sufficiently divisible roots of its boundary coordinates. The resulting unipotent extension is equivariant for the finite deck group. A common positive power kills the characters of the stabilizers on its line fibers and descends to a line bundle in the original chart. The descents agree on overlaps: compare them on a common root cover and use the uniqueness of the unipotent extension. This defines a class in \operatorname{Pic}(P)\otimes\Q and proves compatibility with further pullbacks and powers. One can perform the comparison globally on a smooth adapted alteration; auxiliary branch divisors cause no difficulty because the original variation extends over them. Nefness descends under a proper surjective morphism, since every curve downstairs is dominated by a curve upstairs. Finally, nef numerical dimension is unchanged by a generically finite pullback, so the same construction on strata proves (iii).□

Proof. The closed-test-form version of this calculation is proved in Lemma 9.1; that proof uses only the extension facts in Lemma 8.2, not any descent or rank-drop result. Taking the test form to be \omega^{d-k} gives Equation (8.1). Griffiths’ curvature formula gives

If r is the generic differential rank, then \alpha^{r+1}=0 everywhere on the open set, whereas \alpha^r\wedge\omega^{d-r} is positive on a nonempty open subset. The intersection identity and nefness therefore give \nu(L)=r. For quasi-unipotent monodromy, both numerical dimension and differential rank are unchanged by a smooth adapted generically finite cover, and Lemma 8.2 identifies the pulled-back rational line with its unipotent extension.□

Proof. For a general algebraic U, choose a dense quasi-projective open U_0. The map \pi_1(U_0)\to\pi_1(U) is surjective, since loops can be moved off the complement. Thus the monodromy decompositions below are unchanged, and a flat projector of Hodge type (0,0) on U_0 has that type throughout U by continuity. Write \Gamma for the original monodromy image. By semisimplicity [13, Theorem 4.2.6 and the added-in-proof note], the complex local system is a sum of monodromy-isotypic summands V_j. These are complex Hodge subvariations. To see this precisely, the fixed-part theorem equips \operatorname{End}_{\Gamma}(V) with its Hodge algebra structure, and its center has type (0,0) by [13, Corollary 4.2.8(i),(ii)]. Equivalently, the connected Hodge action fixes every primitive central idempotent of this semisimple algebra. The images of those idempotents are exactly the V_j.

The top line belongs to one summand V_{j_0}. Retain that summand and all its rational conjugates. Their sum is a rational subvariation: the central idempotents are defined over a finite splitting field of the rational commutant algebra, and the selected sum is Galois invariant. It inherits the rational polarization and the lattice obtained by intersecting with the original lattice. Denote it by V_0. None of this construction requires the highest line, or its scalar character, to be rational.

Suppose \gamma\in\Gamma fixes the moving line projectively everywhere. Its projective fixed locus is the finite union of the projectivizations of its eigenspaces. The connected line image lies in one of them, so \gamma is scalar on the linear span S of that image. Equivariance makes S a nonzero \Gamma-subrepresentation of V_{j_0}. Write V_{j_0}=E\otimes M, with E irreducible and \Gamma acting trivially on M. Since S contains a copy of E, scalar action on S implies the same scalar action on E and hence on all of V_{j_0}. Rational conjugation gives scalar action on every retained V_j.

It follows that \gamma|_{V_0} preserves the Hodge filtration at any point. It also preserves the rational polarization and the lattice. The stabilizer of a polarized Hodge structure in its real polarization group is compact, since it preserves the positive definite Hodge metric. Its intersection with the integral automorphism group is finite. Therefore \gamma|_{V_0} has finite order. For this particular variation, one may take the exponent of that finite stabilizer at a fixed base point. There is also a choice depending only on R=\rank V_0. Indeed, an eigenvalue of an integral finite-order matrix of rank R has order q with \varphi(q)\le R, because its cyclotomic minimal polynomial divides the characteristic polynomial. The finite set of such q therefore gives the explicit permissible choice

Every scalar on every retained summand has bth power one. This uniform exponent is chosen before constructing the new variation.

Inside V_0^{\otimes b} take

The sum is taken inside V_0^{\otimes b}. The individual terms are distinct direct summands in the tensor expansion, so the sum is direct. It is Galois invariant and hence rational, and each summand is a Hodge subvariation. Restricting the tensor polarization and intersecting with the tensor lattice make it rationally polarized and integral. The summand V_{j_0}^{\otimes b} has top line (L^\circ)^{\otimes b}; all other summands have strictly smaller highest Hodge index. Thus this is the unique top line of V_1.

Every element considered above acts trivially on V_1. Conversely, if an element of the new monodromy image fixes its moving top line, choose a lift in \Gamma. Injectivity of the Veronese map \mathbb P(V_{j_0})\to\mathbb P(V_{j_0}^{\otimes b}) shows that the lift fixes the original moving line. It is therefore killed on V_1. This proves Condition (8.3). Compatibility of extensions with tensor powers finishes the proof.□

Proof. Let \mathcal D be the period domain for the chosen polarized lattice, and choose an arithmetic group \Gamma_{\mathrm{ar}} containing the actual monodromy image \Gamma. By [4, Theorem 1.1], the full period map factors through an algebraic dominant morphism P^\circ\to Y_0, where Y_0 is an irreducible quasi-projective variety and Y_0^{\mathrm{an}}\to\Gamma_{\mathrm{ar}}\backslash\mathcal D is a closed analytic immersion. Normalize Y_0 in the relative algebraic closure of \C(Y_0) in \C(P). This extension is finite, so the normalization is finite over Y_0. Its function field is relatively algebraically closed in \C(P); in characteristic zero this makes the generic fiber geometrically connected. Shrink source and target so that the latter is smooth and the map is a smooth topologically locally trivial fibration with connected fibers. To obtain this shrinking for the possibly nonproper source, stratify a proper algebraic compactification of the map together with its source boundary and restrict over the open base stratum.

A local lift of the full period map along a fiber takes values in one \Gamma_{\mathrm{ar}}-orbit. These orbits are discrete, including when the quotient has finite stabilizers, so that lift is locally constant in fiber directions. Continuation around a fiber loop therefore fixes the lifted full period. Transport the loop to nearby fibers using the local trivialization. In flat coordinates its actual monodromy fixes the top line on a nonempty analytic open subset of the source. This identity extends to its connected universal cover, and Lemma 8.4 kills that monodromy. The homotopy sequence of the fibration now factors the integral representation through the ordinary fundamental group of the base. The filtration descends as well: on a simply connected base chart the pulled-back flat bundle is trivial, the filtration is constant on each connected fiber, and holomorphic local sections of the smooth map show that the descended filtration is holomorphic. Its transversality and polarization follow by pullback.

The descended period map has the dimension of the original full period image and is generically immersive. Base loops lift to the source, so the same argument and Condition (8.3) show that the descended variation retains that condition. Shrinking introduces no change to it: any monodromy element fixing the line on the smaller open fixes it identically by analytic continuation.

Choose a smooth projective compactification of the base, resolve its boundary, and resolve the rational map from P to this compactification. The resulting morphism h:\widetilde P\to Z has connected generic fiber and hence connected fibers, by Stein factorization and normality of Z. Enlarge the source boundary to contain the inverse image of the chosen base boundary and resolve it. The rational extension of the original variation is unchanged where it already extended. Lemma 8.2 identifies its b-th power with h^*A_1.□

Proof. After shrinking around the given point, the submersion theorem provides coordinates

The coordinate z is retained because dq already has rank r on its zero divisor. On the punctured neighborhood, the kernel inclusion and the rank bound imply \rank(d\ell)=r and \ker(dq)=\ker(d\ell). Fix q and u and make one positive turn in the z coordinate. Along every lift of that loop, the derivative of \ell vanishes, so its value is constant. Its endpoints are related by T, which consequently fixes the initial line. Varying the fixed coordinates proves the claim on an open subset. The projective equality T\ell=\ell then extends by the identity theorem to the connected universal cover. For r=0, the rank bound makes \ell locally constant and the same loop argument applies with no q coordinates.□

Proof. If the boundary rank did not drop, suitable single-valued functions of the moving line would have the same differential kernel as that line. A transverse loop in one of their fibers would then force the monodromy to fix the moving line on an open set, contradicting Condition (8.3) after analytic continuation.

Work at a general point of D_i, with coordinates (z,s) on \Delta\times\Delta^{d-1} and D_i=\{z=0\}. Write T=T_sT_u for the Jordan decomposition of the transverse monodromy and put N=\log T_u. Choose e>0 with T_s^e=1 and set z=t^e. In flat coordinates the nilpotent orbit extension supplies a holomorphic line map \Psi on the root polydisk such that

These are equalities of lines; \ell is evaluated on the logarithmic cover when necessary.

Let W=W(N), centered at the weight of the variation. On the boundary there is a unique index k in which the top line has a nonzero weight-graded image. The map N is a strict morphism of limiting mixed Hodge structures of type (-1,-1). Since F^{p+1}=0, it gives

Thus the limiting line survives in C=V/NV.

We also need to preserve its derivative rank. The primitive decomposition of the fixed graded local system identifies

Here k is at least the original weight: lower graded pieces are entirely in the image of N and cannot contain the top line. The second summand has zero F^p, so the graded moving top line lies in the flat primitive summand P_k. Its projective derivative rank is therefore unchanged by the displayed isomorphism. Projecting the boundary line in C further to its relevant graded piece recovers precisely this pure line period map. By Lemmas 8.2 and 8.3, its rank is

This equality may be checked on a global adapted cover: restriction of the pulled-back line to a component dominating D_i is its generically finite pullback, which preserves numerical dimension.

Equation (8.4) makes \Psi(0,s) an eigenline of T_s. Its eigenvalue \lambda is constant on a sufficiently small connected boundary neighborhood. Let E=C_\lambda and let q_0:V\to E be the quotient followed by the eigenspace projection. The projective map

is defined near the chosen boundary point and has boundary rank at least r_i. Since N vanishes on C, it also equals \mathbb P(q_0)(\ell(t,s)). Moreover T_s acts by a scalar on E, so Q(\zeta t,s)=Q(t,s). Hence Q descends holomorphically to the original (z,s)-polydisk, including z=0. This descent uses a projective map; a choice of a fractional scalar frame is unnecessary.

Suppose r_i\ge r. Choose r affine coordinates of Q, denoted q_1,\ldots,q_r, whose differentials are independent along D_i. The map q=(q_1,\ldots,q_r) factors locally through \ell on the punctured chart. Its differential therefore satisfies the kernel inclusion in Lemma 8.6. The rank bound for \ell is r by Lemma 8.3. The transverse-loop lemma shows that T fixes the moving highest line everywhere on the connected universal cover. Condition (8.3) gives T=\id, contrary to the choice of the divisor. If r=0, the same lemma applies with no coordinate functions. If r>d-1, the supposed inequality r_i\geq r is already impossible.□

Proof. If r=0, the restriction hypothesis forces J=0, so K_Z is already big. Suppose r>0. By Kodaira’s lemma write

and set B_a=H+aA_1 for rational a\ge0. Choose one ample integral divisor C_0 such that both C_0 and C_0+J are globally generated, and put H'=H+C_0. This choice is independent of a and of all section degrees below.

For every sufficiently divisible m>0, filter by kJ for 0\le k<m. The restriction sequences give

Since J is reduced, its structure sheaf injects into \bigoplus_i\cO_{D_i}. Also

is globally generated. On each D_i, multiplication by a section which is not identically zero bounds the restricted summand by h^0(D_i,m(H'+aA_1)|_{D_i}). We obtain the exact estimate

Fix a and divide by m^d/d!. Asymptotic Riemann–Roch for the ample divisors involved, along divisible m\to\infty, yields

No uniform Riemann–Roch remainder in a is needed: the exact inequality in Equation (8.6) holds for every a, and the limit is taken with a fixed.

The first polynomial on the right of Equation (8.7) has degree r, with positive leading coefficient \binom dr A_1^rH^{d-r}. Every restriction polynomial has degree at most \nu(A_1|_{D_i})\le r-1. The right side is therefore positive for every sufficiently large a. Finally K_Z+aA_1\sim_{\Q}(B_a-J)+E is big.□

Proof of Theorem 8.1. Apply Lemmas 8.4 and 8.5. We obtain a variation on a dense open of a smooth projective Z satisfying Condition (8.3), with generically immersive full period map, and rational Hodge extension A_1 satisfying b\pi^*L\sim_{\Q}h^*A_1. Resolve the boundary so that it is simple normal crossing. Pullback compatibility retains the line identity. Delete every boundary component with identity transverse monodromy, and call the remaining reduced divisor J.

The variation extends to Z\setminus J. Indeed, at a point lying only on deleted components all local monodromies are the identity, and the nilpotent orbit extension with all monodromy logarithms zero is a pure polarized extension. The full period map is still generically immersive. Thus [7, Theorem 1.1 and Remark 1.2(1)] applies to the polarized variation on Z\setminus J and gives the bigness of K_Z+J; its hypothesis is immersivity at one point, and it imposes no extra condition on boundary monodromy. Lemma 8.7 applies to every component of J, and Lemma 8.8 proves that K_Z+aA_1 is big for all sufficiently large rational a. When \dim Z=0 the assertion holds by convention. Set A=A_1/b. This is nef, satisfies \pi^*L\sim_{\Q}h^*A, and has the same asserted adjoint positivity after rescaling the large parameter.□

Proof. We may compute after a generically finite smooth alteration on which all boundary monodromies are unipotent. To obtain one, pass to a finite level cover with neat monodromy, compactify, and resolve. Quasi-unipotence then gives unipotence, including for the commuting local monodromies on the resolved boundary. The parabolic pullback property of Lemma 8.2 identifies the new line with the pullback of A. Generically finite pullback preserves nef numerical dimension by the intersection and projection formulas. It also preserves the generic differential rank in characteristic zero. We therefore work on the unipotent model, retaining the notation R,A.

Let \alpha be the semipositive Chern form of the highest-line Hodge metric on the open part. Griffiths’ curvature formula [16, Theorem 5.2, Formula 5.3, and Formula 6.18] says that \alpha(v,\bar v) is a positive fixed multiple of the squared norm of the projective highest-line differential in direction v. In particular, its rank is the rank in (9.1).

Consider a crossing chart with coordinates z_1,\ldots,z_n and boundary z_1\cdots z_c=0. Put \rho_i=-\log|z_i|^2 and shrink the chart so that \rho_i>e. The horizontal Schwarz estimate, applied separately to the coordinate disks, gives

Here and below harmless positive normalization constants are absorbed into C. The coordinate estimate follows from the curvature of the horizontal period metric and the singular Ahlfors–Schwarz lemma; see [7, §2.1, Proposition 2.4 and Lemma 2.5]. Positivity and the Hermitian-form Cauchy–Schwarz inequality control the off-diagonal entries and give the uniform product estimate (9.2), even where the period differential is degenerate.

By the nilpotent-orbit theorem [25, Theorem 4.12], a nonvanishing top extension frame, written in flat coordinates on a bounded angular sector, is

where the N_i commute and u is holomorphic and nonvanishing. If \beta is the curvature of a fixed smooth metric on A, write \alpha-\beta=\ddc\varphi. The ratio of the two metrics satisfies

This is the logarithmic growth estimate for canonical Hodge frames; see [10, Theorem 5.1]. Here the Schwarz estimate also gives a direct proof of the bound needed for integration. The differential of the logarithmic Hodge norm of any flat vector is bounded by a constant times the invariant period metric, with a constant independent of the point by homogeneity. Travel from a fixed point along coordinate paths in a bounded angular sector. Their product-Poincaré length is bounded by C(1+\sum_i\log\rho_i), so the Hodge metric and its inverse in a flat frame grow at most as products of powers of the \rho_i. The exponential in (9.3) and its inverse are polynomials in the logarithms, since the N_i are commuting nilpotent operators. Finally, u has ordinary norm bounded above and below on a smaller chart. These three observations give the upper and lower logarithmic bounds in (9.4). We have therefore obtained the two controls needed for the cutoff argument:

Chern-current representation for canonical Hodge extensions is established in [10, Theorem 5.2]. We now justify the highest-line intersection formula with closed test forms, including the absence of boundary contributions. For 0\leq j\leq n and every smooth closed form \gamma of type (n-j,n-j),

The case j=0 is immediate, so assume j>0. Choose a smooth function \chi that is one on (-\infty,1] and zero on [2,\infty), and use products of \chi((\log\rho_i)/M) to cut off the boundary. The \rho_i can be globalized using smooth metrics on the boundary divisor bundles. Denote the resulting cutoff by \chi_M. Its second derivatives are uniformly bounded in the product-Poincaré metric, and their support escapes to the boundary as M\to\infty.

Since all curvature forms are closed, on the open part we have

Multiplying by \chi_M\gamma and integrating by parts makes the difference of the integrals equal to

Its absolute value is bounded on escaping tails by a constant times

This is integrable: in a cusp variable the relevant radial integrals are bounded by \int^{\infty}(1+\log\rho)\rho^{-2}\,d\rho. Thus the error tends to zero. The same domination permits removal of \chi_M from the two curvature integrals, proving (9.6). Rational bundles are handled by first taking an integral tensor power and then dividing the forms and identities by that power.

Let r be the generic rank of the projective line map. Then \alpha^{r+1}=0 on the open part, whereas \alpha^r is nonzero and positive on a nonempty open subset. Apply (9.6) with complementary powers of an ample Kähler form. It gives vanishing of the (r+1)st intersection power and strict positivity of the rth one. The nef intersection characterization proves (9.1).□

Proof. Put r=\nu(A). We first compute a bound for the numerical dimension on a boundary component of a unipotent model. This computation is unchanged on a component dominating D_i under a generically finite alteration, since restriction of the pulled-back line is the generically finite pullback of A|_{D_i}.

The limiting highest line.. Near a general point of a unipotent boundary component write the coordinates as (t,s), with divisor t=0 and monodromy logarithm N. Let w be the weight and p the largest filtration index. The limiting mixed Hodge structure is polarized by N [25, Theorem 6.16]; its graded structures vary as polarized variations along the smooth boundary stratum [10, Proposition 2.10 and the following discussion]. The limiting F^p has dimension one and F^{p+1}=0. Consequently it consists of one Deligne summand I^{p,q}. Write p+q=w+l. In the primitive decomposition for N, a nonzero N^k-image term with k>0 would have to come from F^{p+k}, which is zero. The line is therefore primitive, l\geq0, and, for a nonzero vector u_0(s)=u(0,s) spanning it,

Shrink the boundary stratum so that its Hodge type, and hence l, is fixed.

For a highest Hodge vector in the pure variation, the Hodge norm is the flat polarization pairing with its conjugate, multiplied by the fixed factor determined by its Hodge type. Substituting the exponential frame into that pairing gives a polynomial in y=-\log|t|. Primitive polarization identifies its leading term as a positive constant times y^l times the squared norm h_l(s) of the primitive graded line. Replacing u(t,s) by u_0(s) introduces errors O(|t|) times powers of y; the same holds after tangential differentiation, by holomorphicity of u. It follows, locally uniformly with tangential derivatives, that

for some M and a constant c. Thus the tangential curvature converges to the curvature of the primitive graded highest line. If q_i denotes the generic rank of its projective differential, the limiting tangential curvature has rank at most q_i.

The restriction intersection and crossing points.. We claim that

The following estimate proves the claim without assuming uniform convergence of (9.8) at crossings. Fix an ample Kähler form \omega and an integer j with q_i<j\leq n-1. Choose a smooth metric on \cO(D_i), write its curvature as \beta_i, and put \tau=\log\|s_{D_i}\|^2. Let g be a fixed smooth convex function with g'=0 on (-\infty,0] and g'=1 on [\log4,\infty), and set f_\epsilon(x)=g(x-\log\epsilon^2). Then

is a smooth closed representative of c_1(\cO(D_i)). Here the normalization is \ddc\log|t|^2=[t=0]. The function f_\epsilon(\tau) is constant near D_i, so it extends smoothly there. Outside \|s_{D_i}\|\leq2\epsilon, the two curvature terms cancel and \eta_\epsilon=0.

On a crossing chart put

The tangential form \Omega_s is a product cusp metric, with Euclidean summands in the remaining coordinates. It has finite volume. The bound (9.2) gives 0\leq T\leq C\Omega^{n-1}. Write \tau=\log|t|^2+\psi with \psi smooth, and put L_\epsilon=|\log\epsilon|. The smooth first term of (9.10) contributes at most

For the second term write \partial\tau=a_0+b_0, with a_0=dt/t and b_0=\partial\psi. Its support lies in c\epsilon\leq|t|\leq C\epsilon, with constants independent of s. On this shell, positivity gives

The first estimate follows because wedging with \lambda selects only the tangential part of T. For the second, a smooth positive form dominating i b_0\wedge\bar b_0 is bounded by C\Omega. Cauchy–Schwarz for the positive form T now bounds the mixed terms by O(L_\epsilon^{-1}). The bounded coefficient f_\epsilon'' does not change these estimates.

It remains to consider the pure normal-shell term. Set t=\epsilon u. Its cutoff coefficient becomes

whose support is contained in a fixed annulus, since \psi is bounded. The absolute integrand is bounded, independently of \epsilon, by

This is integrable on that annulus times the tangential chart, including all tangential divisor crossings. For almost every s, (9.8) gives a tangential curvature limit of rank less than j, so the tangential jth power tends to zero. Dominated convergence makes the pure normal-shell integral tend to zero. The estimates are uniform under truncating the tangential cusp variables, so no mass is lost when those truncations are removed. Choose a finite partition of unity subordinate to crossing charts. Multiplication by its bounded smooth functions preserves each absolute estimate. Summing the local estimates proves the same conclusion on all of D_i; no integration by parts is performed after inserting this partition.

For each fixed \epsilon, apply (9.6) with \gamma=\omega^{n-1-j}\wedge\eta_\epsilon. Its cohomology class is independent of \epsilon. The preceding limit gives

Nefness proves (9.9). If q_i=n-1, that inequality is automatic. This also explains why the generic limit calculation alone would not be sufficient: the uniform product bound is what controls the crossing locus.

A fixed quotient and transverse loops.. Return to the original boundary component, with transverse monodromy T_s\exp N, where T_s has finite order. Work first in a transverse root coordinate killing T_s. Set C_N=\operatorname{coker}N, endowed with the quotient of the fixed monodromy weight filtration. The primitive limiting line of weight k=w+l projects nontrivially into W_kC_N. The fixed linear quotient

recovers its primitive graded line. Indeed, in the Lefschetz decomposition of \operatorname{Gr}^W_k, quotienting by the image of N removes exactly the summands that are positive powers of N applied to higher primitives. The induced quotient is the primitive part of weight k. In particular, projective differential rank cannot increase under (9.11). No choice of a moving primitive splitting is involved.

The limiting line in C_N lies in one T_s-eigenspace. Compose the quotient by N with the fixed projection onto this eigenspace. Near the chosen general boundary point this gives a nonzero holomorphic projective map from the untwisted highest line. Under deck rotation the untwisted line is acted on by T_s; on the projected eigenspace that action is scalar. The projective map therefore descends to the original chart. On the punctured chart it factors through the moving highest-line map, since the exponential of N disappears in C_N. Its boundary differential rank is at least the rank q_i of the primitive graded line.

Suppose \nu(A|_{D_i})\geq r. By (9.9), the projected map has at least r independent tangential differentials at a general boundary point. Choose r affine components g_1,\ldots,g_r with that property. Their differentials are independent along the boundary at that point, and the map g=(g_1,\ldots,g_r) extends holomorphically over the original chart. On the punctured chart it factors through the highest-line map. Lemma 9.1 gives the latter rank at most r. The hypotheses of the transverse-loop criterion (Lemma 8.6) are therefore satisfied in the original transverse coordinate, before the finite root cover. It follows that the original transverse monodromy fixes the moving line on the connected universal cover. This contradicts faithfulness and its nonidentity. When r=0, use the same lemma with an empty list of functions; when r>n-1, the supposed inequality \nu(A|_{D_i})\geq r is impossible on dimensional grounds. This proves Equation (9.7).□