Proof. After clearing denominators, a line of zero real class has torsion integral Chern class. Another positive power has zero integral class, so belongs to \(\mathop{\mathrm{Pic}}^0\) by the exponential sequence. The second assertion is the standard bimeromorphic invariance of \(\mathop{\mathrm{Pic}}^0\): a smooth modification preserves \(H^1(\mathcal O)\) and the integral lattice defining this torus. It also follows by factoring smooth bimeromorphic maps into blowups and blowdowns with smooth centers. Finally \(\mu_*\mathcal O_Y=\mathcal O_X\) by normality. The projection formula identifies the sections of every integral multiple of \(L\) with those of its pullback. A base point downstairs would therefore be a base point at every point above it. ◻

Proof. The positive-part assertion of (OpenAI 2026b, Proposition 8.1) gives a smooth model on which the rational positive part of the original adjoint is numerically semiample. Separately, take a small projective \(\mathbb Q\)-factorialization and run a terminating generalized klt program as in (OpenAI 2026b, Theorem 2.3). Its fixed nef data are the pullback of \(M\). The generalized discrepancies initially agree with the ordinary ones because \(M\) descends, and the generalized adjoint is pseudo-effective. The resulting model has nef adjoint \(H_m\). The comparison on a common resolution is (2), with effective exceptional error.

Choose this resolution also to dominate the model furnished by the positive-part theorem. Lemma 4(5) and nefness of \(v^*H_m\) identify \(E\) as the negative part. Pulling up the first model’s decomposition by Lemma 4(4) identifies its numerically semiample positive class with \(v^*H_m\). Algebraic and analytic negative parts agree on these smooth projective models. ◻

Proof. Choose an integer \(m>0\) clearing all divisors and rational line identities. A meromorphic section of the line \(m b^*H\) exists because \(S\) is projective. Its pullback, under the identity \[m(K_X+D_X)\simeq g^*(m b^*H),\] and division by the canonical meromorphic section of \(mD_X\) give a meromorphic section \(\sigma\) of \(K_X^{\otimes m}\). On the open where \(\sigma\) has neither zeros nor poles, take its \(m\)th roots in the fibers of \(K_X\). In local canonical frames this is the equation \(z^m=\sigma\); on overlaps the roots transform by the transition functions of \(K_X\). The local covers therefore glue without a meromorphic trivialization of \(K_X\), and without choosing a root of \(H\).

The Grauert–Remmert extension theorem for finite analytic covers (Grothendieck et al. 2003, Exposé XII, Proposition 5.3 and Theorem 5.4) extends it normally across the divisor of \(\sigma\), giving a finite cyclic cover of \(X\), possibly disconnected. It is compact Kähler. Take a resolution functorial for the cyclic action and for local isomorphisms of the pair, as in (Ambro 2005, sec. 1.1 and Lemma 1.1). Thus both resolutions are functorial. A local flow preserving the original pair lifts through them once its lift to the root cover has been chosen; we will construct that lift explicitly below. The resulting compact manifold \(V\) is Kähler. After deleting from \(S\) the zeros and poles of the chosen base section and the degeneration locus, the map \(h:V^\circ\to S^\circ\) is smooth and proper. We take \(S^\circ\) inside the isomorphism locus of \(b\). Shrinking it further, the original fibers of \(Y\to Z\) are normal effective klt pairs and the chosen resolutions restrict to resolutions of those pairs.

Average a global Kähler class on \(V\) under the cyclic group. Its restriction is a flat real section of \(R^2h_*\mathbb R\) and polarizes the primitive pieces. Their Lefschetz decomposition supplies a flat real polarization of the full degree-\(l\) cohomology. We retain the full cohomology local system, with lattice \(R^lh_*\mathbb Z\) modulo torsion; an irrational primitive summand is not required to carry its own lattice. The finite group action preserves this variation and its polarization.

Locally over the base, the tautological root, divided by a local base volume form, gives a relative meromorphic top form \(\tau\) on the cover. It belongs to a fixed character of the cyclic group. The klt condition makes its squared volume locally integrable, including after resolution, so it extends holomorphically: a meromorphic top form with a divisorial pole cannot be locally square integrable. To see the orders directly, consider a boundary prime on the original normal fiber with coefficient \(u=p/e\in(0,1)\) in lowest terms. At a general point the cover has local coordinate \(x=y^e\), and the pulled-back root form has order \[ e-1-eu=e-1-p\in\{0,\ldots,e-2\}. \tag{12}\] Away from the boundary its order at a nonexceptional prime is zero. The corresponding calculation on a crepant resolution uses coefficients less than one and proves integrability at exceptional divisors as well.

Suppose that \(\tau'\) is another holomorphic top form of the same character on a fiber. The ratio \(\tau'/\tau\) is an invariant meromorphic function and descends to the original normal connected fiber. A pole along a boundary prime downstairs would pull back with order at least \(e\), whereas (12) permits only \(e-2\) zeros of \(\tau\). No such pole is possible. Away from the boundary, \(\tau\) has no divisorial zeros downstairs. Normality extends the ratio across codimension two, and compactness makes it constant. This also handles a disconnected cover, since the full root group acts transitively on its components over a connected fiber. The highest Hodge piece of the chosen character is therefore one-dimensional.

Finally, changing a local frame of \(m b^*H\) changes the root by an \(m\)th root of the corresponding base factor; changing the base volume form contributes the inverse canonical transition. Taking the \(m\)th tensor power removes the finite root ambiguities. These are precisely the transitions of \(m(b^*H-K_S)\), proving the asserted identity on \(S^\circ\). ◻

Proof. Fix a general point of a prime \(Q_0=(t=0)\) on \(S\). We normalize the root form by a nonvanishing local frame of \(b^*H-K_S\). More explicitly, if the meromorphic section of \(m b^*H\) chosen in Lemma 13 is \(a\) times a nonvanishing local frame, divide its root by \(a^{1/m}\) and by the local base volume form. This is an ordinary frame after a finite power substitution; the finite ambiguity is precisely part of the parabolic convention. Thus zeros or poles of the chosen meromorphic base section do not enter the following exponent. Tangential base coordinates may be treated as parameters. On a log resolution, \[t=\prod_{i=1}^k x_i^{a_i}\] up to a nowhere-zero factor, and the squared absolute root form has vertical density factors \(|x_i|^{-2u_i}\). Horizontal boundary factors are integrable because their coefficients are less than one. The squared Hodge norm of the relative form is its fiber integral. Writing \(s_i=-\log|x_i|\), the integral is computed on slices \[\sum_i a_i s_i=-\log|t|\] with exponential weight \(\exp(-2\sum_i(1-u_i)s_i)\). Conversion to the ordinary base area element contributes \(|t|^{-2}\). The minimum in (10) thus gives constants \(C,N>0\) such that, on a sufficiently small punctured disk, \[ C^{-1}|t|^{2(t_{Q_0}-1)}(-\log|t|)^{-N} \leq \|\tau\|^2 \leq C|t|^{2(t_{Q_0}-1)}(-\log|t|)^N. \tag{13}\] Indeed, the slice has at most polynomial volume in \(-\log|t|\), which proves the upper estimate after factoring out the least exponential decay. For the lower estimate take a chart along a component attaining the minimum, away from all other vertical components, and integrate over a fixed compact set in the remaining directions. Compactness allows finitely many charts; the integrable horizontal factors do not alter the power of \(|t|\).

After a local power substitution making monodromy unipotent, extending Hodge frames and their duals have norms bounded by powers of \(-\log|t|\). These are the nilpotent-orbit estimates of Schmid and Cattani–Kaplan–Schmid (Schmid 1973; Cattani et al. 1986), in the form used in (OpenAI 2026c, sec. 4.1). Consequently the comparison between a frame of \(b^*H-K_S\) and an extending Hodge frame has rational order \(t_{Q_0}-1\). Multiplication by \(t^{1-t_{Q_0}}\), on a cover where this power is integral, removes exactly that order. Both the corrected comparison and its inverse have at most logarithmic growth. A holomorphic function on a punctured disk with such growth has no pole; applying this to the inverse excludes a zero. The same argument with tangential parameters extends the identification across each divisor, and normal extension treats codimension two. Thus the extending line is \[b^*H-K_S+\sum_{Q_0}(t_{Q_0}-1)Q_0=L_S.\] For the last assertion, first make the boundary monodromies unipotent. In canonical extending frames a local monomial pullback \(t_i=\text{unit}\cdot\prod_j s_j^{a_{ij}}\) replaces the commuting nilpotent residues \(N_i\) by \(\sum_i a_{ij}N_i\), which are again nilpotent. Both the canonical flat extension and its extending Hodge filtration therefore pull back without an additional divisor. Descending with rational weights incorporates the finite-monodromy parts and gives the same conclusion for the parabolic line, including exceptional base divisors; see (OpenAI 2026c, sec. 4). ◻

Proof. Work in an open patch on which the ranks are constant, and let \(\xi\) be a holomorphic base vector field in the kernel of the line period map. Choose a local frame \(\tau\) of the eigenline, viewed as a relative top form on \(V^\circ\). The infinitesimal period formula says that contraction with \(\tau\) sends the Kodaira–Spencer class of \(\xi\) to zero in the fiber cohomology \(H^1(\Omega^{l-1})\).

Here is the corresponding lifting construction. Pull back the tangent sequence to the line generated by \(\xi\) and push it out along \[T_{V^\circ/S^\circ}\longrightarrow T_{V^\circ/S^\circ}\otimes K_{V^\circ/S^\circ} \simeq\Omega^{l-1}_{V^\circ/S^\circ}, \qquad v\longmapsto\iota_v\tau.\] Write \(U\) for a sufficiently small Stein base patch and put \(\mathcal F=\Omega^{l-1}_{V_U/U}\). The pushed-out extension is \[0\longrightarrow\mathcal F \longrightarrow\mathcal E \longrightarrow\mathcal O_{V_U}\longrightarrow0,\] with class in \(H^1(V_U,\mathcal F)\). Shrink within the generic locus so that \(R^1h_*\mathcal F\) is locally free and coherent base change holds. The contraction formula makes the image of this class in \(H^0(U,R^1h_*\mathcal F)\) zero. The Leray exact sequence and the vanishing \(H^1(U,h_*\mathcal F)=0\) show that \(H^1(V_U,\mathcal F)\to H^0(U,R^1h_*\mathcal F)\) is injective. The extension class is therefore zero, and a holomorphic splitting exists on the entire \(V_U\). Where \(\tau\) is nonzero, divide the vertical part of the splitting by \(\tau\). This gives a meromorphic vector field \(\widetilde\xi\) on the resolved cover projecting to \(\xi\), with poles bounded by the zero divisor of \(\tau\). Average it under the root group. Its projection remains \(\xi\), and the averaged field is invariant.

We check this field at codimension-one points of the original normal total space. At a boundary prime use the ramification coordinate \(x=y^e\) from (12). A tangential coefficient of an invariant vector field has Laurent exponents congruent to zero modulo \(e\). Its permitted pole order is at most \(e-2\), so every exponent is nonnegative. The coefficient of \(\partial/\partial y\) has exponents congruent to one modulo \(e\). Its first possible negative exponent is \(1-e\), which is again excluded. The normal coefficient is therefore divisible by \(y\). Thus the descended vector field is holomorphic and tangent to the boundary. At a prime outside the boundary the root form has no zero, and the conclusion is immediate.

The resolution is an isomorphism at the codimension-one points just used. Remaining exceptional divisors map into codimension at least two in the original normal space. Holomorphic derivations extend across that subset, so the descended field is holomorphic everywhere on the family over our base patch and still projects to \(\xi\). Its local flow preserves the boundary components, their coefficients, and the singular locus. By the functorial choice made above, the flow first lifts to \(X\) and preserves its crepant boundary \(D_X\).

This flow then lifts holomorphically to the root cover. Indeed, on the good base patch the defining pluriform has divisor exactly the negative multiple of the boundary. Pullback by a pair-preserving flow gives a pluriform with the same divisor. Their ratio is a holomorphic unit on the normal space; its \(m\)th root can be chosen starting at one in the flow parameter. The root normalized at flow time zero is unique as a germ. Thus the local cover lifts agree on overlaps of the unbranched open and glue by normality. Functorial resolution then lifts the flow to \(V^\circ\). Consequently the Kodaira–Spencer class of the resolved family vanishes on \(\xi\), and so does its full period differential. The reverse kernel inclusion holds because the line is part of the full Hodge filtration. The two kernels, and therefore the generic ranks, are equal. ◻

Proof of Proposition 11. Apply Lemma 12 to the variation of Lemma 13. Its monodromy is quasi-unipotent by the monodromy theorem for this proper Kähler family (Schmid 1973). Lemma 14 identifies its parabolic line with \(L_S\). Passing to the indicated higher smooth projective model, and keeping the notation \(S\), we have \[L_S\sim_{\mathbb Q}p^*P\] for a nef rational line \(P\) on a smooth projective quotient \(Q\). Let \(r\) be the generic rank of the line period map. Numerical dimension of a nef line is unchanged by a surjective pullback, so \[r=\nu(L_S)=\nu(P)\leq\dim Q \leq\operatorname{rank}(d\Phi)=r,\] where \(\Phi\) is the full period map and the last equality is Lemma 15. Thus \(P\) is big if \(\dim Q>0\). If \(Q\) is a point, \(L_S\) is rationally trivial.

Suppose first that \(\dim Q>0\). By Kodaira’s lemma write \(P\sim_{\mathbb Q}A+E\), with \(A\) ample rational and \(E\geq0\) rational. For every small positive rational \(\varepsilon\), \[ P\sim_{\mathbb Q}A_\varepsilon+\varepsilon E, \qquad A_\varepsilon=(1-\varepsilon)P+\varepsilon A \quad\text{ample}. \tag{14}\] The pair \((S,\Delta_S)\) is sub-klt. On a fixed log resolution of \(\Delta_S+p^*E\), choose \(\varepsilon\) sufficiently small that \((S,\Delta_S+\varepsilon p^*E)\) remains sub-klt. Choose a sufficiently large divisible integer \(N\) and a general member \(G\in|N A_\varepsilon|\). The system \(|Np^*A_\varepsilon|\) is basepoint-free; on that resolution its general member meets the fixed boundary transversely. Taking \(N\) large makes its coefficient \(1/N\) less than one. Bertini therefore gives an effective rational divisor \[E_S=\varepsilon p^*E+\frac1N p^*G \sim_{\mathbb Q}L_S\] with \((S,\Delta_S+E_S)\) sub-klt. In the point case take \(E_S=0\). In either case, \[ K_S+\Delta_S+E_S\sim_{\mathbb Q}b^*H. \tag{15}\]

Set \(D_Z=b_*(\Delta_S+E_S)\). This divisor is effective. Indeed, for each prime of the original normal base, some component of its pullback on \(Y\) dominates it. That component has multiplicity \(a\geq1\) and boundary coefficient \(u\geq0\), so its threshold is at most \((1-u)/a\leq1\). Thus the coefficient of the discriminant at every nonexceptional base prime is nonnegative, and \(E_S\) is effective.

Choose compatible canonical divisors and a rational Cartier divisor representing \(H\). The line identity (15) gives an integer \(m>0\) and a rational function \(\varphi\) in the common function field of \(S\) and \(Z\) such that \[K_S+\Delta_S+E_S-b^*H=\frac1m\operatorname{div}_S(\varphi).\] Pushing down gives \[K_Z+D_Z-H=\frac1m\operatorname{div}_Z(\varphi).\] In particular, \(K_Z+D_Z\) is rational Cartier and rationally linearly equivalent to \(H\). Pulling the latter divisor equality back and comparing with the former proves the crepant identity \[K_S+\Delta_S+E_S=b^*(K_Z+D_Z).\] Since the pair upstairs is sub-klt, the effective pair \((Z,D_Z)\) is klt. This proves the proposition. ◻

Proof. Choose a Kähler class \(\omega\) and \(\varepsilon>0\) such that \(C-\varepsilon\omega\) is pseudo-effective. When \(n\ge2\), the identity \[C^{n-1}-(\varepsilon\omega)^{n-1} =(C-\varepsilon\omega) \sum_{j=0}^{n-2}C^{n-2-j}(\varepsilon\omega)^j\] and nonnegativity of intersections of a pseudo-effective class with nef classes give \[0=\gamma\cdot C^{n-1} \ge \varepsilon^{n-1}\gamma\cdot\omega^{n-1}\ge0.\] The same conclusion is immediate when \(n=1\). A positive current in \(\gamma\) thus has zero mass, so it vanishes.

For the last assertion, write \(f:V\to A\) for the morphism. The pushforward of the nef rational line class is a rational pseudo-effective \((1,1)\)-class on \(A\). Averaging a positive current under translations represents it by a constant semipositive form. The kernel of a rational such form is the tangent space of a subtorus; the induced positive integral form, after clearing denominators, polarizes the quotient. Since \(a(A)=0\), this quotient is a point, and hence \(f_*\gamma=0\). For a Kähler class \(\eta\) on \(A\), the class \(C=f^*\eta\) is nef and big and \[\gamma\cdot C^{n-1}=(f_*\gamma)\cdot\eta^{n-1}=0.\] Apply the first assertion. ◻

Proof. Apply the ordinary good divisorial decomposition to \(K_T+B\). On a smooth modification its semiample part is rationally trivial, because \(a(T)=0\). After the klt modification convention of Section 2, the resulting ordinary adjoint still equals its rational negative divisor as an actual rational line bundle. The contraction of a known negative part (OpenAI 2026a, Theorem 2.13 and Proposition 3.8), with nef part zero, produces an ordinary compact Kähler klt pair \((T_0,B_0)\) such that \(K_{T_0}+B_0\sim_{\mathbb Q}0\).

The decomposition theorem of Matsumura–Wang–Wu–Zhang (Matsumura et al. 2026, Corollary 1.4) applies to this pair: its nef b-part is zero. A finite quasi-étale cover of \(T_0\) is a product of a rationally connected factor, strict Calabi–Yau factors, irreducible holomorphic symplectic factors, and a torus. Algebraic dimension is unchanged by proper generically finite maps and by modifications. No positive-dimensional rationally connected factor can occur, since a Kähler resolution of such a factor is projective by algebraic connectedness (Campana 1981). A strict Calabi–Yau factor of dimension at least three has no holomorphic two-forms. Extension of forms for klt spaces (Kebekus and Schnell 2021) gives the same vanishing on a resolution, which is projective by Kodaira’s criterion. Such factors are also excluded. Dimension-two factors of Calabi–Yau type are counted among the symplectic factors. A resolution of an irreducible symplectic factor has \(h^{2,0}=1\), generated by a generically nondegenerate form.

Resolve the cover and the maps to \(T\) and to smooth models of all its factors. We obtain a smooth compact Kähler manifold \(V\), with the pulled-back nef line class \(\gamma\), mapping generically finitely to \(T\) and to a product \(X_1\times\cdots\times X_r\). If \(r\ge2\), let \(C\) be the pullback of a sum of Kähler classes from the factors. For the projection omitting \(X_i\), every component of a smooth general fiber maps generically finitely to \(X_i\). This fiber has algebraic dimension zero and a nonzero top form, obtained by pulling back either a torus volume form or a power of a symplectic form. Its canonical class is therefore pseudo-effective. Its dimension is less than \(n\), so the induction hypothesis, with zero boundary, makes the restriction of \(\gamma\) zero.

Expand \(C^{n-1}\). A nonzero term has top powers from all factors except one, where precisely one power is missing. Integration over the fibers just considered shows that its pairing with \(\gamma\) is zero. Thus \(\gamma\cdot C^{n-1}=0\), and Lemma 17 gives \(\gamma=0\). Vanishing descends to \(T\) by push–pull for a generically finite map. A single torus factor is covered by the last assertion of that lemma.

It remains to consider one symplectic factor. On a smooth model \(X\) we have \(a(X)=0\) and \(H^0(X,\Omega_X^2)=\mathbb C\sigma\), with \(\sigma\) generically nondegenerate. There is no dominant meromorphic fibration from \(X\) to an intermediate-dimensional compact base. Indeed, on smooth Kähler models such a base has algebraic dimension zero, hence is nonprojective and has a nonzero holomorphic two-form by Kodaira’s criterion. Its pullback would be a nonzero degenerate holomorphic two-form on \(X\), which is impossible.

The minimal-fibration theorem of Campana, in the form (Campana et al. 2010, Theorem 2.3(3) and Corollary 2.5(2)), now says that \(X\) is isotypically semi-simple. This means that \(X\) and a power \(S^r\) of a simple manifold have a common generically finite cover; we may take smooth Kähler models. If \(r\ge2\), repeat the preceding product intersection argument. The smaller-dimensional fibers are simple: simplicity is preserved by generically finite maps between spaces of algebraic dimension zero. Indeed, a covering family of proper subvarieties upstairs or downstairs gives such a family on the other space by images or inverse images at points where the map is finite; the countability of compact cycle components gives the very general formulation. They are not uniruled, and hence their canonical bundles are pseudo-effective by (Ou 2025, Theorem 1.1). The induction hypothesis again applies. If \(r=1\), the common smooth cover is simple and carries the pullback of \(\sigma\), a generically nondegenerate holomorphic two-form. This is precisely the assertion stated in the lemma. ◻

Proof. Fix a generically nondegenerate holomorphic two-form \(\sigma\) on \(X\). A moving correspondence would produce a holomorphic one-form on a fixed finite cover of \(X\). To see this, we identify its first projection with a fixed finite cover over a parameter ball, and then contract the pulled-back two-form in a direction varying the second projection.

For a subvariety through a pair whose coordinates are very general, simplicity makes each projection image either a point or all of \(X\). If both projections dominate, their fibers through general points are either zero-dimensional or the full other factor. A proper such subvariety must therefore be a generically finite correspondence.

Suppose these correspondences sweep \(X^2\). The compact cycle spaces of a compact Kähler manifold have countably many compact irreducible components (Lieberman 1978). One component must have incidence image equal to \(X^2\). Resolve that component and the relevant incidence component, obtaining a compact Kähler parameter space \(S\), a smooth incidence space \(Y\), and maps \[h:Y\longrightarrow S,\qquad F_1,F_2:Y\longrightarrow X.\] For a general parameter, the fiber is smooth and irreducible, and each \(F_i\) restricts to a generically finite map. The map \((F_1,h):Y\to X\times S\) is generically finite. Its normal finite Stein factor is a finite cover of the smooth product.

By purity, the branch locus of this cover is divisorial. Every branch component dominating \(S\) has a proper image in \(X\): otherwise its divisorial fibers over \(S\) would sweep \(X\), contradicting simplicity. Compactness of \(S\) makes these images closed analytic subsets. Remove the images in \(S\) of the remaining branch components and take a small ball \(B\) about a general parameter. There is a fixed proper analytic subset \(A\subset X\) such that the cover is étale over \((X\setminus A)\times B\). Since \(B\) is simply connected, this étale cover is pulled back from a fixed cover of \(X\setminus A\). Normalize \(X\) in that cover. Uniqueness of normal finite extension then identifies the cover over \(X\times B\) with the product of this fixed normal cover and \(B\); see (Grothendieck et al. 2003, Exposé XII, Proposition 5.3 and Theorem 5.4).

Let \(\widetilde X\) be a smooth Kähler resolution of the fixed cover. The second projection gives a meromorphic map \(G:\widetilde X\times B\dashrightarrow X\). The pullback \(G^*\sigma\) is a holomorphic two-form: resolve the meromorphic map and descend holomorphic forms across a modification of a smooth space. Dominance of \((F_1,F_2)\) implies that, at suitable general points, the derivatives of \(G\) in the parameter directions span \(T_X\), while the derivative along \(\widetilde X\) is an isomorphism. Choose a constant tangent vector on \(B\) for which this parameter derivative is nonzero at such a point. Contracting \(G^*\sigma\) with that vector and restricting to \(\widetilde X\times\{t\}\) yields a nonzero global holomorphic one-form on \(\widetilde X\). This contradicts the hypothesis on irregularity.

Each component parametrizing generically finite correspondences thus has proper incidence image. Their countable union misses a very general pair, proving the assertion. ◻

Proof. We give the slope argument from (OpenAI 2026a, Lemma 6.2), indicating why it does not require any hypothesis about canonical sections. Let \(\gamma\) belong to the full dual of the pseudo-effective cone. Ou’s convention calls precisely these classes movable, and Harder–Narasimhan filtrations exist for them (Ou 2025, Definition 4.2 and Lemma 4.3). If the minimum \(\gamma\)-slope of \(\Omega_X\) were negative, the first Harder–Narasimhan subsheaf \(\mathscr T\subset T_X\) would have positive slope and be semistable. Tensor slope inequalities make its bracket into \(T_X/\mathscr T\) zero, so it is a foliation. Its dual has negative maximum slope and is non-pseudo-effective. By (Ou 2025, Theorem 1.4, Lemmas 4.3–4.4 and Proposition 4.5), its leaves have compact closures given by a meromorphic fibration. The rank of \(\mathscr T\) is strictly less than \(\dim X\), since \(c_1(T_X)\cdot\gamma\le0\). A general leaf closure contradicts simplicity.

All Harder–Narasimhan quotient slopes of \(\Omega_X\) are consequently nonnegative, and its maximum slope is at most \(c_1(K_X)\cdot\gamma\). Tensor slopes give \[c_1(\mathscr H)\cdot\gamma \le k c_1(K_X)\cdot\gamma\le kL\cdot\gamma.\] Separation by the dual cone proves (16).

Irregularity zero embeds \(\mathop{\mathrm{Pic}}(X)\) in the countable group \(H^2(X,\mathbb Z)\). For any vector bundle on \(X\), linearly independent global sections are generically pointwise independent: the coefficients expressing one section in a maximal pointwise independent subfamily are meromorphic functions, hence constants because \(a(X)=0\). Evaluation is therefore injective outside a proper analytic subset. Apply this simultaneously to \(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for all lines \(\mathscr H\) and all \(k\).

For a point outside the resulting countable exceptional union, write \(\mathscr H_Y=a^*\mathscr H\otimes\mathcal O_Y(mF)\), where \(F\) is the exceptional divisor. A nonzero map as in the statement extends away from \(F\) to a section on \(X\) by Hartogs. If \(m>0\), that section vanishes at \(x\), because \(a_*\mathcal O_Y(-mF)=\mathcal I_x^m\), contrary to injectivity of evaluation. Thus \(m\le0\), and (16) proves (17). ◻

Proof. The assertion is immediate for a point. A positive-dimensional space of algebraic dimension zero cannot be a curve, so assume \(\dim X\ge2\). If \(q(X)>0\), its Albanese map is generically finite onto its image, by simplicity. Ueno’s structure theorem for subvarieties of complex tori makes this image a translate of a torus: its quotient by the connected stabilizer is of general type, and positive dimension of that quotient would contradict algebraic dimension zero (Ueno 1975). Lemma 17 proves the conclusion.

The same argument applies after any smooth generically finite Kähler cover of \(X\) having positive irregularity. Such a cover is still simple and has algebraic dimension zero; vanishing upstairs descends by push–pull. We may therefore suppose that every such cover has irregularity zero. Lemma 19 then describes the subvarieties through a very general point of \(X^2\).

Clear denominators and let \(N_0\) be the resulting nef holomorphic line bundle. A top wedge of the generically nondegenerate two-form is a nonzero holomorphic section of \(K_X\). In particular \(K_X\) is effective. Set \[\mathscr L=K_X+N_0,\qquad L=c_1(\mathscr L).\] The class \(L\) is pseudo-effective and dominates \(c_1(K_X)\). Lemma 20 supplies both slope inequalities. Every hypothesis of Proposition 21 is now satisfied. Some positive power of \(K_X+N_0\) therefore has a nonzero meromorphic section. Dividing by the corresponding power of a canonical section shows that \(N_0\), as a rational line bundle, is represented by a signed divisor.

Resolve its positive and negative parts. On the resulting smooth Kähler model write \[N_0\sim_{\mathbb Q}E-G,\qquad E,G\ge0,\] with rational coefficients and simple normal crossing union of their supports; we suppress the pullback notation for \(N_0\). The canonical bundle \(K'\) of this model remains effective. Choose a small rational \(u>0\) so that both \(uE\) and \(uG\) are klt boundaries. The ordinary decomposition, Theorem 5, applies to \(K'+uE\) and \(K'+uG\). On a common higher model, denote their pulled back classes by \(\alpha_E\) and \(\alpha_G\). Algebraic dimension zero makes each semiample part rationally trivial, so that the pulled back rational lines equal their negative divisors. Since \(\alpha_E-\alpha_G=u\{N_0\}\) is nef, Lemma 4 (2) gives \[N(\alpha_E)\le N(\alpha_G),\qquad u\{N_0\}=\{N(\alpha_E)-N(\alpha_G)\}.\] Thus \(-\{N_0\}\) is pseudo-effective, whereas \(\{N_0\}\) is nef. Pairing both classes with a Kähler power gives zero mass, and hence \(\{N_0\}=0\). Pullback injectivity yields the same conclusion on \(X\). ◻

Proof of Proposition 16. Lemma 18 reduces the assertion to Proposition 32, which proves it. In particular, once the induction has reached dimension \(n\), the ordinary decomposition of \(K_T+B\) also proves the decomposition assertion of Theorem 2 in algebraic dimension zero: adding the nef line changes no Chern class. Its semiample positive part is rationally trivial, and its negative part is unchanged. ◻

Proof. Choose a positive integer \(q_i\) such that \(q_iL_i\) is an actual line bundle. For every positive multiple \(m\) of \(q_i\), the proper direct-image theorem gives a coherent analytic sheaf \[\mathcal E_{i,m}=h_*\mathcal O_X(mL_i)\] on \(W\). Work over the smooth-fibration locus, where the line bundle is flat over the base. On a dense open set where the dimension of fiberwise sections is locally constant, coherent base change gives the isomorphism \[ \mathcal E_{i,m}\otimes\mathbb C(w) \simeq H^0(X_w,\mathcal O_{X_w}(mL_i|_{X_w})) \tag{60}\] by (Grauert 1960, sec. 7, Satz 5). It suffices here to delete proper analytic subsets within dense open sets of \(W\). Fix an ample line bundle \(H\) on \(W\). By GAGA and Serre’s theorem, \(\mathcal E_{i,m}\otimes H^{k_{i,m}}\) is generated by global sections for some integer \(k_{i,m}\). The projection formula and (60) therefore show that all sections of \(mL_i|_{X_w}\) lift, after choosing a nonzero frame of \(H^{k_{i,m}}\) at \(w\), to global sections of \[\mathcal O_X(mL_i)\otimes h^*H^{k_{i,m}}.\]

There are only countably many pairs \((i,m)\). Choose \(w\) in the smooth locus of \(h\), outside all the base-change exceptions and the further generic exceptions in the statement. If \(\kappa(X_w,L_i|_{X_w})>0\), some divisible \(m\) has two sections whose ratio is nonconstant on the connected fiber \(X_w\). Lifting these sections as above gives a meromorphic function on \(X\) whose restriction to \(X_w\) is that ratio. Every meromorphic function on \(X\) factors through its algebraic reduction. Such a function, when defined on a nonempty open subset of the smooth fiber \(X_w\), is constant there: a local holomorphic section of \(h\) through a point where the quotient is holomorphic shows that the base function extends holomorphically at \(w\). This contradicts the chosen nonconstant ratio.

The countable simultaneous choice is legitimate by Baire’s theorem. If a generic assertion is initially established on a dense open set, a proper analytic exceptional subset of that open is still nowhere dense in \(W\). Thus such generic assertions can also be included in the choice. ◻

Proof. The difference between the two sides of (61) is flat as a rational line bundle. By Lemma 3, there is a flat rational line bundle \(\eta\) on \(F\) such that \[\rho^*(A+\eta)\sim_{\mathbb Q}P+R.\] Choose \(m\) clearing the denominators and such that \(mH_Q\) is very ample. The divisors \[g^*D_Q+mR, \qquad D_Q\in|mH_Q|,\] descend to an analytic family of divisors on \(F\) with class \(m c_1(A)\). Indeed \(\rho_*\mathcal O_{\widehat F}=\mathcal O_F\), so the projection formula identifies sections of \(m(A+\eta)\) with sections of its pullback. On the isomorphism locus of \(\rho\), away from \(\operatorname{Supp}R\), they distinguish general \(g\)-fibers, since hyperplanes distinguish points of \(Q\).

For an arbitrary effective divisor \(D\) of this class, Lemma 4(1) gives \[\rho^*D\geq mR.\] The residual divisor \(D'=\rho^*D-mR\) is effective and has class \(m c_1(P)\). Let \(G\) be a smooth compact connected fiber of \(g\). If \(D'\) does not contain \(G\), its restriction is an effective divisor of class zero on \(G\), hence is empty: in positive dimension this follows by integrating against a Kähler form to the appropriate power. Consequently \(\operatorname{Supp}D'\) either contains \(G\) or misses it. For a point fiber the same alternative is immediate. Taking the image of the isomorphism locus of \(\rho\), and deleting \(\operatorname{Supp}R\) and the nonsmooth locus of \(g\), gives the stated open set \(U\). This choice uses only \(\rho\), \(R\), and \(g\), and works for all the divisors simultaneously. ◻

Proof. We associate a map to each component of the relative divisor space by recording all its divisors through a point. On a fiber with the stated decomposition, Lemma 34 makes these incidence supports constant along the semiample fibration. Integral multiplicities will then give constancy of the cycle maps themselves, while the distinguishing linear system will give the converse.

Put \(d=\dim X-\dim W\), and let \(W^\circ\) be the smooth-fibration locus of \(h\). Consider the relative cycle spaces whose points are pairs \((w,D)\), where \(D\) is an effective \((d-1)\)-cycle in \(X_w\). They are closed subspaces of the product of \(W\) with the corresponding compact cycle components of \(X\). Thus they have countably many compact irreducible components in class \(\mathcal C\).

For each positive integer \(m\) clearing the denominator of \(A\), retain the components meeting \(W^\circ\) on which the divisor class is \(m c_1(A|_{X_w})\). This is a condition on an entire component over \(W^\circ\): in a smooth family the cohomology class of an analytic family of cycles is locally constant, and \(c_1(A)\) supplies a global section of the same cohomology local system. After resolving a component, its open part over \(W^\circ\) is connected, so equality at one point implies equality throughout that part. Components not dominating \(W\) have proper analytic images; exclude all these images from the eventual choice of \(w\).

Fix a retained dominating component and a smooth compact Kähler resolution \(S\) of it. Denote its divisor at \(s\in S\) by \(D_s\) and form the incidence \[\mathcal I_S =\{(x,s)\in X\times S:x\in\operatorname{Supp}D_s\}.\] We take the incidence of the analytic cycle family, including its limits from the open part over \(W^\circ\). Over that open part it is a family of pure \((d-1)\)-dimensional cycles. Hence every irreducible component there dominates \(S\) and has dimension \(\dim S+d-1\). Let \(I_1,\ldots,I_t\) be the incidence components that dominate \(X\). The other components have proper analytic images in \(X\); outside those images they contribute nothing to incidence.

For each \(I_j\to X\), analytic flattening gives a meromorphic map recording its generic fiber as an effective cycle in \(S\). All these cycles have dimension \[e=\dim S+d-1-\dim X.\] Taking the tuple of these maps, together with \(h\), gives a meromorphic map \(\Phi_S\) from \(X\) into \(W\) times a finite product of compact cycle components of \(S\). On a dense open subset of \(X\), the union of the supports of the recorded cycles is precisely \[ \{s\in S:x\in\operatorname{Supp}D_s\}. \tag{63}\] In obtaining this open set we discard the images of the nondominating incidence components and the exceptional sets of the fiber maps. The target components and the image of \(\Phi_S\) belong to class \(\mathcal C\). Resolving the graph, taking Stein factorization, and replacing the base by a smooth compact Kähler model therefore gives a diagram (62). The coordinate \(h\) ensures that its base maps to \(W\). If there is no dominating incidence component, we retain only the coordinate \(h\); such a component will not be needed to detect a positive-dimensional \(Q\).

This construction gives countably many diagrams. Fix their models, their incidence-recording open sets, and their Stein factorizations. We may simultaneously require that their fibers over the eventual \(w\) be smooth, that \(p_i\) restrict to a modification over \(w\), and that each of the dense open sets used meet the corresponding fiber densely. For any one diagram these are generic conditions, by properness, generic smoothness, and the fiber-dimension theorem. In particular, let \(U_i\subset Z_i\) be the locus over which \(f_i\) is smooth. We require that \(k_i^{-1}(w)\) meet \(U_i\) densely. To justify this condition, a component of \(Z_i\setminus U_i\) that does not dominate \(W\) has proper image, whereas a component dominating \(W\) has general fiber of dimension strictly less than \(\dim Z_i-\dim W\). Since the general \(k_i\)-fiber is smooth of that pure dimension, it cannot have a component contained in \(Z_i\setminus U_i\). The same conditions may be imposed after any preselected countable collection of further modifications of these diagrams.

We now choose such a very general \(w\) and a decomposition on \(F=X_w\) as in the statement. By Lemma 34, a projective linear system of divisors of class \(m c_1(A|_F)\) distinguishes the general \(g\)-fibers for some \(m\). Its image in the relative cycle space is irreducible and lies in one irreducible component. That component dominates \(W\), since all images of nondominating components were excluded. It is therefore one of the components used above. Passing to its parameter resolution \(S\) preserves the distinguishing property: every divisor still has a preimage in \(S\).

The map \(\Phi_S|_F\) distinguishes general \(g\)-fibers. Indeed two different general points of \(Q\) are separated by a divisor in the chosen linear system, so their incidence supports (63) differ. Conversely, fix a general smooth connected \(g\)-fiber \(G\). On the common open set of Lemma 34 and the incidence recording, membership in every divisor \(D_s\) is constant along \(G\). Thus the support in (63) is fixed as the point moves in that open part of \(G\).

This support assertion also gives constancy of the recorded cycles. Their fixed union of supports is pure \(e\)-dimensional and has finitely many irreducible components. Since all recorded cycles have dimension \(e\), there are only countably many possible tuples of effective cycles supported there: their multiplicities are nonnegative integers. A holomorphic map from a connected complex manifold with countable image is constant. Removing a proper analytic subset from the smooth connected \(G\) leaves a connected open set, so \(\Phi_S\) is constant on a general \(g\)-fiber. Point fibers require no separate argument.

Constancy on the general connected fibers gives a meromorphic factorization through \(Q\). One can see this by taking the image of the graph in \(Q\) times the target: its projection to \(Q\) has one-point general fibers and is bimeromorphic. The distinguishing property makes the induced map from \(Q\) generically one-to-one onto its image. Hence \(\Phi_S|_F\) defines the same meromorphic fibration as \(g\).

Finally, the restriction of \(f_i\) over \(w\) still has connected fibers: its fibers are exactly the corresponding fibers of \(f_i\). Its base over \(w\) is connected, being the image of the connected source fiber. The finite map in Stein factorization cannot split a connected general \(g\)-fiber: a map from that fiber into a finite set is constant. The preselected resolutions restrict to modifications over \(w\). Thus the same comparison holds for \(f_i\), proving the assertion. ◻

Proof. Set \(A=J+M\). By Theorem 5, fix an effective rational divisor \(E\) with \(J\sim_{\mathbb Q}E\). Apply Lemma 35 to \(h\) and \(A\), and fix its countable collection \((p_i,f_i,k_i)\). Let \(I_0\) be the subset of indices for which \(c_1(p_i^*M)\) vanishes on the general smooth fiber of \(f_i\). For every \(i\in I_0\), apply Corollary 9 to \(p_i^*M\) on \(X_i\). It gives a representative on \(X_i\) differing from \(p_i^*M\) by a flat rational line. By Lemma 3(2), that flat difference comes from \(T\). We therefore obtain a rational line bundle \(M_i^*\) on \(T\), with \[M_i^*\equiv M,\] whose pullback is an actual rational pullback from the intermediate base, after further modifications of the diagram. Fix one such representative and one such diagram for every \(i\in I_0\). In particular, \(M_i^*\) restricts rationally trivially to a general fiber of that modified fibration. This is a countable collection of choices on \(T\).

Choose \(w\in W\) simultaneously satisfying the detection lemma and its generic conditions for all the further diagrams just fixed. Require also the conclusion of Lemma 33 for the countable collection \[(J+M_i^*)_{i\in I_0}.\] We take \(F=T_w\) smooth, with \(B|_F\) an SNC klt boundary. We also require that the defining section of \(E\) restrict nontrivially to \(F\); then \[J_F:=K_F+B|_F\sim_{\mathbb Q}E_F:=E|_F\geq0.\] All the choices preceding \(w\) depend only on data on the total space.

Suppose, towards a contradiction, that \(M|_F\not\equiv 0\). The lower-dimensional decomposition theorem gives a modification \(\rho:\widehat F\to F\) and \[ \rho^*(J_F+M|_F)\equiv P_F+R_F, \qquad R_F=N(c_1(\rho^*(J_F+M|_F))), \tag{64}\] where \(P_F\) is semiample and \(R_F\) is a rational effective divisor. Lemma 4, applied to the nef summand \(\rho^*(M|_F)\), gives \[ R_F\leq N(c_1(\rho^*J_F))\leq\rho^*E_F, \qquad P_F\equiv(\rho^*E_F-R_F)+\rho^*(M|_F). \tag{65}\] Let \(g:\widehat F\to Q\) be the fibration defined by \(P_F\). If \(Q\) were a point, the last equality would make a nef class and an effective divisor sum to zero. Pairing with a Kähler power would give \(\rho^*(M|_F)\equiv 0\), contrary to the assumption. Hence \(\dim Q>0\).

On a general smooth \(g\)-fiber \(G\), the class of \(P_F\) is zero. Restricting the last equality in (65) and using the same positivity argument shows that \[ c_1(\rho^*(M|_F)|_G)=0, \qquad G\cap\operatorname{Supp}(\rho^*E_F-R_F)=\varnothing. \tag{66}\] If \(G\) is a point, the first assertion is automatic and the second holds for a general point. In positive dimension, choose a general fiber not contained in the effective divisor, restrict it, and integrate the effective and nef classes against a Kähler power. Both nonnegative masses must vanish.

Lemma 35 gives an index \(i\) such that the restriction of \(f_i\) over \(w\) and \(g\) have the same general fibers up to modifications. By (66) and injectivity of pullback on cohomology, \(p_i^*M\) has class zero on a smooth \(f_i\)-fiber above \(w\). Such a fiber can be chosen in the smooth-fibration locus of \(f_i\), by the generic conditions already imposed. Over the connected smooth locus in \(Z_i\), the restrictions of this Chern class form a locally constant section of \(R^2(f_i)_*\mathbb R\). Vanishing at one fiber therefore implies vanishing at every fiber in this locus. Thus \(i\in I_0\); its representative \(M_i^*\) was already chosen before \(w\).

We claim that \[ \kappa(F,J_F+M_i^*|_F)>0. \tag{67}\] On a common resolution of the maps on \(F\), the line \(M_i^*|_F\) is rationally trivial on a general \(g\)-fiber because it is a pullback from the intermediate base of \(f_i\). This triviality descends between smooth models of the fiber: for a modification, pullback on line bundles is injective by normality and the projection formula. The difference \[\Lambda=\rho^*(J_F+M_i^*|_F)-P_F-R_F\] is a flat rational line bundle by (64). Its restriction to a general \(G\) is rationally trivial. Indeed \(P_F|_G\) is trivial, \(M_i^*|_G\) is trivial, and (66) makes \(\mathcal O_{\widehat F}(\rho^*E_F-R_F)|_G\) trivial, while \(J_F\sim_{\mathbb Q}E_F\).

Resolve \(Q\) and the induced map, and continue to denote the resulting fibration between smooth models by \(g\). By Lemma 10, \(\Lambda\sim_{\mathbb Q}g^*\lambda\) for a flat rational line bundle \(\lambda\) on the smooth projective base. The line \(P_F\) is the pullback of the ample rational line on the original \(Q\); on its resolution the corresponding base line \(H_Q'\) is nef and big. Twisting by \(\lambda\) preserves bigness. On these models we have an actual rational identity \[\rho^*(J_F+M_i^*|_F) \sim_{\mathbb Q}g^*(H_Q'+\lambda)+R_F.\] Multiplication by the canonical section of a divisible multiple of \(R_F\) embeds the section spaces of the big base line into the section spaces on \(\widehat F\). Since \(\dim Q>0\), this proves (67).

This contradicts the instance of Lemma 33 imposed on \(J+M_i^*\) when \(w\) was chosen. Hence \(M|_F\equiv 0\) for the chosen fiber. Finally, restrictions of \(c_1(M)\) form a locally constant section of \(R^2h_*\mathbb R\) on the connected smooth locus of \(h\). Their vanishing therefore holds on every fiber in that locus, and in particular on a general smooth fiber. ◻

Proof. Proposition 36 supplies the fiberwise vanishing required in Proposition 8. Apply that proposition to \(h\) and \(M\). The modified base remains projective, so the descended rational class is nef. ◻

Proof. Put \(n=\dim V\), \(\beta=c_1(r^*H)\), and \(\alpha=\beta+\{E\}\). Nefness gives \(N(\alpha)\leq E\). Suppose \[U=E-N(\alpha)>0.\] Then \(\beta+\{U\}=Z(\alpha)\) is modified nef. Write \(U=\sum_i u_iU_i\), with \(u_i>0\), and choose \[s=\max_i\dim r(U_i)\leq\dim S-2.\] In particular \(0\leq s\leq n-2\). Let \(\eta\) be the pullback of an ample class on \(S\), and let \(\omega\) be a Kähler class on \(V\). Consider the symmetric bilinear form \[Q(\gamma,\delta)=\int_V\gamma\,\delta\,\eta^s\omega^{n-s-2} \quad\text{on }H^{1,1}(V,\mathbb R).\] Since the image of each \(U_i\) has dimension at most \(s\), any pairing on \(U_i\) with \(s+1\) classes from \(S\) vanishes. Therefore \[ Q(\{U\},\eta)=0,\qquad Q(\{U\},\beta)=0. \tag{68}\] By Lemma 4(3), \(Z(\alpha)\) restricts pseudo-effectively to a resolution of \(U_i\). Pairing there with the nef classes \(\eta^s\omega^{n-s-2}\) and summing over \(i\) gives \[0\leq Q(\{U\},Z(\alpha))=Q(\{U\},\{U\}).\] On the other hand, \[Q(\eta,\eta)=\int_V\eta^{s+2}\omega^{n-s-2}>0,\] because \(s+2\leq\dim S\). The mixed Hodge–Riemann relations (Dinh and Nguyen 2006, Theorem A), applied with \(\eta+\varepsilon\omega\) and then passed to the limit, show that \(Q\) has at most one positive eigenvalue. Thus its restriction to \(\eta^\perp\) is negative semidefinite. Equations (68) force \(Q(\{U\},\{U\})=0\). A zero-square vector for a negative semidefinite form belongs to its radical; decomposing every class into a multiple of \(\eta\) and an element of \(\eta^\perp\) shows that \(\{U\}\) belongs to the radical of the whole form \(Q\).

This contradicts \[Q(\{U\},\omega)= \sum_i u_i\int_{U_i}\eta^s\omega^{n-s-1}>0.\] Indeed every term is nonnegative, and a component with image dimension \(s\) gives a strictly positive integral on its smooth generic locus. Consequently \(U=0\). ◻

Proof. Choose \(A_W\) very ample and an integer \(b>2n\), and set \(C=h^*A_W\). A general member of a sufficiently high multiple of the free system \(|C|\), divided by that multiple and multiplied by \(b\), gives an effective rational representative of \(bC\). Bertini’s theorem, with the multiple chosen large enough, makes its sum with \(B\) an effective simple normal crossing klt boundary. Theorem 5 therefore gives a decomposition of \(J+bC\).

Pass to a higher smooth model \(T_1\) and apply the convention (1) to the original pair \((T,B)\). Write \(J_1=K_{T_1}+B_1\) and \(C_1=h_1^*A_W\). The exceptional error and Lemma 4 retain an actual identity \[ J_1+bC_1\sim_{\mathbb Q}P_1+R_1,\qquad P_1\text{ semiample},\qquad R_1=N(J_1+bC_1)\geq0, \tag{69}\] with rational terms. On this fixed model choose a fresh general fractional representative of \(bC_1\), so that its sum with \(B_1\) is again simple normal crossing and klt. Apply Proposition 6 to this augmented ordinary adjoint and its known negative part in (69). Its finite ordinary scaling contracts or flips detected analytic extremal rays negative for the augmented adjoint. It preserves the actual semiample positive line \(P_1\), with a fixed generated Cartier multiple, and contracts the negative part \(R_1\).

The positive line \(P_1\) and the fixed algebraic-base line \(C_1=h_1^*A_W\) have distinct roles; they need not be equal or proportional. Preservation of \(C_1\) and of the specific morphism to \(W\) follows from a separate ray estimate. We check inductively that each step is over \(W\). Suppose the current model \(T_i\) still has a morphism \(h_i:T_i\to W\) and the unaugmented pair is klt. Put \[J_i=K_{T_i}+B_i,\qquad C_i=h_i^*A_W.\] An extremal ray negative for \(J_i+bC_i\) is \(J_i\)-negative because \(C_i\) is nef. The Kähler klt cone theorem (Hacon and Xie 2026, Theorem 1.3), with zero nef b-part, supplies a rational curve \(\Gamma\) generating this ray and satisfying \[0<-J_i\cdot\Gamma\leq2n.\] If \(C_i\cdot\Gamma>0\), Cartier integrality gives \[(J_i+bC_i)\cdot\Gamma\geq-2n+b>0,\] a contradiction. Hence \(C_i\) is numerically trivial on the contracted ray.

The contracted fibers are projective, and all their curves have class in the contracted ray. If their image under \(h_i\) had positive dimension, a curve on a fiber would have positive degree against \(h_i^*A_W\); thus \(h_i\) is constant on each contracted fiber. Connected-fiber descent to the normal contraction base factors \(h_i\) through it. For a flip, composing the morphism from the flipped space with this base map gives the next \(h_i\). The added line is consequently the same Cartier pullback \(C_{i+1}=h_{i+1}^*A_W\) on the next model. Each step is also \(J_i\)-negative, and the unaugmented ordinary klt pair persists. This proves the induction through the finite program. On a common resolution the two pullbacks of the line from \(W\) agree. Subtracting \(b\) times this pullback from the augmented comparison therefore gives an effective exceptional comparison for the unaugmented adjoints as well. Writing \(P_Y\) for the descended positive line, the final actual rational-line identity is \[K_Y+\Delta+b h_Y^*A_W\sim_{\mathbb Q}P_Y.\] The line \(P_Y\) is semiample by Proposition 6, which proves the required semiampleness of the augmented adjoint. ◻

Proof of Theorem 2. We induct simultaneously on \(n=\dim T\) for both assertions of the theorem. Dimension zero is immediate. Suppose both assertions hold in every dimension less than \(n\). The projective case of the decomposition is Proposition 7. If \(a(T)=0\), Proposition 16 gives \(M\equiv 0\); ordinary decomposition then proves the required statement for \(J+M\). It remains to treat \[0<a(T)<n.\]

Resolve the algebraic reduction. This changes the ordinary adjoint by (1), so we may work on the resulting smooth Kähler model and remove its exceptional error at the end. We obtain a morphism \(h:T\to W\) with connected fibers and smooth projective \(W\), where \(\dim W=a(T)\). By Proposition 36 and Corollary 37, after further modifications if needed, there is a nef rational line \(N_W\) on \(W\) such that \[ M\equiv h^*N_W. \tag{70}\] Apply Proposition 39. On its final model \(Y\), write \(J_Y=K_Y+\Delta\). The map defined by a generated multiple of \(J_Y+b h_Y^*A_W\), together with \(h_Y\), has image in a product of projective varieties. Its Stein factorization is \[f:Y\longrightarrow Z,\qquad j:Z\longrightarrow W,\] where \(Z\) is normal projective, \(f\) has connected fibers, and \(h_Y=j\circ f\). The semiample augmented line comes from \(Z\); subtracting \(b j^*A_W\) gives a rational Cartier divisor \(H\) with \[ J_Y\sim_{\mathbb Q}f^*H. \tag{71}\] Surjectivity to \(W\) and the definition of algebraic dimension give \[a(T)=\dim W\leq\dim Z\leq a(Y)=a(T).\] In particular \(f\) has positive relative dimension.

A divisible multiple of \(J_1\) has a nonzero section by Theorem 5. The effective exceptional comparisons in the ordinary negative program preserve these section spaces. Equation (71) and \(f_*\mathcal O_Y=\mathcal O_Z\) therefore give a nonzero section of a multiple of \(H\); in particular \(H\) is pseudo-effective. Proposition 11 gives \[H\sim_{\mathbb Q}K_Z+\Delta_Z\] for an effective rational boundary \(\Delta_Z\) with \((Z,\Delta_Z)\) klt. The divisor \[H+j^*N_W\] now satisfies exactly the hypotheses of Proposition 7.

Accordingly, take a smooth projective modification \(u:Z'\to Z\) and a birational morphism \(v:Z'\to Z_m\) such that \[ u^*(H+j^*N_W)\sim_{\mathbb Q}v^*H_m+E, \tag{72}\] where \(H_m\) is nef rational Cartier, \(E\geq0\) is rational and \(v\)-exceptional, and \(v^*H_m\) is numerically equivalent to a semiample rational line \(P_{Z'}\).

Take a smooth compact Kähler common resolution \(V\) of the program and of the main transform over \(Z'\). Denote its morphisms by \[q:V\to T_1,\qquad p:V\to Y,\qquad g:V\to Z', \qquad r=v\circ g.\] Thus \(f\circ p=u\circ g\). Pulling back (72) gives \[p^*(J_Y+h_Y^*N_W)\sim_{\mathbb Q}r^*H_m+g^*E.\] Every component of \(g^*E\) maps into a subset of codimension at least two in \(Z_m\). Lemma 38 therefore identifies \[N\bigl(p^*(J_Y+h_Y^*N_W)\bigr)=g^*E.\] Its positive class is represented by the semiample rational line \(g^*P_{Z'}\).

Finally, the ordinary program comparison gives \[q^*J_1\sim_{\mathbb Q}p^*J_Y+F,\qquad F\geq0 \text{ exceptional over }Y.\] The line from \(W\) is unchanged throughout the program. Combining this equality with (70) yields \[q^*(J_1+M_1)\equiv g^*P_{Z'}+g^*E+F,\] where \(M_1\) is the pullback of \(M\). By exceptional translation, \[N\bigl(q^*(J_1+M_1)\bigr)=g^*E+F.\] This is the required decomposition on the model \(T_1\). Removing the effective resolution errors by Lemma 4(5) proves it for the original \(T\). The zero-algebraic-dimension assertion and the decomposition have now both been established in dimension \(n\), completing the simultaneous induction. ◻

Proof of Theorem 1. Apply Theorem 2 to \(D=K_X+B+M\). It gives a smooth compact Kähler modification \(\mu:U\to X\) and \[\mu^*D\equiv P+R,\qquad P\text{ semiample},\qquad R=N(\mu^*D).\] Since \(D\) is nef, \(\mu^*D\) is nef and \(R=0\). Thus \(P-\mu^*D\) has zero real Chern class. By Lemma 3, it is the pullback of a rational line \(F\in\mathop{\mathrm{Pic}}^0(X)\otimes_{\mathbb Z}\mathbb Q\). Set \(L=D+F\). Then \(c_1(L)=c_1(D)\) and \(\mu^*L\sim_{\mathbb Q}P\). The last assertion of Lemma 3 descends semiampleness to \(L\). ◻