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 \(V^{\min}\cap V_{\mathbf Z}\) 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 (Bakker et al. 2025, Definition 2.9). ◻

Proof. The monodromy and fixed-part theorems give that \(G\) is semisimple and normal in the connected reductive group \(P\); see (André 1992, sec. 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 \[a\rho(\gamma)a^{-1}=\rho(b_*\gamma) \qquad (\gamma\in\pi_1(U,e)).\] 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 \(W\subset V_{\mathbf C}\) 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 \(\mathbf Q\), 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 4 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 \(\mathop{\mathrm{End}}(\mathbb V|_{U_1})\). 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 (Schmid 1973, 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 \(a_0=a(0,v)\). 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 \[N^j P_{w+\ell+2j},\qquad j\ge\max(0,-\ell),\] 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 \(\mathop{\mathrm{Gr}}^{W(N)}_{w+\ell}\) along \(E^\circ\). The boundary extension input identifies it with \(J|_E\). Projecting \(N^\ell a_0\) to \(\mathop{\mathrm{Gr}}^{W(N)}_{w-\ell}\) yields the image of this graded top line under the flat isomorphism \[N^\ell:\mathop{\mathrm{Gr}}^{W(N)}_{w+\ell}\longrightarrow \mathop{\mathrm{Gr}}^{W(N)}_{w-\ell}.\] 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 7 to the graded variation on \(E^\circ\) and its smooth projective compactification \(E\). This proves (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 \(v=(x,y)\in\mathbf C^r\times\mathbf C^{n-r}\) 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 \[x\longmapsto w-E(u,x,y)\] 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 \[f(u,\chi(u,w,y),y)=(w,H(u,w,y)).\] 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 \[\begin{align*} z^{-\ell}e^{zN}a(u,v) &=b(v)+\sum_{j<\ell}\frac{z^{j-\ell}}{j!}N^ja_0(v) +u\sum_{j=0}^k\frac{z^{j-\ell}}{j!}N^jq(u,v). \end{align*}\] For every fixed tangential derivative, uniformly on compact subsets of \(B\), the remainder is \[O(|z|^{-1})+O(|u|\,|z|^{k-\ell});\] 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 (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 9. ◻

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 \[\liminf_{m\to\infty} \frac{t!}{m^t}h^0\bigl(T,m(A+D+jJ)\bigr) \ \geq\ (H_1+jJ)^t\ \geq c j^r\] 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 \[\mathcal O_{E_i} \left(m(A+D+jJ)-\sum_h k_hE_h\right), \qquad 0\leq k_h\leq md_h.\] 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 \[m(B_0-A-D)+ \sum_h\bigl((md_h-k_h)B_h+k_h(B_h+E_h)\bigr).\] 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 \[t\sum_i d_i\,(H_2+jJ)^{t-1}\cdot E_i.\] This is \(O(j^{r-1})\), since every intersection containing at least \(r\) factors of \(J|_{E_i}\) vanishes. Consequently \[\liminf_{m\to\infty} \frac{t!}{m^t}h^0\bigl(T,m(A+jJ)\bigr) \ \geq\ c j^r-O(j^{r-1})>0\] 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 \(\mathbf C(T)\). 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 [altord:p:observation]. We may first make the birational modification allowed in the hypothesis. Replace \(\mathbb V\) by the minimal rational subvariation from Lemma 3, 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 \[\pi_1(U)\longrightarrow\Gamma/\Gamma'\] 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 (Bakker et al. 2023, 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 \(\mathbf C(U')\).

Neatness makes the arithmetic action free. The universal local system on the quotient is \[(\mathcal D\times V_{\mathbf Z})/\Gamma',\] 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: \[[x,v]\longmapsto[\gamma x,\gamma v].\] 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 (Bakker et al. 2023, 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 \[G_f=H/\ker(H\longrightarrow\operatorname{Aut}(B))\] 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 6. 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 \[q^*\overline L\sim_{\mathbf Q}L'.\] 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 \[p_0:Y\longrightarrow S_0.\] We claim that \[ M\sim_{\mathbf Q}p_0^*L_0. \tag{6}\] 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 (Bakker et al. 2025, Lemma 5.6), or the unipotent extension theory (Schmid 1973; Cattani et al. 1986). 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 \[\mu^*M\sim_{\mathbf Q}cJ_W \sim_{\mathbf Q}t^*L' \sim_{\mathbf Q}\mu^*p_0^*L_0.\] 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 (6). In particular no boundary twist has been inferred from an equality merely on the open locus.

If \(\dim S_0=0\), equation (6) gives \(M\sim_{\mathbf Q}0\), and the theorem follows. Assume henceforth that \(\dim S_0>0\).

A common equivariant model and log general type. Normalize \(S_0\) in \(\mathbf C(S')\), and take a \(G_f\)-equivariant smooth projective resolution of the resulting model and its birational map to \(S'\). Denote the resulting maps by \[\eta:T\longrightarrow S',\qquad \pi:T\longrightarrow S_0.\] 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 \[ L_T:=cJ_T\sim_{\mathbf Q}\eta^*L' \sim_{\mathbf Q}\pi^*L_0. \tag{7}\]

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 \[U_N=T\setminus D_N.\] 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 (Schmid 1973; Cattani et al. 1986). 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 \[ K_T+D_N\quad\text{big}. \tag{8}\] Here the applicable theorem is (Brunebarbe and Cadorel 2020, 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 12: \(T\) 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 \[ r:=\nu(L_T)>0, \qquad \nu(L_T|_E)<r \tag{9}\] 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 \(\mathbf C(T)/\mathbf C(S_0)\) 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 12.

Removal of the boundary and ramification. Write the effective Jacobian divisor as \[R_\pi=K_T-\pi^*K_{S_0} =R_{\mathrm{div}}+R_{\mathrm{exc}},\] 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 14, take \[J=L_T,\qquad D=D_N+R_{\mathrm{div}},\qquad A=K_T-R_{\mathrm{div}} =\pi^*K_{S_0}+R_{\mathrm{exc}}.\] All components of \(D\) satisfy the rank inequality above, and \(A+D=K_T+D_N\) is big by (8). The volume lemma gives \[A+jL_T \sim_{\mathbf Q}\pi^*(K_{S_0}+jL_0)+R_{\mathrm{exc}} \quad\text{big for }j\gg0.\] Lemma 15 removes the effective divisors exceptional over \(S_0\) and descends bigness through the generically finite map. Therefore \[ K_{S_0}+jL_0\quad\text{is big for every sufficiently large integer }j. \tag{10}\]

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 \(\mathbf C(S)\) is already algebraically closed in \(\mathbf C(Y)\), 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 (6) pulls back to \(M\sim_{\mathbf Q}p^*L\). Since both \(S\) and \(S_0\) are smooth, the Jacobian divisor of \(\sigma\) is effective. Thus \[K_S+jL =\sigma^*(K_{S_0}+jL_0)+R_\sigma\] is big for \(j\gg0\) by (10) 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 \(\mathbf C(C)\) 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 \[\eta^{\otimes b}=h^*s.\] 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 \(\mathop{\mathrm{Supp}}D_C\), 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 \(\mathop{\mathrm{Supp}}D_C\).

If \(Q\subset T\) lies over \(P\subset\mathop{\mathrm{Supp}}D_C\), write \(e=e(Q/P)\) and \(d=\operatorname{ord}_P(D_C)\geq1\). Tameness gives the ramification coefficient \(e-1\), and \[e-1\leq ed=\operatorname{ord}_Q(\pi^*D_C).\] The canonical divisor formula for finite maps, interpreted at codimension-one points, therefore gives \[K_T\leq\pi^*(K_C+D_C).\] On \(W\) the same inequality holds away from the \(\rho\)-exceptional divisors. Hence there is an effective integral \(\rho\)-exceptional divisor \(E\) such that \[ K_W\leq h^*(K_C+D_C)+E. \tag{13}\] 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 (13) imply \[\kappa(W,K_W) \leq \kappa(W,h^*A+E) =\kappa(T,\pi^*A) =\kappa(C,A).\] 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 \(\mathbf C(C)\); 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 \[\kappa(C,A)=\kappa(C,(b+1)K_C)=0.\] Together with the nonzero form \(\eta\), this proves \(\kappa(W)=0\), and then \(h^0(W,K_W)=1\). ◻

Proof. Over \(\mathbf C(Z)\), 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 \(\mathbf C(Z)\); 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 16 to \(X_0\to Y_0\). All its modifications can be performed over \(Z\). Exceptionality over \(X_0\) implies exceptionality over \(X\). Lemma 18 and the subsequent family construction give the required rational variation and its top line. The two comparisons are proved in Lemma 20 below. ◻

Proof. The formula gives \[K_{\widetilde X}+R^- \sim_{\mathbf Q}g^*D+R^+.\] 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 \[H^0\bigl(\widetilde X,m(K_{\widetilde X}+R^-)\bigr) \simeq H^0(X,mK_X).\] 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 \[K_{\widetilde X_{\bar\eta}} \leq_{\mathbf Q} g_{\bar\eta}^*(D|_{Y_{\bar\eta}}) +R^+|_{\widetilde X_{\bar\eta}}.\] Here \(\leq_{\mathbf Q}\) means that the difference is \(\mathbf Q\)-linearly equivalent to an effective divisor. Formation of a proper direct image commutes with this flat base change, so (12) yields, for sufficiently divisible \(m\), \[(g_{\bar\eta})_* \mathcal O_{\widetilde X_{\bar\eta}} (mR^+|_{\widetilde X_{\bar\eta}}) =\mathcal O_{Y_{\bar\eta}}.\] Projection formula now gives \[\begin{align*} h^0(\widetilde X_{\bar\eta},mK_{\widetilde X_{\bar\eta}}) &\leq h^0\bigl(\widetilde X_{\bar\eta}, m g_{\bar\eta}^*(D|_{Y_{\bar\eta}}) +mR^+|_{\widetilde X_{\bar\eta}}\bigr)\\ &=h^0(Y_{\bar\eta},mD|_{Y_{\bar\eta}}). \end{align*}\] It follows that \[d=\kappa(F) \leq\kappa(Y_{\bar\eta},D|_{Y_{\bar\eta}}) \leq\dim Y_{\bar\eta}=d.\] 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 \(\mathbf C(Z)\). 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 (14); 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 \[Y\xrightarrow{r}W,\qquad t:W\to S,\qquad h:W\to Z, \qquad p=t\circ r,\quad q=h\circ r,\] 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 \[K_{Y'}+B'=\mu^*(K_Y+B)+E,\qquad E\geq0\] 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 \[a=\dim W_s,\qquad m=\dim Z.\] 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 \[0\longrightarrow \Omega^1_{S,s}\otimes\mathcal O_{W_s} \longrightarrow \Omega^1_W|_{W_s} \longrightarrow \Omega^1_{W_s}\longrightarrow0.\] 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 \[ h_s^*\omega_Z\longrightarrow \bigwedge\nolimits^{m-a}\Omega^1_{S,s}\otimes\omega_{W_s}. \tag{16}\] 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 (16) 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 \(S\), \(M|_G\sim_{\mathbf Q}0\), and adjunction gives \[D|_G\sim_{\mathbf Q}K_G+B|_G.\] 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 23 gives \[\kappa(Y_s,K_{Y_s}+B|_{Y_s})\geq \dim G+\kappa(W_s)\geq0.\] 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 (The Stacks Project Authors 2026, 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 \[\sigma:S_1\to S,\qquad \mu:Y_1\to Y,\qquad p_1:Y_1\to S_1\] 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 \[ K_{Y_1}+B_1=\mu^*(K_Y+B)+E,\qquad E\geq0,\quad E\ \text{\(\mu\)-exceptional}. \tag{17}\] Generic-fibre nonvanishing and field-extension base change allow us to choose a sufficiently divisible \(k>0\) for which \[\mathcal E= p_{1*}\mathcal O_{Y_1}\bigl(k(K_{Y_1/S_1}+B_1)\bigr)\] has positive rank. Theorem 24 applies to \(\mathcal E\).

Fix an ample Cartier divisor \(A\) on \(S_1\), and choose a positive integer \(\alpha\) so large that \[\sigma^*P-\frac{1}{k\alpha}A\] is big. For some \(\beta>0\), weak positivity makes \[\bigl(\mathop{\mathrm{Sym}}^{\alpha\beta}\mathcal E\bigr)^{**} \otimes\mathcal O_{S_1}(\beta A)\] 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 \[\mathop{\mathrm{Sym}}^\ell\mathcal E|_U\longrightarrow p_{1*}\mathcal O_{Y_1} \bigl(k\ell(K_{Y_1/S_1}+B_1)\bigr)|_U.\] 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 \[k\ell(K_{Y_1/S_1}+B_1)+\beta p_1^*A.\] Here \(U\) is the locally free locus, not merely a smaller open where the chosen sections generate.

The big divisor \[k\ell\sigma^*P-\beta A =k\ell\left(\sigma^*P-\frac{1}{k\alpha}A\right)\] 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 \[D_1:=K_{Y_1/S_1}+B_1+p_1^*\sigma^*P.\] 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 \[ \operatorname{div}(\varphi)+nD_1+T\geq0. \tag{18}\]

Choose compatible canonical divisors and put \[D_0=K_{Y/S}+B+p^*P,\qquad R=K_{S_1}-\sigma^*K_S\geq0.\] Equation (17) gives the exact identity \[D_1=\mu^*D_0+E-p_1^*R.\] Push (18) forward to the smooth variety \(Y\). Since \(\mu_*E=\mu_*T=0\), we obtain \[\operatorname{div}(\varphi)+nD_0 \geq n\mu_*p_1^*R\geq0.\] This proves rational linear effectivity of \(D_0\). ◻

Proof of Theorem [altord:ordinary-subadditivity] over \(\mathbf C\). If either \(\kappa(F)\) or \(\kappa(Z)\) is \(-\infty\), the assertion follows from the convention in the statement. Assume therefore that \[d:=\kappa(F)\geq0,\qquad \kappa(Z)\geq0.\] Proposition 19 and Lemma 20 provide algebraic fibre spaces \[\widetilde X\longrightarrow Y\xrightarrow{q}Z\] and an effective klt pair \((Y,B)\), together with a nef \(\mathbf Q\)-divisor \(M\) having the required Hodge realization. Writing \(D=K_Y+B+M\), their conclusions include \[ \dim(Y/Z)=d,\qquad \kappa(X)\geq\kappa(Y,D),\qquad D|_{Y_{\bar\eta}}\ \text{big}, \tag{19}\] where \(Y_{\bar\eta}\) is the geometric generic \(q\)-fibre.

Apply adjoint positivity, Theorem [altord:p:observation], and the compatible modifications of Lemma 21. We obtain an algebraic fibre space \(p:Y\to S\) with \(S\) smooth projective and \[M\sim_{\mathbf Q}p^*L,\qquad L\ \text{nef}.\] If \(\dim S=0\), then \(M\sim_{\mathbf Q}0\). Theorem 23, applied directly to \(q\), and (19) give the result. We may consequently suppose \(\dim S>0\). Adjoint positivity also gives \[K_S+jL\ \text{big for all sufficiently large integers }j.\]

Openness of the big cone on \(Y_{\bar\eta}\) permits a rational number \(v\) with \(0<v<1\) such that \[ (K_Y+B+vM)|_{Y_{\bar\eta}}\ \text{is big}. \tag{20}\] Lemma 25 shows that the geometric generic \(p\)-fibre has nonnegative log Kodaira dimension. Choose a sufficiently large integer \(j>1\) and an ample \(\mathbf Q\)-divisor \(H\) on \(S\) such that \[P_0:=K_S+jL-H\] is big. Lemma 26 gives \[ E_1:=K_Y+B+jM-p^*H \sim_{\mathbf Q}K_{Y/S}+B+p^*P_0 \ \sim_{\mathbf Q}\ \text{an effective divisor}. \tag{21}\]

Choose \[\lambda=\frac{1-v}{j-v}\in(0,1)\cap\mathbf Q, \qquad \lambda j+(1-\lambda)v=1.\] This balances the coefficients of \(M\); to cancel the corresponding term \(-\lambda p^*H\), put \[A_S=vL+\frac{\lambda}{1-\lambda}H,\qquad E_2=K_Y+B+p^*A_S.\] Then \(A_S\) is ample and \[ D\sim_{\mathbf Q}\lambda E_1+(1-\lambda)E_2. \tag{22}\] 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 \[\Gamma=\frac{1}{N}T_N\sim_{\mathbf Q}p^*A_S\] 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 \(E_2\sim_{\mathbf Q}K_Y+B+\Gamma\) is the log canonical class of an effective klt pair. Its restriction to \(Y_{\bar\eta}\) is the big divisor in (20) plus the nef divisor \(\lambda p^*H/(1-\lambda)\), and hence is big. Theorem 23 now yields \[\kappa(Y,E_2)\geq d+\kappa(Z).\] In sufficiently divisible degrees, multiplication by powers of a nonzero section in (21) embeds the corresponding linear systems of \((1-\lambda)E_2\) into those of \(D\). Positive rational scaling leaves Iitaka dimension unchanged, so \[\kappa(X)\geq\kappa(Y,D)\geq\kappa(Y,E_2) \geq\kappa(F)+\kappa(Z),\] as required. ◻

Proof. Flat base change for proper coherent cohomology gives, in every positive degree \(n\), \[H^0(V,nK_V)\otimes_K K' \simeq H^0(V_{K'},nK_{V_{K'}}).\] 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 (Schmid 1973; Cattani et al. 1986); precise formulations for the top line are given in (Bakker et al. 2025, sec. 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 \(\mathop{\mathrm{rank}}F^p=1\) 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 \[\exp\left(-\frac{\log u(s)}{2\pi\sqrt{-1}}N\right).\] 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, (Bakker et al. 2025, sec. 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 \(\operatorname{Pic}(P)\otimes\mathbf Q\) 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 37; that proof uses only the extension facts in Lemma 29, not any descent or rank-drop result. Taking the test form to be \(\omega^{d-k}\) gives Equation (23). Griffiths’ curvature formula gives \[ \ker\alpha=\ker(d\ell). \tag{24}\] 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 29 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 (Deligne 1971, 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 (Deligne 1971, 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 \(R=\mathop{\mathrm{rank}}V_0\). 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 \[b=\operatorname{lcm}\{q\geq1:\varphi(q)\leq R\}.\] Every scalar on every retained summand has \(b\)th power one. This uniform exponent is chosen before constructing the new variation.

Inside \(V_0^{\otimes b}\) take \[V_1=\bigoplus_{j\ \mathrm{retained}}V_j^{\otimes b}.\] 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 (25). 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 (Bakker et al. 2023, 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 \(\mathbf C(Y_0)\) in \(\mathbf C(P)\). This extension is finite, so the normalization is finite over \(Y_0\). Its function field is relatively algebraically closed in \(\mathbf C(P)\); 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 31 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 (25) 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 29 identifies its \(b\)-th power with \(h^*A_1\). ◻

Proof. After shrinking around the given point, the submersion theorem provides coordinates \[(z,q_1,\ldots,q_r,u_1,\ldots,u_{d-r-1}).\] 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 \(\mathop{\mathrm{rank}}(d\ell)=r\) 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 (25) 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 \[ \ell(t,s)= \exp\left(\frac{e\log t}{2\pi\sqrt{-1}}N\right)\Psi(t,s), \qquad \Psi(\zeta t,s)=T_s\Psi(t,s), \quad \zeta=\exp(2\pi\sqrt{-1}/e). \tag{26}\] 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 \[ F^p\cap NV=N(F^{p+1})=0. \tag{27}\] 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 \[\operatorname{Gr}^W_k V =P_k\oplus N\operatorname{Gr}^W_{k+2}V, \qquad P_k\xrightarrow{\ \sim\ }\operatorname{Gr}^W_k C.\] 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 29 and 30, its rank is \[r_i=\nu(A_1|_{D_i}).\] 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 (26) 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 \[Q(t,s)=\mathbb P(q_0)(\Psi(t,s))\] 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 33. The rank bound for \(\ell\) is \(r\) by Lemma 30. The transverse-loop lemma shows that \(T\) fixes the moving highest line everywhere on the connected universal cover. Condition (25) gives \(T=\mathop{\mathrm{id}}\), 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 \[K_Z+J\sim_{\mathbf Q}H+E,\qquad H\text{ ample},\quad E\ge0,\] 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 \[h^0\bigl(Z,m(B_a-J)\bigr) \ge h^0(Z,mB_a) -\sum_{k=0}^{m-1}h^0\bigl(J,(mB_a-kJ)|_J\bigr).\] Since \(J\) is reduced, its structure sheaf injects into \(\bigoplus_i\mathcal O_{D_i}\). Also \[mC_0+kJ=(m-k)C_0+k(C_0+J)\] 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 \[ h^0\bigl(Z,m(B_a-J)\bigr) \ge h^0(Z,mB_a) -m\sum_i h^0\bigl(D_i,m(H'+aA_1)|_{D_i}\bigr). \tag{28}\] Fix \(a\) and divide by \(m^d/d!\). Asymptotic Riemann–Roch for the ample divisors involved, along divisible \(m\to\infty\), yields \[ \mathop{\mathrm{vol}}(B_a-J) \ge B_a^d-d\sum_i(H'+aA_1)^{d-1}\cdot D_i. \tag{29}\] No uniform Riemann–Roch remainder in \(a\) is needed: the exact inequality in Equation (28) holds for every \(a\), and the limit is taken with \(a\) fixed.

The first polynomial on the right of Equation (29) 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 \(K_Z+aA_1\sim_{\mathbf Q}(B_a-J)+E\) is big. ◻

Proof of Theorem 28. Apply Lemmas 31 and 32. We obtain a variation on a dense open of a smooth projective \(Z\) satisfying Condition (25), with generically immersive full period map, and rational Hodge extension \(A_1\) satisfying \(b\pi^*L\sim_{\mathbf Q}h^*A_1\). 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 (Brunebarbe and Cadorel 2020, 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 34 applies to every component of \(J\), and Lemma 35 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 \(\pi^*L\sim_{\mathbf Q}h^*A\), 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 29 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 (Griffiths 1970, 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 (30).

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 \[ 0\leq\alpha\leq C\Omega, \qquad \Omega= \sum_{i=1}^{c}\frac{i\,dz_i\wedge d\bar z_i} {|z_i|^2\rho_i^2} +\sum_{i=c+1}^{n}i\,dz_i\wedge d\bar z_i. \tag{31}\] 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 (Brunebarbe and Cadorel 2020, sec. 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 (31), even where the period differential is degenerate.

By the nilpotent-orbit theorem (Schmid 1973, Theorem 4.12), a nonvanishing top extension frame, written in flat coordinates on a bounded angular sector, is \[ v(z)=\exp\left(\sum_{i=1}^{c} \frac{\log z_i}{2\pi i}N_i\right)u(z), \tag{32}\] where the \(N_i\) commute and \(u\) is holomorphic and nonvanishing. If \(\beta\) is the curvature of a fixed smooth metric on \(A\), write \(\alpha-\beta=dd^c\varphi\). The ratio of the two metrics satisfies \[ |\varphi(z)|\leq C\left(1+\sum_{i=1}^{c}\log\rho_i\right). \tag{33}\] This is the logarithmic growth estimate for canonical Hodge frames; see (Cattani and Kaplan 1989, 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 (32) 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 (33). We have therefore obtained the two controls needed for the cutoff argument: \[ 0\leq\alpha\leq C\Omega, \qquad |\varphi|\leq C\Bigl(1+\sum_i\log\rho_i\Bigr). \tag{34}\]

Chern-current representation for canonical Hodge extensions is established in (Cattani and Kaplan 1989, 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 \((n-j,n-j)\), \[ \int_{R\setminus D_R}\alpha^j\wedge\gamma =c_1(A)^j\cdot[\gamma]. \tag{35}\] 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 \[\alpha^j-\beta^j =dd^c\left(\varphi \sum_{k=0}^{j-1}\alpha^k\wedge\beta^{j-1-k}\right).\] Multiplying by \(\chi_M\gamma\) and integrating by parts makes the difference of the integrals equal to \[\int\varphi\,dd^c\chi_M\wedge \sum_{k=0}^{j-1}\alpha^k\wedge\beta^{j-1-k}\wedge\gamma.\] Its absolute value is bounded on escaping tails by a constant times \[\left(1+\sum_i\log\rho_i\right)\Omega^n.\] 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 (35). 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 (35) with complementary powers of an ample Kähler form. It gives vanishing of the \((r+1)\)st intersection power and strict positivity of the \(r\)th one. The nef intersection characterization proves (30). ◻

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 \(A|_{D_i}\).