Proof. We give the construction and then prove the assertion for the full coefficient lattice.

Step 1: the cyclic root and the pure vector. The cyclic-cover construction and its Higgs iteration follow the Viehweg–Zuo method (Viehweg and Zuo 2001, sec. 3); here we must also retain every coefficient in the original base lattice. Put \(N=K_X+E\). Normalize the cyclic cover defined by an \(m\)-th root of \(s\), and resolve it to obtain a projective generically finite map \(\psi:Z\to X\). The source may be disconnected. The tautological root gives a regular pairing \[ \psi^*\mathcal O_X(-N)\longrightarrow\mathcal O_Z. \tag{14}\] Choose a reduced simple-normal-crossing divisor on \(Z\) containing the inverse image of \(E\) and the exceptional divisors needed for the chosen compactification. Shrink the base to \(Y^\circ\subset V\) so that \(Z\to Y\) and every stratum of this divisor are smooth there. Write \(H\) for its restriction over \(Y^\circ\) and \(q:Z\setminus H\to Y^\circ\) for the resulting open family. Properness of the compactification and smoothness of all its strata give local topological triviality of this family.

The logarithmic mixed Hodge construction gives a variation on \(R^d q_*\mathbb Q\) with weight filtration \(W\) and Hodge filtration \(F\). We recall the precise ingredients used here. For a smooth projective compactification with a simple-normal-crossing boundary, residues identify the weight quotients of the logarithmic complex with complexes on the closed boundary intersections, shifted and Tate twisted. The associated weight spectral sequence degenerates at \(E_2\), the Hodge-to-de Rham sequence degenerates at \(E_1\), and the same residue filtration on a fixed form sheaf degenerates at \(E_2\); see (Deligne 1971, sec. 3.1.5, Theorem 3.2.5 and Corollary 3.2.13). These statements apply relatively on \(Y^\circ\): cohomology and base change give the Hodge bundles, and the proper smooth boundary strata give the pure Gauss–Manin systems occurring on the residue page. Their Gysin maps, including the Tate twists, are morphisms of pure variations. Their kernels and images split by polarization. Thus the weight quotients are polarizable rational variations, and their integral cohomological constructions give invariant lattices after discarding torsion. Their monodromy is quasi-unipotent. The proper stratum Gauss–Manin connections are algebraic and regular singular by the regularity theorem for Gauss–Manin connections (Deligne 1970, II, §6.14 and Theorem 7.9); residues and their cohomological maps are algebraic. Consequently all the Hodge and weight maps used below have their usual algebraic structures.

Over \(Y^\circ\) there is a canonical identification \[\Omega^d_{X/Y}(\log E)\otimes\mathcal O_X(-N) =f^*\mathcal O_Y(-L_Y).\] Pulling back the top relative logarithmic form and applying (14) therefore gives \[ u^{\mathrm{mix}}:\mathcal O_{Y^\circ}(-L_Y) \longrightarrow F^d(R^d q_*\mathbb C\otimes\mathcal O_{Y^\circ}). \tag{15}\] The form is nonzero: the root is nonzero generically, and pullback of a top relative differential under a generically finite map in characteristic zero is injective at the generic point. A nonzero global top logarithmic form represents a nonzero top Hodge class by Hodge-to-de Rham degeneration. Choose the smallest \(w\) for which the image of (15) lies generically in \(W_w\). Projection to \(\mathop{\mathrm{Gr}}^W_w\) is nonzero. Let \(\mathbb V=\mathop{\mathrm{Gr}}^W_w\) and let \(u\) be this projection. Since the fibers of \(q\) have dimension \(d\), \(F^{d+1}=0\). Thus \(u\) is in the highest nonzero Hodge grade of this pure variation, even when it does not span that grade.

For use with this construction, the Higgs field has the following description. Filter absolute logarithmic forms by the number of base differentials. The short exact sequence consisting of base degrees zero and one induces on relative de Rham cohomology the Gauss–Manin connection. This follows either from the logarithmic de Rham comparison or by lifting a flat class to an absolute class on a topologically trivial base disk. Taking its Hodge symbol gives the connecting map for the corresponding short exact sequence of individual form sheaves. Hodge-to-de Rham degeneration identifies the resulting cohomology sheaves with the Hodge grades. Since \(W\) is flat and all the filtrations are strict, these connecting maps preserve weight and induce the pure Higgs field on \(\mathop{\mathrm{Gr}}^W_w\).

Step 2: coefficient maps on higher models. We now work on a higher model \(Y'\). Resolve the main component of \(X\times_Y Y'\) to obtain \(f':X'\to Y'\), and let \(E'\) be the strict transform of \(E\) together with the reduced divisors exceptional over \(X\). Resolve the pair, preserving the smooth relative pair over \(Y'\setminus D'\). Then \[ \mathop{\mathrm{Supp}}f'^*D'\subset\mathop{\mathrm{Supp}}E'. \tag{16}\] Indeed, over \(D\) this follows from the original support inclusion. Outside \(D\), the image in \(Y\) of a \(\beta\)-exceptional divisor has codimension at least two. Smoothness of \(f\) there implies that its inverse image has codimension at least two in \(X\). Every divisor above it is consequently exceptional over \(X\) and belongs to \(E'\). The logarithmic discrepancy formula gives a pulled-back section \[s'\in H^0(X',\mathcal O_{X'}(mN')), \qquad N'=K_{X'}+E',\] whose rational identification with \(s\) is fixed by pullback. We may dominate both cyclic-root diagrams on a common resolution, preserving the chosen family over a common dense open.

Fix a prime \(A\subset Y'\). On the inverse image of a sufficiently small Zariski neighborhood of its generic point, the natural map \[ f'^*B'\longrightarrow\Omega^1_{X'}(\log E') \tag{17}\] is an injection of subbundles. If \(A\not\subset D'\), this is the relative smooth-pair condition. For a component of \(D'\), take its parameter \(t\) and tangential base coordinates \(y_2,\ldots,y_b\). At a point over a general point of \(A\), write \(f'^*t=v\prod z_\alpha^{n_\alpha}\) in boundary coordinates, with \(v\) a unit and at least one \(n_\alpha>0\). Its logarithmic differential has the nonzero residue vector \((n_\alpha)\). On each boundary stratum dominating \(A\), generic smoothness gives independent pullbacks of \(dy_2,\ldots,dy_b\) in the ordinary tangential directions. Strata not dominating \(A\), and the nonsmooth loci on the dominating strata, have images missing a dense open of \(A\); properness permits their removal on the base. The residue vector and these tangential differentials are linearly independent. This proves (17) at every point over that open of \(A\), without any smoothness assumption over the original boundary.

Let \(\Omega^i_{\mathrm{rel}}\) denote the exterior powers of the locally free quotient in (17). Its top determinant gives \[ \Omega^d_{\mathrm{rel}}\otimes\mathcal O_{X'}(-N') =f'^*\mathcal O_{Y'}(-L_{Y'}). \tag{18}\] For \(i\ge1\), the exterior-power filtration gives the exact sequence \[ \begin{split} 0\longrightarrow f'^*B'\otimes\Omega^{i-1}_{\mathrm{rel}} \longrightarrow \Omega^i_{X'}(\log E')/\mathrm{Fil}_{\mathrm{base}}^2 \longrightarrow\Omega^i_{\mathrm{rel}} \longrightarrow0. \end{split} \tag{19}\] The following connecting-map construction is the logarithmic cotangent procedure of Kovács (Kovács 2000, sec. 1, Lemma 1.1), applied before passing to the root family so that the coefficient lattice remains fixed. Tensor this sequence by \(\mathcal O_{X'}(-N')\), apply higher direct images, and use the projection formula. Starting with the unit map into the direct image of (18), its successive connecting maps give honest morphisms of coherent sheaves \[ \alpha_j:\mathcal O_{Y'}(-L_{Y'})\longrightarrow A_j\otimes(B')^{\otimes j},\qquad A_j=R^j f'_*(\Omega^{d-j}_{\mathrm{rel}}\otimes\mathcal O_{X'}(-N')). \tag{20}\] They are defined before passing to the root family. On the good open, differential pullback and root multiplication map each short exact sequence (19) to the corresponding sequence on that family. Naturality of connecting homomorphisms identifies the images of \(\alpha_j\) with \((\theta^{\mathrm{mix}})^j(u^{\mathrm{mix}})\). This argument uses morphisms of short exact sequences of \(\mathcal O\)-modules; it does not require root multiplication to commute with the de Rham differential or a connection on \(\mathcal O(-N')\).

Step 3: the semistable coefficient test. Take a general algebraic curve transverse to \(A\) and then a disk about its intersection point. Resolve the root diagram over that disk, keeping its smooth pair over the puncture, and include the special fiber in the resolved boundary. One-parameter semistable reduction, after finite ramification, gives a projective semistable family with smooth total space \[g:Z_\Delta\longrightarrow\Delta\] with reduced simple-normal-crossing special fiber \(V_0\), and a horizontal divisor \(H\) such that \(V_0+H\) has simple normal crossings. We use the toroidal construction with the horizontal divisor included in the boundary (Kempf et al. 1973, IV, §3); an explicit statement allowing the prescribed boundary is (Kollár and Mori 1998, Theorem 7.17). After resolving the special fiber, its parameter is monomial; the resolution centers over the bad fiber, followed by base change and toroidal subdivisions for the vertical monomial preserve the transverse horizontal coordinate hyperplanes. Thus the local equations on the resulting model are \[ \tau=z_1\cdots z_r, \qquad H:\ z_{r+1}\cdots z_{r+h}=0. \tag{21}\] In particular every closed horizontal intersection is itself a proper semistable family over \(\Delta\). The model maps to \(X'\) and to the chosen root family. Its boundary contains the inverse image of \(E'\).

Let \(i:\Delta\to Y'\) denote the ramified testing map. Pullback of forms, followed by the root pairing, gives \[p^*(\Omega^{d-j}_{\mathrm{rel}}\otimes\mathcal O_{X'}(-N')) \longrightarrow \Omega^{d-j}_{Z_\Delta/\Delta}(\log(V_0+H)),\] where \(p:Z_\Delta\to X'\) is the diagram map. This descends to relative forms because every pulled-back base logarithmic form from \(Y'\) is a form from \(\Delta\) and vanishes in the relative quotient. Its pole at \(0\) is at most logarithmic. The root pairing is regular on this model; alternatively, its monic power equation and normality prove this regularity. The natural coherent-cohomology pullback maps therefore give \[ i^*A_j\longrightarrow R^j g_*\Omega^{d-j}_{Z_\Delta/\Delta}(\log(V_0+H)). \tag{22}\] No cohomology base-change isomorphism at \(0\) is needed.

To see exactly what (22) tests, choose a local source generator \(e\) and an entire local frame \((b_\nu)_\nu\) of \((B')^{\otimes j}\), and write \[\alpha_j(e)=\sum_\nu a_\nu\otimes b_\nu.\] Every \(i^*a_\nu\) maps to a regular relative log-form cohomology class on \(\Delta\). We keep each \(i^*b_\nu\) as a basis vector of the external bundle \(i^*(B')^{\otimes j}\). On \(\Delta^*\), smooth-family comparison identifies these classes with the individual coefficients of the original mixed Higgs iterate. Each coefficient belongs to \(W_w\) there, because the mixed Higgs field preserves \(W_w\).

5 shows that its weight projection is a regular section of the Schmid-extended pure Hodge grade. This proves the required bound for every coefficient of (12). In particular, a tangential factor annihilated by the differential of \(i\) is still retained as an external frame vector. At an additional root-cover discriminant outside \(D'\), the transverse frame is \(dt\). Although \(d(\tau^e)=e\tau^e d\log\tau\), no coefficient is divided by \(e\tau^e\), since this differential substitution is never made on the external frame. This proves the bound with the original \(D'\), without adding that discriminant to it.

For completeness, these tests detect divisorial regularity. Take a finite level on which the variation is unipotent and a smooth compactification with normal-crossing boundary. Its canonical Hodge extensions are algebraic, and the rational coefficient maps are meromorphic in their frames. A pole at a prime persists on a general transverse disk; the preceding argument excludes it after divisible ramification. Further ramified tests preserve regularity by compatibility of the unipotent canonical extension with pullback. Since \(Y'\) and \(A\) were arbitrary, this proves the assertion on every stated higher model.

Step 4: homogeneous replacements. Finally, the tensor product Higgs field satisfies the Leibniz rule. Each coefficient of \(\theta^j(u^{\otimes a})\) is a finite sum of tensor products of coefficients of iterates of \(u\), with total iteration degree \(j\). Tensor products and flat Hodge-compatible projections commute with unipotent canonical extension. The same applies to the conjugate factors in a finite-orbit tensor construction on a common finite level. Thus every homogeneous tensor construction used in 7 preserves the asserted regularity, with source \(\mathcal O(-aL_{Y'})\). Those constructions use polynomial tensor operations and projections, so no division introduces a pole. This proves (13). ◻

Proof. Filter \(\Omega^i_{Z/\Delta}(\log(V_0+H))\) by the number \(k\) of horizontal logarithmic factors. In (21), taking a horizontal residue commutes with quotienting by \(d\log\tau=\sum_{a=1}^r d\log z_a\). Therefore the residue quotient of filtration degree \(k\) is \[ \bigoplus_{|I|=k} (\iota_I)_*\Omega^{i-k}_{H_I/\Delta}(\log(V_0|_{H_I})), \qquad H_I=\bigcap_{\alpha\in I}H_\alpha. \tag{23}\] The terms are zero when their form degree is negative. The maps \(g_I:H_I\to\Delta\) are proper projective semistable families. The semistable logarithmic de Rham comparison (Steenbrink 1977, sec. 2, Theorem 2.7, Corollaries 2.9–2.10 and Theorem 2.11), in the canonical-extension form explained in (Fujino and Fujisawa 2014, sec. 4.9 and the proof of Lemma 4.10, Step 1), with its strictness argument completed in (Fujino and Fujisawa 2017), identifies \[R^q(g_I)_*\Omega^p_{H_I/\Delta}(\log(V_0|_{H_I})) \quad\text{with}\quad \mathop{\mathrm{Gr}}_F^p\overline{R^{p+q}(g_I|_{\Delta^*})_*\mathbb C},\] where the bar means unipotent canonical extension with its extended Hodge filtration. In particular these are locally free sheaves.

Consider the spectral sequence of the finite residue filtration of this single relative form sheaf. We use the indexing \[E_1^{-k,j+k}= \bigoplus_{|I|=k}R^j(g_I)_* \Omega^{i-k}_{H_I/\Delta}(\log(V_0|_{H_I})).\] On \(\Delta^*\), \(d_1\) is the Hodge grade of the signed Gysin map from the \(k\)-fold intersections to the \((k-1)\)-fold intersections, with the Tate twists included. Source and target have the same pure weight \(i+j+k\). A morphism of polarizable pure variations has a flat Hodge kernel and image and admits flat Hodge complements. Consequently it factors as a projection to a complement, an isomorphism onto its image, and an inclusion. Canonical extension and extension of \(F\) preserve this direct-sum factorization. Its extended Hodge-graded map has constant rank at \(0\).

The coherent \(d_1\) and that extended Gysin map are holomorphic maps between the same locally free \(E_1\) terms and agree on \(\Delta^*\). They therefore agree everywhere. It follows that \(E_2\) is locally free and is precisely the canonical Hodge-graded extension of the pure residue cohomology. By the fixed-form version of Deligne’s residue theorem (Deligne 1971, Corollary 3.2.13(iii)), every \(d_r\) for \(r\ge2\) vanishes on \(\Delta^*\). Inductively its target is locally free on \(\Delta\), so a holomorphic map vanishing on the puncture vanishes everywhere. Thus \(E_2=E_\infty\) on the disk. This is also the constant-rank extension mechanism in (Fujino and Fujisawa 2014, Proposition 3.12 and the proof of Lemma 4.10, Step 1); here the residue calculation (23) supplies its particular geometric complex.

The abutment has a finite filtration with these locally free successive quotients. It and every filtration step are therefore locally free, and the filtration steps are subbundles. On \(\Delta^*\) this filtration, with the usual cohomological shift, is the induced weight filtration on \(\mathop{\mathrm{Gr}}_F^i H^{i+j}(Z_t\setminus H_t)\): this is exactly the compatibility of the two filtrations in the logarithmic mixed Hodge complex, or equivalently the fixed-form assertion in the cited corollary. Its extended quotients were identified on \(E_2\) above. Finally, the image of a regular class in the quotient by a weight subbundle is a section of a locally free sheaf. If it is generically zero it is zero. The class hence lies in that subbundle on the entire disk and has a regular image in its successive quotient, as claimed. ◻

Proof. Near \(y\) the original morphism is smooth and its boundary is relative simple normal crossing. Its fiber is reduced. Consequently, in any local frame of \(\mathcal O_X(m(K_X+E))\), the coefficient of \(s\) belongs to \(\mathfrak m_y\mathcal O_X\). On the blowup the ideal \(\mathfrak m_y\) becomes the invertible ideal of \(A\). Along a transverse parameter \(t\) at its general point the pulled coefficient is therefore divisible by \(t\).

Use throughout the pulled-back old line \(\mathcal O_X(K_X+E)\) and the old relative logarithmic forms. After \(t=\tau^e\), the tautological root \(r\) satisfies, in this old line frame, \[ \left(r/\tau^{e/m}\right)^m=s/\tau^e. \tag{24}\] The right-hand side is regular. The left-hand root is a rational section integral over the normal root model and is therefore regular there. It stays regular on the semistable model used above. Thus the map from old relative top logarithmic forms into the semistable relative forms has the factor \(\tau^{e/m}\).

Passing to coherent cohomology retains that factor. Divide it out first: the resulting regular class still lies generically in \(W_w\), since multiplication by a nonzero scalar on the puncture does not change membership in a weight step. By 5, its projection to \(\mathop{\mathrm{Gr}}^W_w\) is regular. Multiplying the projected class back by \(\tau^{e/m}\) proves the claimed order for \(u\). Every degree-\(a\) homogeneous tensor operation multiplies the order by \(a\), and every flat Hodge-compatible projection preserves this divisibility. The result follows for \(u_*\). This calculation never replaces the old pulled-back source line by \(\mathcal O(-L_{Y'})\) on the blown-up base; the asserted positive order is in the original source frame. ◻

Proof. André’s monodromy normality theorem (André 1992, sec. 5, Theorem 1) applies to this polarizable pure variation over a smooth connected algebraic variety: the connected monodromy is normal in the derived generic Mumford–Tate group. It is therefore semisimple. The same statement applies after a finite étale cover and after restriction to a smooth algebraic subvariety, with its own generic Mumford–Tate group.

The theorem of the fixed part for polarized variations says that the invariant part of each tensor variation is a constant Hodge structure; see (Schmid 1973, Theorem 7.22 and Corollary 7.23). These tensor invariants determine the reductive group \(G_c\). To see the latter assertion in the form needed here, a \(G_c\)-stable subspace in a tensor representation has a \(G_c\)-equivariant complementary subspace, so it can be encoded by its invariant projector. The tensor-stabilizer description of an algebraic subgroup then applies (Milne 2017, Theorem 9.2 and §§9.6–9.7). Thus, in a fixed flat frame, the Hodge circles \(h_x\) normalize \(G_c\), and \[h_x(z)h_{x_0}(z)^{-1}\in G_c(\mathbb R).\] In particular their images in the quotient by \(G_c\) are constant. For completeness, local conjugacy follows directly from compactness of the circle. Along a smooth path of these homomorphisms, the logarithmic derivative is a Lie-algebra-valued cocycle for the circle action. Averaging a cocycle over the circle writes it as a coboundary. Integrating the resulting time-dependent element of \(\mathop{\mathrm{Lie}}G_c(\mathbb R)\) conjugates the circles along the path. One can do this first in the adjoint group and lift through the central isogeny; the remaining central discrepancy is locally constant.

The tangent to the filtration orbit is obtained from the negative degrees in \(\mathfrak g_{c,\mathbb C}\). Griffiths transversality makes its only possible degree equal to \(-1\), giving \(t(v)\) and its indicated action in every representation. Higgs integrability gives the commutation relation in (28).

The Lie algebra sits as a rational flat subbundle of \(\mathop{\mathrm{End}}\mathbb V\). Normalization by the Hodge circles makes it a Hodge subvariation, and restriction of the endomorphism polarization polarizes it. Intersecting its rational fiber with the endomorphism lattice supplies an invariant full lattice. These assertions also hold for the stated rational sums of ideals. Finally, preservation of the polarization and normalization by the Hodge grading imply stability under Hodge-metric adjoint. On the real Lie algebra the negative adjoint is the Cartan involution obtained from the Weil operator. Its complex antilinear extension is the claimed compact conjugation. A compact real form of a semisimple complex algebra is the direct sum of compact real forms of its simple ideals, so this conjugation preserves each ideal. ◻

Proof. At connected monodromy level decompose the complex representation into \(G_{c,\mathbb C}\)-isotypical summands. Each is stable under the Hodge circle, since that circle acts on the connected group by inner automorphisms. An isotypical summand is the tensor product of an irreducible representation with its multiplicity space. After a finite covering of the circle, lift its inner action to the first factor; the remaining circle action is on the multiplicity space. Its eigenspace decomposition permits a decomposition into group-irreducible summands stable under the circle. The corresponding flat projectors are Hodge-compatible at a basepoint and, by 8, everywhere. At least one projector has generically nonzero value on \(u\).

Conjugate this projector by the full monodromy group. Its orbit is finite because \(G_c\) acts trivially on the projector. Tensor the projected vectors over that finite orbit. On the normal connected level all factors are generically nonzero: the deck transformations carry one projected vector to the others. Permutation of the factors makes the tensor construction equivariant for full monodromy, so it descends. Its source is a positive tensor power of \(\mathcal L\).

For the connected group take the Cartan component, the irreducible summand of highest weight equal to the sum of the highest weights of the factors. It has multiplicity one, and its projection is preserved by simultaneous automorphisms and permutations of the factors. This projection never kills a pure tensor of nonzero vectors. Indeed, for a nonzero vector in an irreducible representation its coefficient along an extremal weight line after translation by \(g\in G_{c,\mathbb C}\) is a nonzero regular function of \(g\). Finitely many such functions are simultaneously nonzero on a nonempty open set. Their product is the corresponding extremal coefficient in the Cartan component. Since the projection commutes with translation, its value on the original tensor was nonzero as well. The projector preserves the Hodge grading. The projected vector is in the highest Hodge degree of the tensor, which, because its projection is nonzero, is also the highest degree of the retained Cartan component.

Retain each rational simple adjoint factor having at least one complex simple ideal acting nontrivially on this irreducible representation. This collection is full-monodromy invariant. Let \(q:M\to\operatorname{Aut}(\mathfrak g)\) be the projected adjoint action, and put \(K=\ker q\). Every element of \(K\) commutes with the represented Lie algebra. Schur’s lemma implies that its action on the irreducible representation is scalar. Every ideal in \(\mathop{\mathrm{Lie}}K\) is unrepresented by the definition of the retained collection, so \(K^\circ\) acts trivially. Consequently the image of \(K\), although \(K\) itself need not be finite, is a finite group of scalars. Choose a positive integer killing that image and take the Cartan component in the corresponding tensor power. The resulting representation factors through \(q(M)\), and the preceding nonvanishing argument applies again. In particular its connected representation factors through the adjoint group \(G\).

These constructions, as well as subsequent invariant projections, are homogeneous of positive degree in \(u\). The number of retained rational factors is a nonnegative integer, so it has a minimum among these choices. Fix a minimizing choice. Its purpose is to prevent a further nonzero invariant projection from discarding a factor. When a factor fixes the highest space throughout its orbit, this minimality will force the instability that supplies an additional flag.

Connected irreducibility does not require the highest Hodge step \(E_0\) to have rank one; the construction keeps the entire step. The adjoint representation is faithful on \(G\). Tensor generation for a reductive group, (Milne 2017, Theorem 4.14), realizes each of its representations as a summand of finite sums of mixed tensors of this faithful representation. For a factor of \(G\) the Hodge grading on its Lie algebra is a cocharacter of the adjoint group. Its action in an irreducible representation and the grading inherited from \(\mathbb A\) induce the same action on the Lie algebra; their quotient is scalar. They thus differ only by a constant Hodge shift. Equivariant projectors on the Lie tensors preserve the grading, and the induced compact-form metrics have the asserted compatibility with adjoints. Applying this separately to the factors gives the tensor decomposition of \(E_0\). Finally the parabolic associated with the Hodge grading preserves its maximal-weight subspace. The stabilizer of \(E_0\) therefore contains a parabolic, proving that its orbit is closed and projective.

All projectors used above are flat Hodge morphisms. They and all tensor operations extend on unipotent canonical extensions, by the functoriality recalled in [sec:preliminaries]. The Leibniz rule for the Higgs field proves the last assertion of 7 about iterates. ◻

Proof. On a nontrivially represented complex ideal in an invisible rational factor the maximal Hodge subspace is invariant under the whole simple group. Irreducibility makes it the whole representation factor. The grading on that representation is scalar, and therefore its induced Lie grading is zero. The Hodge metric on this Lie ideal is consequently flat and positive definite. Monodromy acts isometrically for this compact structure.

Fix an orbit of invisible rational factors under the full monodromy, and let \(U\) be the complexification of their product. Consider the action of \(U\) on a general nonzero value of \(u_*\). If this vector were not unstable for the origin, some homogeneous invariant polynomial of positive degree would be nonzero on it. Equivalently, its tensor power would have a nonzero projection to the \(U\)-invariant subspace. This elementary invariant-theoretic equivalence follows because invariants separate the disjoint closed sets \(\{0\}\) and a closed orbit in the orbit closure; averaging over a maximal compact subgroup gives the invariant projection. The projection is Hodge-compatible: \(U\) is normalized by the Hodge circle. It is also compatible with full monodromy, because the chosen product of rational factors is invariant under it. After repeating 9, the nonzero invariant projection eliminates all factors in this orbit and introduces none. This contradicts minimality. Thus the vector is generically unstable for each such \(U\).

Choose on the cocharacter lattice of \(U\) an integral positive length form invariant under its Weyl group and under the finite factor automorphisms in question. One obtains it by summing positive invariant forms over those automorphisms. Kempf’s optimal one-parameter subgroup theorem (Kempf 1978, sec. 3), in its origin-instability form (Kempf 1976, Theorem 6), applied to the affine representation and the closed origin, assigns to an unstable vector a unique optimal parabolic. It is equivariant under \(U\) and under every automorphism preserving the representation and the length form. It is unchanged by multiplying the vector by a nonzero scalar. It is proper, since \(U\) is semisimple and an optimal positive slope requires a noncentral cocharacter.

We spell out why this pointwise assignment supplies an algebraic section on a dense open. Fix a maximal torus. There are finitely many subsets of the weight set of the representation. For each subset, maximizing the positive normalized minimum pairing is the closest-point problem for its convex hull, with respect to the chosen rational positive form. It determines a rational optimal ray when the minimum is positive. Thus there are only finitely many optimal cocharacter directions and slopes up to conjugacy. The conditions that a translate of a vector have a specified nonzero weight support are constructible. Their images, and the conditions selecting the maximal slope, are constructible by Chevalley’s theorem. The resulting graph of optimal parabolics on the unstable locus is constructible and single valued. Use the algebraic principal monodromy reduction of 10; its associated projective representation and flag bundles are algebraic. Equivariance makes the graph an algebraic constructible subset of their product over the base. Pull this graph back along the actual algebraic section \([u_*]\), on the dense open where that section is unstable. The resulting constructible graph has exactly one point over each point of this open base. Over an irreducible base in characteristic zero, its dominant component is birational to the base: after restricting to dense opens, its projection is a generically injective morphism and therefore has degree one. It consequently defines a rational section, regular after shrinking. Applying this argument to the actual section avoids any assumption that it misses the indeterminacy locus of a rational map on the ambient projective representation bundle. The construction uses algebraic associated bundles, not algebraic local trivializations by holomorphic flat frames.

An optimal cocharacter has zero component on every ineffective factor, since deleting such a component preserves its slope numerator and decreases its length. All nontrivial flags therefore occur on the isometric complex ideals identified above. On the irreducible connected level the generic parabolic types are fixed. The component action permutes these types along with the rational factors. Properness on at least one factor in an orbit implies properness on some complex factor of each rational factor in that orbit. Repeating this for all invisible orbits gives \(j_{\mathrm u}\).

The highest-space marks already see every other rational factor. Stabilizers of highest spaces and of the new flags are parabolic, and the two constructions use disjoint factors. Thus their joint orbit is the asserted product \(J\). A tensor product of highest subspaces determines each factor subspace, so this description is also valid when \(E_0\) is presented as a single subspace of \(\mathbb A\). Embed a parabolic flag by the top exterior power of its parabolic subalgebra. The dual tautological Plücker line is positive, with the metric induced by the flat compact metric. Tensoring these lines gives a positive line invariant also under the relevant factor permutations. ◻

Proof. A rational endomorphism preserving the filtration of a pure Hodge structure also preserves its conjugate filtration, hence is a Hodge endomorphism. It commutes with the Mumford–Tate group. At a Hodge-generic point this is the generic Mumford–Tate group, which contains the connected monodromy by (André 1992, sec. 5, Theorem 1). If \(\alpha\) is also a Lie automorphism, then for every \(x\in\mathfrak a\), \[\operatorname{ad}(\alpha x) =\alpha\operatorname{ad}(x)\alpha^{-1} =\operatorname{ad}(x).\] The Lie algebra has zero center, so \(\alpha x=x\). Such points may be chosen while imposing finitely many generic differential rank conditions: in a flat chart the proper loci where any one of the countably many rational tensors acquires an additional Hodge condition can be avoided together with those conditions. ◻

Proof. Use the full filtration on the rational integral polarized Lie variation, not only the highest subspace of \(\mathbb A\). Let \(\Gamma\) be an arithmetic group of isometries of its polarized lattice containing its monodromy, and let \(\Omega\) be its period domain, or the associated generic Mumford–Tate domain. The algebraicity theorem for pure integral period maps (Bakker et al. 2023, Theorem 1.1) factors the period map through a dominant algebraic map to a quasi-projective algebraic image. Its hypotheses hold here by 8; in particular a lattice has been retained in the full Lie system. Normalize this image in the relative algebraic closure of its function field in \(\mathbb C(B)\). The resulting map \(b_0:B^\circ\dashrightarrow T^\circ\) has geometrically connected generic fiber. Shrink to smooth opens where it is a morphism. The dimension of \(T^\circ\) is the generic rank of the full period map. Indeed, the real period stabilizer is compact, arithmetic orbits are discrete, and the factorization preserves the local analytic image dimension.

Here is the local-system descent, including possible finite monodromy. Compactify the graph properly over \(T^\circ\), resolve the complement, and shrink \(T^\circ\) so that the compactification and all strata of its SNC complement are smooth over it. Lifting vector fields tangent to the strata gives a local differentiable trivialization of this proper smooth pair and hence of its open part. We may thus assume that \(b_0\) is a topological fiber bundle with connected fibers. Removing a proper algebraic subset of a smooth complex variety induces a surjection on its fundamental group, so these shrinkings do not decrease the relevant monodromy images.

Choose a Hodge-generic point of \(B^\circ\). Along the universal cover of its \(b_0\)-fiber the lifted Lie-period map has values in one discrete arithmetic orbit; it is therefore constant. Every vertical loop acts by a rational Lie automorphism fixing that lifted period. 12 makes its action trivial. The exact sequence \[\pi_1\bigl(b_0^{-1}(t)\bigr)\longrightarrow \pi_1(B^\circ)\longrightarrow\pi_1(T^\circ) \longrightarrow1\] now descends the Lie representation. The representation on \(\mathbb A\) descends as well: by 9 it factors through the full projected Lie action, including its component group. This is where removal of the finite scalar image is necessary.

The Lie filtration is constant in flat coordinates along each connected fiber. The derivative of the filtration on \(\mathbb A\) is the action of (28); it too vanishes on those fibers. Consequently both filtrations descend. Holomorphicity, Griffiths transversality, and polarization can be checked after the smooth surjective pullback. The descended local systems have quasi-unipotent local monodromy: for a boundary valuation of \(T\) choose a valuation above it on a proper model of \(B\); a positive power of its monodromy is a monodromy of the original variation. Quasi-unipotence of that power implies quasi-unipotence of the original operator. Canonical extension and GAGA, or regular-singular Riemann–Hilbert, therefore give the algebraic flat bundles and algebraic Hodge bundles on these opens; see (Deligne 1970, II) and [sec:preliminaries].

Over \(T^\circ\) consider the algebraic flag bundle of the descended compact factors. The section \(j_{\mathrm u}\) gives an algebraic map from \(B^\circ\) into that bundle. Take its image and again normalize in the relative algebraic closure inside \(\mathbb C(B)\). This is the desired \(S^\circ\). Variations pull back from \(T^\circ\), and the compact flag descends by its tautological definition. On a dense open the finite map from \(S^\circ\) to its image is étale, and the period-image factorization preserves tangent rank. Hence the full period and compact marks have joint differential of rank \(\dim S\). Resolve projective compactifications and the map from \(B\) to obtain \(b:B\to S\) with connected fibers. The monodromy images on the open source and target agree by the fundamental-group sequence just used.

For a monodromy-stable collection of rational factors the projected Lie system again has its rational polarization and lattice. The identical argument gives its quotient and descent. At connected level every rational normal factor collection is permitted. ◻

Proof. Taking a Cartier multiple and dividing its weights and curvature by that multiple reduces to a line bundle. We use \(dd^c\) normalized so that the curvature of a metric of weight \(\varphi\) is \(dd^c\varphi\).

First we extend its local plurisubharmonic weights. For \(\epsilon>0\), the function \[\varphi_\epsilon=\varphi+ \epsilon\sum_{i=1}^{\ell}\log|z_i|\] is plurisubharmonic off the coordinate divisor and is locally bounded above near it, by (32). The removable singularity theorem extends it plurisubharmonically to the polydisk. On a smaller polydisk, the maximum principle, applied successively in the coordinate variables, bounds it above by its values on a fixed distinguished boundary torus disjoint from \(\Delta\). These bounds are uniform as \(\epsilon\downarrow0\). Its increasing limit, with upper semicontinuous regularization, is therefore a plurisubharmonic extension of \(\varphi\). The extensions agree under the pluriharmonic frame changes and give a closed positive current \(T\in c_1(L)\) on \(Z\).

This current has zero Lelong number at every point. At a boundary point approach along a path on which all the vanishing boundary coordinates have absolute values comparable to a common parameter \(r\downarrow0\). The two-sided bound in (32) is then \(O(\log\log(1/r))=o(|\log r|)\). The growth characterization of the Lelong number gives an upper bound of zero; positivity gives the reverse bound. Away from \(\Delta\) the current is smooth. Demailly’s regularization theorem, in its zero-Lelong-number consequence (Demailly 1992, Corollary 6.4), now shows that \(c_1(L)\) is nef.

We next prove the mass assertion, since zero Lelong numbers alone would not suffice for it. Fix \(\epsilon>0\). Nefness makes \(\alpha_\epsilon=c_1(L)+\epsilon[\eta]\) a Kähler class. Choose a Kähler representative \(\omega_\epsilon\) and write \[T+\epsilon\eta=\omega_\epsilon+dd^c\phi.\] Here \(\phi\) is a global \(\omega_\epsilon\)-plurisubharmonic function, smooth on \(Z^\circ\); changing the smooth background has not changed its logarithmic growth bound. Write the irreducible components of \(\Delta\) as \(\Delta_i\), choose their defining sections \(s_i\) with smooth metrics, and rescale these metrics so that \[u_i=-\log\|s_i\|^2\geq1\quad\text{on }Z^\circ.\] For any fixed \(0<b<1\) and sufficiently small \(\delta>0\), the function \[ \psi=-\delta\sum_i u_i^b \tag{35}\] is \(\omega_\epsilon\)-plurisubharmonic. Indeed, on \(Z^\circ\) each \(dd^c u_i\) is a fixed smooth form and \[dd^c(-\delta u_i^b) =-\delta b u_i^{b-1}dd^c u_i +\delta b(1-b)u_i^{b-2}du_i\wedge d^c u_i.\] The second summand is positive and the first is bounded below by \(-\delta C\omega_\epsilon\), because \(u_i\geq1\). Decrease \(\delta\) so that the sum of these lower bounds is at least \(-\tfrac12\omega_\epsilon\). The same extension argument applies across \(\Delta\). Since \(\log u=o(u^b)\), the lower growth bound for \(\phi\) also gives a constant \(C_\epsilon\) such that \[ \phi\geq\psi-C_\epsilon. \tag{36}\]

We verify directly that \(\psi\) has full Monge–Ampère mass. Take a smooth decreasing function \(\chi:[0,\infty)\to[0,1]\) that is \(1\) on \([0,1]\) and \(0\) on \([2,\infty)\). For \(R>1\) set \[f_R(t)=1+\int_1^t b s^{b-1}\chi(s/R)\,ds\qquad(t\geq1).\] Then \(f_R=t^b\) for \(t\leq R\), \(f_R\) is increasing and concave, and it is constant for \(t\geq2R\). Its derivative bounds, uniformly in \(R\), are \[ 0\leq f_R'(t)\leq C t^{b-1},\qquad 0\leq-f_R''(t)\leq C t^{b-2}. \tag{37}\] Moreover \(f_R\uparrow t^b\) as \(R\to\infty\). The functions \(\psi_R=-\delta\sum_i f_R(u_i)\) extend smoothly over \(Z\), are \(\omega_\epsilon\)-plurisubharmonic for the same choice of \(\delta\), and equal \(\psi\) locally on \(Z^\circ\) once \(R\) is sufficiently large.

In a fixed boundary coordinate chart, comparison of \(du_i=-d\log|z_i|^2+\) a smooth form and (37) gives a bound of positive forms \[ 0\leq\omega_\epsilon+dd^c\psi_R \leq C\left(\eta_0+ \sum_{i=1}^{\ell} \frac{\sqrt{-1}\,dz_i\wedge d\bar z_i} {|z_i|^2(-\log|z_i|)^{2-b}}\right), \tag{38}\] where \(\eta_0\) is a smooth positive coordinate form and \(C\) is independent of \(R\). The right-hand side has an integrable \(n\)th power. Each singular summand has rank one and its square is zero, and its normal integral is bounded by a constant times \[\int_0^{r_0}\frac{dr}{r(-\log r)^{2-b}}<\infty.\] Products involving different normal coordinates are integrable by Fubini. A finite chart covering and dominated convergence now imply \[\begin{align*} \int_{Z^\circ}(\omega_\epsilon+dd^c\psi)^n &=\lim_{R\to\infty} \int_Z(\omega_\epsilon+dd^c\psi_R)^n\\ &=\int_Z\omega_\epsilon^n. \end{align*}\] The last equality is Stokes’ theorem for the smooth approximants. Since the non-pluripolar Monge–Ampère product does not charge \(\Delta\), this is exactly full mass for \(\psi\).

The full-mass class is preserved on passing to a less singular plurisubharmonic potential. Apply (Guedj and Zeriahi 2007, Proposition 1.6) to the Kähler form \(\omega_\epsilon\) and (36). All the hypotheses are satisfied: \(Z\) is compact Kähler, both potentials are \(\omega_\epsilon\)-plurisubharmonic, and the lower potential has just been shown to have full mass. It follows that \[ \int_{Z^\circ}(\Theta+\epsilon\eta)^n =(c_1(L)+\epsilon[\eta])^n \qquad(\epsilon>0). \tag{39}\] All mixed integrands on the left are nonnegative. The identity for one positive \(\epsilon\) therefore proves that each has finite integral. Expansion of (39) and comparison of polynomial coefficients proves (33).

Finally, a smooth semipositive \((1,1)\)-form has \(\Theta^k\wedge\eta^{n-k}>0\) at a point exactly when its rank there is at least \(k\). If this happens at one point it happens on an open neighborhood, so its integral is positive. Combining this observation with (33) and (5) proves (34). For a real-analytic form the nonzero maximal minors cannot vanish on an open set, so the maximal-rank locus is dense. This proves the final assertion. When \(n=0\) the statements follow from the empty-product convention. ◻

Proof. Let \(d=\dim S\) and work first on a smooth unipotent level \(p:\widetilde S\to S\), resolving further when necessary. On its open part the joint differential has image \[\mathcal J\subseteq \mathop{\mathrm{Gr}}_F^{-1}\mathfrak g_{\mathbb C}\oplus j_{\mathrm u}^*T_{J_{\mathrm u}}.\] The differential here is \((t,dj_{\mathrm u})\), where the second component is the vertical derivative in flat flag charts. It is globally defined because the transition maps are flat. Its rank is \(d\) by 13. Resolve the map to the corresponding relative Grassmannian so that \(\mathcal J\) extends as a subbundle of the indicated extended ambient bundle.

Equip it with the subbundle metric. We give the curvature calculation because it uses the full period differential, whereas the nef line \(\mathcal H\) only uses the marks \(j\). Fix a tangent vector \(v\), and choose an orthonormal frame \((a_\mu,b_\mu)\) of \(\mathcal J\) at the point. Each \(a_\mu\) lies in the image of \(t\), so integrability gives \([t(v),a_\mu]=0\). The Hodge curvature formula on the degree \(-1\) Lie space consequently gives \[\sum_\mu \bigl\langle\Theta_{\mathop{\mathrm{Gr}}_F^{-1}\mathfrak g}(v,\bar v) a_\mu,a_\mu\bigr\rangle =-\sum_\mu\bigl\|[t(v)^*,a_\mu]\bigr\|^2.\] The convention is the curvature normalization of (29). The flag tangent curvature is bounded above by \(C\omega_{\mathrm u}(v,\bar v)\) times the identity, where \(\omega_{\mathrm u}\) is the positive compact-flag form pulled back along the section. Such a uniform \(C\) exists because the flag variety is compact and the flat transition actions are isometric. Subbundle curvature subtracts a squared second fundamental form. After increasing \(C\) by the fixed rank \(d\), this proves \[ \Theta_{-\det\mathcal J+c\widetilde{\mathcal H}_{\mathrm u}} (v,\bar v) \geq \sum_\mu\bigl\|[t(v)^*,a_\mu]\bigr\|^2 +(c-C)\omega_{\mathrm u}(v,\bar v). \tag{44}\] The right side is positive whenever the joint differential is nonzero. Indeed, if its compact component is nonzero, the last term is positive for \(c>C\). Otherwise \(t(v)\ne0\). Since \((t(v),dj_{\mathrm u}(v))\) belongs to \(\mathcal J\), vanishing of all the commutators in the first term would imply \([t(v)^*,t(v)]=0\). But \(t(v)\) acts as a nonzero lowering nilpotent in the faithful adjoint representation. A normal nilpotent endomorphism is zero, a contradiction. Thus the line \[R=-\det\mathcal J+c\widetilde{\mathcal H}_{\mathrm u}\] has semipositive curvature and full generic curvature rank. The extended ambient Hodge metrics and their duals have logarithmic bounds, the compact metric is smooth, and the resolved subbundle metric has the same bounds. By 14, \(R\) is nef with numerical dimension \(d\), hence big.

It remains to compare \(R\) with the claimed adjoint on \(S\). The differential gives a generically invertible map \[ p^*T_S(-\log D_0)\dashrightarrow\mathcal J. \tag{45}\] It is regular at every prime upstairs which dominates a prime of \(S\). At a marked prime, the finite cover is generically a power substitution \(t=z^e\). The lift of \(t\partial_t\) is \(e^{-1}z\partial_z\). The logarithmic Higgs field extends on the unipotent canonical extension, so its Lie component is regular; one recovers the Lie-valued \(t\) from the adjoint Higgs field by the split adjoint inclusion into endomorphisms. The resolved compact flag is regular upstairs, so its derivative on \(z\partial_z\) is regular as well. The tangential generators cause no poles. At an unmarked prime the monodromy level is generically unramified, since it was defined using only the projected adjoint monodromy. The local Lie monodromy is the identity and the extended period has no logarithmic term. The same regularity assertion follows using the ordinary tangent generator there.

These regular ambient maps land in \(\mathcal J\) at the primes in question. Indeed, \(\mathcal J\) is a subbundle, so its quotient in the ambient bundle is torsion-free; a regular ambient section whose generic value lies in \(\mathcal J\) has zero image in that quotient. Taking determinants in (45) and dualizing gives a generically nonzero map \[ R\dashrightarrow p^*(K_S+D_0+c\mathcal H_{\mathrm u}) \tag{46}\] regular above every prime of \(S\).

Possible poles on divisors exceptional over \(S\) do not affect the conclusion, as follows. Factor \(p\) through the finite normalization \(S'\to S\) of this level. Choose an ample line \(A_S\) on \(S\) and a positive rational \(\epsilon\) sufficiently small that \(R-\epsilon p^*A_S\) is big. For a sufficiently divisible positive integer \(m\), it has a nonzero section. Its image under (46) is a rational section of \[p^*m(K_S+D_0+c\mathcal H_{\mathrm u}-\epsilon A_S)\] regular at every divisor of the normal variety \(S'\). Normality extends it over \(S'\). Taking its field norm to \(S\) gives a nonzero section of a positive multiple of \(K_S+D_0+c\mathcal H_{\mathrm u}-\epsilon A_S\). Adding the ample term proves bigness. Since \(\mathcal H_{\mathrm u}\) is nef, the conclusion persists when \(c\) is increased. ◻

Proof. Put \(\mathfrak h=\mathop{\mathrm{Lie}}H\), and let \(\xi\in H^0(\mathcal P,\Omega^1_{\mathcal P}\otimes\mathfrak h)\) be the absolute connection form. It has kernel the horizontal distribution, is the identity on fundamental vertical vectors, and satisfies \[ d\xi+\tfrac12[\xi,\xi]=0,\qquad R_g^*\xi=\operatorname{Ad}(g^{-1})\xi\quad(g\in H). \tag{48}\] Restrict to a nonempty smooth open subset \(R^\circ\subset R\) on which \(\xi|_{TR}\) has constant rank \(r\). The germ \(U\) meets this subset, because it is Zariski dense in \(R\). We may replace \(U\) by a small connected smooth germ there without changing its Zariski closure. Set \(\mathcal K=\ker(\xi|_{TR^\circ})\), and consider the algebraic Grassmann map \[\beta:R^\circ\longrightarrow\operatorname{Gr}(r,\mathfrak h), \qquad x\longmapsto\xi(T_xR).\] For a local holomorphic vector field \(V\) in \(\mathcal K\), Cartan’s formula and (48) give \[\mathcal L_V(\xi|_{R^\circ}) =\iota_Vd\xi+d(\iota_V\xi)=0.\] The local flow of \(V\) consequently preserves both the restricted connection form and its image subspace; hence \(d\beta(V)=0\). Since \(TU\subset\mathcal K|_U\), the map \(\beta\) is constant on \(U\). Its algebraicity and the Zariski density of \(U\) make it constant on \(R^\circ\). Denote its value by \(\mathfrak b\subseteq\mathfrak h\).

This fixed subspace is a Lie subalgebra. In fact, for \(b_1,b_2\in \mathfrak b\), locally choose tangent fields \(V_1,V_2\) on \(R^\circ\) with \(\xi(V_i)=b_i\). Such lifts exist because \(\xi|_{TR^\circ}\) is a surjection onto the constant bundle \(\mathfrak b\). Evaluating (48) yields \[\xi([V_1,V_2])=[b_1,b_2],\] so \([b_1,b_2]\in\mathfrak b\).

Suppose \(\mathfrak b\ne\mathfrak h\). Let \(A\) be the connected analytic subgroup of \(H\) with Lie algebra \(\mathfrak b\), and let \(B\) be its Zariski closure. Then \(B\) is a connected algebraic subgroup and is proper in \(H\). To prove the latter point, if \(B=H\), the fact that \(A\) preserves \(\mathfrak b\) under its adjoint action would imply that \(H\) preserves \(\mathfrak b\): the stabilizer of this subspace is algebraic. Thus \(\mathfrak b\) would be an ideal of the semisimple Lie algebra \(\mathfrak h\). Every such ideal is a sum of simple factors and integrates to a closed connected normal algebraic subgroup of \(H\). That subgroup is \(A\), by uniqueness of a connected analytic subgroup with a given Lie algebra. A proper \(\mathfrak b\) therefore cannot have Zariski-dense \(A\), a contradiction.

Consider the irreducible algebraic subset \[R_B=\overline{R\cdot B}\subseteq\mathcal P.\] It dominates \(Z\), since \(p(U)\) is dense, and it is invariant under the right action of \(B\). The equivariance in (48) shows that, at a general smooth point, \[ \xi(T R_B)\subseteq\mathop{\mathrm{Lie}}B. \tag{49}\] Indeed, on the image of the multiplication map \(R^\circ\times B\to \mathcal P\), tangent vectors coming from \(R^\circ\) have connection values in \(\operatorname{Ad}(B)\mathfrak b\subseteq\mathop{\mathrm{Lie}}B\), and tangent vectors coming from \(B\) have values in \(\mathop{\mathrm{Lie}}B\). Generic smoothness of this dominant map onto \(R_B\) proves (49).

The \(B\)-orbits in a fiber have dimension \(\dim B\). Domination of \(Z\) therefore gives \(\dim R_B\geq\dim Z+\dim B\). Conversely, (49) and \(\dim\ker\xi=\dim Z\) give the reverse inequality. Equality holds, and at general smooth points \[TR_B=\xi^{-1}(\mathop{\mathrm{Lie}}B).\] In particular the whole horizontal distribution is tangent to \(R_B\). This is an algebraic tangency statement on all of \(R_B\): if a local algebraic horizontal vector field differentiates an equation of \(R_B\), the result vanishes on its dense smooth locus and hence on \(R_B\). Thus the ideal sheaf of \(R_B\) is preserved by horizontal vector fields. Their local holomorphic flows preserve \(R_B\) in both directions.

Choose a nonempty smooth open \(Z_0\subseteq Z\) over which the general fiber of \(R_B\to Z\) is nonempty of dimension \(\dim B\). Horizontal transport along a path in \(Z_0\) is defined throughout the path: locally it is the fundamental solution of a holomorphic linear system after a faithful representation of \(H\). The preceding ideal invariance shows that it preserves \(R_B\). In particular the monodromy of \(Z_0\) preserves the nonempty closed subset \((R_B)_z\) of the \(H\)-torsor \(\mathcal P_z\). Deleting a proper algebraic subset of a smooth complex variety does not decrease the original monodromy image: the map \(\pi_1(Z_0,z)\to\pi_1(Z,z)\) is surjective, since loops can be perturbed off a subset of real codimension at least two. This monodromy is therefore Zariski dense in \(H\). A closed subset of an \(H\)-torsor invariant under a Zariski-dense translation subgroup is invariant under \(H\), and is either empty or the whole torsor. But \((R_B)_z\) is nonempty and has dimension \(\dim B<\dim H\). This contradiction proves \(\mathfrak b=\mathfrak h\).

Finally the generic kernel of \(\xi|_{TR}\) contains \(TU\) and hence has dimension at least \(\dim U\). Its generic image has dimension \(\dim H\), so rank–nullity gives (47). ◻

Proof. We may choose the initial point outside the countably many exceptional tensor loci and in the maximum-rank loci of the maps associated with every rational normal-factor subset of \(G\). There are only finitely many such subsets. The restriction to \(Z\) has the same generic Mumford–Tate group as the ambient variation: a tensor condition holding on \(Z\) holds at the chosen Hodge-generic point. The connected monodromy normality theorem, in the pure polarized integral setting specified in 8, makes \(H\) normal in that Mumford–Tate group. Since \(H\subseteq G\) and \(G\) is semisimple adjoint, \(H\) is a product of rational simple factors and has the complementary factor \(Q\).

The restricted \(Q\) monodromy is finite. Kill it by a finite cover of a smooth open of \(Z\). The fixed-part theorem then says that the full \(Q\) variation is constant. Its flat frames are algebraic frames: the algebraic connection has regular singularities and its trivial local system is the trivial regular-singular connection. The \(Q\) compact marks are algebraic in these frames. They are constant on the lifted \(T\), since \(j\) was constant there, and hence on the lifted \(Z\). Indeed, a lift of a nonempty open subgerm of \(T\) is Zariski dense in an irreducible component of this finite cover. Its closure has dimension at least \(\dim Z\), because its finite image contains a Zariski-dense subset of \(Z\); an irreducible component of the cover also has dimension \(\dim Z\). This proves the required density assertion after finite levels. In the line case we only need the constancy of the \(Q\) period.

Use the algebraic flat \(H_{\mathbb C}\)-frame torsor over this smooth open of \(Z\). It is the reduction specified by the parallel tensor invariants; its connection has Zariski-dense monodromy \(H\). Set \(a_0=a(T)\) in a flat trivialization along \(T\). Consider the algebraic incidence consisting of frames in which \(a\) has value \(a_0\). Over a point where it is nonempty its fiber is a coset of the stabilizer of \(a_0\) in \(H_{\mathbb C}\). Its image contains \(T\) and is constructible, so contains a dense open of \(Z\). If \(d=\dim(H_{\mathbb C}\cdot a_0)\), the incidence over that open therefore has dimension \[\dim Z+\dim H-d.\] It contains the horizontal lift of \(T\), whose Zariski closure therefore has at most this dimension. Applying 17 to that horizontal germ gives \[\dim T+\dim H\leq\dim Z+\dim H-d, \quad\hbox{or equivalently}\quad \dim Z-\dim T\geq d.\] This use of the incidence works equally for a locally closed projective-line orbit; projectivity of the orbit is not needed.

Move within \(T\) to a smooth point of \(Z\) lying over the dense open just obtained. Nonempty open subgerms are still Zariski dense in \(Z\). In flat frames the values of \(a\) on a neighborhood in \(Z\) lie in \(H_{\mathbb C}\cdot a_0\). Moreover \[\ker(da|_{T_zZ})=T_zZ\cap\ker da=T_zT,\] because the ambient map has constant rank near \(T\). Thus \(\dim Z-\dim T\leq d\). Equality proves (50). For the joint marks, the factor-by-factor description of \(J\) identifies this orbit with \(J_H\).

We now work with \(a=j\) on a level of \(S\). The \(Q\) joint data is constant on \(Z\), so \(T_zZ\subseteq\ker dq\), while \(\ker dj=T_zT\subseteq\ker dq\) at the chosen points. The orbit calculation gives \[dj_H(\ker dq)=T_{j_H(z)}J_H.\] Since \(j_Q\) descends through \(q\), one has \(\ker dq\subseteq\ker dj_Q\). Conversely, if \(dj_Q(v)=0\), choose \(w\in\ker dq\) with \(dj_H(w)=dj_H(v)\). Then \(dj(v-w)=0\), hence \(v-w\in\ker dq\), and therefore \(v\in\ker dq\). It follows that \(\ker dj_Q=\ker dq\). Computing the rank first along \(\ker dq\) and then on its quotient proves (51). Our initial choice in the maximum-rank loci makes these generic rank identities, even though the argument was carried out at points of \(T\).

The representation decomposes at connected level as \(\mathbb A=A_H\otimes A_Q\), and its highest step as \(E_0=E_H\otimes E_Q\). The factors are the compatible Hodge representations constructed from the separate adjoint Lie tensor systems; a constant scalar shift of a grading has no effect on the following determinant calculation. If \(e_H=\mathop{\mathrm{rk}}E_H\) and \(e_Q=\mathop{\mathrm{rk}}E_Q\), then \[\det E_0=(\det E_H)^{\otimes e_Q} \otimes(\det E_Q)^{\otimes e_H}.\] Add the compact-mark line on the appropriate factors to define \(\mathcal H_H\) and \(\mathcal H_Q\). Their semipositive curvature forms have kernels \(\ker dj_H\) and \(\ker dj_Q\). The first has rank \(d_H\), by orbit filling and the upper bound \(\dim J_H\); the second has rank \(s_Q\). 14 gives their numerical dimensions.

The \(Q\) representation and flags descend through \(q\) by 13, applied to precisely the rational normal factors in \(Q\). Its integral lattice inherits the congruence-level condition, so its quasi-unipotent boundary monodromies are already unipotent. Choose smooth projective models resolving \(q\) and all the flag sections. Unipotent canonical extension commutes with this pullback: in a monomial boundary chart the new logarithmic residues are integral sums of the commuting old nilpotent residues, and the untwisted filtration pulls back. Thus the equality of the \(Q\) lines on the open set extends as the equality in (52). The metric rank on the quotient is \(s_Q\), so the same metric lemma gives \(\nu(\underline{\mathcal H}_Q)=s_Q\). ◻

Proof. Put \(r=\nu(\mathcal H)=\mathop{\mathrm{rk}}dj\) and use 18. Write \(B\) for its smooth unipotent level, and put \(n=\dim B=\dim S\). On \(B\) choose a smooth boundary component \(D'\) mapping generically finitely onto \(D_i\). Further resolutions and restrictions to the strict transform are allowed. For any nef line \(L\) on a smooth projective variety, restriction to a divisor cannot increase its numerical dimension: choose an ample \(A\) with \(mA-D'\) effective and compare the nonnegative intersections \(L^kD'A^{n-k-1}\) and \(mL^kA^{n-k}\). Consequently \[\nu(\mathcal H_H|_{D'})\leq d_H.\] If \(D'\) does not dominate \(S_Q\), then \[\nu(\mathcal H_Q|_{D'})\leq\dim q(D')<s_Q.\] The binomial expansion of intersections of nef classes gives \(\nu(P+Q)\leq\nu(P)+\nu(Q)\) for nef \(P,Q\). Applied on \(D'\), this proves the strict inequality in this case. We may therefore assume that \(D'\) dominates \(S_Q\). At its generic point the \(Q\) system is pulled back from the interior of \(S_Q\).

Infinite local monodromy. The logarithm \(N\) of a sufficiently divisible power of the original monodromy is then nonzero and belongs to the rational Lie algebra of \(H\). Its \(Q\) component is zero because the \(Q\) variation extends from the interior at the generic point of \(D'\). It acts nontrivially on the \(H\) orbit of \(E_H\). To check this last assertion, a nonzero rational element in a rational simple factor has nonzero projection to every Galois-conjugate complex simple factor. A unipotent isometry of a positive Hermitian space is the identity. Thus a rational factor detected only by compact marks cannot support a nonzero component of \(N\). Every other retained rational factor has a visible first-kind flag factor, and the action of a complex simple adjoint group on a nontrivial flag variety is faithful. A nonzero component of \(N\) therefore moves such a flag.

Let \(p_H\) be the highest Hodge index of \(A_H\) and set \(p_0=e_Hp_H\). There is no restriction on the rank \(e_H\) of \(E_H\). Form \[\mathbb B=\bigwedge^{e_H}A_H, \qquad L_H=\det E_H=F^{p_0}\mathbb B, \qquad F^{p_0+1}\mathbb B=0.\] The highest filtration step of \(\mathbb B\) is exactly the one-dimensional line \(L_H\). Denote the weight of \(\mathbb B\) by \(w\). On the open stratum of \(D'\), untwist by \(\exp(-\log(z)N/(2\pi\sqrt{-1}))\) in a local normal coordinate \(z\) and let \([v]\) be the limiting highest line. Schmid’s nilpotent-orbit theorem and limiting mixed Hodge theorem (Schmid 1973, Theorems 4.12 and 6.16) apply to the polarized unipotent variation. The same assertions hold in \(\mathbb B\): it is a Hodge-compatible summand of rational adjoint tensor systems, whose flat projectors commute with \(N\) and preserve the two limiting Hodge filtrations and the weight filtration.

Write \(W_\bullet=W(N)_\bullet\), centered at \(w\). The line \(F^{p_0}\) occupies a single summand \(I^{p_0,q}\) of the Deligne splitting, since there are no summands of first index greater than \(p_0\). Set \(p_0+q=w+k\). The primitive decomposition of the monodromy weight grades shows that \(k\geq0\) and that the class of \(v\) in \(\mathop{\mathrm{Gr}}^W_{w+k}\) is primitive. Here is the specific reason. A nonprimitive term of first Hodge index \(p_0\) has the form \(N^j u\) with \(j>0\) and with \(u\) of first index \(p_0+j\), which is impossible. Likewise a weight grade below \(w\) is entirely obtained from positive powers of \(N\) on higher primitive grades, and so cannot contain the highest line. Monodromy Lefschetz gives a nonzero \(N^k[v]\) in \(\mathop{\mathrm{Gr}}^W_{w-k}\) and zero \(N^{k+1}[v]\). In fact \[ N^kv\ne0,\qquad N^{k+1}v=0 \tag{54}\] for the actual vector: \(N\) has bidegree \((-1,-1)\) in the Deligne splitting, so each \(N^jv\) lies in a single summand; its vanishing on the corresponding weight grade is its vanishing as a vector. The primitive decomposition and Lefschetz isomorphisms used here are those of the monodromy weight filtration, as in (Schmid 1973, Lemma 6.4 and Theorem 6.16).

Let \(O_H\) be the closed orbit of the highest line of \(\mathbb B\), namely the exterior-power embedding of the \(H\) orbit of \(E_H\). The untwisting acts through \(H_{\mathbb C}\), so \([v]\in O_H\). By (54), \[\lim_{\tau\to\infty}\exp(\tau N)[v]=[N^kv] \in O_H\cap\mathbf P(\ker N).\] The intersection on the right is a proper closed subset of the irreducible variety \(O_H\), because \(N\) acts nontrivially on \(O_H\). Project \(N^kv\in W_{w-k}\) to \(\mathop{\mathrm{Gr}}^W_{w-k}\) and then apply the fixed inverse of \[N^k:\mathop{\mathrm{Gr}}^W_{w+k}\xrightarrow{\ \sim\ } \mathop{\mathrm{Gr}}^W_{w-k}.\] The resulting projective point is the highest graded line \([v]_{\mathop{\mathrm{Gr}}}\). These maps are fixed linear maps in local flat limiting coordinates. Projection is taken on the open set where its value is nonzero; this set contains all the points under consideration by Lefschetz. The local variation rank of the highest graded line is therefore at most \(\dim O_H-1\). Together with all the \(H\) compact marks its rank is at most \[ \dim O_H-1+\dim J_{{\rm u},H}=d_H-1. \tag{55}\]

From limiting rank to numerical dimension. We next justify that the graded rank just obtained bounds the numerical dimension of the restricted line, including at crossings. One cannot simply restrict the original singular Hodge metric to \(D'\). Instead, the pure weight grades of the limiting mixed Hodge structures give polarized pure variations on the open stratum of \(D'\) (Cattani and Kaplan 1989, Proposition 2.10). Their filtrations are holomorphic and transverse, by the extended filtration and the tangential logarithmic connection. Their primitive parts have the monodromy polarizations, and their remaining parts are polarized through the Lefschetz decomposition.

There is a canonical extension description on the whole component. In a Deligne frame at a crossing \(z_1\cdots z_a=0\), with \(D'=(z_1=0)\), the transverse residue is the constant nilpotent \(N_1\) and the remaining residues \(N_2,\ldots,N_a\) commute with it. The filtration \(W(N_1)\) therefore consists of subbundles of the restricted extended flat bundle. Every remaining residue preserves this filtration and induces a nilpotent operator on each weight grade. Changing the normal coordinate or the tangential lift changes the tangential connection by a multiple of \(N_1\); this acts trivially on the weight grades. Their flat connections are thus well defined. The remaining logarithmic exponentials are exactly the Deligne extensions of these flat graded variations. Separately, each polarized pure graded variation has its Schmid-extended Hodge subbundles inside that canonical flat bundle. We do not identify a possibly jumping intersection with a weight step with an extended Hodge subbundle. Instead we use the independent Hodge extension, and below map into it by continuation from the open stratum. This distinction proves the needed comparison also at crossings.

On the stratum, \(L_H|_{D'}\) lies in \(W_{w+k}\) and maps nontrivially to its highest graded line. Inclusion in the subbundle \(W_{w+k}\) extends everywhere because it holds generically. Projection to \(\mathop{\mathrm{Gr}}^W_{w+k}\) is regular. The highest line of the graded variation extends as its independent Schmid Hodge subbundle. The projection lands in it: its image in the locally free quotient of the canonical flat graded bundle vanishes on the dense open stratum and hence vanishes everywhere. The resulting map can have zeros at further boundary strata; these produce an effective difference in the direction needed below. If necessary resolve the additional algebraic filtration-rank loci on \(D'\). The loops newly omitted from the interior have trivial monodromy, and all boundary monodromies remain unipotent; the preceding description is preserved on this resolution. We obtain a generically nonzero morphism of line bundles \[ L_H|_{D'}\longrightarrow L_{H,\mathop{\mathrm{Gr}}}, \qquad L_{H,\mathop{\mathrm{Gr}}}-L_H|_{D'}\ \text{effective}. \tag{56}\] Here and below restriction notation includes the pullback to that resolution when one was needed.

Both sides of (56) are nef. For the right side this follows from the pure highest-line metric and 14; for the left side it follows by restricting the original nef line. Tensor the comparison by \(e_Q\) and add the same compact-mark line on \(D'\). If \(P,Q\) are the resulting nef lines, then \(Q-P\) is effective, and for an ample \(A\) on this \((n-1)\)-dimensional variety, \[(Q^k-P^k)A^{n-1-k} =(Q-P)\sum_{a=0}^{k-1}Q^{k-1-a}P^a A^{n-1-k}\geq0.\] Thus \(\nu(P)\leq\nu(Q)\). The metric lemma and (55) now give \[\nu(\mathcal H_H|_{D'})<d_H.\] Adding \(\mathcal H_Q|_{D'}\), whose numerical dimension is at most \(s_Q\), proves the desired strict drop upstairs.

Finite nonidentity local monodromy. Let \(\gamma\ne1\) denote the original monodromy. On the unipotent level its logarithm is zero, so the pure period extends ordinarily across a general point of \(D'\). Its limiting flags and compact marks are fixed by \(\gamma\). Indeed, the local cover is \(t=z^e\); the continuation around \(t=0\) acts on its sheets by \(z\mapsto e^{2\pi\sqrt{-1}/e}z\). Equivariance of the extended period and marks then gives the fixed-point assertion at \(z=0\) in common flat frames. The comparison in (56), now with a pure ordinary limit, shows that the numerical dimension on \(D'\) is bounded by the differential rank of these limiting marks. That rank is at most \(r\): in the ordinary extending flat frame, all \((r+1)\)-minors of the mark differential vanish on the interior and hence on its extension, including after restriction to \(D'\).

Suppose that rank were \(r=d_H+s_Q\). Since \(D'\) dominates \(S_Q\), the limiting \(Q\) data comes from the interior of \(S_Q\). In particular \(dj_Q\) on \(D'\) has rank \(s_Q\) and kernel \(\ker(dq|_{D'})\). Equality of the total rank forces the \(H\) flags along a local general fiber of \(q|_{D'}\) to fill an open subset of \(J_H\). The \(Q\) flags are fixed on this fiber.

The rational automorphism \(\gamma\) normalizes the connected group, and acts on the product \(J\) factorwise up to permutation of simple factors. An open subset with all \(H\) flag coordinates varying independently cannot be fixed if any visible \(H\) coordinate is moved to a different coordinate. If the destination were a \(Q\) coordinate it would be constant; if it were another \(H\) coordinate the fixed-point equations would impose a relation between two independent coordinates. Thus every visible complex \(H\) factor is preserved and \(\gamma\) acts identically on its whole flag variety. Equivariance and faithfulness of the simple adjoint action imply identity on its Lie algebra. Rationality then implies identity on every Galois-conjugate complex factor, hence on all of \(H\). Consequently \(\gamma\) preserves \(Q\). Its action on \(Q\) fixes a generic lifted full \(Q\) period, because \(D'\) dominates \(S_Q\). 12 gives identity on \(Q\) also, a contradiction. The limiting rank is strictly less than \(r\) in this case as well.

Pullback by the generically finite map \(D'\to D_i\) preserves the numerical dimension of a nef line, so the strict drop descends to \(D_i\). Finally, if \(r=0\), a local fiber of \(j\) is open in the base, so its Zariski closure is the entire base and \(H=G\). Orbit filling gives \(\dim J=0\). Visibility on every retained rational factor forces \(G=1\); the full projected adjoint action then has no nonidentity boundary monodromy. Thus \(D_0\) is empty. ◻

Proof. Choose fixed ample integral lines \(A_i\) on \(T_i\) sufficiently positive that all successive restriction terms obtained while imposing order \(le\) along each \(T_j\) inject into \[H^0\bigl(T_i,l(A|_{T_i}+tH|_{T_i}+A_i)\bigr).\] This choice is uniform in \(l\) and \(t\). Explicitly, choose ample \(C_{ij}\) so that \(C_{ij}\) and \(C_{ij}+T_j|_{T_i}\) have nonzero sections, and put \(A_i=e\sum_j C_{ij}\), enlarging it if needed. For \(0\leq n_j\leq el\) the line \[lA_i+\sum_j n_jT_j|_{T_i} =\sum_j\bigl((el-n_j)C_{ij} +n_j(C_{ij}+T_j|_{T_i})\bigr)\] has a nonzero section. Multiplication by it supplies exactly the asserted injections, including the normal twists when \(j=i\).

The restriction exact sequences, applied one copy of a component at a time, give \[\begin{align*} h^0\bigl(M,l(A+tH-eT)\bigr) &\geq h^0\bigl(M,l(A+tH)\bigr)\\ &\quad-le\sum_i h^0\bigl(T_i,l(A|_{T_i}+tH|_{T_i}+A_i)\bigr). \end{align*}\] For each fixed divisible \(t\), all the positive lines in this formula are ample. Their Hilbert polynomials give, in the coefficient of \(l^n\), a positive supply \[\frac{(A+tH)^n}{n!}\] and a loss at most \[\frac{e}{(n-1)!}\sum_i (A|_{T_i}+A_i+tH|_{T_i})^{n-1}.\] The supply is a polynomial in \(t\) of degree exactly \(r\) with positive leading coefficient. Each term in the loss has degree at most \(r-1\). For \(n=1\) the restriction terms are simply section dimensions on points and the same statement holds. Choose \(t\) sufficiently large that the difference of these leading coefficients is positive, and then let \(l\to\infty\). This proves bigness. No asymptotic estimate uniform in both \(l\) and \(t\) is used. ◻

Proof. By 16, \(K_S+D_0+c\mathcal H_{\rm u}\) is big for sufficiently large \(c\). Since \(\mathcal H-\mathcal H_{\rm u}\) is nef, \(K_S+D_0+c\mathcal H\) is big. Choose an integer \(e>0\) clearing denominators and an ample integral \(A\) with \[e(K_S+c\mathcal H+D_0)-A\] effective. [prop:boundary-drop,lem:boundary-remove] show that \(A+t\mathcal H-eD_0\) is big for large divisible \(t\). Adding the displayed effective divisor proves bigness of \(eK_S+(ec+t)\mathcal H\). When \(\nu(\mathcal H)=0\), 19 says \(D_0=0\), and the initial bigness already gives the conclusion. Finally the sum of a big line and a nef line is big, so every larger rational \(b\) works as well. ◻

Proof of Proposition 22. The connected-fiber hypothesis ensures that, after shrinking to a smooth locally trivial locus, the monodromy image of the pullback is the same as that on \(S\). Passing to its connected level therefore leaves Zariski-dense monodromy \(G\). The variation and the orbit calculation of 18 apply on \(\widehat W\).

Each \(U_j\) is Higgs invariant because \(\theta G_i\subseteq G_{i+1}\otimes\Omega^1\); this follows from Higgs integrability and the definition by all contractions. On the open locus the Hodge decomposition is orthogonal. Write \(\mathcal D\) for its Chern connection. For a tangent vector \(v\) put \(A=\theta(v)\), let \(\Pi_j\) be orthogonal projection onto \(U_j\), and set \[B_j(v)=(1-\Pi_j)\mathcal D'_v|_{U_j}.\] The Hodge curvature equation and the subbundle curvature formula give, in a common positive normalization, \[ \Theta_{-\det U_j}(v,\bar v) =\bigl\|(1-\Pi_j)A^*|_{U_j}\bigr\|_{\rm HS}^2 +\|B_j(v)\|_{\rm HS}^2. \tag{60}\] To see the sign directly, use \(A U_j\subseteq U_j\) and write \(A\) in upper block-triangular form for \(U_j\oplus U_j^\perp\). The traces of the two internal products cancel. The remaining trace is minus the square of the off-diagonal block of \(A^*|_{U_j}\) for \(\det U_j\), and subbundle curvature subtracts the square of \(B_j(v)\). Negating the determinant gives (60).

The metrics on the extended subbundles have the two-sided logarithmic norm estimates needed in 14. Indeed they are restrictions of the extended Hodge metrics; after resolution their frames have full rank in the extended Hodge grades, so the dual estimates also restrict. Thus every \(-\det U_j\), and hence \(P\), is nef.

Suppose now that \(\Theta_P(v,\bar v)=0\). Every nonnegative summand in (60) vanishes. Applying its first term for \(U_i\) to \(G_i\), the vector \(A^*G_i\) belongs to the preceding Hodge grade, which is orthogonal to \(U_i\). Hence \[ A^*G_i=0\quad\hbox{for every }i. \tag{61}\] Since \(AG_i\subseteq G_{i+1}\), for \(g\in G_i\) one then has \(\|Ag\|^2=\langle g,A^*Ag\rangle=0\). Thus \(AG_i=0\) as well. Moreover \(B_0(v)=0\) and \(\mathcal D'\) preserves the Hodge grading, so \(\mathcal D'_v\) preserves \(G_0=\ell\). The flat connection satisfies \(\nabla^{1,0}=\mathcal D'+\theta\) in the smooth Hodge splitting. Its projective derivative on \(\ell\) is consequently zero. In other words, \[ v\in\ker d\ell. \tag{62}\] The compact marks, which by hypothesis are the same pointwise functions of \(\ell\) as before, are constant on a local \(\ell\)-fiber. At its constant-rank points (62) also gives \(dj_{\rm u}(v)=0\).

It remains to prove \(dE_0(v)=0\). Apply the line version of 18 to a local \(\ell\)-fiber \(T\), with Zariski closure \(Z\) and decomposition \(G=H\times Q\). Work at one of the smooth generic points \(z\) supplied by that lemma. The preceding calculation places every \(v\in\ker\Theta_{P,z}\) in \(T_zT\); fix this same \(T\), \(Z\), and \(H\) for all such vectors. The full \(Q\) period is constant on \(Z\). Hence the infinitesimal period element of every such \(v\) satisfies \[t(v)\in\mathfrak h^{-1,1}.\] Write \(\ell_z=\mathbb Cs\). Evaluating the tangent map of \(\ell\) at \(s\) and projecting to the next Hodge grade sends \(d\ell(T_zZ)\) into \(G_1\): its flat derivative is \(\mathcal D'_w s+\theta(w)s\), and only the second term contributes to that grade. Orbit filling in (50) therefore gives \[ \mathfrak h^{-1,1}s\subseteq G_1. \tag{63}\] Use the metric supplied by the adjoint tensor construction, so that representation adjoints agree with Lie adjoints. Set \(x=t(v)^*\). By (61), \(xG_1=0\). Also \(xs=0\) because \(s\) is in the highest Hodge grade and \(x\) raises that grade. It follows from (63) that \[[x,\mathfrak h^{-1,1}]s=0.\] 23, with \(\mathfrak g_k=\mathfrak g^{k,-k}\), gives \(t(v)E_0=0\). The flat-chart differential of the whole highest step is precisely Higgs evaluation on \(E_0\), so \(dE_0(v)=0\).

We have proved, at the points where the orbit calculation applies, the kernel inclusion \[ \ker\Theta_P\subseteq \ker\Theta_{\pi^*h^*\mathcal H}. \tag{64}\] For completeness, these points can be chosen to compute the global curvature ranks. The Hodge metrics and subspace data are real analytic on the variation locus. Choose the initial point in the open maximum-rank loci of \(\Theta_P\) and \(\Theta_P+\Theta_{\pi^*h^*\mathcal H}\), as well as the finitely many relevant period, flag, and projective-line rank loci and the Hodge-generic locus. Moving slightly within its \(\ell\)-fiber reaches the smooth Zariski-open locus of \(Z\) required by 18, while staying in the two curvature-rank opens. Indeed an open subgerm of that fiber is Zariski dense in \(Z\). The argument then establishes (64) at a point where both curvature ranks are maximal. Semipositivity implies equality of those ranks. Apply 14 to \(P\) and \(P+\pi^*h^*\mathcal H\) to obtain (58). ◻

Proof of Lemma 23. Choose a Cartan in the compact form adapted to the grading, and choose positive roots so that each simple root has nonnegative degree. Metric adjoints exchange opposite root spaces. Let \(\Delta_0\) be the degree-zero simple roots, and let \(v_\lambda\) be a highest-weight vector for \(V\). Write \(\mathfrak g_0^-\) for the sum of the negative root spaces of degree zero. The highest graded piece is \[ E=U(\mathfrak g_0^-)v_\lambda. \tag{65}\] In particular it is irreducible for the full degree-zero reductive algebra. Indeed, reaching the extremal grading from the highest weight allows only degree-zero lowerings; the highest-weight theory for that Levi algebra then gives (65).

Every degree-one positive root contains exactly one degree-one simple root \(\alpha\), with coefficient one, and otherwise only simple roots of degree zero. Let \(\mathfrak b^+_\alpha\) be the sum of its root spaces and \(\mathfrak b^-_\alpha\) the sum of the opposite root spaces. These are \(\mathfrak g_0\)-modules, and \(*\) exchanges them. Roots of degree one in \(\mathfrak h\) are exactly the blocks belonging to its simple factors. For \(\alpha\ne\beta\), \[ [\mathfrak b^+_\alpha, \mathfrak b^-_\beta]=0. \tag{66}\] Indeed, the difference of the relevant roots has a positive coefficient at \(\alpha\) and a negative one at \(\beta\), and hence is neither a root nor zero.

Write \(x=\sum_\alpha x_\alpha\) using these blocks in \(\mathfrak h_1\). For every nonzero \(x_\alpha\), (59) and (66) imply \([x_\alpha,\mathfrak b^-_\alpha]s=0\). Hence the incidence \[I_\alpha= \left\{([z],[u])\in \mathbf P(\mathfrak b^+_\alpha)\times\mathbf P(E): [z,y]u=0\text{ for every }y\in\mathfrak b^-_\alpha\right\}\] is nonempty, closed, and projective. It is stable under the full connected degree-zero group. Apply the Borel fixed-point theorem to the Borel of this group determined by the chosen positive roots and containing the full Cartan. The theorem used here says that a connected solvable algebraic group acting on a nonempty complete variety has a fixed point (Milne 2017, Corollary 17.3). A fixed pair in \(I_\alpha\) has the form \(([e_\rho],[v_\lambda])\): the first coordinate is a root line since the full Cartan acts, and the second is the unique highest line of the irreducible degree-zero module \(E\). Irreducibility of \(\mathfrak b^+_\alpha\) is not required.

Let \(C_\alpha\) be the connected component containing \(\alpha\) in the Dynkin subdiagram on \(\Delta_0\cup\{\alpha\}\). Connectedness of root support gives \(\operatorname{Supp}(\rho)\subseteq C_\alpha\). If the inclusion were strict, there would be a missing zero-degree node \(\delta\) adjacent to the support. Its coefficient in \(\rho\) is zero, while some adjacent support coefficient is positive, so \(\langle\rho,\delta^\vee\rangle<0\). The root-string property implies that \(\rho+\delta\) is a root. Thus \(e_\delta\) does not preserve the line \(\mathbb Ce_\rho\), contrary to the Borel-fixed condition. Therefore \[\operatorname{Supp}(\rho)=C_\alpha.\] Taking \(y=e_{-\rho}\) in the incidence equation gives \(\langle\lambda,\rho^\vee\rangle=0\). The simple-coroot expansion of \(\rho^\vee\) has strictly positive coefficients on its support. Dominance of \(\lambda\) therefore gives \(\langle\lambda,\delta^\vee\rangle=0\) for every \(\delta\in C_\alpha\). Every root indexing \(\mathfrak b^+_\alpha\) has support in \(C_\alpha\), so its coroot also pairs trivially with \(\lambda\). Finite-dimensional root \(\mathfrak{sl}_2\) theory implies that every vector in \(\mathfrak b^-_\alpha\) kills \(v_\lambda\). Since \(\mathfrak b^-_\alpha\) is a \(\mathfrak g_0\)-module, commuting it past the degree-zero lowerings in (65) shows that it kills all of \(E\). These root-string and highest-weight facts are the usual semisimple representation theory; see (Humphreys 1972, secs. 9–10,20–21). Finally \(x_\alpha^*\in\mathfrak b^-_\alpha\) for every nonzero component, and summing proves \(x^*E=0\). ◻

Proof. Over the generic point of \(S\) the coefficient bundle of \(u_*\) is constant. At any divisor horizontal over \(S\), its lattice is the ordinary pullback lattice from the interior of \(S\). The bounds of 4 consequently give regular logarithmic coefficients on \(Y_{0,\eta_S}\). If the bounds were verified after ramification, regularity descends: the order of a pulled-back ordinary coefficient is the ramification index times its original order. Choosing a basis of \((E_0)_{\eta_S}\) and trivializing the base canonical line therefore gives a nonzero element of \[H^0\bigl(Y_{0,\eta_S},a(K_{Y_{0,\eta_S}}+D_{Y_0,\eta_S})\bigr) \otimes_{\mathbb C(S)}(E_0)_{\eta_S}.\] At least one scalar component is nonzero, proving \(k\ge0\).

Work temporarily over \(K=\mathbb C(S)\). Choose a complete logarithmic pluricanonical system whose image has dimension \(k\), and resolve its base ideal. If its degree is \(b\), its morphism \(\phi:Z\to Z_b\) satisfies \[ b(K_Z+D_Z)\sim\phi^*H+F_b, \qquad F_b\ge0, \tag{70}\] where \(H\) is the hyperplane divisor on its projective image. The extra logarithmic discrepancy in a source resolution is effective, so the indicated inequality remains valid after further resolution. Take the Stein factor of \(\phi\), resolve it and the source, and spread the resulting diagram over a dense open subset of \(S\). Compactification and resolution give \(Y\xrightarrow{x}W \xrightarrow{h}S\). The field \(\mathbb C(W)\) is the relative algebraic closure of the system-image field in \(\mathbb C(Y)\); in particular it is algebraically closed in \(\mathbb C(Y)\). Since \(\mathbb C(S)\) is algebraically closed in \(\mathbb C(Y)\), it is also algebraically closed in \(\mathbb C(W)\). In characteristic zero these facts give geometrically connected generic fibers for both maps.

Here is the section-ratio argument for \(\kappa(J,K_J+D_J)=0\). The restriction of \(u_*\) still gives a nonzero section in degree \(a\), exactly as above. Suppose two independent logarithmic pluriforms existed on \(J\) in some common degree \(c\). Their ratio \(r\) would not lie in \(\mathbb C(W)\), and would be transcendental over that field, by its relative algebraic closure. Spread the two sections to rational sections of \(c(K_Z+D_Z)\), trivializing the canonical directions from the base rationally. Their pole divisors are vertical over \(W_{\eta_S}\). There are finitely many such prime divisors; the images of all of them in \(Z_b\) are proper, since \(W_{\eta_S}\to Z_b\) is generically finite. Hypersurfaces on \(Z_b\) containing these images, with sufficient multiplicity, give a section of some \(nH\) whose pullback cancels all their poles. Multiply also by the effective section of \(nF_b\) in (70). The two resulting sections belong to \(H^0(Z,(c+nb)(K_Z+D_Z))\) and retain ratio \(r\). Together with the \(k\) independent ratios of the original system, they give \(k+1\) independent ratios of logarithmic pluriforms. All finitely many ratios can be placed in one degree by multiplying numerators and denominators by the other denominators. This contradicts the definition of \(k\).

The same conclusion holds for the geometric generic fiber. In every fixed degree, proper cohomology in degree zero commutes with field extension; two sections over an algebraic closure would therefore already give section-space dimension at least two over \(\mathbb C(W)\). Adjunction and generic smoothness identify the fiber line with \(K_J+D_J\), and (68) justifies all the log source replacements. This proves the claimed geometric Iitaka dimension without an abundance assumption.

The degree-\(a\) space on \(J\) is now one-dimensional and nonzero. Choose its generator \(\omega\) and extend it as a rational relative canonical tensor. Divide the vector \(u_*\) by it on the generic fiber. Its scalar coefficients are in \(\mathbb C(W)\), so they define \(u_W\) and prove (69) in the stated degree \(a\). The Higgs operator is \(\mathcal O\)-linear; thus its iterates satisfy the same factorization under differential pullback. There is no derivative of \(\omega\) in this assertion. In particular the projective direction \([u_W]\) gives precisely the same compact flags as \([u_*]\) at general corresponding points. ◻

Proof. Write \(n=\dim Y_1\), \(w=\dim W_1\), and \(r=n-w\). We explain the valuation assertion needed for the modification. Let \(I_1\) be a prime of \(Y_1\) whose image has codimension at least two in \(W_1\), and let \(v=\operatorname{ord}_{I_1}\). Its restriction to \(\mathbb C(W_1)\) is nontrivial: a local function vanishing on the image has positive order at \(I_1\). The nonzero value subgroup is \(m\mathbb Z\) for an integer \(m>0\). Let \(v_0\) be the normalized restricted valuation and let \(\kappa(v)\) and \(\kappa(v_0)\) denote their residue fields.

Residues algebraically independent over \(\kappa(v_0)\) lift to elements algebraically independent over \(\mathbb C(W_1)\). Indeed, in a proposed polynomial relation among lifts of value zero, divide all coefficients by a coefficient of smallest valuation. Reduction then gives a nonzero polynomial relation among their residues. It follows that \[\operatorname{trdeg}_{\kappa(v_0)}\kappa(v)\le r, \qquad \operatorname{trdeg}_{\mathbb C}\kappa(v_0)\ge(n-1)-r=w-1.\] The reverse inequality is elementary: lifts of independent residues, together with one element of positive valuation, are algebraically independent over \(\mathbb C\), by examining the term of smallest valuation in a polynomial relation. Thus the residue transcendence degree is exactly \(w-1\).

Choose \(w-1\) such independent residues and rational-function lifts on \(W_1\). Resolve the graph of their map to \((\mathbf P^1)^{w-1}\). The center of \(v_0\) on this proper model has dimension at least \(w-1\), since its image contains the generic point of the product; it is proper since the valuation is nontrivial. It is therefore a prime divisor. Resolving further makes the base smooth and does not contract that center. This proves that the restriction of \(v\) is a positive integral multiple of a divisorial valuation of the base.

Only finitely many primes \(I_1\) need this treatment. Such a prime lies in the locus where the local fiber dimension of \(x_1\) is at least \(r+1\). This is a proper closed subset of \(Y_1\), and a prime divisor contained in it must be one of its finitely many divisorial components. Extract all the restricted valuations simultaneously. After any further base modification their centers are still divisors. On a compatible source resolution, the strict transform of each \(I_1\) survives and dominates its extracted center. Every prime on the fixed model \(Y_1\) now either dominates the base or has strict transform dominating a base prime.

There are finitely many divisors in \(\operatorname{div}_{aK_{Y/W}}(\omega)+aD\). Outside the images of their vertical components, all the numerators in (71) vanish. This proves finiteness of \(T_W\). For a nonzero section \(s\) on the right of (72), pullback and (67) first make \(s\) a logarithmic pluriform on \((Y,D)\). If \(a\mid l\), the section space on \(J\) in degree \(l\) is exactly the line generated by \(\omega^{l/a}\): it contains that element, and two independent sections would have a nonconstant ratio, contradicting \(\kappa(J,K_J+D_J)=0\). Hence, at the function-field level, there is a unique rational \(l\)-canonical tensor \(\sigma\) on \(W\) such that \(s=\omega^{l/a}x^*\sigma\).

Fix a base prime \(A\), take a regular local frame of \(lK_W\) at its generic point, and let \(c\) be the coefficient of \(\sigma\) in that frame. For every \(I\) dominating \(A\), regularity of \(s\) says \[ m_I\operatorname{ord}_A(c) +\frac la\operatorname{ord}_I(\omega)+l d_I\ge0. \tag{73}\] These inequalities are equivalent to \(\operatorname{ord}_A(c)+lt_A\ge0\). After clearing denominators, they say precisely that \(\sigma\in H^0(W,lL_W)\).

Conversely, suppose \(\sigma\in H^0(W,lL_W)\), and form the rational pluriform \(s=\omega^{l/a}x^*\sigma\). A horizontal prime of the fixed source \(Y_1\) satisfies the logarithmic regularity test because \(\omega^{l/a}\) is a logarithmic pluriform on \(J\). The strict transform of every other prime of \(Y_1\) dominates a prime \(A\) of \(W\), by the extraction just performed. The minimum rule implies (73) there. The boundary coefficient and the order of an absolute rational pluriform at a surviving prime agree with those on \(Y_1\). Thus \(s\) is regular as a section of \(l(K_{Y_1}+D_1)\) at every prime of the fixed model. Smoothness, or equivalently the normality extension criterion for a line bundle, extends it across the remaining codimension-two subset.

This argument does not discard the primes contracted by \(x_1\): their tests are exactly the reason for extracting their restricted valuations. It also explains why no indefinite extraction of all new source exceptional divisors is needed. Once a section is regular on \(Y_1\), (67) makes its pullback regular on every higher log source. Both maps in (72) are inverse rational identifications, which proves the last assertions. ◻

Proof. The exact lattice and relative bigness. Fix an initial smooth source model \(Y_1\to Y_0\) on which the just constructed Iitaka map is a morphism. Apply 25 to \(Y_1\), and thereafter use only models dominating its extracted base. That lemma, composed with (68), proves (74) with the original fixed \(Y_0\). Its order argument also applies over \(\eta_S\). The logarithmic systems on \(Y_{\eta_S}\) have image dimension \(k\), and their ratios descend to systems of \(L_W|_{W_{\eta_S}}\). After replacing a degree by a multiple to clear denominators, those descended ratios still have transcendence degree \(k=\dim W_{\eta_S}\). Thus \(L_W\) is big over \(S\). This remains true when \(k=0\), with the usual meaning of bigness on a zero-dimensional generic fiber.

The auxiliary subpair. On the current smooth diagram define \[ \Delta=-\frac1a\operatorname{div}_{aK_{Y/W}}(\omega)+x^*T_W. \tag{77}\] Choose compatible rational canonical forms, and write \(\varphi\) for the rational function expressing \(\omega\) in the corresponding relative frame. Then \[ K_Y+\Delta+\frac1a\operatorname{div}(\varphi)=x^*L_W. \tag{78}\] At a horizontal prime the inequality \(\Delta\le D\) follows from the logarithmic regularity of \(\omega|_J\). At every prime dominating a fixed base prime \(A\), it is exactly the minimum inequality (71). Divisors mapping into codimension two disappear after shrinking around \(\eta_A\). It follows that \(\Delta\le D\) near \(x^{-1}(\eta_A)\) for every \(A\), and also over the generic point of \(W\). Since the reduced log smooth pair \((Y,D)\) is log canonical, these inequalities imply that \((Y,\Delta)\) is sub-log-canonical there. Negative coefficients of \(\Delta\) cause no difficulty.

We verify the discrepancy-sheaf rank condition rather than inferring it from generic log canonicity. On a resolved geometric generic fiber \(\rho:J'\to J\), use the higher reduced boundary \(D_{J'}\) and put \[Z_{J'}=\operatorname{div}_{aK_{J'}}(\omega)+aD_{J'}\ge0.\] The crepant transform \(\Delta_{J'}\) of the auxiliary subpair is \[\Delta_{J'}=D_{J'}-\frac1a Z_{J'}.\] Moreover \(Z_{J'}\sim a(K_{J'}+D_{J'})\), so \(\kappa(J',Z_{J'})=0\). For the discrepancy b-divisor \(\mathbf A^*\), the coefficient \(-1\) is omitted. On this resolution its rounded trace \(R=\lceil\mathbf A^*_{J'}\rceil\) is effective and is supported on \(\mathop{\mathrm{Supp}}Z_{J'}\): where the logarithmic zero divisor vanishes, the auxiliary coefficient is either \(0\) or \(1\), and its contribution to \(R\) is zero. Therefore \(R\le N Z_{J'}\) for some integer \(N\), and \[h^0(J',\mathcal O(R))\le h^0(J',\mathcal O(NZ_{J'}))=1.\] If \(Z_{J'}=0\), this simply says \(R=0\). Sections of the discrepancy sheaf satisfy in particular the order conditions on this resolution, so the displayed bound is also an upper bound for its generic rank. Conversely, on every higher model sub-log-canonicity gives discrepancies at least \(-1\); after omitting \(-1\), their ceilings are nonnegative. The constant function \(1\) consequently belongs to the discrepancy sheaf. We have proved \[ \operatorname{rank}x_*\mathcal O_Y \bigl(\lceil\mathbf A^*(Y,\Delta)\rceil\bigr)=1. \tag{79}\]

The morphism \(x\) has connected fibers: its Stein factor is finite birational over the normal variety \(W\), since its geometric generic fiber is connected. Thus \(x\) is projective and surjective with connected fibers, its source and base are normal, \(\Delta\) is rational, the subpair is sub-log-canonical generically, and (78) and (79) give its remaining lc-trivial conditions. Thus it is an lc-trivial fibration in the precise sense of (Fujino and Gongyo 2014, Definition 3.2). By (Fujino and Gongyo 2014, Theorem 3.6), applied with the structural morphism \(W\to\operatorname{Spec}\mathbb C\), its moduli b-divisor is b-nef. This application requires neither effectivity of the auxiliary subpair nor abundance or semiampleness of the moduli part.

The discriminant and higher models. Following (Fujino and Gongyo 2014, sec. 3.4), define \[c_A=\sup\{c\in\mathbb Q:(Y,\Delta+c x^*A) \text{ is sub-log-canonical over }\eta_A\}, \qquad b_A=1-c_A.\] The comparison with \(D\) already proved gives \(c_A\ge0\), hence \(b_A\le1\). If \(I\) attains the minimum in (71), its coefficient in \(\Delta\) is exactly \(d_I\). The discrepancy condition at that prime forces \(d_I+c_A m_I\le1\). Therefore \[1\ge b_A\ge1-\frac{1-d_I}{m_I}\ge0,\] where the last inequality uses \(d_I\in\{0,1\}\) and \(m_I\ge1\). Put \(B_W=\sum_A b_AA\) and \(M_W=T_W-B_W\), using the normalization (78). This is the moduli trace for that normalization.

It remains to reconcile b-nefness on a higher model with recomputation of our minimum rule. Let \(\beta:W'\to W\) be a smooth birational base modification, take a compatible source resolution \(\rho:Y'\to Y\), and use the higher reduced boundary \(D'\). Write \(L'=K_{W'}+T_{W'}\) for the recomputed minimum divisor and \(\Delta'\) for (77) on that diagram. Let \(\Delta^{\rm cr}\) be the crepant transform, defined by \(K_{Y'}+\Delta^{\rm cr}=\rho^*(K_Y+\Delta)\). Comparing (78) on the two models, with the same rational trivialization, gives \[ \Delta'=\Delta^{\rm cr}+x'^*Q, \qquad Q=L'-\beta^*L_W. \tag{80}\] This identity includes the change of the relative canonical frame; omitting that change would give an incorrect formula for \(T_{W'}\).

Let \(B^{\rm cr}_{W'}\) be the discriminant of the crepant induced fibration, as in (Fujino and Gongyo 2014, sec. 3.3). At a prime of \(W'\), adding \(x'^*Q\) decreases its threshold by the local coefficient of \(Q\). Thus the recomputed discriminant and moduli divisors satisfy \[ \begin{split} B_{W'}&=B^{\rm cr}_{W'}+Q,\\ M_{W'}&=L'-K_{W'}-B_{W'} =\beta^*L_W-K_{W'}-B^{\rm cr}_{W'}. \end{split} \tag{81}\] The latter is precisely the old moduli b-divisor’s trace. Hence we may first dominate a model on which it is nef and descends, and then make any necessary further modifications; its new trace is its nef pullback. Repeating the minimum and threshold argument on each new diagram still proves \(0\le B_{W'}\le1\).

For completeness, at a base prime not exceptional for \(\beta\) the minimum does not change. Indeed, the crepant transform of the old auxiliary divisor is bounded by \(D'\) over its generic point, by (67); this gives the old minimum as a lower bound for all new candidates. An old minimizing prime survives strictly and supplies equality. The discriminant is likewise unchanged there. Thus any new discriminant support is contained in the exceptional locus together with the transform of the old support. Resolve that union. Recomputing the minimum on the resulting model therefore gives simple-normal-crossing support and keeps the moduli part nef. The extraction in 25 survives all these operations, so the exact section comparison continues to refer to the same fixed \(Y_0\). This proves all the assertions. ◻

Proof of 27 when \(S\) is a point. For a point base there is a nonzero constant base section and the zero test has no nonzero input satisfying its premise. Assume \(\dim Y>0\). Apply 4 and 7 to the chosen \(s\), obtaining a homogeneous vector with source \(-aL_Y\).

Since \(S\) is a point, the descended coefficient system is constant. In a constant frame its coordinates are rational sections of \(aL_Y\). The divisorial lattice bounds make each coordinate regular at every prime of \(Y\): this can be tested after finite ramified substitution, since a negative order remains negative after multiplying it by the ramification index. On the smooth variety \(Y\), regularity in codimension one gives global regularity. At least one coordinate is nonzero because the homogeneous vector is nonzero; choose it as \(\tau\).

If \(s|_{X_y}=0\) for \(y\in V\), 6 says that all these coordinates have positive order in the pulled-back original \(aL_Y\) frame along the divisor of the blowup of \(y\) (along \(y\) itself if \(Y\) is a curve). Finite ramification multiplies orders by its positive index. Hence each original regular coefficient has positive exceptional order, equal to its vanishing order at \(y\). Thus the same nonzero coordinate vanishes at \(y\), as asserted. ◻

Proof. Put \(r=\nu(P)\). The polynomial \[(A+tP)^n\] has degree \(r\) and strictly positive leading coefficient. In the polynomial \(n b(A+tP)^{n-1}H\), all coefficients of \(t^i\) for \(i\ge r\) vanish. Indeed \[(P+H)^{i+1}A^{n-1-i}=0\quad(r\le i\le n-1),\] and each mixed nef intersection in this expansion is nonnegative; in particular \(P^iHA^{n-1-i}=0\). When \(r=n\), this index range is empty. For \(r=0\) this says that the entire second polynomial is zero. Thus, for large \(t\), \[ (A+tP)^n>n(A+tP)^{n-1}bH. \tag{82}\]

We recall directly why this strict nef Morse inequality implies bigness here. Write \(U=A+tP\) and \(V=bH\). The first divisor is ample and the second is nef. Replace \(V\) by \(V+\delta A\), with rational \(\delta>0\) sufficiently small that the strict inequality persists. After multiplying by a common positive integer, \(U\) is integral ample and the new \(V\) is very ample. Choose a general member \(H_V\in|V|\). Successive restriction exact sequences give \[h^0\bigl(Z,l(U-V)\bigr) \ge h^0(Z,lU) -\sum_{j=0}^{l-1} h^0\bigl(H_V,(lU-jV)|_{H_V}\bigr).\] Multiplication by a nonzero section of \(jV|_{H_V}\) bounds each summand by \(h^0(H_V,lU|_{H_V})\). On a curve choose this section nonvanishing at the finitely many points of \(H_V\); in higher dimension take \(H_V\) smooth and irreducible. Ample Hilbert asymptotics therefore give \[h^0\bigl(Z,l(U-V)\bigr) \ge\frac{U^n-nU^{n-1}V}{n!}\,l^n+O(l^{n-1}).\] The coefficient is positive. Hence \(U-V\) is big, and adding back \(\delta A\) proves bigness before perturbation as well. ◻

Proof of Proposition 28. Choose \(b\geq0\) with \(K_S+bH\) big and put \(D_b=L+b h^*H\). By hypothesis (i) and bigness of \(K_S+bH\), there are an ample rational \(A_S\) and \(\delta>0\) rational such that \(D_b\ge_{\mathrm{pe}}\delta h^*A_S\). The divisor \(D_b\) is also big over \(S\). Hence \(D_b+c h^*A_S\) is big for some rational \(c>0\).

To justify the latter assertion, fix an ample \(A_W\) on \(W\). Relative bigness gives a nonzero generic-fiber section of \(lD_b-A_W\) for some divisible \(l\). Its direct image is nonzero; after twisting by \(kA_S\) for large \(k\) it has a nonzero global section. Thus \(lD_b-A_W+k h^*A_S\) is effective. Taking \(c=k/l\) proves the assertion. Now \[D_b=\frac{\delta}{c+\delta}(D_b+c h^*A_S) +\frac{c}{c+\delta}(D_b-\delta h^*A_S)\] expresses \(D_b\) as a positive multiple of a big class plus a pseudoeffective class. Therefore \(D_b\) is big.

Pull back and choose an ample rational \(A'\) on \(\widehat W\) with \(\pi^*L+b\widetilde H\ge_{\mathrm{pe}} A'\). For every \(t>0\), hypothesis (iii) gives \[(1+tQ)\pi^*L\ge_{\mathrm{pe}} A'+tP-b\widetilde H.\] The right side is big for large rational \(t\) by the numerical-dimension equality in (iii) and Lemma 29. Thus \(\pi^*L\), and hence \(L\), is big. Bigness descends under a dominant generically finite projective map, for example by the norm of a section after subtracting a small ample class. No equality of numerical and Iitaka dimensions for a general pseudoeffective divisor is being assumed. ◻

Proof. The extending lattices and their determinant divisors are invariant under the Galois group. For every prime \(A_0\) of \(W\), the orders of the pullback of \(D_i^G\) therefore agree with those of \(\widehat D_i^G\) at all primes over \(A_0\). Indeed, if \(e\) is the common ramification index and \(v\) their common order, then the coefficient downstairs obtained by pushforward and division by \(\deg\pi\) is \(v/e\).

Factor \(\pi\) through the finite normalization \(W^{\mathrm{fin}}\to W\). The difference \[Z=\pi^*P_W-\sum_i(i+1)\widehat D_i^G\] is exceptional over \(W^{\mathrm{fin}}\). Its negative is relatively nef: the pullback term has degree zero on contracted curves, and the remaining term represents the nef line \(P\). The negativity lemma (Kollár and Mori 1998, Lemma 3.39) gives \(Z\ge0\). This proves (86). Pushforward of the pseudoeffective class \(P\) gives \(\deg(\pi)P_W\), proving the first assertion.

We prove the lattice assertion at an arbitrary prime \(A_0\subset W\). Use the notation of 26: \[t_{A_0}=\min_{I\to A_0} \frac{a^{-1}\operatorname{ord}_I(\omega)+d_I}{m_I}, \quad m_I=\operatorname{ord}_I(x^*A_0), \quad d_I=\operatorname{coeff}_I(D).\] Choose a component \(I\) achieving this minimum. The discriminant coefficient \(b_{A_0}=\operatorname{coeff}_{A_0}(B_W)\) satisfies \[ t_{A_0}=\frac{a^{-1}\operatorname{ord}_I(\omega)+d_I}{m_I}, \qquad b_{A_0}\ge1-\frac{1-d_I}{m_I}. \tag{88}\] Both statements use this same \(I\).

At general points of \(A_0\) and \(I\), choose a parameter \(t\) downstairs and a parameter \(z\) upstairs. After an etale local change absorbing a unit, \(t=z^{m_I}\); the other base coordinates pull back to independent coordinates along \(I\). Work on a local cover \(t=v^e\) sufficiently divisible to make the Hodge monodromy unipotent and clear all relevant denominators. Write a coefficient \(c\) of \(\theta^i(u_W)\) in Schmid frames, using an ordinary frame of \(aK_W\) and a cotangent tensor having \(j\) factors \(dt/t\), with the other factors tangential. Define \(\operatorname{ord}(c)\) using the parameter \(v\).

The identity \(u_*=\omega x^*u_W\) holds for Higgs iterates as well: the Higgs field is linear over functions, so there is no derivative of \(\omega\). Make a further divisible substitution in \(z\) dominating the chosen \(t\)-cover. The lattice bounds of [prop:logarithmic-vector,lem:weight-extension] then apply to each coefficient in the independent pulled-back coordinate tensors. Each transverse factor pulls back as \(m_I\,dz/z\) and costs \(1-d_I\) units of \(I\)-order. Consequently \[m_I\frac{\operatorname{ord}(c)}{e} +\operatorname{ord}_I(\omega)+ad_I-j(1-d_I)\ge0.\] The order of \(\omega\) is measured in \(aK_{Y/W}\), so the canonical change of variables is already included. Using (88), we obtain \[ \frac{\operatorname{ord}(c)}e \ge-at_{A_0}+\frac{j(1-d_I)}{m_I} \ge-at_{A_0}+j(1-b_{A_0}). \tag{89}\] This is exactly the required regularity after twisting the source by \(-aL_W\): the transverse orbifold cotangent frame is \(t^{1-b_{A_0}}dt/t\), interpreted after ramification, and the tangential frames are ordinary differentials. For example, when \(b_{A_0}=0\) this frame is \(dt\), and when \(b_{A_0}=1\) it is \(dt/t\).

Generically the contractions of this map surject onto \(G_i\). On the unipotent resolution \(G_i\) has its subbundle lattice inside the Schmid grade, so the regularity just proved holds with this target lattice. Dualize, take the top exterior power of \(G_i^\vee\), and use the natural inclusion of an exterior power in a tensor power. The resulting source is \(\det G_i^\vee-ar_i\pi^*L_W\), which gives (87) by the order identification for \(D_i^G\) at primes over \(W\).

Here is also a global interpretation of the cover assertion. Fix an integer \(N_0>0\) clearing all the source divisors \(D_i^G-ar_iL_W\). On a smooth adapted cover \(\rho:W_{\rm ad}\to W\) for \(B_W+\Lambda\), the \(N_0\)th tensor power is a rational map from \(\rho^*\mathcal O_W(N_0(D_i^G-ar_iL_W))\) to the tensor power of degree \(N_0ir_i\) of its orbifold cotangent bundle. At each prime it is regular after a further common local ramification that is also unipotent for the Hodge system. Fractional cotangent frames pull back compatibly under that ramification; an inequality of integral orders after such a finite substitution implies the original inequality. Thus the map was already regular on \(W_{\rm ad}\). Regularity in codimension one extends across codimension two because source and target are locally free on its smooth model.

Increasing the boundary coefficient enlarges the cotangent lattice: the original transverse frame is the new one multiplied by \(t^{\operatorname{coeff}_{A_0}(\Lambda)}\). At auxiliary branch divisors with zero boundary coefficient the lattice is the pullback of ordinary cotangent forms. Hence the same argument applies at every prime of any adapted cover for the larger boundary. Only the denominators of this cover change with \(\Lambda\); the integer \(N_0\) is the fixed integer chosen above. ◻

Proof. Fix an ample rational divisor \(A\) on \(W\) and write \(n=\dim W>0\). For a positive rational \(\epsilon\) for which \(L_W+\epsilon A\) is pseudoeffective, the divisor \(M_W+\epsilon A\) is ample. Choose a sufficiently divisible \(k\) and a general smooth member of \(|k(M_W+\epsilon A)|\), transverse to every stratum of \(B_W\), and divide it by \(k\). This gives \[\Lambda_\epsilon\sim_{\mathbb Q}M_W+\epsilon A, \quad 0\le B_W+\Lambda_\epsilon\le1, \quad K_W+B_W+\Lambda_\epsilon\sim_{\mathbb Q}L_W+\epsilon A.\] The support is SNC. Thus the pair is log canonical and has pseudoeffective adjoint, as required in (Campana and Păun 2019, Theorem 1.3).

Choose a smooth Galois adapted cover for this pair, as in (Campana and Păun 2019, Definition 5.1 and Lemma 5.2). Such covers can be obtained by the covering construction: add general transverse auxiliary divisors to make the root divisors divisible in the Picard group, and normalize the resulting finite root covers. Use sufficiently many general auxiliary choices that, at each point and for each prescribed boundary component, one auxiliary equation is a unit. Locally the normalization then extracts roots of independent SNC coordinates and is smooth. The auxiliary ramifications are assigned boundary coefficient zero. On this cover the determinant of the orbifold cotangent bundle is the pullback of \(K_W+B_W+\Lambda_\epsilon\).

Put \(d_i=ir_i\). For \(d_i>0\), the tensor in 30 has rank \(n^{N_0d_i}\) and first Chern class \[N_0d_i\,n^{N_0d_i-1} \rho^*(L_W+\epsilon A).\] Apply the preceding quotient statement to its nonzero source line. Divide the resulting inequality by \(N_0\deg\rho\). For every movable curve class \(\alpha\) on \(W\) this gives \[\bigl(c_i(L_W+\epsilon A)+ar_iL_W-D_i^G\bigr)\cdot\alpha\ge0, \qquad c_i=d_i n^{N_0d_i-1}.\] For \(d_i=0\), the map is into the trivial line; take \(c_i=0\) and use its effective zero divisor. All \(c_i\) are independent of \(\epsilon\) and of the chosen representative \(\Lambda_\epsilon\).

The duality between the pseudoeffective divisor cone and the movable curve cone on the smooth projective variety \(W\) (Boucksom et al. 2013, Theorem 2.2) now gives \[ C(L_W+\epsilon A)+RL_W\ge_{\mathrm{pe}}P_W, \qquad C=\sum_i(i+1)c_i\ge0, \quad R=a\sum_i(i+1)r_i>0. \tag{91}\] The strict positivity of \(R\) follows from \(G_0\ne0\).

For completeness, consider \[\tau=\inf\{t\ge0:L_W+tA\text{ is pseudoeffective}\}.\] This is finite. Since \(P_W\) is pseudoeffective, (91) implies, for every rational \(\epsilon>\tau\), \[L_W+\frac{C}{C+R}\epsilon A \text{ is pseudoeffective}.\] If \(\tau>0\), closedness of the cone and \(\epsilon\downarrow\tau\) would give \(\tau\le C\tau/(C+R)<\tau\), a contradiction. Thus \(\tau=0\) and \(L_W\) is pseudoeffective. We may now let \(\epsilon\downarrow0\) in (91); pullback and (86) yield (90) with \(Q=C+R\). ◻

Proof. If \(h\) has relative dimension zero, it is birational by connectedness of its generic fiber, and the conclusion follows from \(K_W-h^*K_S\ge0\), \(B_W\ge0\), and nefness of \(M_W\). Assume henceforth that its relative dimension is positive. Use \(A\) and \(\Lambda_\epsilon\) from the preceding proof. We have established pseudoeffectivity of \(K_W+B_W+\Lambda_\epsilon\sim_{\mathbb Q}L_W+\epsilon A\) for every rational \(\epsilon>0\). Apply (Campana and Păun 2019, Theorem 3.4 and Remark 3.3) to the fibration and this smooth SNC log canonical pair. We spell out the model issue because the fixed \(W\) need not itself be the good model used in that theorem.

Take a commutative birational diagram \[\begin{array}{ccc} W'&\xrightarrow{h'}&S'\\ {\scriptstyle p}\downarrow&&\downarrow{\scriptstyle q}\\ W&\xrightarrow{h}&S, \end{array}\] with smooth projective varieties, resolved morphism, SNC discriminant and reduced inverse discriminant, and such that every divisor of \(W'\) contracted by \(h'\) to codimension at least two is exceptional over \(W\). These are the good-model requirements in (Campana and Păun 2019, Remark 3.3). They may be arranged by extracting on \(S\) the restricted valuations of the finitely many divisors of \(W\) with exceptional image, as in 25, and then resolving the map and the boundary. Once those original valuations have divisorial center, further base modifications preserve that property for their strict transforms; all remaining offending source divisors are exceptional over \(W\).

One can require the discriminant here to be the actual critical-value locus, rather than merely an SNC divisor containing it. To check this point in the preparation, first take an SNC divisor on the resolved base containing the critical values and all the loci over which the modifications occur; resolve its reduced inverse image, preserving the smooth complement. If a component of this base divisor is not generically a critical-value divisor and the relative dimension is positive, blow up a smooth component of its inverse image intersected with a general very ample hypersurface. Choose a component of this intersection dominating the base divisor, which exists since its generic fiber has positive dimension. The center has normal crossings with the other boundary components. Locally it has equations \(t=z=0\), where \(t\) is the base parameter and \(z\) a fiber coordinate; after the blowup a chart has \(t=uv\). The resulting critical locus therefore dominates that base divisor. Repeating for the finitely many remaining components makes the actual critical-value locus equal to the chosen SNC divisor. All these new source divisors are exceptional over \(W\), and the reduced inverse image remains SNC.

Set \(\Delta_\epsilon=B_W+\Lambda_\epsilon\) and take on \(W'\) the strict transform of \(\Delta_\epsilon\) plus the reduced \(p\)-exceptional divisor, resolving its support together with the discriminant. Denote this boundary by \(\Delta'_\epsilon\). Log canonicity gives \[K_{W'}+\Delta'_\epsilon =p^*(K_W+\Delta_\epsilon)+E_\epsilon, \qquad E_\epsilon\ge0 \text{ exceptional over }W.\] Its adjoint is therefore pseudoeffective. Theorem 3.4 of (Campana and Păun 2019) is applied with the full boundary \(\Delta'_\epsilon\) and gives pseudoeffectivity of \[K_{W'/S'}+(\Delta'_\epsilon)_{\rm hor}-D(h'),\] where \(D(h')\) counts ramification along primes dominating divisors of \(S'\). Its generic-fiber hypothesis, as used in the proof of that theorem, holds because the restriction of the pseudoeffective total adjoint to a very general fiber is pseudoeffective. The vertical boundary restricts to zero there, so the fiber sees exactly \((\Delta'_\epsilon)_{\rm hor}\). This does not require \(K_{W'}+(\Delta'_\epsilon)_{\rm hor}\) to be pseudoeffective globally; see also (Campana and Păun 2019, Remark 3.6). Adding the effective vertical boundary and \(D(h')\) proves that \(T'_\epsilon=K_{W'/S'}+\Delta'_\epsilon\) is pseudoeffective.

Write \(K_{S'}=q^*K_S+F\) with \(F\ge0\) exceptional; this is the canonical Jacobian divisor of a birational morphism between smooth varieties. Pushforward of the preceding pseudoeffective divisor gives \[p_*T'_\epsilon =K_W+\Delta_\epsilon-h^*K_S-p_*h'^*F.\] Consequently \(L_W+\epsilon A-h^*K_S\) is the sum of a pseudoeffective class and the effective class \(p_*h'^*F\). Letting \(\epsilon\downarrow0\) proves (92) on the original fixed \(W\). ◻

Proof. Apply 28 with \(L=L_W\) and \(H=\mathcal H\). Relative bigness is 26; relative pseudoeffectivity is 32; adjoint bigness is 21. The nef line \(P\) and numerical-rank equality are supplied by 22; its fixed comparison with \(L_W\) is 31. Thus \(L_W\) is big.

The divisible-degree comparison of 26 sends its complete section spaces to actual sections on the original fixed base and preserves their ratios. The images of sufficiently large divisible systems therefore have dimension \(\dim W\). Hence \(\kappa(Y,L_Y)\geq\dim W\geq\dim S>0\). ◻

Proof of 27. The point-quotient case, including its original-point zero assertion, was proved at the start of the section. For \(\dim S>0\), 24 supplies the nonnegative generic-fiber Iitaka dimension and the diagram used above. 33 then gives \(\kappa(Y,L_Y)\ge\dim W\ge\dim S>0\). These two alternatives prove all the assertions. ◻

Proof of 3. Apply 27 to the specified nonzero section \(s\). It gives \(\kappa(Y,L_Y)\geq0\). If that dimension is zero, the positive-dimensional quotient case is excluded. The point-quotient case supplies a nonzero original base section vanishing at every chosen \(y\in V\) for which \(s|_{X_y}=0\). This is the second assertion. Together with the reduction in 2, it completes the independent upper inequality. ◻