Proof. The difference in (8) is exceptional, since \(p\) is an isomorphism at the generic point of every target prime. To compute its coefficient at an exceptional prime \(E\), choose compatible local canonical forms. Near a general point of its center on \(Z\), let \(z_1,\ldots,z_n\) be coordinates in which the boundary components containing that center are \(z_i=0\) for \(1\leq i\leq r\), with coefficients \(\gamma_i\). Put \(t_i=\mathop{\mathrm{ord}}_E(p^*z_i)\) and let \(k_E\) be the order of the Jacobian of \(p\) along \(E\). The pullback of \[\frac{dz_1}{z_1}\wedge\cdots\wedge\frac{dz_r}{z_r} \wedge dz_{r+1}\wedge\cdots\wedge dz_n\] has at most a simple pole along \(E\). Indeed, at a general smooth point of \(E\), each logarithmic factor has the form \(t_i\,du/u\) plus a holomorphic one-form, and a nonzero wedge product contains at most one factor \(du/u\). Consequently \(k_E+1\geq\sum_i t_i\). The coefficient in question is therefore \[k_E+1-\sum_i\gamma_i t_i \ \geq\ \sum_i(1-\gamma_i)t_i\ \geq\ 0.\] For an effective integral exceptional divisor \(mA\), normality of \(Z\) gives \(p_*\mathcal O_{Z'}(mA)=\mathcal O_Z\): a meromorphic function with poles only on \(mA\) descends outside the codimension-two exceptional image and extends by Hartogs’ theorem. The projection formula proves (9), compatibly with products and ratios.

Finally, an exceptional prime has infinite boundary multiplicity. A nonexceptional prime is a strict transform, with pullback order one and unchanged coefficient. These two cases prove (2). ◻

Proof. On the complement of the exceptional image the form is a holomorphic section of the prescribed line bundle on \(Z\): divisorial regularity suffices on a smooth space. A local trivialization reduces extension across the remaining codimension-two analytic subset to Hartogs’ extension theorem. The tensor-twisted assertion has the same proof. ◻

Proof. For (i), choose the parameter so that the relevant smoothness and dimension statements hold for the finitely many exceptional components and their images. A component dominating the base restricts to a reduced divisor, and its exceptional image restricts to codimension at least two. Components not dominating the base do not meet this fiber. The restricted map has the same boundary rule, so 2 applies on the fiber. For (ii), every prime exceptional for the composite is either newly exceptional or the transform of an earlier exceptional prime; in both cases its coefficient is one.

For (iii), fix a prime \(D\subset T\). Every old prime dominating \(D\) has a strict transform, with the same pullback order and boundary multiplicity. Every new prime in the minimum defining \(m(g',\Gamma';D)\) is exceptional for \(p\), so its boundary multiplicity is infinite. Adding these entries leaves the old minimum unchanged, including when that minimum was already infinite. ◻

Proof. Flatten first, before resolving the source. After resolving the new base by further base change, denote the equidimensional strict transform by \(\bar g:\bar Z\to T\), and its modification to \(Z_0\) by \(\bar p\). Resolve \(\bar Z\) and all boundary data, and write \[Z\xrightarrow{r}\bar Z\xrightarrow{\bar p}Z_0, \qquad p=\bar p\circ r, \qquad g=\bar g\circ r.\] Let \(N=\dim Z_0\), \(b=\dim T\), and \(d=N-b\). If a prime \(E\) on \(Z\) has image contained in an analytic subset \(C\subset T\) of codimension at least two, equidimensionality gives \[\dim\bar g^{-1}(C)\leq\dim C+d\leq N-2.\] It follows that \(r(E)\), and therefore also \(p(E)\), has dimension at most \(N-2\). Thus \(E\) is exceptional over \(Z_0\). The boundary and orbifold-morphism assertions follow from 2. The new map has connected general fiber. Its Stein factorization has a finite bimeromorphic map to the normal base \(T\), which is an isomorphism, so all its fibers are connected.

For any further base modification \(T'\to T\), pull back the flat intermediate family and take a common resolution dominating the previous smooth source. The same dimension estimate proves exceptionality over \(Z_0\). This includes principalizations of prescribed bad loci and their SNC resolutions on the base. Iterating proves the persistence assertion; 4(ii) identifies the stepwise boundary with that computed directly over \(Z_0\). ◻

Proof. Assertion (i) is 5. For (ii), a source prime is either newly exceptional, with coefficient one, or the strict transform of an old prime with unchanged image in \(Y\). The hypothesis handles the latter case. Assertions (iii) and (iv) are 4; (11) is the defining minimum for \(C\). Flattening the original map supplies the initial coefficient-one condition by making its contracted primes exceptional. ◻

Proof. Consider all degrees clearing the denominators of \(\Delta\). Their relative adjoint direct images are coherent. First shrink to the dense analytic open where the map and boundary strata are smooth. There the family and its adjoint line bundles are flat over the base; generic constancy of the fiber section dimension permits cohomology and base change [33], with its corrigendum [34]. Together with the evaluation ranks this gives a dense analytic open for each degree; outside the resulting countable collection of exceptional sets, the maximum image dimension is \(k\), attained in one fixed degree.

Choose a sufficiently divisible degree giving the stabilized maps, resolve the relative meromorphic map to the projective bundle of its direct image, and take Stein factorization onto the image. The Iitaka fibration theorem for line bundles on compact complex spaces identifies the resulting map on very general fibers [59]. No finite generation is required. The image has relative dimension \(k\); its Stein factor is in class \(\mathcal C\), being a proper image of a space in that class. Resolve it and the resulting source map. Connectedness of the fibers of \(h\) follows from that of \(f\), since their images under \(g\) are precisely the fibers of \(h\).

The assertion that the restriction has Kodaira dimension zero concerns the original line bundle. Its standard proof applies here: coherent direct images on the projective Iitaka image acquire sections after an ample twist. If the restriction had positive Iitaka dimension, multiplying those sections by powers of the defining system would enlarge the Iitaka image. A nonzero defining section restricts nontrivially to a very general Iitaka fiber, so the restriction has dimension exactly zero.

Apply this to the resolved log adjoint on a very general \(f\)-fiber. Its divisible section ring is unchanged by 2, and adjunction identifies the restriction with \(K_G+\Delta_G\). Further source modifications preserve the same section ring, hence the relative Iitaka factorization and its asserted fiber dimensions. ◻

Proof. Let \(D\subset Y\) and \(V\subset W\) be prime divisors with \(h(V)=D\). Write \(b_V=\mathop{\mathrm{ord}}_V(h^*D)\). If \(E\subset X\) is a prime divisor dominating \(V\), put \(a_E=\mathop{\mathrm{ord}}_E(g^*V)\). At their generic points orders multiply, so \[\mathop{\mathrm{ord}}_E((h\circ g)^*D)=b_Va_E.\] The hypothesis on \(C\) implies \(b_Va_E m_\Delta(E)\geq m_C(D)\) for every such \(E\). Taking the minimum over them and then using the hypothesis on \(T\) gives \[b_Vm_T(V)\ \geq\ b_Vm(g,\Delta;V)\ \geq\ m_C(D).\] Taking the minimum over \(V\) proves the first assertion. All these inequalities are valid with the prescribed convention for infinity.

For the second assertion, let \(V\) be mapped into codimension at least two in \(Y\). Every prime \(E\) dominating it has \(\Delta_E=1\), so \(m(g,\Delta;V)=\infty\). It follows that \(B(g,\Delta)_V=1\), and hence \(T_V=1\). ◻

Proof. We may check the assertions after a finite cover and resolution on which local monodromy is unipotent. This does not require an integral lattice. Indeed, \(\pi_1(T\setminus E)\) is finitely generated. Put the entries of finitely many monodromy generators and their inverses in a finitely generated \(\mathbf Z\)-algebra \(R\subset\mathbf C\). Adjoin the finite group of roots of unity generated by the eigenvalues of the finitely many boundary monodromies, and invert every difference \(\zeta-1\) for a nonidentity element of this group. Reduction at a maximal ideal with finite residue field gives a finite-index normal congruence subgroup. Every element of a local boundary monodromy group that lies in this subgroup is unipotent: its eigenvalues belong to that finite group and reduce to one, so the inverted differences force them to equal one. This also applies to products of commuting boundary monodromies at crossings. The corresponding finite cover extends over the compactification, and a Kähler resolution gives the desired unipotent model; see [61]. These covers preserve nefness, numerical dimension, and bigness of the rational line. Let \(h\) be its Hodge metric and \(\Theta\) its curvature on \(T\setminus E\), normalized to represent \(c_1(A)\). Griffiths’ curvature formula gives \(\Theta\geq0\); its rank is the rank of the line map, since its value on a tangent vector is the squared norm of the second fundamental form of the highest step. The same formula for the Griffiths line detects the differential of the full period map.

The Hodge norm estimates in extending frames give, in boundary coordinates \(t_1,\ldots,t_k\), weights whose absolute values are bounded by \[ C\left(1+\sum_{j=1}^k \log(-\log|t_j|)\right). \tag{29}\] The semipositive metric therefore extends as a positive current in \(c_1(A)\) with zero Lelong numbers. Regularization of positive currents then gives nefness; see [22]. This extends the nefness argument of [7] from a proper log smooth algebraic space to the present compact Kähler setting.

To identify numerical dimension, we must also show that no intersection mass is lost at \(E\). Write \(\alpha=c_1(A)\) and fix \(\varepsilon>0\). Since \(\alpha\) is nef, \(\alpha+\varepsilon[\omega]\) has a smooth Kähler representative \(\theta_\varepsilon\). Write \[\Theta+\varepsilon\omega=\theta_\varepsilon+dd^cu, \qquad dd^c=\frac{\sqrt{-1}}{2\pi}\partial\bar\partial.\] The global potential \(u\) satisfies the local bounds (29). Put \(n=\dim T\). If \(E\) is empty, the potential is smooth and there is no lost mass. Otherwise write \(E=E_1+\cdots+E_N\) and choose smooth divisor metrics, with defining sections \(s_j\), such that \[u_j=-\log|s_j|^2,\qquad dd^c(-u_j)=[E_j]-\eta_j,\] where the \(\eta_j\) are smooth curvature forms. Choose \(c>0\) with \(c\eta_j\leq\theta_\varepsilon\) for all \(j\), and rescale the metrics so that \(cu_j\geq1\). Then \(\phi_j=-cu_j\) is \(\theta_\varepsilon\)-plurisubharmonic and \(\phi_j\leq-1\). For any fixed \(0<b<1\), the attenuation theorem [36] gives \[\psi_j=-(-\phi_j)^b=-c^bu_j^b\] with full Monge–Ampère mass in the Kähler class \([\theta_\varepsilon]\). The full-mass class is convex and contains the zero potential [36]. Thus, for \(0<\delta\leq c^b/N\), the potential \[ v=-\delta\sum_{j=1}^N u_j^b =\sum_{j=1}^N\frac{\delta}{c^b}\psi_j +\left(1-\frac{N\delta}{c^b}\right)0 \tag{30}\] is \(\theta_\varepsilon\)-plurisubharmonic and has full mass. A positive power grows faster than a logarithm, so (29) implies \(v-C\leq u\) globally for some constant \(C\). Full mass is preserved on passing to less singular potentials [13]. Hence \(u\) has full mass. The non-pluripolar measure gives no mass to the analytic set \(E\) and agrees with the smooth wedge product on its complement. Consequently \[\int_{T\setminus E}(\Theta+\varepsilon\omega)^n =(\alpha+\varepsilon[\omega])^n.\] Expand both sides as polynomials in \(\varepsilon\). Their coefficients are finite, since all summands on the left are nonnegative. Equality for all \(\varepsilon>0\) gives \[ \int_{T\setminus E}\Theta^k\wedge\omega^{n-k} =\alpha^k\cdot[\omega]^{n-k} \qquad(0\leq k\leq n). \tag{31}\] If the generic curvature rank is \(r\), the left side is positive for \(k\leq r\) and zero for \(k>r\). This proves the numerical assertion. For an immersive Griffiths line the top integral is positive, and the volume criterion for a positive current on a compact Kähler manifold gives bigness; see [12]. ◻

Proof. The extended filtration and \(W(N_i)\) give the limiting mixed variation along the open stratum. Since the highest step has rank one, exactly one weight grade contains it. Changes of the normal parameter act by \(\exp(cN_i)\) and act trivially on the associated graded. At a further crossing the relative monodromy filtration identifies the extension of this graded variation with the graded of the original extending filtration. Thus the identification holds on the entire component. For log smooth algebraic spaces it is stated in [7]; its local proof uses the multivariable nilpotent orbit theorem [19] and applies here as well. The filtrations and strictness also restrict to a complex summand. Apply 18 on \(E_i\). ◻

Proof. Semisimplicity and the theorem of the fixed part hold for polarizable real and complex variations on a Zariski open subset of a compact Kähler manifold. We may use a Kähler modification of \(B\) to apply them; see [54] and [5]. Thus the algebraic monodromy group is reductive. Pass temporarily to a finite cover on which it is connected. The irreducible complex constituents of the local system carry polarizable complex variations. They can be defined over a number field: the representation is rational and its algebraic monodromy is reductive. The determinant of each constituent is unitary, as are all its algebraic conjugates, which are again constituents of the rational representation. Its monodromy values are algebraic units by the integral lattice. Kronecker’s theorem makes these values roots of unity in a fixed number field, hence makes the determinant character finite. It is consequently trivial on \(G\). This is Deligne’s finite-determinant argument; its classical algebraic formulation is [21]. The connected center of \(G\) acts by scalar characters on the irreducible constituents; their determinants detect these characters. Since the original representation is faithful, this connected center is trivial. Therefore \(G\) is semisimple.

We next equip the monodromy Lie algebra with a rational polarization. In every tensor construction, the subspace of \(G\)-invariant tensors is the fixed part and has a constant Hodge structure. In a flat trivialization the Hodge circles \(h_x\) and \(h_{x_0}\) therefore induce exactly the same linear operator on each such subspace at each parameter \(z\in S^1\). Reductivity identifies \(G\) with the pointwise stabilizer of all its invariant tensors. The circles preserve these subspaces, so normalize \(G\), and the equality of their actions gives \[h_x(z)h_{x_0}(z)^{-1}\in G(\mathbf R).\] Here is a direct local conjugacy argument. Along a smooth path in the base, write \(a_t=\operatorname{Ad}\circ h_t:S^1\to G^{\rm ad}(\mathbf R)\) and \(u_t(z)=(\partial_t a_t(z))a_t(z)^{-1}\). Differentiating the homomorphism identity gives \[u_t(zz')=u_t(z)+\operatorname{Ad}(a_t(z))u_t(z').\] Normalized Haar averaging, with \(X_t=\int_{S^1}u_t(z')\,dz'\), yields \(u_t(z)=X_t-\operatorname{Ad}(a_t(z))X_t\). The solution of \(g_t'g_t^{-1}=X_t\), \(g_0=1\), therefore gives \(a_t=g_ta_0g_t^{-1}\). Lift this path through the central isogeny \(G(\mathbf R)^+\to G^{\rm ad}(\mathbf R)^+\). The ratio of \(h_t\) and \(g_th_0g_t^{-1}\) lies in \(G(\mathbf R)\) by the invariant-tensor equality, and centralizes \(G\) by the equality of their adjoint actions. It lies in the finite center of \(G\); continuity in the circle parameter makes it the identity. Thus \(h_t=g_th_0g_t^{-1}\), proving that the lifted period map lies in a single \(G(\mathbf R)^+\)-orbit.

It follows that \(\mathfrak g_{\mathbf Q}=\operatorname{Lie}(G)\) is a Hodge sub-local system of \(\mathop{\mathrm{End}}(\mathbb V_{\mathbf Q})\), with its weight-zero grading. The Weil involution \(\theta\) preserves \(\mathfrak g_{\mathbf R}\) and is the restriction of the Cartan involution of the real polarization isometry group. To check the restriction, use the positive Hodge metric: on its Lie algebra \(\theta(X)=-X^*\). If \(\mathfrak g_{\mathbf R}=\mathfrak k\oplus\mathfrak p\) is its eigenspace decomposition, then \(\mathfrak k\oplus i\mathfrak p\) is a compact real form of \(\mathfrak g_{\mathbf C}\), represented by skew-Hermitian endomorphisms. Thus \(-B_{\mathfrak g}(X,\theta X)\) is positive definite, proving that \(\theta\) is Cartan on \(\mathfrak g_{\mathbf R}\). The appropriately signed Killing form therefore polarizes this weight-zero variation. The form is rational. Moreover \[\mathfrak g_{\mathbf Q}\cap\mathop{\mathrm{End}}(\mathbb V_{\mathbf Z})\] is a monodromy-invariant lattice. This gives the rationally polarized integral adjoint variation.

Let \(\ell\) be the highest line at a reference point of the chosen complex summand. The non-lowering parabolic of the Hodge grading preserves \(\ell\), so a Borel subgroup preserves it. Its closed orbit \[J=G_{\mathbf C}\ell\] is a flag variety, a product of flag varieties for the complex simple factors. The span \(W\) of this orbit is irreducible. Indeed, in a semisimple decomposition of the representation, the nonzero components of a Borel-eigenline all have the same highest weight; their span is a single diagonal copy of that irreducible representation. The line is its unique highest Hodge step. Indeed, at every point the Hodge circle normalizes \(G\) and preserves the highest line there; it consequently preserves the span of its \(G_{\mathbf C}\)-orbit. Thus \(W\) is a Hodge subvariation. The lifted line map takes values in \(J\), and the original line is the pullback of the tautological line there, equivariantly for monodromy.

Call a complex simple factor visible if it acts nontrivially on \(J\). Retain every rational simple adjoint factor containing a visible factor. This collection is invariant under the possibly disconnected monodromy group. On its Lie algebra we obtain the variation \(\mathbb A\); its monodromy is contained in the automorphisms of a lattice and is discrete. Notice that an entire rational factor is kept. Projection to only one real factor would not ensure discreteness.

We now recover the line from this adjoint system. Choose a positive integer \(N\) for which the highest weight \(N\lambda\) of \(W\) belongs to the root lattice. The Cartan component \(W_N\subset W^{\otimes N}\) is generated by the orbit of \(\ell^{\otimes N}\). It factors through the connected adjoint group, and its highest Hodge line is \(\ell^{\otimes N}\). The same orbit-span argument just used shows that \(W_N\) is a Hodge subvariation.

We must also account for disconnected monodromy before applying tensor generation. Deck transformations take the lifted line to another point of the same connected orbit, so they preserve \(W\) and its Cartan components. The kernel of the action on the retained adjoint factors centralizes the \(G\)-action on \(W\), and hence acts scalarly by irreducibility. Its connected part is semisimple and has no nontrivial scalar character. Its component group is finite. Increasing \(N\) therefore kills the entire kernel action on \(W_N\). Consequently \(W_N\) is a representation of the full projected algebraic monodromy group, not just of its identity component. This step rules out a discarded nontorsion scalar local system.

The adjoint representation of that projected group is faithful and self-dual. By tensor generation and complete reducibility, \(W_N\) embeds as a flat direct summand in a tensor construction \(\mathbb T(\mathbb A)_{\mathbf C}\); see [48]. To make the embedding compatible with Hodge structures, apply the theorem of the fixed part to \[H^0\bigl(B^\circ, \mathcal Hom(W_N,\mathbb T(\mathbb A)_{\mathbf C})\bigr).\] This nonzero space of flat maps has a constant complex Hodge structure. A nonzero Hodge-homogeneous component of any nonzero flat map is still flat. Its kernel is \(G\)-invariant, so the map is injective because \(W_N\) is irreducible. A constant one-dimensional complex Hodge shift makes it a morphism of variations; that shift changes neither the underlying holomorphic line nor its boundary extension. This is the constituent and fixed-part argument of [5].

The complex summand carries a flat Hermitian polarization. In a real tensor variation of weight \(w\) polarized by \(Q\), it is induced by \(i^wQ_{\mathbf C}(v,\bar u)\), up to the conventional overall sign. Its restriction to a Hodge subvariation is nondegenerate because it is definite with the prescribed sign on each Hodge component. There is no need for the real bilinear form itself to restrict nondegenerately to the chosen complex summand: it may pair that summand with its conjugate.

Finally, on a factor-preserving finite cover an irreducible representation is a tensor product over the simple factors. The dimension of its highest step is the product of the corresponding dimensions. Since that dimension is one, each factor has a unique rank-one highest step. This gives the asserted factor lines. Their extensions are nef and functorial by [period:numerics,period:boundary]. ◻

Proof. Suppose \(\gamma\) fixes those periods. By equivariance, every monodromy conjugate of \(\gamma\) is everywhere of Hodge type \((0,0)\) in the endomorphism variation. Their rational span is therefore a rational subvariation of type \((0,0)\). Its induced polarization is positive definite. Intersection with the endomorphism lattice is a full monodromy-invariant lattice: clear one denominator of \(\gamma\); integral monodromy conjugation preserves that denominator. A positive definite integral monodromy group is finite. Consequently the connected group \(G\) fixes \(\gamma\), so \(\gamma\) centralizes every inner automorphism of \(\mathfrak g\). The centralizer of the inner group in \(\operatorname{Aut}(\mathfrak g)\) is trivial: on each simple ideal commutation with all adjoint operators makes an endomorphism scalar, and compatibility with the Lie bracket makes this scalar one; permutations of distinct ideals do not commute with their separate inner actions. Thus \(\gamma=1\). ◻

Proof. Choose a compact Kähler modification of the class-\(\mathcal C\) source. Then pass to a finite level at which the projected monodromy is neat and preserves the factors. Finite covers of the open locus extend to finite ramified covers of compactifications and then to Kähler resolutions; see [61]. Resolve the boundary to simple normal crossings. Extend the period map over all finite-local-monodromy strata. At neat level their monodromy is trivial and the nilpotent orbit theorem gives this extension.

The extended period map on this locus is proper; see [35]. Indeed, if a sequence tending to the boundary has period images in a compact subset of the quotient, translate period lifts into a compact set. The punctured-disk Schwarz estimate makes the displacement of each vanishing-coordinate monodromy tend to zero. The discrete group acts properly, since its isotropy in the real period domain is compact. The local monodromies must therefore be trivial at neat level, so such a sequence cannot escape the extended locus.

Take the Stein factorization of this proper map. Its first map has compact connected fibers and its base is finite over the period image. It extends after modifications to a map of compact class-\(\mathcal C\) spaces with Kähler resolutions by [61] (version 1). That theorem applies to proper connected-fiber maps from Zariski open subsets of compact class-\(\mathcal C\) spaces. In terms of cycles, one first flattens the proper map, sends its irreducible general fibers to a component of the Douady space of the compact source, and takes the closure of that family. The component is compact and in class \(\mathcal C\). Near a general fiber the map to the cycle space is an open embedding, since an irreducible compact cycle of the same dimension in a sufficiently small neighborhood maps to a compact analytic subset of a small Stein neighborhood in the base, hence to a point, and must be the whole irreducible fiber. The incidence map is consequently a modification of the source. This produces the required compact quotient and also shows its uniqueness as the quotient by the connected general period fibers.

On a dense open subset of this quotient the period and its integral variation descend. At neat level a constant period has trivial stabilizer; local slices to the connected-fiber map give the descent data. After shrinking to a proper smooth fibration, the map on fundamental groups is surjective, so the descended variation has the same connected algebraic monodromy. Its period differential is generically injective by construction. On a Kähler compactification its Griffiths line is big by 18; equivalently, one can apply [14]. The quotient therefore has a smooth projective model. This is also the argument in [61].

The finite deck group acts on the connected-period-fiber quotient. Take its finite quotient and resolve. This again has a smooth projective model. A general fiber before this finite quotient is the union of connected level fibers indexed by one deck-group orbit. The group acts transitively on these fibers, so the quotient is the continuous image of any one of them and is connected. Thus the induced map from the original source has connected general fibers, hence connected fibers after its Stein factorization. The latter does not change its function field. All maps can be resolved using modifications of the original source and of this projective model.

There is an actual adjoint variation on a dense Zariski open subset of the quotient before changing level. Write \(\Gamma'\) for the normal level subgroup of \(\Gamma\). The induced action of \(\Gamma/\Gamma'\) on the level quotient is faithful. Indeed, if \(\delta\in\Gamma\) acts trivially there, discreteness shows locally that its action on lifted periods agrees with a fixed \(\eta\in\Gamma'\). Then \(\eta^{-1}\delta\) fixes a period germ, hence the whole generic lifted period image by analytic continuation. By 21 it is the identity, so \(\delta\in\Gamma'\). Resolve the finite group action equivariantly and remove its proper algebraic fixed loci and the branch locus. On the resulting dense Zariski open subset the level quotient is free and unramified. Its equivariant integral adjoint local system and Hodge filtration descend there. In particular, this descent uses an open subset, not merely very general points.

The highest-weight construction in 20 gives a line on that open subset whose pullback is a fixed positive power of \(L\). Take its parabolic extension on a log smooth projective compactification \(S\) and divide by that power to obtain \(P\). Nefness follows from 18. Pullback of the unipotent extensions, followed by rational descent, gives \(\tau^*L=p^*P\) on the compact models, including the boundary. Equivalently, this can be checked at level and then descended by the norm: a line whose finite pullback is trivial is torsion in the rational Picard group. The identity therefore includes the boundary and persists under further modifications.

If no factor is retained, the orbit is a point and the only possible scalar monodromy is the finite scalar group treated in 20. Its parabolic line is rationally trivial, as asserted. ◻

Proof. Write \(n=\dim S\) and \(r=\nu(P)\), the generic rank of the line map \(j\) by 18. The full retained period map has rank \(n\); the argument does not require \(r=n\). Work first on a finite Galois cover of the open locus with connected algebraic monodromy and no factor permutations. Its connected adjoint group is denoted by \(G\).

The Ax–Schanuel decomposition. Choose a Hodge-generic point, also general for the ranks of the factor period maps and their joint differentials. There are only finitely many rational normal-factor decompositions of \(G\), so these requirements can be imposed simultaneously. Let \(T\) be a local fiber germ of \(j\) there, and \(Z\) its irreducible Zariski closure in the source of the variation. Thus \(\dim T=n-r\). Move within the germ to a smooth point of \(Z\) and use a smooth open subset of a resolution when applying Hodge theory.

The restricted variation on \(Z\) has the same generic Mumford–Tate group as the ambient variation, since \(Z\) contains the chosen Hodge-generic point. The monodromy normality theorem therefore makes its connected monodromy \(H\) normal in \(G\); see [4]. In adjoint notation write \[G=H\times Q.\] The \(Q\)-period is constant on the lifted \(Z\), by the fixed-part theorem after killing finite monodromy. The varying monodromy domain on \(Z\) is therefore the domain for \(H\). Write \(J_H\) for its line-orbit projection and \(d_H=\dim J_H\). In local flat coordinates write \(j=(j_H,j_Q)\), with targets \(J_H\) and \(J_Q\), the products of line orbits for the two groups. We will show that the \(H\)-line map has full rank \(d_H\) on \(Z\), whereas the \(Q\)-line map and the full \(Q\)-period map have equal generic rank on the ambient base.

For the first assertion, put \(m=\dim Z\) and \(d=\dim\check{\mathcal D}_H\), where \(\check{\mathcal D}_H\) is the compact dual of the connected-monodromy period domain. The projection \(\check{\mathcal D}_H\to J_H\) is a surjective morphism of homogeneous projective flag varieties. All its fibers have dimension \(d-d_H\). Let \(V\subset Z\times\check{\mathcal D}_H\) be the algebraic locus where the projected line equals its value on \(T\). It follows that \(\dim V=m+d-d_H\). The period graph has dimension \(m\) in an ambient variety of dimension \(m+d\), so its expected intersection dimension with \(V\) is \(m-d_H\).

By [8], an intersection component of larger dimension projects into a proper algebraic weak Mumford–Tate subvariety of \(Z\). This is impossible for the component containing the lifted period graph over \(T\), since \(T\) is Zariski dense in \(Z\). Conversely, a map to \(J_H\) has fibers of dimension at least \(m-d_H\). At the chosen smooth point the restricted fiber is exactly \(T\), because the ambient local fiber is \(T\subset Z\). Equality follows, proving the claim. The cited theorem applies to the rationally polarized integral adjoint variation on a smooth algebraic base and uses precisely its connected-monodromy domain.

Let \(q\) denote the full \(Q\)-period map and set \(s_Q=\mathop{\mathrm{rank}}(dq)\). Choose the smooth point \(x\in T\cap Z_{\mathrm{reg}}\) inside the ambient constant-rank neighborhood fixed above. Since \(T\) is Zariski dense in \(Z\), such a point exists. The constant-rank theorem gives \(\ker(dj_x)=T_xT\subset T_xZ\), and hence \[\mathop{\mathrm{rank}}(dj_x|_{T_xZ})=\dim Z-\dim T=d_H.\] The \(Q\)-period is constant on \(Z\), so \(T_xZ\subseteq\ker(dq_x)\). Thus, at this same point, \[\ker(dj)\subseteq\ker(dq),\qquad dj_H(\ker(dq_x))=T_{j_H(x)}J_H.\] The first inclusion uses \(T_xT\subset T_xZ\); the second is the just-established rank calculation on \(T_xZ\). The differential \(dj_Q\) factors through \(dq\). If \(dj_Q(v)=0\), choose \(w\in\ker(dq)\) with \(dj_H(w)=dj_H(v)\). Then \(v-w\in\ker(dj)\), and therefore \(dq(v)=0\). We conclude \[ \mathop{\mathrm{rank}}(dj_Q)=s_Q,\qquad r=d_H+s_Q. \tag{33}\] These are generic identities by our choice of the point.

The factor-line construction, after a common positive rescaling and a further finite cover, now gives nef lines with \[ \pi^*P=P_H+P_Q,\qquad \nu(P_H)=d_H, \qquad P_Q=q^*B_Q. \tag{34}\] Here \(q:\widetilde S\to S_Q\) is a resolved projective quotient for the full \(Q\)-period, constructed by 22, and \(\dim S_Q=s_Q\). The highest-line rank on \(S_Q\) is \(s_Q\) by (33). Only the quotient construction, not adjoint positivity, has been used to construct \(S_Q\).

Reduction for a boundary component. For a fixed marked \(D_i\), use a common finite Galois cover that preserves the factors and makes boundary monodromy unipotent. Resolve it and the map \(q\) equivariantly, and let \(E_i\) be the strict transform of a component above \(D_i\) that is generically finite over \(D_i\). The inertia group at its generic point fixes \(E_i\) pointwise. Further rescaling does not change any numerical dimension. Nefness gives \[\nu(P_H|_{E_i})\leq d_H.\] For example, every intersection involving \(E_i\) is bounded by the corresponding intersection with a sufficiently ample effective divisor; the vanishing of higher powers of \(P_H\) then gives this inequality. If \(E_i\) does not dominate \(S_Q\), \[\nu(P_Q|_{E_i})\leq s_Q-1,\] and the binomial expansion for the two nef lines in (34) proves the strict drop. We may therefore suppose \(E_i\) dominates \(S_Q\). At its generic point the \(Q\)-variation is pulled back from the interior of \(S_Q\), so its local unipotent logarithm vanishes.

Infinite local monodromy. If the original local monodromy is infinite, its logarithm at unipotent level is a nonzero rational element \(N\in\mathfrak h\). Every nonzero rational element of a rational simple factor has nonzero projection to every conjugate complex factor. Each retained rational factor has a visible complex factor. Hence \(N\) acts nontrivially on \(J_H\).

Let \(w\) be the pure weight of the representation supplying \(P_H\), and let \(F^a=\mathbf Cv\) be its limiting highest line at a general point of \(E_i\). This line has a unique weight contribution, say of weight \(w+k\). It is primitive and \(k\geq0\): a nonprimitive contribution, or a contribution below the middle weight, would come by monodromy Lefschetz from a nonzero filtration piece above \(F^a\). Moreover \[ N^{k+1}v=0,\qquad N^kv\ne0. \tag{35}\] To see that this statement holds for the actual vector rather than only its weight-graded image, use the Deligne splitting of the limiting mixed Hodge structure. The one-dimensional highest step occupies a single summand, and \(N\) has degree \((-1,-1)\) for this functorial splitting. Primitive vanishing on the graded piece therefore gives the first equality in (35); the Lefschetz isomorphism gives the second. For a complex summand the same argument is made in its conjugate-paired real variation and restricted to the summand.

In nilpotent-orbit coordinates \([v]\in J_H\). Since this orbit is closed in projective space and invariant under \(\exp(tN)\), \[ [N^kv]=\lim_{t\to\infty}\exp(tN)[v] \in J_H\cap\mathbb P(\ker N). \tag{36}\] This fixed locus is a proper closed subset of the irreducible variety \(J_H\), because \(N\) acts nontrivially on it. Its dimension is less than \(d_H\). Restrict to a simply connected open part of the boundary stratum on which \(k\) is constant. In a flat trivialization, both \(N\) and \(W(N)\) are constant. Project \(N^kv\) to \(\mathop{\mathrm{Gr}}^{W(N)}_{w-k}\) and apply the inverse of the fixed Lefschetz isomorphism \[N^k:\mathop{\mathrm{Gr}}^{W(N)}_{w+k}\longrightarrow\mathop{\mathrm{Gr}}^{W(N)}_{w-k}.\] This recovers the weight-graded limiting highest line. Every actual vector \(N^kv\) lies in \(W(N)_{w-k}\), and its image in that graded piece is nonzero. Thus the graded line map factors through the rational map on \[J_H\cap\mathbb P(\ker N)\cap\mathbb P(W(N)_{w-k})\] induced by this fixed linear quotient and inverse. This locus is contained in the proper fixed locus in (36), and the rational map is defined at every point under consideration. Its rank is less than \(d_H\). By 19, \[\nu(P_H|_{E_i})<d_H.\] Together with \(\nu(P_Q|_{E_i})\leq s_Q\) this proves the required inequality in the infinite-monodromy case.

Finite nonidentity local monodromy. Let \(\gamma\) be the original finite local monodromy. The disconnected-monodromy argument in 20 shows that \(\gamma\) preserves the orbit span \(W\). It therefore acts on the highest-line orbit \(J\), permuting its nontrivial simple-factor flag varieties. Equivariance preserves the kernel of \(G_{\mathbf C}\to\operatorname{Aut}(J)\), which is the product of the invisible simple factors. Thus visible and invisible factors cannot be exchanged. At unipotent level its logarithm is zero, so the variation extends with pure limiting flags. The inertia generator fixes the generic boundary point; equivariance therefore makes these flags fixed by \(\gamma\). Suppose that the boundary line retained rank \(r=d_H+s_Q\). Since \(E_i\) dominates \(S_Q\), its full \(Q\)-period and its \(Q\)-line map both have rank \(s_Q\). Along a generic fiber of this \(Q\)-period, the \(H\)-line map must consequently have rank \(d_H\). Its coordinates fill an open subset of \(J_H\) while the full \(Q\)-flags remain fixed.

The fixed-point condition for \(\gamma\) rules out permutations involving any visible \(H\)-factor, including a permutation with a factor whose coordinate is held fixed. On each visible \(H\)-factor of the highest-line orbit it forces the identity action. The action of a simple adjoint group on a nontrivial flag variety is faithful; equivariance then makes the induced automorphism of that simple group the identity as well. Rationality forces the same conclusion on all conjugate factors. Thus \(\gamma\) acts identically on \(H\) and preserves \(Q\).

Its \(Q\)-action fixes every generic \(Q\)-period, by the domination of \(S_Q\) and analytic continuation from the resulting open set of lifted periods. Apply 21 to the descended \(Q\)-variation. This action is the identity. Hence \(\gamma\) is the identity on the whole retained adjoint system, contradicting the marking.

In both cases nef binomial expansion and (33) give \(\nu(\pi^*P|_{E_i})<r\). Numerical dimension of a nef line is unchanged by a generically finite pullback, so this is (32). If \(r=0\), the local line fiber is the whole source germ, so its Zariski closure is the whole source and \(H=G\), \(Q=1\). The Ax–Schanuel calculation gives \(d_H=0\). Every retained rational factor contains a factor acting nontrivially on the line orbit; hence there can be no retained factor. Generic immersivity of the full retained period map then forces \(S\) to be a point, with no marked component. ◻

Proof. If \(r=0\), the condition forces \(D=0\), so \(A\) is already big; adding a nef divisor preserves bigness. This also covers dimension zero with the usual convention. Assume \(r>0\) and put \(n=\dim Z\). Choose an ample rational divisor \(H_1\) with \(A+D-H_1\) rationally effective. For fixed rational \(t\geq0\) and sufficiently divisible \(m\), \[h^0(Z,m(A+D+tJ))\geq h^0(Z,m(H_1+tJ)).\] Since \(H_1+tJ\) is ample, asymptotic Riemann–Roch makes the leading coefficient on the right \((H_1+tJ)^n/n!\). As a polynomial in \(t\), this intersection has degree \(r\) and positive leading coefficient.

Remove \(mD\) one component at a time by divisor exact sequences. Every lost quotient is a line bundle on an \(E_i\) of the form \[\mathcal O_{E_i}\left(m(A+D+tJ)-\sum_h k_hE_h\right), \qquad 0\leq k_h\leq md_h.\] Choose an ample integral divisor \(B_0\) so large that a fixed denominator-clearing multiple of \(B_0-A-D\) is globally generated. For each \(h\), choose an ample integral divisor \(B_h\) such that both \(B_h\) and \(B_h+E_h\) are globally generated. Put \(H_2=B_0+\sum_h d_hB_h\). The difference between \(m(H_2+tJ)\) and the displayed quotient divisor is \[m(B_0-A-D)+ \sum_h\bigl((md_h-k_h)B_h+k_h(B_h+E_h)\bigr).\] It is globally generated in these divisible degrees. A section not vanishing identically on \(E_i\) gives an injection into \(\mathcal O_{E_i}(m(H_2+tJ)|_{E_i})\). There are exactly \(md_i\) quotient terms on \(E_i\). Hence \[h^0(Z,m(A+tJ)) \geq h^0(Z,m(H_1+tJ)) -m\sum_i d_i h^0(E_i,m(H_2+tJ)|_{E_i}).\] For each fixed \(t\), divide by \(m^n/n!\) and let \(m\) tend to infinity through the specified multiples. All the comparison divisors are ample, so the resulting lower bound is \[(H_1+tJ)^n -n\sum_i d_i(H_2+tJ)^{n-1}\cdot E_i.\] Each restriction polynomial has degree at most \(\nu(J|_{E_i})\leq r-1\), because the restricted divisor is nef. The positive degree-\(r\) term of the first polynomial therefore dominates the loss for every sufficiently large \(t\). The section growth of \(A+tJ\) is then positive in dimension \(n\), which is exactly bigness. ◻

Proof of 14. Apply 22. If its target is a point there is nothing further to prove. Otherwise mark the divisor \(D\) as above. The adjoint variation extends across every unmarked boundary component, including their intersections away from \(D\), because its local monodromies there are trivial. Its period differential is generically injective. Consequently [14] gives \[K_S+D\text{ big}.\] If an initial choice omitted codimension-two loci, apply the theorem on a log resolution and push the sections down; the resulting logarithmic canonical divisor pushes forward to \(K_S+D\). By 23, each marked component has strictly smaller numerical rank. Apply 24 with \(A=K_S\) and the marked reduced divisor \(D\). It gives \(K_S+aP\) big. Together with the rational line identity in 22, this proves (21). Since \(P\) is nef, the same bigness holds for every larger rational coefficient \(a\). ◻

Proof. Some \(\Lambda_j\) is not contained in a countable union of proper closed analytic subsets; otherwise their union would be so contained. Resolve the graph of \(\phi_j\) and shrink \(U_j\), if necessary. This discards only a proper analytic subset of \(Y\) and preserves the stated properties of the general fibers. Write \(G_j=\bigcup_iG_{ji}\). If no \(G_{ji}\) is dense, the inverse images \[E_{ji}=\phi_j^{-1}(\overline{G_{ji}})\] are proper closed analytic subsets of the resolved \(Q_j\). For \(y\in\Lambda_j\), the genuine frame fiber is covered by these subsets. Baire’s theorem implies that one \(E_{ji}\) contains a relatively open part of the fiber. Irreducibility then implies that \(E_{ji}\) contains the whole compact fiber.

The locus \[A_{ji}=\{y\in U_j: \dim(E_{ji}\cap q_j^{-1}(y))\geq r_j\}\] is analytic, by the fiber-dimension theorem for proper maps. It is proper: otherwise \(E_{ji}\) would contain the general fibers and hence all of \(Q_j\). The same theorem before restriction to \(U_j\) shows that \(A_{ji}\), together with \(Y\setminus U_j\), is contained in a proper closed analytic subset of \(Y\). This would place \(\Lambda_j\) in countably many proper closed analytic subsets, a contradiction. Finally, a dense constructible subset of an irreducible algebraic variety contains a nonempty open subset. ◻

Proof. The construction has three outputs. Relative pluricanonical images produce algebraic parameter data even though the base is analytic. A classifying map then supplies a projective stable family after a finite base cover. Finally, an isomorphism cover matches that family with the relative image; fiberwise canonical models determine the horizontal discrepancy signs.

Relative images and frames. Set \(\mathcal L=\mathcal O_V(K_{V/Y}+A)\), understood as a rational line bundle. For the factorial multiples \(m\) clearing its denominator, the sheaves \[\mathcal E_m=h_*\mathcal L^{\otimes m}\] are coherent. First restrict to a dense analytic open where \(h\) and the horizontal boundary strata are smooth, so the proper family and its line bundles are flat over the base. Generic constancy of the fiber section dimension and cohomology and base change then identify the fibers of \(\mathcal E_m\) with the complete systems, separately for each \(m\); see [33] and its corrigendum [34]. After a modification of \(Y\), the strict transform of \(\mathcal E_m\), modulo torsion, is locally free. The evaluation map gives a relative meromorphic map into its projective bundle. Resolve this map and take its proper analytic image \(\mathcal Z_m\). We also take the proper images of the horizontal boundary components and retain their divisorial cycle parts. Analytic flattening makes these embedded image and cycle data flat on a dense analytic open, with fixed Hilbert and cycle polynomials. These uses of coherent direct images and analytic flattening are independent of any minimal-model theorem; see [39].

Write \(s_m=\mathop{\mathrm{rank}}\mathcal E_m\) on this open; indices with \(s_m=0\) are omitted. Its projective frame bundle has a natural compactification \[\overline Q_m= \mathbb P\!\left(\mathop{\mathrm{Hom}}(\mathcal O_Y^{\,s_m},\mathcal E_m)\right).\] Here \(Y\) denotes the modification chosen for this degree; its map to the original base remains proper and bimeromorphic. The locus of invertible matrices modulo scalars is a principal \(\operatorname{PGL}(s_m)\)-bundle. A genuine frame embeds the corresponding fiber of \(\mathcal Z_m\) in a fixed \(\mathbb P^{s_m-1}\). There is therefore a holomorphic Hilbert–Chow parameter map on the genuine frame locus. The Hilbert spaces here are the algebraic Hilbert schemes of projective space, viewed analytically; equivalently one may use their Douady realization [24].

To extend this map meromorphically, use the existing compact relative image and the projective linear action in local trivializations. The graph of that action extends over the projective matrix space: inversion is given by the adjugate matrix. Apply this construction to the proper family \(\mathcal Z_m\) and its marked cycles, take proper analytic images, and flatten by strict transform. One obtains a proper flat family of embedded subspaces over a modification of \(\overline Q_m\). Its Hilbert–Chow map is holomorphic. The local graph constructions agree on the genuine frames and hence glue. Thus the original parameter map is meromorphic. After resolving it, its image is compact analytic in a fixed projective Hilbert–Chow space, and is consequently algebraic. Denote the irreducible image by \(Z_m\). This construction uses a frame bundle, not a meromorphic trivialization of \(\mathcal E_m\).

The good parameter locus. In \(Z_m\), call an embedded pair \((Z,D)\) good if it is klt and stable, has the prescribed boundary weights, has a complete nondegenerate embedding, and satisfies \[ \mathcal O_Z(1)\simeq \mathcal O_Z\bigl(m(K_Z+D)\bigr). \tag{37}\] One may also impose the generic Hilbert and marking data. This is a countable union of algebraically constructible loci. Here is the parameter-space justification. Stratify the universal embedded families by their Hilbert polynomials, Cartier indices, and base-change loci. On each such stratum, [generaltype:embedded-adjoint] can be imposed by parameterizing the map \[\omega_Z^{[m]}\longrightarrow\mathcal O_Z(1)\] with its prescribed divisor. These parameter spaces are of finite type. The stable-pair and klt conditions are constructible there, and their images are constructible by Chevalley’s theorem. This is the embedded construction used for stable log-variety moduli in [46]. In particular no assertion about analytic minimal models is hidden in the good locus.

Every very general fiber is projective: bigness makes it Moishezon, and it is Kähler. We use Moishezon’s projectivity criterion, which is also the smooth case of [49]. The projective klt finite-generation theorem [11] therefore gives its log canonical model. Every sufficiently large divisible pluricanonical degree embeds that model and satisfies [generaltype:embedded-adjoint]. Exclude at once the countably many generic base-change loci, relative-rank loci, and generic-degree loci for the maps constructed above. Let \(\Lambda_m\) consist of the remaining parameters for which the degree-\(m\) map actually constructs the fiber’s log canonical model. Their union contains a very general subset of \(Y\), and all genuine frames over a point of \(\Lambda_m\) give good data. Apply 25. We obtain one fixed \(m\) for which \(\Lambda_m\) is not contained in a countable union of proper analytic subsets and good data hold on an algebraic open of \(Z_m\).

For this degree the relative image map is generically birational. Indeed its generic finite degree agrees with its fiberwise degree outside the generic-degree exceptional set already excluded. Choose a point of \(\Lambda_m\) there; that degree is one. Retain \(\Lambda_m\): its actual canonical models will determine the horizontal discrepancy signs at the end of the construction.

The invariant classifying map. Fix the dimension, the volume read from [generaltype:embedded-adjoint], and a finite coefficient set containing the coefficients of \(A\) and closed under sums at most one. The good algebraic parameter locus has a classifying map to the projective coarse space \(M\) of stable pairs with these data. If an additional parameterization of the map in [generaltype:embedded-adjoint] was used, its moduli graph descends: the embedded pair determines a unique isomorphism class, so the closure of that graph is generically a singleton over the good parameter locus. Restricting a dense algebraic open therefore gives the same rational classifying map on \(Z_m\). It is invariant under change of projective frame.

Compose with the meromorphic frame parameter map. The resulting map \(\overline Q_m\dashrightarrow M\) is constant on general frame fibers. Its graph has a proper analytic image in \(Y\times M\). This image is generically a singleton over \(Y\), so its projection to \(Y\) is bimeromorphic. It is the graph of a meromorphic map \[\mu\colon Y\dashrightarrow M.\] Let \(M_0\) be the closure of its image; it is projective.

A stable parameter cover and the matching cover. Stable moduli supplies a finite parameter cover \(S\to M\) from a normal scheme carrying a stable family [46]. The coarse space \(M\) is projective [46], so its finite cover \(S\) is projective as well. The moduli construction for the specified coefficient set is given in [46]. Resolve \(\mu\), and normalize a component of the pullback of \(S\to M\) dominating \(Y\). Let \(R_0\) be its image in \(S\). The restricted stable family over \(R_0\) has maximal variation, because \(R_0\to M_0\) is finite.

At this point the projective family exists over \(R_0\). Equality of coarse moduli points identifies its general fibers with those of the original analytic image, but does not identify the families. We construct a further generically finite cover carrying such an identification.

On a dense analytic open of the comparison base, embed the original flat image by the complete system of \(\mathcal O_{\mathcal Z_m}(k)\), and the pulled-back stable family by its \(km\)-adjoint system, for a common sufficiently large and divisible \(k\). The two sheaves of sections are coherent direct images of compact families; generic cohomology and base change make them locally free after shrinking. On a common compact modification of the base, make their torsion-free strict transforms locally free, resolve the evaluation maps, and flatten the embedded image families and their weighted boundary cycles. This gives holomorphic relative Hilbert–Chow data. The modification concerns the base itself, not a space of choices of frames.

On every general good fiber, [generaltype:embedded-adjoint] identifies the two polarizations. Thus the projective linear maps identifying the two embedded varieties and their weighted boundary cycles form a nonempty Isom set. This set is finite, since stable pairs have finite automorphism groups [46]; completeness and nondegeneracy of the embeddings exclude additional ambient stabilizers. At this stage we use only the embedded analytic image and its polarization. Its \(\mathbf Q\)-Gorenstein family structure will be transported from the stable family by the identification.

To compactify these Isom sets, work in local trivializations of the two section bundles. In the product of their projective parameter spaces with the projective matrix space, take the algebraic closure of the incidence relation for invertible matrices identifying the embedded varieties and weighted boundary cycles, after clearing the fixed coefficient denominators. Pull back this closure by the two Hilbert–Chow maps. Equivariance under changes of both frames glues the local incidence spaces into a proper analytic space in the projective bundle of homomorphisms between the section bundles; no meromorphic trivialization is used. Retain its compact irreducible components that dominate the base and meet the invertible locus, and normalize them. The invertible locus has the finite Isom fibers just described, so the resulting cover is proper and generically finite. Its tautological isomorphism identifies the generic families, including the stable family’s \(\mathbf Q\)-Gorenstein structure; its graph therefore extends meromorphically over the compact base.

Finally resolve \(R_0\) projectively and resolve the graph of the map from the comparison cover to it. Denote the resulting spaces by \(R\) and \(Y'\). Stable families and their divisible relative adjoint line bundles commute with base change, with boundary understood by divisorial restriction [46]. Thus the pullback family over \(R\) remains stable; it has klt general fibers and maximal variation. All covers and modifications used above stay in class \(\mathcal C\).

On a common resolution we can now compare the source adjoint divisor with the pullback of the stable relative adjoint divisor, which is rational Cartier. There are only finitely many horizontal prime coefficients in their difference. On fibers above \(\Lambda_m\), away from the finitely many generic comparison exceptions, they are the log canonical model discrepancy coefficients. They are therefore nonnegative and supported on divisors exceptional over the stable model. The same statements hold horizontally in the family. Remove the images of the remaining vertical discrepancy divisors. On the resulting dense analytic open this is the log canonical model comparison, with its divisorial pushforward boundary. This proves the asserted meromorphic comparison without relative analytic finite generation. If \(M_0\) is a point, the construction uses the fixed stable pair and \(R\) is a point. ◻

Proof. On general fibers, pullback from the log canonical model lies in the original log pluricanonical system. The effective canonical-model discrepancy on a common resolution therefore proves the horizontal bound.

For the vertical assertion, first fix a smooth modification of the comparison base and a prime divisor on it. The stable family and its divisible relative adjoint line bundles commute with this base change. Work transversely to a general point of that divisor. The resulting one-parameter family is still stable. This order of operations is what permits the same bound at divisors exposed only on a higher base model. Its total pair with the reduced central fiber added is log canonical. More explicitly, choose the disk with klt fibers away from its origin. Its total space is normal: flatness over the smooth disk and the \(S_2\) property of the fibers give \(S_2\). A codimension-one point on the central fiber is a generic point of one of its reduced components. Its local ring is one-dimensional, and its quotient by the disk parameter is a field; its maximal ideal is consequently generated by that parameter, so this local ring is regular. Away from the central fiber the total space is normal, because its fibers are klt. Serre’s criterion now gives normality. Compatibility of the relative adjoint divisor with the stable fiber identifies the adjunction different on the normalization of the central fiber with its stable boundary and conductor. The resulting pair is log canonical because the central fiber is stable; see [55] for this normalization-and-conductor identity. The analytic Cartier-divisor inversion theorem [28] consequently applies. The argument also applies after any finite ramified disk change.

Let \(t\) be the disk parameter and let \(\xi\) denote the pulled-back relative section in degree \(b\). Then \(\xi\wedge(dt/t)^b\) is a section of the total log canonical multiple with the central fiber added. On a log resolution the log canonical discrepancy bound says that this form has pole order at most \(b\) at every vertical prime. A further common resolution with the original family preserves this bound. This is [generaltype:relative-log-bound]. Smooth transverse parameters do not change any of these orders. ◻

Proof. Use 26. If \(\dim R>0\), the stable-family bigness theorem [46] applies: the base is normal projective, the family is stable and has maximal variation, and its general fiber is klt. Thus \(K_{\mathcal V/R}+\mathcal A\) is big. Choose finitely many sections of one sufficiently divisible multiple whose map is generically finite. Its image has dimension \(\dim F+\dim R\), and its restriction to a general \(\pi\)-fiber has rank \(\dim F\). If \(R\) is a point, choose such sections of the ample log canonical divisor of the fixed stable pair instead. Pullback and the meromorphic comparison preserve these image dimensions and give the pole bounds of 27.

Replace the comparison cover by a Galois cover and resolve the diagrams equivariantly. To do this analytically, first take its Stein factorization over \(Y\); its finite factor has the same cover over a dense open set. Over the finite étale locus, the Galois closure is a component of a sufficiently large fiber product of this finite factor. Take the corresponding compact component, normalize, and then resolve the graph of its map to the comparison cover. Equivariant resolution gives the required diagram; write \(G\) for its finite group. This construction uses finite analytic maps and does not require the meromorphic function field of \(Y\) to have transcendence degree \(\dim Y\).

Let \(\mathcal B\) be the graded \(\mathbf C\)-algebra generated by the chosen finite set of relative sections and all its \(G\)-conjugates. Every homogeneous element has the required horizontal and vertical linear pole bounds: products add the orders and sums cannot decrease the lower bounds. The algebra is integral and finite over \(\mathcal B^G\). Indeed each homogeneous generator satisfies its orbit polynomial with invariant coefficients, and invariants of a finite group acting on a finitely generated algebra are finitely generated. The corresponding map \(\operatorname{Proj}\mathcal B\to\operatorname{Proj}\mathcal B^G\) is finite. Before taking invariants, the system contains the sections pulled back from \(\mathcal V\), whose restriction to a general source fiber has image dimension \(\dim F\). A finite map preserves the dimension of the image of that fiber, even though \(G\) may permute the fibers above one point of \(Y\). Choose a Veronese degree for \(\mathcal B^G\) generated by finitely many elements of that single degree. Their ratios therefore still have rank \(\dim F\) on general fibers. Denote their pluricanonical degree by \(b_0\). The same finite quotient also preserves the total image dimension, which is at least \(\dim F+\dim R\).

Use the natural identification of relative canonical bundles on the common smooth locus for every base change or modification. On the dense étale locus, invariant relative tensors therefore descend to \(V\). They extend meromorphically across the remaining locus by finite analytic descent. The horizontal bounds descend because the comparison cover is generically étale along each horizontal divisor.

For completeness, the vertical order calculation is exact. Let \(E'\) lie over \(E\), with ramification index \(e\). For a degree-\(b_0\) top pluriform \(\alpha\), \[ \mathop{\mathrm{ord}}_{E'}(\alpha_{\mathrm{pullback}})+b_0 =e\bigl(\mathop{\mathrm{ord}}_E(\alpha)+b_0\bigr). \tag{39}\] A logarithmic top form on the base pulls back to a nonvanishing logarithmic top form at a prime dominating its divisor. Apply [generaltype:ramified-order] to the relative form wedged with this base logarithmic form. The bound upstairs is therefore equivalent to [generaltype:relative-log-bound] downstairs. For (iii), pull the chosen Galois comparison cover and its original sections to a common refinement over the modified base. Lemma 27 applies to these same pullbacks at primes above the newly exposed base divisor; its bounds pass to their conjugates, products, and invariant combinations. The ramification calculation therefore gives the bound for the pullback of each fixed descended member of \(\mathcal S\). Products add the order inequalities, and taking a full product system is a Veronese operation on its image, so products preserve the relative rank as well. ◻

Proof. Apply analytic flattening to \(h\), by strict transform, and resolve the modified base. Pullback preserves the flatness of the intermediate family, whose fibers have the generic relative dimension \(d\). This intermediate source is proper and bimeromorphic over \(V\). On a resolution dominating it, the strict transform of \(E\) remains a divisor, because a proper modification of the smooth \(V\) is an isomorphism at the generic point of \(E\). Its image in the flat intermediate source has dimension \(\dim V-1\): it still dominates \(E\). The fiber-dimension bound consequently gives an image on the new base of dimension at least \(\dim V-1-d=\dim Y-1\). That image is proper, since its projection to \(Y\) is contained in \(h(E)\), and properness makes it a closed analytic set. It is therefore a prime divisor. Further source resolutions retain the same generic image and do not change the valuation of \(E\). ◻

Proof of 15. We may suppose \(\kappa(Y,K_Y+C)\geq0\). Choose a Kähler modification \(\nu\colon\widetilde V\to V\), resolve the boundary, and put \[\widetilde T=\nu_*^{-1}T+\operatorname{Exc}(\nu)_{\mathrm{red}}.\] By [setup:log-modification,setup:modification-compatibility], pullback and descent identify the divisible log pluricanonical rings on the total spaces and on their very general fibers. In particular, fiberwise bigness is preserved, and \(B(h\circ\nu,\widetilde T)=B(h,T)\). Every prime over base codimension at least two still has coefficient one: it is either a strict transform of such a prime or is newly exceptional. Write \((V,T)\) for this Kähler model; its sections descend to the original pair.

If \(\dim F=0\), the map is bimeromorphic. Pull back sections of \(K_Y+C\). At a prime dominating a base divisor the multiplicity is one and the condition \(C\leq B(h,T)\) gives the required pole bound. At an exceptional prime the log canonical discrepancy bound for \((Y,C)\), together with \(T_E=1\), gives the same conclusion. Thus pullback preserves all the base systems and proves the assertion. If \(Y\) is a point, the assertion is simply the assumed bigness. We may therefore assume both dimensions are positive.

Write \(T^{\mathrm h}\) for the horizontal part of \(T\). Choose a rational \(\epsilon>0\) sufficiently small and set \[A=(1-\epsilon)T^{\mathrm h}.\] Its coefficients are strictly less than one. To make \(K_F+A_F\) big uniformly on very general fibers, consider \(\epsilon=1/j\) and all denominator-clearing degrees. Choose a fiber outside their countably many generic base-change and relative-rank exceptional sets. Openness of the big cone on that projective fiber gives one choice of \(\epsilon\) and one degree with full image dimension. The generic-rank choice makes the same degree have full rank on general fibers. Notice that no vertical coefficient has been lowered, and \(C\) has not been changed.

Apply 28 to \(A\), and write \(\sigma\) for a member of its system in degree \(b\), including any product degree. Let \[\eta\in H^0\bigl(Y,\mathcal O_Y(b(K_Y+C))\bigr).\] The relative-times-base product \(\sigma\wedge h^*\eta\) is a meromorphic section of \(\mathcal O_V(bK_V)\). We check that its poles are bounded by \(bT\) at every prime.

There is nothing further to check at horizontal primes, since \(A\leq T^{\mathrm h}\). Suppose next that \(E\) maps onto a prime divisor \(D\subset Y\), and put \[a_E=\mathop{\mathrm{ord}}_E(h^*D).\] Relative to a local nonvanishing logarithmic base volume \(\omega_D\), the section \(\eta\) vanishes to order at least \(b(1-C_D)\). The relative logarithmic bound gives \[ \mathop{\mathrm{ord}}_E(\sigma\wedge h^*\eta) \geq -b+b\,a_E(1-C_D). \tag{40}\] The inf-multiplicity convention is precisely \[B(h,T)_D =1-\max_{E\mapsto D}\frac{1-T_E}{a_E}.\] Consequently \(C_D\leq B(h,T)_D\) implies \[a_E(1-C_D)\geq1-T_E,\] and [generaltype:orbifold-order] is at least \(-bT_E\). This uses no integrality condition on the boundary multiplicities.

Finally, suppose \(\operatorname{codim}_Y h(E)\geq2\). Expose a base divisor \(D'\) below the strict transform of \(E\) by 29, and resolve the diagrams. The pullback of \(\eta\) has at most a logarithmic pole along \(D'\): this is the divisorial discrepancy inequality for the log canonical smooth pair \((Y,C)\). Property (iii) of 28 therefore gives order at least \(-b\) at the strict transform of \(E\). The order is unchanged, since source modifications are isomorphisms at the generic point of \(E\). The assumption \(T_E=1\) supplies exactly this allowance. We have proved, at all prime divisors, that \[ \sigma\wedge h^*\eta \in H^0\bigl(V,\mathcal O_V(b(K_V+T))\bigr). \tag{41}\] On a smooth space these divisorial bounds imply holomorphic extension of the corresponding log pluricanonical section.

Choose a degree in which the complete base system has image dimension \(\kappa(Y,K_Y+C)\), and pass to a common multiple with \(b_0\) in 28. The product system of relative sections still has image dimension \(d=\dim F\) on a general fiber. The corresponding product of the chosen base system has image dimension \(\kappa(Y,K_Y+C)\), since a Veronese embedding preserves its image. In this common degree write the nonzero relative generators as \(\sigma_i\) and the base generators as \(\eta_j\). The total system in [generaltype:product-sections] contains all \(\sigma_i\wedge h^*\eta_j\). On the open set where \(\sigma_0\) and \(\eta_0\) are nonzero, ratios in this system include \[\frac{\sigma_i}{\sigma_0}, \qquad h^*\!\left(\frac{\eta_j}{\eta_0}\right).\] Its image therefore maps onto the base Iitaka image, and over a general point of that image it contains the \(d\)-dimensional image of a general \(h\)-fiber. The dimension-of-fibers inequality gives image dimension at least \(d+\kappa(Y,K_Y+C)\). This proves [generaltype:inequality] on the Kähler model. The initial log pluricanonical descent then proves it on the given smooth source. ◻

Proof. Choose a neat model as in Lemma 5, with \(p:X'\to X\), \(q:Y'\to Y\), \(f':X'\to Y'\), and \(\Delta'=p_*^{-1}\Delta+\operatorname{Exc}(p)_{\mathrm{red}}\). We may arrange that \(E_Y=(q^{-1}\operatorname{Supp}D_Y)_{\mathrm{red}}\) has simple normal crossings. Every prime of \(X'\) dominating a component of \(E_Y\) has coefficient one in \(\Delta'\): an exceptional prime has coefficient one by definition, and a nonexceptional prime is the strict transform of a component of \(f^*D_Y\). Thus its weighted multiplicity is infinite, and the inf-multiplicity rule gives \[B(f',\Delta')\geq E_Y.\] Pullback by \(q\) takes logarithmic top forms with poles along \(D_Y\) to logarithmic top forms with poles along \(E_Y\). In local coordinates this follows by pulling back each factor \(dz_i/z_i\); the same assertion holds for all tensor powers. Consequently, pullback embeds every plurilogarithmic system on \((Y,D_Y)\) into the corresponding system of \(K_{Y'}+B(f',\Delta')\), preserving all ratios of sections. The neat-model identity (6) now proves the claim. ◻

Proof. Theorem 1 applies to \((X,D_X)\), and Lemma 30 bounds its base term from below. If either term on the right of (48) is \(-\infty\), the assertion follows from our convention for that value. ◻

Proof. Over \(\mathbb C\), the ordinary assertion follows by taking \(D_X=D_Y=0\) in Corollary 31. In a projective family over \(\mathbb C\), the Iitaka dimension of the geometric generic fiber equals that of a very general fiber. To see this, apply cohomology and base change in each positive divisible pluricanonical degree. After shrinking the base separately in each degree, the corresponding relative rational map also has constant fiberwise image dimension. Excluding the resulting countable union of proper closed subsets identifies all these image dimensions with those of the generic fiber.

For the field extension, let \(K\subseteq K'\) be fields and let \(V\) be a smooth projective geometrically integral \(K\)-variety with a rational divisor \(D\). For every \(m\) for which \(mD\) is integral, flat base change gives \[H^0(V,\mathcal O_V(mD))\otimes_K K' \simeq H^0(V_{K'},\mathcal O_{V_{K'}}(mD_{K'})).\] The complete linear system on the right is the scalar extension of the one on the left, so nonvanishing and image dimension are unchanged. Hence Iitaka dimension is invariant under extension of the ground field. Applying this observation to the generic fiber over the function field of the base proves the corresponding invariance for geometric generic fibers.

Descend \(f\), its projective embeddings, and its boundary divisors to a finitely generated subfield \(K\subset k\) over \(\mathbb Q\). The descended varieties are smooth and geometrically integral; the boundary conditions descend as well. The equality \(f_*\mathcal O_X=\mathcal O_Y\) descends by proper flat base change and faithful flatness, and therefore also holds after an embedding \(K\hookrightarrow\mathbb C\). Apply the complex logarithmic inequality to this base change. The preceding invariance identifies its three Iitaka dimensions with those over \(k\), proving both assertions. ◻

Proof. Resolve the core map, using the boundary convention of Lemma 2. The total and fiber Iitaka dimensions are preserved. Theorem 1 gives the lower bound in (50), since the invariant base dimension is \(\dim C\).

For completeness, the reverse bound is the elementary upper addition inequality. For each divisible \(m\), let \(\phi_m\) be the meromorphic map defined by \(|m(K_X+\Delta)|\), when this system is nonempty. The map \((c,\phi_m)\) has image dimension at most \(\dim C+\kappa(G,K_G+\Delta_G)\): on a very general \(c\)-fiber its second component is given by a subspace of \(H^0(G,m(K_G+\Delta_G))\). Since projection from this image onto the image of \(\phi_m\) is surjective, the same upper bound holds for \(\dim\phi_m(X)\). Taking the maximum over \(m\) proves the desired inequality. If the fiber Iitaka dimension is \(-\infty\), restriction to a very general fiber shows that no global pluricanonical section can be nonzero, so both sides of (50) are \(-\infty\).

Finally, if the total Iitaka dimension is nonnegative, a nonzero section restricts nontrivially to a very general core fiber. Its Iitaka dimension is therefore nonnegative. The dimension bound and its equality case follow from (50). When the total dimension is zero, the core base is a point, which is equivalent to specialness. ◻

Proof. Theorem 1 establishes the hypothesis of Campana’s Corollary 6.14, which supplies the \(\kappa\)-rational quotient. Its scope is exactly that used here: smooth compact class-\(\mathcal C\) pairs with rational coefficients in \([0,1]\) [16]. Its stable base has nonnegative Kodaira dimension, so \(M\) is defined and has very general orbifold fiber of Kodaira dimension zero. The new smooth base remains in class \(\mathcal C\), with rational boundary coefficients in \([0,1]\). Thus the construction can be repeated.

Campana’s Lemma 10.1 states that each composite \(M\circ r\) has special general orbifold fiber and decreases dimension unless its source pair is of general type. His Theorems 10.2–10.3 identify the resulting stabilized composite with the core. Their sole conjectural input is the existence of the \(\kappa\)-rational quotient, now supplied above. Strict dimension decrease allows at most \(n\) nontrivial iterations; identity steps give the displayed formula. A point is fixed throughout. These are precisely the decomposition and fiber assertions of [16]. ◻

Proof. Choose a local frame \(\eta\) of \(K_W\) that does not vanish at the generic point of \(P\). The canonical line-bundle identification \[\mathcal O_{X'}(pK_{X'/W})\otimes x^*\mathcal O_W(pK_W) \simeq\mathcal O_{X'}(pK_{X'})\] identifies the order of \(s\) with the order of \(s\wedge x^*\eta^{\otimes p}\). Thus the quantity \(l_E\) in the rank-one order formula is exactly \(r_E\). This uses the actual relative canonical line bundle: the Jacobian contribution is already present in \(r_E\), so no ramification term is added or subtracted here. Consequently its two order formulas give \[\alpha_P=t_P, \qquad \beta_P=\min_E\frac{r_E+1-a_E}{a_E}.\] Put \[\Delta_0=-p^{-1}\operatorname{div}_{K_{X'/W}}(s), \qquad \lambda_P=\min_{E\mapsto P}\frac{1+r_E}{a_E},\] where the minimum is over actual vertical components. The indicated SNC preparation and the generic logarithmic bound give \(\Delta_0\) SNC support and horizontal coefficients at most one. In \(\Delta_0+\lambda_Px^*P\), each vertical coefficient is \(-r_E+\lambda_Pa_E\leq1\). Thus this subpair is sub-log-canonical over the generic point of \(P\), and every higher divisorial valuation satisfies \[1+r_E-\lambda_Pa_E\geq0.\] An actual component attaining \(\lambda_P\) gives equality. Hence the infimum over all such valuations equals \(\lambda_P\), and \(\beta_P=\lambda_P-1\).

For completeness, the first minimum also remains unchanged on higher source models. A coefficient realizing \(t_P\) makes \(t_Pa_E-r_E\) no greater than the original boundary coefficient at every actual component over \(P\). Equivalently, after clearing denominators, the normalized generating form satisfies every logarithmic order test over the generic point of \(P\). Its pullback remains logarithmic with strict plus reduced exceptional boundary. No new valuation can lower the first minimum, while an old minimizing valuation persists.

At any divisorial valuation above the generic point of \(P\), the crepant coefficient of \(\Delta^{\mathrm{aux}}\) is \(-r_E+a_Et_P\). The threshold formula therefore gives \[\operatorname{lct}_{\eta_P}(X',\Delta^{\mathrm{aux}};x^*P) =\inf_E\frac{1+r_E-a_Et_P}{a_E}=c_P.\] Since \(T_P=\alpha_P-\beta_P\), we obtain \(T_P=1-c_P\). Finally, if \(T_P=0\) and \(E\) attains the first minimum, then \[1=c_P\le\frac{1+r_E}{a_E}-t_P =\frac{1-d_E}{a_E}\le1.\] All terms are equal. Since \(a_E\) is a positive integer and \(d_E\in\{0,1\}\), this forces \(a_E=1\) and \(d_E=0\), as claimed. ◻

Proof. Log canonicity of \((W,T)\) gives \[K_{W'}+T'+M'=\mu^*(K_W+T+M)+E,\] where \(E\) is effective and \(\mu\)-exceptional. For divisible \(m\), normality gives \[\mu_*\mathcal O_{W'}(mE)=\mathcal O_W.\] The projection formula therefore identifies the section spaces. The modification is an isomorphism at each old generic divisorial point, so the old local order calculations are unchanged. The final assertion is the definition of \(T'\). ◻

Proof. The first assertion is the description of a rank-one reflexive sheaf as the intersection of its lattices at codimension-one points. For a finite morphism between normal varieties, a prime divisor upstairs lies over a prime divisor downstairs. The last assertion follows from \((\pi_*\mathcal O_{V'})^{\operatorname{Gal}(V'/V)}=\mathcal O_V\) and the projection formula. ◻

Proof. Stein factorization reduces the first assertion to a finite morphism, since a proper birational morphism onto a normal variety has direct image \(\mathcal O_V\). After replacing \(D\) by a multiple, use the natural inclusion of graded section rings \[R(V,D)\ \subseteq\ R(V',\pi^*D).\] Each homogeneous section upstairs is integral over the ring downstairs. Indeed its monic characteristic polynomial in the finite function-field extension has coefficients which are rational sections of the appropriate multiples of \(D\); they are regular in the required lattices at every prime divisor, and hence are global sections by Lemma 40. Thus the two nonzero graded rings have the same transcendence degree, which gives equality of Iitaka dimensions. If the ring downstairs has no positive-degree section, the same norm argument excludes one upstairs. Finally, two independent sections in one degree have a nonconstant ratio and define a positive-dimensional rational image. This proves the last assertion. ◻

Proof. A logarithmic differential on a smooth snc pair pulls back to a logarithmic differential along the reduced inverse image. In local coordinates this follows from \(d(\rho^*z)/\rho^*z\), whose divisorial poles are simple. Taking top exterior powers and then tensor powers proves the order inequalities. Equivalently, at a prime of ramification index \(e\) over a boundary component, the Jacobian contributes \(e-1\), and the coefficient-one boundary supplies the remaining one. At exceptional-image primes the same logarithmic differential calculation applies. Injectivity follows from dominance and separability. ◻

Proof. Apply the Iitaka fibration theorem over the function field of \(B\) [40], take the relative algebraic closure of its section field in \(\mathbf C(V)\), and spread out. Resolve the resulting rational maps and take their Stein factorizations. The dimension of the relative section field is \(k\); the restriction of the full divisor to the Iitaka fiber has Iitaka dimension zero by the Iitaka fibration theorem. For the logarithmic replacement, the proof of Lemma 2 gives the sheaf identity \[\mu_*\mathcal O_{V'}\bigl(m(K_{V'}+B_{V'})\bigr) =\mathcal O_V\bigl(m(K_V+B_V)\bigr)\] in every sufficiently divisible degree. Pushing to the original base identifies the relative section systems, even if there are no global sections. Apply the same full-divisor Iitaka theorem to the replacement divisor; its relative section field identifies its fibration with the one already constructed and gives the asserted Iitaka dimension zero on the fiber. Generic smoothness and adjunction give the last assertion. The dimensions at very general complex fibers agree with the geometric generic values: for each divisible degree use cohomology and base change, and remove the union of the exceptional closed sets over the countably many degrees. ◻

Proof. A nonzero section cannot vanish identically on all fibers over a dense open subset. For each divisible \(m\) with sections, let \(\phi_m\) be its rational map. The image of \((p,\phi_m)\) has dimension at most \(\dim B+\kappa(G,D|_G)\), and dominates the image of \(\phi_m\). Taking the maximum over \(m\) proves the inequality. If the fiber term is \(-\infty\), the last assertion implies that the total term is \(-\infty\) as well. ◻