Proof. This follows directly from [Reference KollárKol23, Corollary 4.55].

Proof. Follows directly from the definition of generalised Property $(*)$ .

Proof. By [Reference Lazić and TsakanikasLT22, Theorem 2.9], we may show the claim up to extracting some log canonical places of $(X,B)$ . Therefore, we are free to replace $(X, B)$ by a dlt modification, which is guaranteed to exist by [Reference Kollár and KovácsKK10, Theorem 3.1], and so we may assume that $(X, B)$ is dlt. Furthermore, since all log canonical places are horizontal over $Z$ , by [Reference Hacon and XuHX13, Theorem 1.1], it suffices to show the claim over a non-empty open subset of $Z$ . Therefore, throughout the proof we are free to shrink $Z$ as needed.

Let $\pi \colon X^{\prime} \rightarrow X$ be a log resolution of $(X,B)$ that only extracts valuations with positive log discrepancy, which exists by [Reference Kollár and MoriKM98, Theorem 2.44]. We denote the strict transform of $B$ by $B^{\prime}$ , and we let $E^{\prime}$ be the reduced $\pi$ -exceptional divisor. Up to shrinking $Z$ , we may assume that $(X^{\prime},B^{\prime}+E^{\prime}) \rightarrow Z$ is log smooth. By construction, a good minimal model for $(X,B)$ over $Z$ is also a good minimal model for $(X^{\prime},B^{\prime}+E^{\prime})$ over $Z$ , and vice versa.

By assumption, for $z \in Z$ general, $(X_z,B_z)$ has a good minimal model. Since $\pi$ is a fibrewise log resolution that only extracts valuations of positive log discrepancy, this good minimal model is also a good minimal model for $(X^{\prime}_z,B^{\prime}_z+E^{\prime}_z)$ . Then, by [Reference Hacon, Mckernan and XuHMX18, Theorem 1.2], $(X^{\prime},B^{\prime}+E^{\prime})$ has a good minimal over $Z$ . In turn, this is a good minimal model for $(X,B)$ over $Z$ , and the claim follows.

Proof. By Lemma 2.4, we see that for $0\lt t\ll 1$ every log canonical centre of $(X, B-tf^*\Sigma )$ dominates $Z$ and $B-tf^*\Sigma \geqslant 0$ . Then, by Lemma 2.5, $(X, B-tf^*\Sigma )$ admits a good minimal model over $Z$ . Since $K_{X}+B \sim _{\mathbb Q} K_{X}+B-tf^*\Sigma$ we see that this is a good minimal model for $(X, B)$ over $Z$ as well.

Proof. From the equality $B = B^h$ we see that if $\Sigma _Z$ is the reduced divisor associated with $(X/Z, B)$ then $\Sigma _Z = 0$ . The definition of Property $(*)$ together with Proposition 2.2 immediately implies our claim.

Proof. The invariance of a divisor may be checked on a big open subset and so we may freely assume that $X$ and $\mathcal F$ are smooth.

Let $\{U_i\}$ be a trivialising open cover for $L$ and let $t_i$ be local generators for $L$ . Write $s = f_it_i$ , where $f_i = 0$ is a local equation for $D$ . We may compute

\begin{align*} 0 = \nabla (s) = \nabla (f_it_i) = df_i\otimes t_i + f_i\nabla (t_i) \mod L \otimes N^*_{\mathcal F}. \end{align*}

Since $\nabla (t_i) = \omega \otimes t_i$ for a holomorphic one form $\omega$ , we see that $df_i +f_i\omega = 0 \mod N^*_{\mathcal F}$ , from which we may conclude that after restricting to $D$ we have $df_i\vert _D = 0 \mod N^*_{\mathcal F}\vert _D$ . This implies that $\{f_i = 0\}$ is $\mathcal F$ -invariant, as required.

Proof. Since $P_z$ is everywhere transverse to $\mathcal F$ , by definition, the natural map $T_P\vert _{P_z} \rightarrow N_{P_z/P}$ is surjective, in particular, $P_z$ is smooth.

We recall the graphic neighbourhood construction of [Reference Bogomolov and McQuillanBM16, § 2.1]. We remark that [Reference Bogomolov and McQuillanBM16] only considers the graphical neighbourhood in the case where $P_z$ is a curve, but the construction works for general smooth varieties just as well. Let $\delta \colon P_z \rightarrow P_z \times P$ be the diagonal embedding. We may find a germ $\Gamma$ of an analytic variety containing $\delta (P_z)$ so that for all $t \in P_z$ we have that $q(p^{-1}(t)) \subset P$ is a germ of the leaf of $\mathcal F$ through $t$ , where $p\colon \Gamma \rightarrow P_z$ and $q\colon \Gamma \rightarrow P$ are the projections.

Observe ${\operatorname {dim}} \Gamma = {\operatorname {dim}} P$ and so in fact $q$ gives an isomorphism between an open neighbourhood $V$ of $P_z$ and $\Gamma$ . By the properness of the fibration, perhaps shrinking this neighbourhood, we may assume that $V = f^{-1}(U)$ for some open Euclidean neighbourhood of $z$ and so we get a morphism $(p\circ q^{-1}, f)\colon V \rightarrow P_z \times U$ which is easily seen to be an isomorphism.

Since the fibres of $p\circ q^{-1}$ are exactly the leaves of $\mathcal F$ we see that if $Y$ is an invariant subvariety, then $(p\circ q^{-1})(Y) = Y_z$ , and we may conclude.

Proof. Item (i) is a standard fact and we omit the proof.

To prove item (ii) let $U \subset X$ be a small open subset such that $T_X$ is generated by $\partial /\partial x_1, \ldots , \partial /\partial x_n$ and let $s_i$ be a basis of flat sections of $E$ . Let $\partial _i$ be the lift of $\partial /\partial x_i$ to a vector field on $P$ given by $\nabla$ .

Write $p^{-1}(U) = \cup _i U_i$ where $U_i = \{\overline {s}_i \neq 0\}$ , where we denote by $\overline {s}_i$ the section of $\mathcal O_P(1)$ corresponding to $s_i$ .

We define a partial connection $\overline {\nabla }$ on $\mathcal O_P(1)$ by requiring $\overline {\nabla }(\overline {s}_i) = 0$ for all $i$ . This is well defined because $dg_{ij} = 0 \mod N^*_{\mathcal F}$ where $g_{ij} = \overline {s_i}/\overline {s_j}$ .

The existence of a connection on $p_*E$ is a straightforward consequence of the fact that $p_*\Omega ^1_{\mathcal F} = \Omega ^1_X$ . The fact that $p_*E$ is locally free follows from [Reference AndréAnd01, Corollaire 2.5.2.2].

Proof. Step 1. In this step we construct a foliation.

For $m \gg 0$ and sufficiently divisible, we have an embedding $Y \rightarrow P \xrightarrow {p} Z$ , where $P:= \mathbb P(E_m)$ . Set $L = \mathcal O_P(1) \vert _Y$ . The connection $\nabla _m$ on $E_m$ guaranteed to exist by [Reference SimpsonSim92, § 3] is in fact a holomorphic connection (see [Reference DengDen21] or [Reference OuOu24, Theorem 3.5]) and therefore by Lemma 2.10 gives a smooth foliation $\mathcal H$ on $P$ such that $T_{\mathcal H} \cong p^*T_Z$ , in particular $T_{\mathcal H}$ is locally free. Observe that $K_{\mathcal H} \cong p^*K_Z$ . Moreover, $\mathcal H$ is everywhere transverse to the fibres of $P \rightarrow Z$ .

It follows from [Reference OuOu24, Corollary 3.10] that for all $k\gt 0$ the kernel of $\operatorname {Sym}^kE_m \rightarrow E_{mk} \rightarrow 0$ is a $\operatorname {Sym}^k\nabla _m$ -invariant subbundle, and therefore $Y$ is invariant by $\mathcal H$ . We may then restrict $\mathcal H$ to $Y$ to get a foliation $\mathcal F_Y$ on $Y$ .

Step 2. In this step we show local triviality of $(Y, B_Y+D) \rightarrow Z$ .

By considering $Y$ as a subscheme of $P$ which is invariant under $\mathcal H$ and applying Lemma 2.9 we see that $Y \rightarrow Z$ is locally trivial. It remains to show that $(Y, B_Y+D)$ is locally trivial. This will follow by another application of Lemma 2.9 by considering ${\rm Supp}(B_Y+D)$ as a subscheme of $P$ if we can show ${\rm Supp}(B_Y+D)$ is $\mathcal F_Y$ -invariant (and therefore $\mathcal H$ invariant as a subscheme of $P$ ).

We may find $\tilde {D} \sim _{\mathbb Q} D$ such that ${\rm Supp}(D) \subset {\rm Supp}(\tilde {D})$ and such that $(Y, B_Y+\tilde {D})$ is klt. It suffices to prove all the claims of the lemma for $(Y, B_Y+\tilde {D})$ , so without loss of generality we may freely replace $D$ by $\tilde {D}$ and so may assume that $(Y, B_Y+D)$ is klt. Let $s\colon (Y^{\prime}, B_Y^{\prime}+D^{\prime}) \rightarrow (Y, B_Y+D)$ be a small $\mathbb Q$ -factorialisation, which exists by [Reference Birkar, Cascini and HaconBCHM10, Corollary 1.4.4], and let $\sigma \colon (\overline {Y}, \overline {B}_Y+\overline {D}) \rightarrow (Y, B_Y+D)$ be the ample model of $K_{Y^{\prime}}+(1+\eta )(B^{\prime}_Y+D^{\prime})$ over $Y$ for $0\lt \eta \ll 1$ , which exists by [Reference Birkar, Cascini and HaconBCHM10, Theorem 1.2]. Note that $\overline {B}_Y+\overline {D}$ is $\sigma$ -ample by construction.

Set $\overline {\mathcal F}_Y = \sigma ^{-1}\mathcal F_Y$ and note that since $\sigma$ is small, $K_{\overline {\mathcal F}_Y} = \sigma ^*K_{\mathcal F_Y}$ . By Lemma 2.10 $\mathcal O_{\overline {Y}}(m(K_{\overline Y/Z}+\overline {B}_Y+\overline {D}))$ admits a partial $\overline {\mathcal F}_Y$ -connection.

By the Bott partial connection we know that $\operatorname {det} N^*_{\overline {\mathcal F}_Y} \cong K_{\overline {Y}} - K_{\overline {\mathcal F}_Y} = K_{\overline {Y}/Z}$ also admits a partial connection. It follows that $\mathcal O_{\overline {Y}}(m(\overline {B}_Y+\overline {D}))$ admits a partial $\overline {\mathcal F}_Y$ -connection, and so $\mathcal O_{\overline {Y}}(m(K_{\overline Y/Z}+\overline {B}_Y+D)+k(\overline {B}_Y+\overline {D}))$ admits a partial $\overline {\mathcal F}_Y$ -connection for all $m \gg k \gg 0$ .

By Lemma 2.10, $\tilde {E}_m := p_*\mathcal O_{\overline {Y}}(m(K_{\overline Y/Z}+\overline {B}_Y+\overline {D})+k(\overline {B}_Y+\overline {D}))$ is a vector bundle which admits a holomorphic connection. It follows that $\operatorname {det} \tilde {E}_m$ admits a holomorphic connection and so $c_1(\tilde {E}_m) = 0$ . By [Reference PatakfalviPat14, Corollary 4.5] (see also [Reference FujinoFuj18, Theorem 1.11]) $\tilde {E}_m$ is nef. Since $\tilde {E}_m$ is nef and $-c_1( \tilde {E}_m)$ is nef we conclude that $\tilde {E}_m$ numerically flat. Let $\tilde {\nabla }_m$ be the connection guaranteed to exist by [Reference SimpsonSim92, § 3]. Let $s \in H^0(\overline {Y}, \mathcal O_{\overline {Y}}(k(\overline {B}_Y+\overline {D}))$ be a global section such that $\{s = 0\} = k(\overline {B}_Y+\overline {D})$ , and let $\varphi \colon E_m \rightarrow \tilde {E}_m$ be the morphism induced by multiplication by $s$ . By the proof of [Reference OuOu24, Corollary 3.10] we see that $\varphi$ is a morphism of vector bundles equipped with connections, i.e., if $t$ is any flat local section of $E_m$ , we see that $t\otimes s$ is a flat local section of $\tilde {E}_m$ .

Without loss of generality we may assume that $m \gg k \gg 0$ are sufficiently large so that $m(K_{\overline Y/Z}+\overline {B}_Y+\overline {D})+k(\overline {B}_Y+\overline {D})$ is $\bar {p}$ -very ample. Let $\tilde {P} := \mathbb P(\tilde {E}_m)$ and by repeating the construction of Step 1, we produce a foliation $\tilde {\mathcal F}$ on $\overline {Y}$ which is everywhere transverse to the fibres of $\overline {Y} \rightarrow Z$ . Let us continue to denote by $\tilde {\nabla }_m$ the $\tilde {\mathcal F}$ -partial connection on $m(K_{\overline Y/Z}+\overline {B}_Y+\overline {D})+k(\overline {B}_Y+\overline {D})$ guaranteed by Lemma 2.10. As noted, if $t$ is any flat local section of $E_m$ , then $\tilde {\nabla }_m(t\otimes s) = 0$ , and so by Lemma 2.8 it follows that $\{t\otimes s = 0\}$ is invariant, from which we may conclude that ${\rm Supp}(\overline {B}_Y+\overline {D}) = {\rm Supp}(\{s = 0\})$ is invariant.

Since $\varphi$ was a morphism of vector bundles equipped with connections it follows that $\mathcal F_Y = \sigma _*\tilde {\mathcal F}$ , and so $\mathcal F_Y$ leaves $B_Y+D$ invariant.

Our final claim follows by first observing that $(Y_0, (B_Y+D)_0)$ is klt. We then recall that the automorphism group of a log canonical pair of general type is finite, and so our locally trivial family is in fact trivial after a finite étale base change.

Proof. Following [Reference AmbroAmb05, § 2], we may find a generically finite morphism $\sigma \colon \widetilde {Y} \rightarrow Y$ , a morphism $\tau \colon \widetilde {Y} \rightarrow Y^!$ and a big and nef divisor ${\bf M}_{Y}$ such that $\overline {\tau ^* {\bf M}_{Y^!}} = \sigma ^* {\bf M}$ as equality of b-divisors. Notice that we use b-divisors to express this identity, since ${\bf M}$ may not descend onto $Y$ , and so ${\bf M}$ may not agree with the Cartier closure of ${\bf M}_{Y}$ . It follows that for some $k \gg 0$ some sublinear system of $\,|k {\bf M}_{Y}|\,$ defines a rational map $\phi \colon Y \dashrightarrow W$ such that, at the generic point $\eta$ of $Y$ , $\,\operatorname {Ker} {\rm d}\phi _{\eta }$ agrees with $\operatorname {Ker} \kappa$ . Here, ${\rm d}\phi _{\eta }$ is the differential $T_{Y, \eta } \rightarrow (f^*T_W)_\eta$ restricted to the generic fibre, $\kappa \colon T_{Y, \eta } \rightarrow H^1(T_{\overline {X}_\eta }(-\log \overline {B}))$ is the Kodaira–Spencer map, see [Reference AmbroAmb05, Theorem 2.2] and $(\overline {X}, \overline {B}) \rightarrow (X, B)$ is a log smooth model as guaranteed by [Reference AmbroAmb05, Lemma 1.1]. Observe that we may not take the full linear series $\,|k {\bf M}_{Y}|\,$ , since ${\bf M}$ may not descend onto $Y$ . Let $\bar {h}\colon \overline {X} \rightarrow Y$ be the natural morphism.

The general fibre of $Y \dashrightarrow W$ is generically an intersection of divisors $D \sim k {\bf M}_{Y}$ . By Lemma 2.11, these divisors are all $\mathcal F_Y$ invariant, and so the general fibre is $\mathcal F_Y$ invariant. It follows that, at the generic point of $Y$ , $T_{\mathcal F_Y}$ is contained in the kernel of the Kodaira–Spencer morphism $T_{Y} \rightarrow R^1\bar {h}_*T_{\overline {X}/Y}(-\log \overline {B})$ , as the latter generically coincides with $\operatorname {Ker} {\rm d}\phi$ . Thus, at the generic point of $Y$ we have a lift $\rho \colon h^*T_{\mathcal F_Y} \rightarrow T_{\overline {X}}(-\log \overline {B})$ , which induces a foliation $\mathcal F_{\overline {X}}$ on $\overline {X}$ . Let $U$ be the largest open subset over which we have a lift $h^*T_{\mathcal F_Y} \rightarrow T_{\overline {X}}(-\log \overline {B})$ .

By construction $T_{\mathcal F_{\overline {X}}}\vert _{\overline {X}_y} \to N_{\overline {X}_y/\overline {X}}$ is a subbundle where $\overline {X}_y$ is a fibre of $\overline {X} \rightarrow Y$ over $U$ . Since $\rho$ takes values in $T_{\overline {X}}(-\log \overline {B})$ we see that $\mathcal F_{\overline {X}}$ leaves the support of $\overline {B}$ invariant. This proves item (i).

We now prove item (ii). Fix a general point $t \in Z$ , and let $t^{\prime} \in Z$ be a nearby point. By Lemma 2.11, we have isomorphisms $\psi \colon Y_t \rightarrow Y_{t^{\prime}}$ given by the flows of the foliation $\mathcal F_Y$ . Perhaps shrinking $U$ , we may assume this gives an isomorphism $\psi \colon U_t \rightarrow U_{t^{\prime}}$ .

For any closed point $s \in U_t$ , let $L$ denote a germ of a leaf of $\mathcal F_Y$ through $s$ , which up to shrinking, we may assume is contained in $U$ . By construction, we have $L \cap U_{t^{\prime}} = \psi (s)$ . Consider the family $(\overline {X}_L, \overline {B}_L) := (\overline {X}, \overline {B}) \times _Y L$ . By construction, $\overline {X}_L$ is $\mathcal F_{\overline {X}}$ -invariant, and so we have a restricted foliation $\mathcal F_{\overline {X}_L}$ which is everywhere transverse to the fibres of $\overline {X} \rightarrow L$ and leaves $B$ invariant. Applying Lemma 2.9 to $\mathcal F_{\overline {X}_L}$ and $(\overline {X}_L, \overline {B}_L) \rightarrow L$ , it follows that $(\overline {X}_s, \overline {B}_s) \cong (\overline {X}_{\psi (s)}, \overline {B}_{\psi (s)})$ , as required.

By considering the relative ${\rm Isom}$ functor ${\rm Isom}_{Y_0}((X_{t}, B_{t}), (X_{t^{\prime}}, B_{t^{\prime}}))$ for general $t, t^{\prime}$ , we see that standard arguments imply the geometric generic fibres $(X_t, B_t) \rightarrow Y_0$ and $(X_{t^{\prime}}, B_{t^{\prime}}) \rightarrow Y_0$ are isomorphic, from which we may conclude item (iii).

Item (iv) is a direct consequence of the previous arguments. We remark that by the last point in Lemma 2.11 the leaves of $\mathcal F_Y$ are all algebraic.

Proof. Suppose first that $K_{X/Z}+B$ is relatively ample. In this case the result follows from [Reference Xu and ZhuangXZ20, Lemma-Definition 2.26] (see also [Reference Codogni and PatakfalviCP21, Proposition 5.4] and [Reference Codogni and PatakfalviCP21, Proposition 6.4]) by taking $L = l(K_{X/Z}+B)$ .

In general, let $g\colon X \rightarrow Y$ be the ample model of $K_{X/Z}+B$ over $Z$ , and, perhaps replacing $l$ by a multiple we may find a Cartier divisor $L^{\prime}$ be such that $g^*L^{\prime} \sim L$ . By [Reference AmbroAmb05, Theorem 0.2], we may write $L^{\prime} \sim _{\mathbb Q} K_{Y/Z}+B_Y$ where $(Y, B_Y)$ is klt over the generic point of $Z$ . We have equalities $\lambda _+(X/Z, B) = \lambda _+(Y/Z, B_Y)$ and $\lambda _-(X/Z, B) = \lambda _-(Y/Z, B_Y)$ and so we may freely reduce to the previous case in order to conclude.

Proof. Part (i) is a direct consequence of [Reference Kovács and PatakfalviKP17, Lemma 7.7] (and its proof) and the fact that $M = K_{X/Z}+B$ .

To show (ii) and (iii) consider the following commutative diagram, which is guaranteed to exist by [Reference Kovács and PatakfalviKP17, Corollary 6.20]:

(1)

where all three of our vertical morphisms are families of stable log varieties, $\tau$ is finite, and the moduli map associated with $(X_0, B_0) \rightarrow Z_0$ is finite.

By [Reference Kovács and PatakfalviKP17, Corollary 7.3] and its proof for $m \gg 0$ sufficiently divisible, $\lambda ^0_m:= \operatorname {det} f_{0*}\mathcal O_{X_0}(mM_0)$ is ample, where $M_0$ is the moduli part of $(X_0/Z_0, B_0)$ . This implies that $M_0+\epsilon f_0^*c_1(\lambda ^0_m)$ is ample for all $\epsilon \in \mathbb Q_{\gt 0}$ , since $M_0$ is globally nef and relatively ample, see [Reference Kovács and PatakfalviKP17, Lemma 7.7 and proof of Theorem 7.1.1]. Next, note that we have $\sigma ^*M = s^*M_0$ and by (i) we know $\tau ^*\lambda _m = t^*\lambda ^0_m$ , which implies both (ii) and (iii).

Proof. It follows immediately from [Reference Liu and XieLX23, Theorem 1.6] and the fact that Du Bois singularities are semi-normal.

Proof. By Lemma 2.7, $(X, B)\rightarrow Z$ satisfies Property $(*)$ , and so by [Reference AmbroAmb04, Theorem 3.1] we see that $(Y, B_Y,{\bf M})\rightarrow Z$ satisfies generalised Property $(*)$ . Observe also that if $D \subset Z$ is any reduced divisor, then $g^*D$ is reduced as well.

Let $p\colon \overline {Y}\rightarrow Y$ be a generalised dlt modification, and let $(\overline {Y}, \overline {B}_Y,{\bf M})$ be the transformed pair. By Lemma 3.3, we see that $\bar {g}\colon (\overline {Y}, \overline {B}_Y,{\bf M}) \rightarrow Z$ satsifies generalised Property $(*)$ , and so it follows that $(\overline {Y}/Z, \overline {B}_Y)$ satisfies Property $(*)$ , and let $\Sigma \subset Z$ be the associated boundary. Since $(X, B) \rightarrow Z$ is locally stable we see that $\overline {B}_Y^h = \overline {B}_Y$ , and so by Lemma 2.7 $(\overline {Y}, \overline {B}_Y) \rightarrow Z$ is locally stable.

Since we are assuming the b-semi-ampleness conjecture, for a general choice of $0 \leqslant H \sim _{\mathbb Q} {\bf M})_{\overline {Y}}$ , by Bertini’s theorem (applied on a log resolution where ${\bf M}$ descends) we may guarantee that $(\overline {Y}_z, \overline {B}_{Y, z}+H_z)$ is slc. By [Reference KollárKol23, Corollary 4.45] there is an open neighbourhood of $z$ such that $(\overline {Y}/Z, \overline {B}_Y+H)$ is locally stable as required.

Proof. Item (ii) immediately follows from item (i) by Grauert’s theorem and cohomology and base change, see [Reference HartshorneHar77, Corollary III.12.9 and Theorem III.12.11]. Similarly, item (iv) follows from item (iii). Thus, we are left with showing items (i) and (iii).

Let us first suppose that ${\operatorname {dim}} Z = 1$ . Let $c \colon (X,\Delta ) \rightarrow Y$ be the relative Iitaka fibration of $f$ , which exists by the assumptions and [Reference Hacon and XuHX13]. Since $X$ is normal, so is $Y$ . Let $(Y/Z,B_Y,{\bf M})$ be the generalised pair induced by the canonical bundle formula. Since $Z$ is a curve, $g \colon Y \rightarrow Z$ is flat, see [Reference HartshorneHar77, Proposition III.9.7]. By semi-continuity [Reference HartshorneHar77, Theorem III.12.8], it suffices to show that, for a special closed point $0 \in Z$ , we have $Y_0 = \operatorname {Proj}(R(X_0,\Delta _0))$ , where we set $R(X_0,\Delta _0)=\oplus _{m \in \mathbb N}\Gamma (X_0,\mathcal{O}_{X_0}(m(K_{X_0} + \Delta _0)))$ . Since $f$ is a locally stable family, $(X,\Delta +X_0)$ is log canonical. Then, by inversion of adjunction for LC-trivial fibrations [Reference AmbroAmb04, Theorem 3.1], the induced generalised pair $(Y/Z,B_Y+g^*0,{\bf M})$ is generalised log canonical. In particular, we have $Y_0=g^*0$ as divisors. In particular, the scheme theoretic fibre $Y_0$ is $R_0$ . Since $Y$ is normal and $g^*0$ is Cartier, it follows that $\mathcal{O}_Y$ and $\mathcal{O}_Y(-g^*0)$ are $S_2$ . Then, by [Reference KollárKol13, Corollary 2.61], $Y_0$ is $S_1$ . But then, by [Sta22, Tag 0344], $Y_0$ is reduced. Furthermore, by Lemma 4.2, $Y_0$ is semi-normal.

Now, we have a tower of morphisms as follows:

\begin{align*} X_0 \xrightarrow {\alpha } \operatorname {Proj}(R(X_0,\Delta _0)) \xrightarrow {\beta } Y_0. \end{align*}

For $m_0$ sufficiently large and divisible, we have the following:

• • the morphism $\alpha$ is given by the full linear series $|\Gamma (X_0,\mathcal{O}_{X_0}(m_0(K_{X_0} + \Delta _0)))|$ ; and

• • the morphism $c_0 \colon X_0 \rightarrow Y_0$ is given by the restricted linear series corresponding to

  (2) \begin{align} \textrm {Im} (\Gamma (X,\mathcal{O}_{X}(m_0(K_{X/Z} + \Delta ))) \rightarrow \Gamma (X_0,\mathcal{O}_{X_0}(m(K_{X_0} + \Delta _0)))) . \end{align}

Since $c_0$ is given by a sub-series of the one giving $\alpha$ , and since $c_0$ contracts exactly the same curves as $\alpha$ , the induced morphism $\beta$ is finite. By construction, we have $\alpha _* \mathcal{O}_{X_0} = \mathcal{O}_{\operatorname {Proj}(R(X_0,\Delta _0))}$ . Similarly, we have $c_* \mathcal{O}_X = \mathcal{O}_Y$ and so $X_0 \rightarrow Y_0$ has connected fibres. Then, by Lemma 4.3, we have $(c_0)_*\mathcal{O}_{X_0}=\mathcal{O}_{Y_0}$ . In turn, this gives $\beta _* \mathcal{O}_{\operatorname {Proj}(R(X_0,\Delta _0))}=\mathcal{O}_{Y_0}$ . But then, as $\beta$ is finite, we have $\operatorname {Proj}(R(X_0,\Delta _0))=Y_0$ . Since $g$ is flat, its fibres are pure-dimensional. This settles items (iii) when ${\operatorname {dim}} Z = 1$ .

A priori, (2) may not be an isomorphism, since a proper linear sub-series may still define an isomorphism. To rule this case out, we observe that we just showed $(c_0)_* \mathcal{O}_{X_0}=\mathcal{O}_{Y_0}$ . Therefore, by the projection formula, in order to prove (i), it suffices to show that $h^0(Y_z,L|_{Y_z})$ is independent of $z \in Z$ , where $L=\mathcal{O}_Y(m(K_Y+B_Y+{\bf M}_Y))$ and $m$ is sufficiently divisible. But this latter fact readily follows by the flatness of $Y \to Z$ and relative Serre’s vanishing applied to the relatively ample line bundle $L$ . This settles (i) when ${\operatorname {dim}} Z = 1$ .

We now handle the general case. Let $l_0$ denote the Cartier index of $K_{X/Z} + \Delta$ . By generic flatness applied to $g \colon Y \rightarrow Z$ and relative Serre’s vanishing, we may find a non-empty open subset $U \subset Z$ and $m_0 \in \mathbb N$ divisible by $l_0$ such that (i) holds over $U$ . Then, for every closed point $z^{\prime} \in Z$ , by restricting to a general curve passing through $z^{\prime}$ and which meets $U$ , we may find $m_{z^{\prime}} \in \mathbb N$ divisible by $m_0$ such that, for every $k \in \mathbb N$ , $h^0(X_{z^{\prime}},\mathcal{O}_{X_z} (km_{z^{\prime}}( K_{X_z^{\prime}} + \Delta _{z^{\prime}})))$ attains the generic value achieved over $U$ . Then, by semi-continuity and Grauert’s theorem [Reference HartshorneHar77, Theorem III.12.8 and Corollary III.12.9], $z^{\prime}$ is in the open subset where $f_*\mathcal{O}_{X} (km_{z^{\prime}}( K_{X} + \Delta ))$ is a vector bundle, for every $k \in \mathbb N$ . Thus, by replacing $m_0$ with a multiple, we may assume that any fixed closed point $z^{\prime}$ belongs to the open set $U$ . Since the $\mathcal{O}_Z$ -algebra $\oplus _{k \in \mathbb N}f_*\mathcal{O}_X(kl_0(K_{X/Z}+\Delta ))$ is finitely generated, after finitely many iterations, the above procedure extends the open subset $U$ to the whole $Z$ .

Since (i) and (ii) are settled and (iii) is known over a curve, item (iii) now follows immediately. Similarly, the additional properties of the fibres of $g$ from (iii) hold since they may be checked in the case the base is a curve. Lastly, the flatness of $g$ follows from [Reference KollárKol23, Theorem 3.20].

Proof. Since $(X, \Delta ) \rightarrow Z$ is locally stable, we know that every log canonical centre of $(X, \Delta )$ dominates $Z$ . By applying [Reference Hacon and XuHX13, Theorem 1.1] to a dlt modification of $(X, \Delta )$ , it follows that $K_{X/Z} + \Delta$ is relatively semi-ample over $Z$ .

By Theorem4.5, the coherent sheaf $f_* \mathcal{O}_X(m(K_{X/Z} + \Delta ))$ is a vector bundle and its formation commutes with base change. In particular, it follows that the line bundle $\lambda _m:= \operatorname {det} f_* \mathcal{O}_X(m(K_{X/Z} + \Delta ))$ is compatible with base change. Lastly, we recall that, given a line bundle $\mathcal{L}$ on a normal variety $U$ and a surjective projective morphism $\phi \colon V \rightarrow U$ where $V$ is normal, then $\mathcal{L}$ is semi-ample if an only if so is $\phi ^* \mathcal{L}$ . Thus, when needed, we are free to replace $Z$ with a surjective generically finite cover and all the relevant varieties and morphisms by base change (recall that the base change of a locally stable family is again a locally stable family). In particular, we may assume that $Z$ is smooth.

Let $c \colon X \rightarrow Y$ denote the relative Iitaka fibration of $(X,\Delta )$ , and let $(Y,B_Y,{\bf M})$ be the generalised pair induced on $Y$ . By Noetherianity and Proposition 4.4, we may find finitely many effective divisors $D_1,\ldots ,D_k \in |{\bf M}_{Y}|_{\mathbb Q}$ and open sets $U_i \subset Z$ such that $Z = \cup _{i= 1}^k U_i$ and $g \colon (Y,B_Y+D_i) \rightarrow Z$ is a stable family of pairs over $U_i$ .

Now, fix $i \in \{1, \ldots , k\}$ . We will show that for integers $m_i \gg 0$ and $\ell \gt 0$ and $\epsilon \in \mathbb Q_{\gt 0}$ , the base locus of $\lambda _{\ell m_i}$ (respectively ${\bf M}_{Y} +\epsilon f^*c_1(\lambda _{\ell m_i})$ ) is contained in $Z \setminus U_i$ (respectively $f^{-1}(Z \setminus U_i)$ ). Supposing this, we see that by taking $m = m_1m_2\ldots m_k$ the claims of the theorem hold.

For ease of notation set $D:= D_i$ and $U := U_i$ .

By [Reference Kovács and PatakfalviKP17, Corollary 6.18], we may find a generically finite cover $\sigma \colon \overline {U} \rightarrow U$ such that $g \colon (Y,B_Y+D)\times _U\overline {U} \rightarrow \overline {U}$ can be recompactified to a family of stable pairs over a normal projective variety $\overline {Z} \supset \overline {U}$ . Call this compactified family $(\overline {Y}, \overline {B}_Y+\overline {D}) \rightarrow \overline {Z}$ . Perhaps replacing $\overline {Z}$ by a higher model, we may assume that $\sigma$ extends to a morphism $\sigma \colon \overline {Z} \rightarrow Z$ .

Let $\tau \colon \overline {Z} \rightarrow \tilde {Z}$ be the Stein factorisation of $\sigma$ , and set $\tilde {Y} := Y\times _Z\tilde {Z}$ . By part (iv) of Theorem4.5, $\tilde {Y}$ is normal. Since $\psi \colon \tilde {Y} \to Y$ is finite, pullback is well defined also for divisors that are not necessarily $\mathbb Q$ -Cartier. We define $\tilde {B}$ via $\psi ^*(K_Y+B_Y) = K_{\tilde {Y}}+\tilde {B}$ and we set $\tilde {D}=\psi ^*D$ .

Let $m_0$ be the integer guaranteed to exist by Theorem4.5. Up to replacing $m_0$ with a multiple, we may further assume that $f_* \mathcal O_X(\ell m_0(K_{X/Z}+\Delta ))$ is a vector bundle that commutes with base change. Note that $\operatorname {det} h _{*} \mathcal O_{\overline {Y}}(km_0(K_{\overline Y /\overline Z} + \overline B + \overline D ))\vert _{\overline {U}}$ coincides with $\sigma ^*\operatorname {det} f_* \mathcal O_X(km_0(K_{X/Z}+\Delta ))\vert _{U}$ .

Now, we claim that we have an inclusion of coherent sheaves

(3) \begin{align} 0 \rightarrow h_{*} \mathcal O_{\overline {Y}}(km_0(K_{\overline Y /\overline Z} + \overline B + \overline D )) \rightarrow \sigma ^*f_* \mathcal O_X(km_0(K_{X/Z}+\Delta )) \end{align}

that is an isomorphism over $\overline {U}$ . The latter claim is immediate by the previous observations. Furthermore, as the two sheaves are torsion free, to conclude it suffices to show there is a non-zero morphism between the two.

Now, let $\hat {Y}$ be the normalisation of the closure of the graph of $\tilde {Y} \dashrightarrow \overline {Y}$ in $\tilde {Y} \times _{\tilde Z} \overline {Y}$ , and let $p \colon \hat {Y} \rightarrow \tilde Y$ and $q \colon \hat {Y} \rightarrow \overline {Y}$ be the corresponding morphisms. As $\tilde Y$ and $\overline Y$ are normal and isomorphic over the generic point of $\tilde Z$ , $p$ and $q$ are isomorphisms over the generic point of $\tilde Z$ . To conclude, it is then sufficient to show that

(4) \begin{align} q^*(K_{\overline {Y}} + \overline {B} + \overline {D}) +\Theta = p^*(K_{\tilde Y} + \tilde B +\tilde D), \end{align}

where $\Theta \geqslant 0$ . Note that, by construction, $p_*\Theta$ is contained in the preimage of $Z \setminus U$ in $\tilde Y$ .

To this end, we follow the proof of [Reference KollárKol23, Theorem 11.40]. Then, we may write

\begin{align*} p^*(K_{\tilde Y} + \tilde B + \tilde D)= K_{\hat {Y}} + \hat {B} + \hat {D} + A -B, \end{align*}

where $\hat {B}$ and $\hat {D}$ denote the strict transforms of $\tilde B$ and $\tilde D$ , respectively, and $A$ and $B$ are effective divisors with no common components. Since $h \colon (\overline {Y}, \overline {B} + \overline {D}) \rightarrow \overline {Z}$ is a stable family, we may write

\begin{align*} q^*(K_{\overline {Y}} + \overline {B} + \overline {D})= K_{\hat {Y}} + \hat {B} + \hat {D} -E, \end{align*}

where $E$ is an effective divisor, cf. [Reference KollárKol23, Proposition 2.15]. Then, it follows that $-(E+A-B)$ is $p$ -nef and $p_{*}(E+A-B)=p_{*}E \geqslant 0$ . Thus, by the negativity lemma [Reference Kollár and MoriKM98, Lemma 3.39], we have $E+A-B \geqslant 0$ . Thus, in particular, $A-B \geqslant -E$ , and (4) follows.

By taking determinants of (3), we get a morphism

(5) \begin{align} \bar {\lambda }_{\ell m_0}:= \operatorname {det} h_{*} \mathcal O_{\overline {Y}}(\ell m_0(K_{\overline Y /\overline Z} + \overline B + \overline D )) \rightarrow \sigma ^*\operatorname {det} f_* \mathcal O_X(\ell m_0(K_{X/Z}+\Delta )) \end{align}

that is an isomorphism over $\overline {U}$ , and so $\sigma ^*c_1(\lambda _{\ell m_0}) = c_1(\bar {\lambda }_{\ell m_0})+G$ where $G \geqslant 0$ is supported on $\overline {Z} \setminus \overline {U}$ . Likewise, we see that $q^*(\overline {M}+\epsilon \bar {f}^*c_1(\bar {\lambda }_{\ell m_0}))+\Theta +\epsilon q^*\bar {f}^*G = p^*(\tilde M+\epsilon \tilde f^*c_1(\tilde {\lambda }_{\ell m_0}))$ , where $\tilde M$ and $\tilde \lambda _{\ell m_0}$ are the pullbacks of $M$ and $\lambda _{\ell m_0}$ to $\tilde X = X \times _{\tilde Z} Z$ and $\tilde Z$ , respectively.

By Lemma 4.1, we know that $\bar {\lambda }_{\ell m_0}$ (respectively $(\overline {M}+\epsilon \bar {f}^*c_1(\bar {\lambda }_{\ell m_0}))$ ) is semi-ample, which from our above equalities implies that the base locus of $\tilde {\lambda }_{\ell m_0}$ (respectively $\tilde M+\epsilon \tilde f^* c_1(\tilde \lambda _{\ell m_0}))$ ) is contained in $G$ (respectively $p_*(\Theta +\epsilon q^*\bar {f}^*G)$ ) as required. As explained at the beginning of the proof, these statements are well behaved under base change, and thus we can descend the corresponding statements regarding $\lambda _{\ell m_0}$ and $M + \epsilon f^* c_1(\lambda _{\ell m_0})$ to $Z$ and $X$ , respectively. This allows us to conclude.

Proof. By [Reference Ambro, Cascini, Shokurov and SpicerACSS21, Theorem 1.1] we may find a model $(X^{\prime}, B^{\prime}) \rightarrow Z^{\prime}$ which satisfies (i)–(iv). By Lemma 2.6 we see that that the assumption on termination of flips in [Reference Ambro, Cascini, Shokurov and SpicerACSS21, Theorem 1.1] is unneeded. Theorem4.6 implies that (v) and (vi) hold.

To prove our final claim, observe that $(X^{\prime\prime}, B^{\prime\prime}) \rightarrow Z^{\prime\prime}$ also satisfies (i)–(vi) and so our final claim is a consequence of Lemma 3.6.

Proof. Consider the model $(X^{\prime}/Z^{\prime}, B^{\prime})$ with moduli part $M^{\prime}$ guaranteed to exist by Theorem4.7, and let $p\colon \tilde {X} \rightarrow X$ and $q\colon \tilde {X} \rightarrow X^{\prime}$ resolve the rational map $\alpha \colon X^{\prime} \dashrightarrow X$ . Similarly, let $\beta \colon Z^{\prime} \rightarrow Z$ be the corresponding morphism as in Theorem4.7.