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Log adjunction: moduli part V. V. Shokurovabc a Department of Mathematics, Johns Hopkins University, Baltimore, MA, USA b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia c Fundamental mathematics center, Moscow Institute of Physics and Technology
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 3, DOI: https://doi.org/10.4213/im9279 
Bibliographic databases:
    § 1. IntroductionLet $(X/Z,D)$ be a pair with a morphism $X/Z$ of algebraic varieties (or spaces) and an $\mathbb{R}$-divisor $D$ on $X$. GLC denotes its generic lc property: GLC: $(X/Z,D)$ is lc over the generic point $\eta$ of $Z$, that is, $X_\eta$ is normal and $(X_\eta,D_\eta)$ is lc. Theorem 1. Let $(X/Z,D)$ be a pair under GLC such that, generically over $Z$, $D$ is a boundary and $(X/Z,D)$ has a weakly log canonical (wlc) model. Suppose that LMMP holds in dimensions $\leqslant \dim X$. Then there exists a b-$\mathbb{R}$-divisor $\mathcal{D}^{\mathrm{mm}}$ of a completion $\overline{X}$ of $X$, the maximal moduli part of adjunction of $(X/Z,D)$. The b-$\mathbb{R}$-divisor $\mathcal{D}^{\mathrm{mm}}$ is unique up to the linear equivalence and depends only on the generic fiber $(X_\eta,D_\eta)$. Moreover, $\mathcal{D}^{\mathrm{mm}}$ is b-nef. Log Minimal Model Program (LMMP) means here LMMP for relative pairs $(X/Z,D)$ with proper $X/Z$, $\mathbb{Q}$-factorial $X$, a boundary $B$, log canonical (lc) $(X,D)$ and the termination of some sequence of elementary flips. Thus essentially LMMP means such a termination. Moreover, we can suppose that the singularities of $(X,D)$ are toroidal divisorially log terminal (tdlt) with respect to $X/Z$, that is, the pair $(X/Z,D)$ is formally toroidal with the invariant divisor $D$ and with respect to $X/Z$ near every scheme point of $X$ with the minimal log discrepancy $0$. Note that if the numerical log Kodaira dimension of $(X_\eta,D_\eta)$ is $0$, then the LMMP assumption can be removed (cf. [1]) that gives a slight generalization of the Ambro positivity [2], Theorem 0.2, and what is more important the proof does not use the positivity of variation of Hodge structures [1], Theorem 1.3. Our result is more general than [1], Theorem 1.3, because we do not assume that $X/Z$ is a contraction (see Example 4). The theorem also states the b-nef property to compare with [3], Proposition-Definition 1. If $Z$ is a point, then $\mathcal{D}^{\mathrm{mm}}$ corresponds to the positive part in Zariski decomposition of $K_X+D$. More precisely, for every sufficiently high crepant model $(Y,D_Y)$ of $(X,D)$, the trace (restriction) $\mathcal{D}^{\mathrm{mm}}_Y$ on $Y$ is the positive part in Zariski decomposition of $K_Y+D_Y$ and, moreover, $\mathcal{D}^{\mathrm{mm}}_Y$ is nef. Similarly, in general, the b-$\mathbb{R}$-divisor $\mathcal{D}^{\mathrm{mm}}$ is presented by its trace $D^{\mathrm{mm}}_Y=\mathcal{D}^{\mathrm{mm}}_Y$ on every model $Y$ of $\overline{X}$. On every sufficiently high model $Y$, $D^{\mathrm{mm}}_Y$ is nef, in particular, $\mathbb{R}$-Cartier and, for every model $Y'$ over $Y$ with the birational morphism $g\colon Y'\to Y$, $D^{\mathrm{mm}}_{Y'}=g^*D^{\mathrm{mm}}_Y$ holds. The maximal property of $\mathcal{D}^{\mathrm{mm}}$ will be established in Proposition-Definition 1 in § 4. So, $\mathcal{D}^{\mathrm{mm}}$ is the special extremal case of a more general moduli part of adjunction $\mathcal{D}^{\mathrm{mod}}$. Related terminology will be recalled and introduced in § 2. However, more general b-$\mathbb{R}$-divisors $\mathcal{D}^{\mathrm{mod}}$ are usually not b-nef and even not $\mathbb{R}$-Cartier as show Examples in § 3. Surprisingly we get the following result which with Theorem 1 will be proved in § 4. Theorem 2. Let $(X/Z,D)$ be a log pair with a $\mathbb{Q}$-divisor $D$, proper surjective $X/Z$ and under GLC. Then $\mathcal{D}^{\mathrm{mod}}$ is $\mathbb{R}$- and so $\mathbb{Q}$-Cartier. § 2. Notation and terminologyUsually we assume that the base field $k$ is algebraically closed of characteristic $0$, e. g., $k=\mathbb{C}$. In this section we recall basics and introduce some new concepts related to b-divisors. For details and more examples see [4]. Let $X$ be a (normal) algebraic variety (or space) over $k$. A model of $X$ is a normal algebraic variety (respectively, space) $Y$ with a proper birational isomorphism $X\,{\dashrightarrow}\, Y$, that is, a composition of two proper birational morphisms $X\leftarrow Z\to Y$. So, there exists a proper canonical birational isomorphism between any two models of $X$. A prime b-divisor of $X$ is a discrete divisorial valuation of the field of rational function $k(X)$ of $X$. Equivalently, the valuation is given by a prime Weil divisor $W$ on a model $Y$ of $X$ and is the multiplicity function
$$
\begin{equation*}
\operatorname{mult}_W\colon k(X)^\times\to \mathbb{Z}.
\end{equation*}
\notag
$$
The canonical birational isomorphism between two models of $X$ allows to identify Weil divisors corresponding to the same valuation. Thus the valuation can be identify with the equivalence class of those Weil divisors. However, for simplicity, we denote a prime b-divisor as a corresponding Weil divisor $W$ assuming that it is defined up to a canonical birational isomorphism of models. An integral b-divisor or b-divisor $\mathcal{D}$ of $X$ is a function on prime b-divisors $P$ taking values or multiplicities in $\mathbb{Z}$:
$$
\begin{equation*}
P\mapsto \operatorname{mult}_P\mathcal{D}\in \mathbb{Z},
\end{equation*}
\notag
$$
such that, for every model $Y$ of $X$, the set of prime Weil divisors $P$ on $Y$ with $\operatorname{mult}_P\mathcal{D}\ne 0$ is finite. Hence, for every model $Y$ of $X$,
$$
\begin{equation*}
\mathcal{D}_Y=\sum (\operatorname{mult}_P\mathcal{D}) P
\end{equation*}
\notag
$$
is a well-defined Weil divisor on $Y$, where the summation is taken only for prime Weil divisors $P$ on $Y$, that is, prime b-divisors nonexceptional on $Y$. The divisor $\mathcal{D}_Y$ is called the trace of $\mathcal{D}$ on $Y$. Similarly, we can define b-$\mathbb{Q}$- and b-$\mathbb{R}$-divisors of $X$ with multiplicities in $\mathbb{Q}$ and $\mathbb{R}$, respectively: $\operatorname{mult}_P\mathcal{D}\in\mathbb{Q}$ and $\mathbb{R}$. Their trace on $Y$, defined as above, is, respectively, a Weil $\mathbb{Q}$- and $\mathbb{R}$-divisor on $Y$. The b-divisors of $X$ form an Abelian group of b-divisors $\operatorname{\mathcal{D}iv}_\mathbb{Z} X$ with a natural addition of b-divisors. Respectively, b-$\mathbb{Q}$- and b-$\mathbb{R}$-divisors form $\mathbb{Q}$-linear and $\mathbb{R}$-linear vector spaces $\operatorname{\mathcal{D}iv}_\mathbb{Q} X$ and $\operatorname{\mathcal{D}iv}_\mathbb{R} X$ when $\dim X\geqslant 2$. Notice that neither $\operatorname{\mathcal{D}iv}_\mathbb{Z} X$ is a free group generated by prime b-divisors, nor the prime b-divisors form a basis of $\operatorname{\mathcal{D}iv}_\mathbb{Q} X$ and $\operatorname{\mathcal{D}iv}_\mathbb{R} X$. Nonetheless a presentation of a prime b-divisor as linear combination $\mathcal{D}=\sum d_P P$ with $d_P=\operatorname{mult}_P\mathcal{D}$ is used in literature but the sum is typically infinite. Notice also that if $\dim X\geqslant 2$, then $\operatorname{\mathcal{D}iv}_\mathbb{Q} X$ and $\operatorname{\mathcal{D}iv}_\mathbb{R} X$ are not tensor products $\operatorname{\mathcal{D}iv}_\mathbb{Z} X\otimes_\mathbb{Z}\mathbb{Q}$ and $\otimes_Z\mathbb{R}$, respectively. However, we have inclusions:
$$
\begin{equation*}
\operatorname{\mathcal{D}iv}_\mathbb{Z} X\subset\operatorname{\mathcal{D}iv}_\mathbb{Q} X\subset\operatorname{\mathcal{D}iv}_\mathbb{R} X.
\end{equation*}
\notag
$$
Example 1. (1) A prime b-divisor $W$ of $X$ is actually a b-divisor with Usually we apply the Roman typeface to prime b-divisors. (2) Let $\varphi\in k(X)^\times$ be a nonzero rational function of $X$. Then
$$
\begin{equation*}
(\varphi)=\sum \operatorname{mult}_P\varphi\in \operatorname{\mathcal{D}iv}_\mathbb{Z} X
\end{equation*}
\notag
$$
is its b-divisor. For every model $Y$ of $X$, the trace $(\varphi)_Y$ is the usual Weil divisor of $\varphi$ on $Y$ with the usual multiplicity of $\varphi$ in $P$ when $P$ is a prime Weil divisor on $Y$. A principal b-divisor of $X$ is a b-divisor $\mathcal{D}=(\varphi)$ for some $\varphi\in k(X)^\times$. The principal divisors of $X$ form a subgroup that defines the linear equivalence on b-divisors and b-$\mathbb{Q}$-, b-$\mathbb{R}$-divisors of $X$. Every principal b-divisor $\mathcal{D}$ of $X$ is a birational invariant of $X$, in particular, $\mathcal{D}$ is a principal b-divisor of every model $Y$ of $X$. (3) Let $\omega$ be a nonzero rational differential form on $X$ of the top degree. Then
$$
\begin{equation*}
(\omega)=\sum \operatorname{mult}_P\omega\in \operatorname{\mathcal{D}iv}_\mathbb{Z} X
\end{equation*}
\notag
$$
is its b-divisor. For every model $Y$ of $X$, the trace $(\omega)_Y$ is the usual canonical Weil divisor $K_Y$ of $\omega$ on $Y$. The multiplicity $\operatorname{mult}_P \omega$ of $\omega$ at $P$ is well-defined when $P$ is a prime Weil divisor on $Y$ because $Y$ is nonsingular at the generic point of $P$. A canonical b-divisor of $X$ is a b-divisor $\mathbb{K}=(\omega)$ for some form $\omega$ as above. A canonical divisor $\mathbb{K}$ of $X$ is defined up to a linear equivalence. Usually a canonical divisor of $X$ we denote by $\mathbb{K}_X$. To avoid a confusion with the trace $(\mathbb{K}_X)_Y=(\omega)_Y$ of $\mathbb{K}_X$ on a model $Y$ of $X$ we denote the trace by $K_Y$ as a canonical divisor of $Y$. Every canonical b-divisor $\mathbb{K}$ of $X$ is a birational invariant of $X$, in particular, $\mathbb{K}$ is a canonical b-divisor of every model $Y$ of $X$: $\mathbb{K}=\mathbb{K}_Y$. (4) Let $X$ be an algebraic curve, that is, $\dim X=1$. Then every model $Y$ of $X$ is a normalization of $X$ and the b-divisors of $X$ are the usual divisors on $Y$. (5) Let $Y$ be a model of $X$ and $C$ be a Cartier divisor on $Y$. Then there exists a unique b-divisor $\overline{C}$ of $X$ such that, for every model $Y'$ of $X$ over $Y$, where $g\colon Y'\to Y$ is the canonical birational morphism. Here and on every other normal variety (or space) we identify the Cartier divisors with Weil ones. The b-divisor $\overline{C}$ is called the Cartier closure of $C$. Similarly, we can define the Cartier closure $\overline{C}$ for a $\mathbb{Q}$- and $\mathbb{R}$-Cartier divisor $C$ on $Y$: $\overline{C}\in\operatorname{\mathcal{D}iv}_\mathbb{Q} X$ and $\operatorname{\mathcal{D}iv}_\mathbb{R} X$, respectively.A b-divisor $\mathcal{D}$ of $X$ is called Cartier if it is a Cartier closure: $\mathcal{D}=\overline{C}$ for some model $Y$ of $X$. Actually a Cartier divisor $C$ on $Y$ in this case is unique: $C=\mathcal{D}_Y$. We say also that the Cartier b-divisor is stable or Cartier stable over $Y$. We also can attribute many properties of Cartier divisors to Cartier b-divisors. For example, we say that a b-divisor $\mathcal{D}$ is nef or b-nef if $\mathcal{D}=\overline{C}$, where $C$ is a nef divisor on a model $Y$ of $X$. Notice that the nef property of $C=\mathcal{D}_Y$ is independent on a model $Y$ of $X$ over which $\mathcal{D}$ is Cartier stable. For instance, every principal b-divisor of $X$ is Cartier, nef, moreover, numerically trivial and stable over every model $Y$ of $X$. Similar notions apply to b-$\mathbb{Q}$- and b-$\mathbb{R}$-divisors. For example, an NQC b-$\mathbb{R}$-divisor $\mathcal{D}$ of $X$ is a b-$\mathbb{R}$-divisor such that where the sum is finite, every $r_i\in\mathbb{R}$, $r_i\geqslant 0$, and every $\mathcal{D}_i$ is a nef b-$\mathbb{Q}$- or b-divisor of $X$.Usually we apply the mathcal typeface to general b-$\mathbb{R}$-divisors, to $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor or to b-$\mathbb{R}$-divisors expected to be $\mathbb{R}$-Cartier. (6) Let $Y$ be a model of $X$ and $D$ be an $\mathbb{R}$-divisor on $Y$ such that $(Y,D)$ is a log pair, that is, $K_Y+D$ is $\mathbb{R}$-Cartier. Then there exists a unique b-$\mathbb{R}$-divisor $\mathbb{D}$ of $X$ such that, where $\mathbb{K}$ is a canonical b-divisor of $X$ and $K_Y=(\mathbb{K})_Y$ is its trace on $Y$ (see Example (3) above). The b-$\mathbb{R}$-divisor $\mathbb{D}=\mathbb{B}(Y,D)$ is called the codiscrepancy of $(Y,D)$. Actually $\mathbb{D}=\mathbb{B}(Y,D)$ depends only on $Y$ and $D$ but independent of a choice of $\mathbb{K}$. Notice also that $\mathbb{B}(Y,D)=-\mathbb{A}(X,D)$, where $\mathbb{A}(Y,D)$ is the discrepancy b-$\mathbb{R}$-divisor of $(Y,D)$: for every prime b-divisor $P$ of $X$ or of $Y$, $\operatorname{mult}_P\mathbb{A}(Y,D)$ is the discrepancy of $(Y,D)$ at $P$. Thus we can measure singularities of the pair $(Y,D)$ in terms of the b-$\mathbb{R}$-divisors $\mathbb{B}(Y,D)$ and $\mathbb{A}(Y,D)$. By definition $(Y,D)$ is lc if and only if every multiplicity of $\mathbb{D}$ is $\leqslant 1$ or of $\mathbb{A}=-\mathbb{D}$ is $\geqslant -1$. More generally, a pair $(X,\mathcal{D})$ with an arbitrary b-$\mathbb{R}$-divisor is lc if and only if every multiplicity of $\mathcal{D}$ is $\leqslant 1$ or of $-\mathcal{D}$ is $\geqslant -1$.If $D$ is rational, that is, a $\mathbb{Q}$-divisor on $Y$, then $\mathbb{B}(Y,D)$ and $\mathbb{A}(Y,D)$ are rational too, that is, b-$\mathbb{Q}$-divisors of $X$. If $Y'$ is another model of $X$ and $D'$ is an $\mathbb{R}$-divisor on $Y'$ such that $(Y',D')$ is also a log pair, then by definition $(Y',D')$ is a crepant pair of $(Y,D)$ if and only if $\mathbb{D}'=\mathbb{B}(Y',D')=\mathbb{B}(D,Y)=\mathbb{D}$. Recall for this that two log pairs $(X,D),(X',D')$ are called crepant if they have the same discrepancies: $\mathbb{A}(X,D)=\mathbb{A}(X',D')$. The b-$\mathbb{R}$-divisors $\mathbb{K},\mathbb{D},\mathbb{B}(Y,D),\mathbb{A}(Y,D)$ are not $\mathbb{R}$-Cartier if $\dim X\geqslant 2$. Their behaviors are related by a Cartier b-$\mathbb{R}$-divisor $\overline{K_Y+D}$. So, we denote those b-$\mathbb{R}$-divisors or ones with expected similar behavior by the mathbb typeface. Definition 1. We say that a b-$\mathbb{R}$-divisor $\mathbb{D}$ of $X$ satisfies BP, the (sub)boundary property if it is a codiscrepancy: $\mathbb{D}=\mathbb{B}(Y,D)$ for some log pair $(Y,D)$ with a model $Y$ of $X$. Actually an $\mathbb{R}$-divisor $D$ on $Y$ in this case is unique: $D=\mathbb{D}_Y$. We say also that the b-$\mathbb{R}$-divisor $\mathbb{D}$ is stable or BP stable over such $Y$. By definition $\mathbb{D}$ satisfies BP (and BP stable over $Y$) if and only if $\mathbb{K}+\mathbb{D}$ is $\mathbb{R}$-Cartier (and, respectively, $\mathbb{R}$-Cartier stable over $Y$). Notice that the BP stable property of $\mathbb{D}$ is independent of a crepant pair of $(Y,D)$ where $Y$ is a model of $X$, $D=\mathcal{D}_Y$ and $\mathcal{D}$ is BP stable over $Y$. Let $f\colon X\to Z$ be a proper surjective morphism of algebraic varieties (or spaces). Then we can define mappings $f_\circ$, $f^\circ$, b-divisoral analogs of mappings $f_*$, $f^*$ for divisors. Recall that a prime Weil divisor $W$ of $X$ is called horizontal over $Z$ or with respect to $f$ if $f(W)=Z$. Similarly, we can define horizontal prime b-divisors because to be horizontal is a birational property. More precisely, let $X\dashrightarrow X'$, $Z\dashrightarrow Z'$ be birational isomorphisms. This gives a unique dominant rational morphism $f'\colon X'\dashrightarrow Y'$ in the commutative diagram which is called birationally equivalent to $f$. We say that $f'$ is a model of $f$ if additionally $f'$ is regular and $X'$, $Z'$ are models of $X$ and $Z$, respectively. In the last case $f'$ is also proper surjective. A prime b-divisor $W$ of $X$ is called horizontal over $Z$ or with respect to $f$ if there exists a birationally equivalent to $f$ (possibly rational) morphism $f'\colon X'\to Z'$ such that $W$ is a prime Weil divisor on $X'$ dominant over $Z'$, that is, $f'W$ is dense in $Z'$. If additionally $f'$ is a model of $f$, then $f'W=Z'$. Notice that every horizontal prime b-divisors of $X$ over $Z$ is nonexeptional on $X$ and dominant over $Z$, that is, its rational image is well-defined and its closure is a Weil divisor on $X$ which dominates $Z$. All other prime b-divisors of $X$ are called vertical over $Z$ or with respect to $f$. The image $f(W)$ of a vertical prime Weil divisor $W$ of $X$ over $Z$ is a proper under inclusion subvariety of $Z$ but is not necessarily a prime Weil divisor. So, the divisorial image $f_*W$ is not defined even for all vertical $W$. The image $f_*W$ is not a prime Weil divisor and is not defined as a divisor or $=0$ exactly when $W$ is truly exceptional [5], Definition 3.2, that is, vertical and $f(W)$ has the codimension $\geqslant 2$ in $Z$. However, we can define a prime b-divisor $f_\circ W$ of $Z$ for every vertical prime b-divisor $W$ of $X$. Indeed, there exists a model $f'\colon X'\to Z'$ of $f$ such that $W$ is a prime divisor on $X'$ and its image $f'W$ is a prime divisor on $Z'$ [6], § 7.2. The image as a prime b-divisor of $Z$ is independent of $f'$.Let $(X/Z,D)$ be a pair with an $\mathbb{R}$-divisor $D$ on $X$. We say that another pair $(X'/Z',D')$ is a model of $X/Z$ if $X'/Z'$ is a model of $X/Z$ and $(X',D')$ is crepant to $(X,D)$, that is, $D'=\mathbb{D}_{X'}$. The usual pull-back $f^*C$ is defined only on $\mathbb{R}$-Cartier $\mathbb{R}$-divisors. However, its analog, an $\mathbb{R}$-linear homomorphisms
$$
\begin{equation*}
f^\circ\colon \operatorname{\mathcal{D}iv}_\mathbb{R} Z\to \operatorname{\mathcal{D}iv}_\mathbb{R} X,
\end{equation*}
\notag
$$
is defined for all b-$\mathbb{R}$-divisors. It induces, respectively, a homomorphism of Abelian group on b-divisors and a $\mathbb{Q}$-linear homomorphism on $\mathbb{Q}$-divisors. Indeed, take $\mathcal{D}\in\operatorname{\mathcal{D}iv}_\mathbb{R} Z$ and a prime b-divisor $P$ of $X$. Put
$$
\begin{equation*}
\operatorname{mult}_Pf^\circ\mathcal{D}= \begin{cases} (\operatorname{mult}_{f_\circ P}\mathcal{D})\operatorname{mult}_Pf &\text{if } P \text{ is vertical over } Z, \\ 0 &\text{otherwise}, \end{cases}
\end{equation*}
\notag
$$
where $\operatorname{mult}_P f=\operatorname{mult}_P f'^*(f_\circ P)$ and $f'\colon X'\to Z'$ is such a model of $f$ that $P$, $f'P=f_\circ P$ are prime divisors on $X'$ and $Z'$, respectively. Since the generic point of $f'P$ has codimension $1$ in $Z'$ and is nonsingular in $Z'$, $f'P$ is Cartier near it and $f'^*(f_\circ P)=f'^*(f'P)$ is a well-defined Cartier divisor near the generic point of $P$. Thus $\operatorname{mult}_P f'^*(f_\circ P)$ is well-defined. The multiplicity $\operatorname{mult}_P f$ independent of the model $f'$. The $\mathbb{R}$-linear homomorphism $f^\circ$ is a natural and unique extension on the b-$\mathbb{R}$-divisors of the divisorial pull-backs $f'^\circ$ (see [7], § 2.4) on $\mathbb{R}$-divisors
$$
\begin{equation*}
f'^\circ\colon \operatorname{Div}_\mathbb{R} Z'\to \operatorname{Div}_\mathbb{R} X'.
\end{equation*}
\notag
$$
Each latter $\mathbb{R}$-linear homomorphism of the Weil $\mathbb{R}$-divisors on $Z'$ is the unique $\mathbb{R}$-linear extension from the prime Weil divisors on $Z'$. In its turn, for a prime Weil divisor $W$ on $Z'$, the $\mathbb{R}$-divisor $f'^\circ W$ is the trace on $X'$ of the b-divisor $f^\circ W$ and is $\stackrel{\mathrm{def}}{=} f'^*W$ over the generic point of $W$, that is, is equal to $f'^*W$ for prime Weil divisors of $X'$ over the generic point of $W$ and $0$ outside. Thus, for every $\mathcal{D}\in \operatorname{\mathcal{D}iv}_\mathbb{R} Z$, $(f^\circ \mathcal{D})_{X'}=f'^\circ(\mathcal{D}_{Z'})$ holds in codimension $1$ over $Z'$. More precisely, the equality holds over the largest open subset $U\subseteq Z'$ such that the fibers of $f'$ over $U$ are of the same dimension $\dim X'-\dim Z'=\dim X-\dim Z$. In other words, the difference $f'^\circ(\mathcal{D}_{Z'})-(f^\circ \mathcal{D})_{X'}$ is truly exceptional over $Z'$. The truly exceptional prime divisors of $X'$ over $Z'$ are lying over the closed subset $Z'\setminus U$ in $Z'$ of the codimension $\geqslant 2$. In general, $(f^\circ \mathcal{D})_{X'}=f'^\circ(\mathcal{D}_{Z'})$ does not hold everywhere over $Z'$ even for a Cartier b-$\mathbb{R}$-divisor $\mathcal{D}$ of $Z$, e. g., on prime truly exceptional divisors. The divisorial pull-back $f'^\circ (\mathcal{D}_{Z'})$ and its b-divisorial analog (extension) $f^\circ\mathcal{D}$ are vertical, that is, their horizontal parts are zero. On the $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisors the pull-back $f^\circ$ coincides with the commonly known pull-back $f^*$ (see [5], § 7.2): Equivalently, for every $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor $\mathcal{D}$ of $Z$, $f^\circ\mathcal{D}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor of $X$ and, for every model $f'$ of $f$ such that $\mathcal{D}$ is Cartier stable over $Z'$, $f^\circ\mathcal{D}$ is Cartier stable over $X'$ and Indeed, both b-$\mathbb{R}$-divisors are vertical. Thus it is enough to compare their vertical multiplicities. This can be done for every vertical prime b-divisor $P$ such that $f'P=W$ is a prime Weil divisor on $Z'$ for some model $f'$ of $f$ with $\mathcal{D}$ Cartier stable over $Z'$.The divisorial part of adjunction is one of the central concepts of modern birational geometry. It plays the central role in our paper too. There are two equivalent approaches to define it: due to Kawamata [8] and due to Ambro [9]. We use the Ambro approach and define the multiplicities of $\mathbb{D}_{\mathrm{div}}$ in terms of lc thresholds. Definition 2. Let $(X/Z,D)$ be a pair under GLC where $f\colon X\to Z$ is a surjective (in codimension $1$ over $Z$) morphism of normal algebraic varieties (or spaces). Then the divisorial part of adjunction or discriminant of $(X/Z,D)$ is the $\mathbb{R}$-divisor on $Z$, where the sum runs over prime divisors $P$ on $Z$ and $c_P$ is the lc threshold of $(X,D)$ with respect to $f^\circ P$ over the generic point of $P$, that is, Notice that $f^\circ P=f^*P$ over the generic point of $P$ and $D_{\mathrm{div}}$ is actually an $\mathbb{R}$-divisor. Under GLC every $c_W$ is a real number and $\sup$ can be replaced by $\max$ because $(X,D+f^\circ P)$ has a log resolution over the generic point of $P$. Supposes additionally that $X/Z$ is proper and $(X,D)$ is a log pair. Then we can define the b-divisorial version [7], § 7.1 and Remark 7.7: a b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ of $Z$ is the divisorial part of adjunction or discriminant of $(X/Z,D)$ if, for every model $(X'/Z',D')$ of $(X/Z,D)$, the trace of $\mathbb{D}_{\mathrm{div}}$ on $Z'$ is the divisorial part of adjunction of $(X'/Z',D')$.Under assumptions of the definition $\mathbb{D}_{\mathrm{div}}$ exists and is unique. The moduli part of adjunction is another central concept of modern birational geometry. Usually it is considered for log trivial fibrations [7], Construction 7.5. Here we use its upper more general version introduced in [3]. Definition 3. Let $(X/Z,D)$ be a log pair with a proper surjective morphism $f\colon X\to Z$ and under GLC. Then an upper moduli part of adjunction $\mathcal{D}^{\mathrm{mod}}$ of $(X/Z,D)$ is easily defined as a b-$\mathbb{R}$-divisor where $\mathbb{D}=\mathbb{B}(X,D)$ is the codiscrepancy of $(X,D)$ and $\mathbb{K}=\mathbb{K}_X,\mathbb{K}_Z$ are canonical b-divisors of $X$ and of $Z$, respectively. As a canonical b-divisor $\mathbb{K}$ the moduli part $\mathcal{D}^{\mathrm{mod}}$ is defined up to a linear equivalence on $X$. So, the log adjunction holds:
$$
\begin{equation*}
\mathbb{K}+\mathbb{D}=f^\circ(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}})+\mathcal{D}^{\mathrm{mod}}.
\end{equation*}
\notag
$$
Since $\mathcal{D}^{\mathrm{mod}}$ is not unique and defined up to a linear equivalence $\sim$, the last formula holds with $=$ only for an appropriate choice of $\mathbb{K}$ and with $\sim$ in general. The mathbb typeface vs the mathcal ones denotes very special behavior of a codiscrepancy and (anti)similar for a canonical b-divisor (see Examples 1, (5), (6)). All other b-$\mathbb{R}$-divisors, except for, the prime b-divisors but including, b-$\mathbb{R}$-Cartier ones will be denoted by the mathcal capital letters. On every model $(X'/Z,D')$ of $(X/Z,D)$,
$$
\begin{equation*}
M'=(\mathcal{D}^{\mathrm{mod}})_{X'}=D^{\prime \mathrm{mod}}\stackrel{\mathrm{def}}{=} (\mathcal{D}^{\prime \mathrm{mod}})_{X'}= K_{X'}+D' -f^\circ(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}})_{X'}
\end{equation*}
\notag
$$
holds, where $\mathcal{D}^{\prime \mathrm{mod}}=\mathcal{D}^{\mathrm{mod}}$ is also the moduli part of $(X'/Z',D')$, $D'=\mathbb{D}_{X'}$ is the trace of $\mathbb{D}$ on $X'$ and is the crepant divisor to $D$ on $X'$. Moreover,
$$
\begin{equation*}
M'=K_{X'}+D'-f'^\circ(K_Z+D_{\mathrm{div}}')= K_{X'}+D'-f'^*(K_{Z'}+D_{\mathrm{div}}')
\end{equation*}
\notag
$$
holds in codimension $1$ over $Z'$, where $f'\colon X'\to Z'$ is the model of $f$. This allows to calculate the moduli part $M'$ on $X'$ immediately in codimension $1$ over $Z'$. In general, for truly exceptional divisors with respect to $f'$, none of pull-backs $f'^\circ$, $f'^*$ give an exact formula everywhere over $Z'$ unless, for $f^*$, $\mathbb{D}_{\mathrm{div}}'=\mathbb{D}_{\mathrm{div}}$ is BP stable over $Z'$ or $\overline{M'}=\mathcal{D}^{\mathrm{mod}}$ is Cartier stable over $X'$ (see Proposition 2 below). However, subsequent blowups of $X'$ and of the base $Z'$ allows to determine the moduli part $\mathcal{D}^{\mathrm{mod}}$. That is the meaning of b-($\mathbb{R}$-)divisors. If there exists an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor $\mathcal{L}$ on $Z$ such that $\mathbb{K}+\mathbb{D}\sim_\mathbb{R} f^*\mathcal{L}$, then a descent of the moduli part $\mathcal{D}^{\mathrm{mod}}$ on the base $Z$ is well-defined. It is a moduli part on base:
$$
\begin{equation*}
\mathcal{D}_{\mathrm{mod}}\stackrel{\mathrm{def}}{=}\mathcal{L}-\mathbb{K}_Z-\mathbb{D}_{\mathrm{div}}.
\end{equation*}
\notag
$$
So,
$$
\begin{equation*}
f^\circ\mathcal{D}_{\mathrm{mod}}=f^*\mathcal{L}-f^\circ(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}) \sim_\mathbb{R} \mathbb{K}+\mathbb{D}-f^\circ(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}) =\mathcal{D}^{\mathrm{mod}},
\end{equation*}
\notag
$$
that gives log adjunction
$$
\begin{equation*}
\mathbb{K}+\mathbb{D}\sim_\mathbb{R} f^*(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}+\mathcal{D}_{\mathrm{mod}}).
\end{equation*}
\notag
$$
The last equivalence also known as the canonical class formula. Of course, the pull-back $f^*\mathcal{L}$ in the definition of $\mathcal{D}_{\mathrm{mod}}$ can be replaced on more general $f^\circ\mathcal{L}$. However, this does not give anything new. Indeed, if $f^\circ\mathcal{L}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor, then $\mathcal{L}$ is also an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor and $f^\circ\mathcal{L}=f^*\mathcal{L}$. Additionally, according to our assumptions, the b-$\mathbb{R}$-divisor $\mathbb{K}+\mathbb{D}$ is $\mathbb{R}$-Cartier and the $\mathbb{R}$-linear equivalence preserves the last property. (The $\mathbb{R}$-Cartier property is preserved also for the numerical equivalence $\equiv$ over $Z$ that allows to define a numerical version of log adjunction.) Proposition 1. The moduli part $\mathcal{D}_{\mathrm{mod}}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor if and only if the divisorial part $\mathbb{D}_{\mathrm{div}}$ satisfies BP. Proof. By definition $\mathcal{D}_{\mathrm{mod}}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor if and only if $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ is $\mathbb{R}$-Cartier. The last property is equivalent to BP for the b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$. $\Box$ Unlike $\mathcal{D}^{\mathrm{mod}}$, the moduli part $\mathcal{D}_{\mathrm{mod}}$ on the base is defined up to an $\mathbb{R}$-linear equivalence. However, $\mathcal{D}_{\mathrm{mod}}$ is defined up to a $\mathbb{Q}$-linear equivalence if $\mathbb{K}+\mathbb{D}\sim_\mathbb{Q} f^*\mathcal{L}$ for an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor $\mathcal{L}$ (possibly, with nonrational multiplicities). More precisely, $\mathcal{D}_{\mathrm{mod}}$ is defined up to an $n$-linear equivalence if $\mathbb{K}+\mathbb{D}\sim_n f^*\mathcal{L}$. Respectively, in the log adjunction, the equivalence $\sim_\mathbb{R}$ can be replaced by $\sim_\mathbb{Q}$ or, more precisely, by $\sim_n$:
$$
\begin{equation*}
\mathbb{K}+\mathbb{D}\sim_n f^*(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}+\mathcal{D}_{\mathrm{mod}}).
\end{equation*}
\notag
$$
In general, according to Examples 2, (5), (6), and Proposition 2 b-$\mathbb{R}$-divisors $\mathcal{D}^{\mathrm{mod}}$, $\mathcal{D}_{\mathrm{mod}}$ are not $\mathbb{R}$-Cartier. § 3. GeneralitiesGeneral properties of divisorial part of adjunction (cf. [7], Lemma 7.4)Let $(X/Z,D)$ be a log pair as in Definition 2 and $W$ be a prime divisor on $Z$. The $\mathbb{R}$-divisor $D_{\mathrm{div}}$ and b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ satisfy the following properties: (1) birationality: $d_W=\operatorname{mult}_W D_{\mathrm{div}}=\operatorname{mult}_W\mathbb{D}_{\mathrm{div}}$ is independent of a model $(X'/Z,D')$ of $(X/Z,D)$ and $\mathbb{D}_{\mathrm{div}}$ is also independent of a model $(X'/Z',D')$ of $(X/Z,D)$; (2) semiadditivity: for any $\mathbb{R}$-Cartier divisor $\Delta$ on $Z$ the log pair $(X/Z,D')$ with $D'=D+f^*\Delta$ satisfies GLC and equalities $D_{\mathrm{div}}'=D_{\mathrm{div}}+\Delta,\mathbb{D}_{\mathrm{div}}' =\mathbb{D}_{\mathrm{div}}+\overline{\Delta}$ hold; (3) $(X,D)$ is lc (respectively, klt) over the generic point of $W$ if and only if $d_W\leqslant 1$ (respectively, $<1$); (4) effectiveness: if $D$ is effective over the generic point of $W$, then $d_W\geqslant 0$; so, $D_{\mathrm{div}}\geqslant 0$ if $D\geqslant 0$; (5) rationality: if $D$ is a $\mathbb{Q}$-divisor, then $D_{\mathrm{div}},\mathbb{D}_{\mathrm{div}}$ are, respectively, $\mathbb{Q}$-, b-$\mathbb{Q}$-divisors; (6) boundary: if $D$ is an $\mathbb{R}$- (respectively, $\mathbb{Q}$-) boundary, then $D_{\mathrm{div}}$ is an $\mathbb{R}$- (respectively, $\mathbb{Q}$-) boundary; a similar statement holds for subboundaries (cf. (3) above). Notice that in (3) the relative klt property over the generic point of $W$ means the klt property in prime b-divisors of $X$ with the center $W$. Corollary 1. Let $(X/Z,D)$ be a log pair under GLC. Suppose that BP holds for the divisorial part of adjunction $\mathbb{D}_{\mathrm{div}}$. Then the divisorial part of adjunction exactly preserves lc singularities: (1) $(X,D)$ is lc if and only if so does $(Z,D_{\mathrm{div}})$; (2) $(X,D)$ is klt over $Z$ if and only if so does $(Z,D_{\mathrm{div}})$. We can omit the BP assumption on $\mathbb{D}_{\mathrm{div}}$. Then lc and klt properties on $Z$ are determined directly by the b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ (see Example 1, (6)). Proof. The last phrase of the statement means that the pair $(Z,\mathbb{D}_{\mathrm{div}})$ is lc, klt, if, respectively, $\operatorname{mult}_P\mathbb{D}_{\mathrm{div}}\leqslant,< 1$ for every prime b-divisor $P$ of $Z$; in particular, $\mathbb{D}_{\mathrm{div}}$ is a b-subboundary. Respectively, the klt property over $Z$ means: $\operatorname{mult}_P\mathbb{D}<1$ for every prime b-divisor $P$ of $X$, vertical over $Z$, where $\mathbb{D}=\mathbb{B}(X,D)$. So, if $\mathbb{D}_{\mathrm{div}}$ satisfies BP, stable over $Z$, then the lc property holds for $(Z,D_Z)$ with the trace $D_Z=(\mathbb{D}_{\mathrm{div}})_Z$ and the klt property holds for $(X,D)$ over $Z$.
The proof follows immediately from General property (3). $\Box$ Proposition 2. b-$\mathbb{R}$-Divisor $\mathbb{D}_{\mathrm{div}}$ satisfies BP, or, equivalently, $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor if and only if the moduli part $\mathcal{M}$ of $(X/Z,D)$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor. More precisely, let $(X'/Z',D')$ be a model of $f$ with $f'\colon X'\to Z'$. In this situation if $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ is $\mathbb{R}$-Cartier stable over $Z'$, then $\mathcal{M}$ is $\mathbb{R}$-Cartier stable over $X'$, and $\mathcal{M}=\overline{M'}$, where Conversely, if $\mathcal{M}$ is $\mathbb{R}$-Cartier stable over the model $X'$ and additionally $f'$ has the equidimensional fibers, then BP holds for $\mathbb{D}_{\mathrm{div}}$ with stability over $Z'$.In the proposition we have two kinds of stabilities for models, e. g., of $X$: the BP one for $\mathbb{D}$ and the Cartier stability for $\mathbb{K}+\mathbb{D}$ and $\mathcal{M}$. Notice that either of stabilities over $X'$ implies the same over every model of $X$ over $X'$. In particular, each stability holds over every sufficiently high model of $X$ if the stability holds over some model. So, by Hironaka if a proper morphism $f'\colon X'\to Z'$ is a model of $f$, then replacing $X'$ by a higher model of $X$ we can suppose that $(X'/Z',D')$ with $D'=\mathbb{D}_{X'}$ is a model of $(X/Z,D)$ and $\mathbb{D}$ satisfies BP with stability over $X'$. Equivalently, $\mathbb{K}+\mathbb{D}$ is Cartier stable over $X'$. Proof of Proposition 2. According to the last remark, required $f'$ exists for every model $Z'$ of $Z$. For the converse statement, the Raynaud–Gruson flattening [10], I.5.2.2, allows to construct $f'$ with equidimensional fibers for some sufficiently high models $Z'$ over $Z$.
If $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ is $\mathbb{R}$-Cartier, stable over $Z'$, then immediately by definition and our assumptions
$$
\begin{equation*}
\begin{aligned} \, \mathcal{M} &=\mathbb{K}+\mathbb{D}-f^\circ(\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}})= \overline{K+D}-f^*(\overline{K_{Z'}+D_{\mathrm{div}}'}) \\ &=\overline{K_{X'}+D'}-\overline{f'^*(K_{Z'}+D_{\mathrm{div}}')}= \overline{K_{X'}+D'-f'^*(K_{Z'}+D_{\mathrm{div}}')}. \end{aligned}
\end{equation*}
\notag
$$
Conversely, let $\mathcal{M}$ be $\mathbb{R}$-Cartier stable over $X'$, and $f'$ be with equidimensional fibers. Then the $\mathbb{R}$-Cartier property of the b-$\mathbb{R}$-divisor $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ and its stability over $Z'$ follows from the following two statements.
For a proper morphism $f\colon X\to Z$ of normal varieties, with equidimensional fibers, the equality of pull-backs $f^\circ=f^*$ holds on $\mathbb{R}$-Cartier divisors, wherein $D$ on $Z$ is $\mathbb{R}$-Cartier if and only if $f^\circ D$ is $\mathbb{R}$-Cartier on $X$. General hyperplane sections of $X$ reduce the proof to the case of a finite morphism $f$. Let $f\colon X\to Z$ be a composition of surjective morphisms of normal varieties $g\colon X\to Y$ and $h\colon Y\to Z$, and $D$, $E$ be $\mathbb{R}$-divisors on $Y$ and $Z$, respectively, such that $E$ is $\mathbb{R}$-Cartier and $g^\circ D=f^*E$ in codimension $1$ over $Y$. Then $D=h^*E$, in particular, $D$ is also $\mathbb{R}$-Cartier and $g^*D=f^*E$. Reduce to the case with $E=0$ and $D:=D-h^*E$. Then $g^\circ D=f^*0=0$ in codimension $1$ over $Y$ and $D=0$ too, that is, $D-h^*E=0$. Indeed, $\mathbb{R}$-divisors $D-h^*E$ on $Y$ and $0$ on $Z$ satisfy the above assumptions: $0$ is $\mathbb{R}$-Cartier and $g^\circ(D-h^*E)=f^*E-g^*(h^*E)=0=f^*0$ in codimension $1$ over $Y$. $\Box$ Example 2. (1) Let $(X/Z,D)$ be a log pair with a birational contraction $f\colon X\to Z$, that is, $f$ is a proper birational map of normal varieties. Then $D_{\mathrm{div}}=f_*D$ and $D^{\mathrm{mod}}=0$. Moreover, $\mathcal{D}^{\mathrm{mod}}=0$ and is a b-divisor and the divisorial part $\mathbb{D}_{\mathrm{div}}$ satisfies BP. The naive moduli part $K+D-f^*(K_Z+D)$ is the discrepancy $\mathbb{R}$-divisor of $(Z,D)$ on $X$, if $(Z,D)$ is a log pair and $f_*D=D$ as b-$\mathbb{R}$-divisors, equivalently, the $\mathbb{R}$-divisor $D$ does not have exceptional components and is a proper birational preimage of $f_*D$ with respect to $f$. This situation corresponds to the definition of discrepancies for the pair $(Z,D)$ and the discrepancies are determined for prime divisors on the blowup $X/Z$. In general, however, more natural to present the codiscrepancy b-$\mathbb{R}$-divisor $\mathbb{D}=\mathbb{B}(X,D)=-\mathbb{A}(X,D)$ as the divisorial part of adjunction for $(X/Z,D)$: In particular, BP for $\mathbb{D}_{\mathrm{div}}$ is stable over $Z$, that is, $\mathbb{K}_Z+\mathbb{D}_{\mathrm{div}}$ is an $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor, stable over $Z$, if $K+D\equiv 0$ over $Z$ and $K_Z+f_*D$ is an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor on $Z$, equivalently, $(X/Z,D)$ is a $0$-pair or $(X,D)$ and $(Z,f_*D)$ are crepant over $Z$. In this situation $\mathbb{D}_{\mathrm{div}}=\mathbb{B}(Z,f_*D)=-\mathbb{A}(Z,f_*D)$. Notice that the birational map $f$ should be proper to define a moduli part of adjunction. Indeed, a birational map $f$ is surjective (at least in codimension $1$) for any base change if and only if the map is proper. (2) Let $(C/C',D)$ be a log pair with a surjective map $f\colon C\to C'$ of normal curves. In this example we can consider also curves over an algebraically closed field $k$ with a positive characteristic assuming that $f$ is separable. For every point $p\in C'$ the geometric fiber $f^*p$ can be identified with the divisor of the fiber $f^*p=\sum m_i p_i$, where the sum runs over all point $p_i$ of the fiber $f^{-1}p$ and $m_i=\operatorname{mult}_{p_i}f$ denotes the multiplicity of $f$ in $p_i$. Denote, respectively, by $r_i$ the ramification index of $f$ in $p_i$. (If the characteristic of $k$ does not divide $m_i$, then $r_i=m_i-1$.) Let $d_i=\operatorname{mult}_{p_i}D$ be the multiplicity of the $\mathbb{R}$-divisor $D$ in $p_i$. Then the multiplicity $d'=\operatorname{mult}_pD_{\mathrm{div}}$ of the divisorial part of adjunction can be determined by the formula In characteristic $0$, if $D$ has the same multiplicities $d_i=d\leqslant 1$ over $p$, the multiplicities of divisorial and moduli parts are
$$
\begin{equation*}
d'=\frac{m-1+d}{m} \quad\text{and}\quad \operatorname{mult}_{p_i}D^{\mathrm{mod}}=(d-1)\biggl(1-\frac{m_i}{m}\biggr),
\end{equation*}
\notag
$$
respectively, where $m=\max\{m_i\}$. Since the moduli part is defined only up to an $\mathbb{R}$-linear equivalence, the last formula is meaningful only for a complete curve. In particular, if $D=0$, then $d'=(m-1)/m$ and is equal to $0$ if and only if all $m_i=m$, for instance: $C/C'$ is a Galois covering. Next Example (3) generalizes the last statement. Notice that $D^{\mathrm{mod}}=0$ also if all $d_i=1$ for $m\geqslant 2$ or all $d_i=d$ for $m=1$ (cf. the maximal property in Proposition-Definition 1). (3) Let $f\colon X\to Z$ be a (finite) Galois covering and $(X/Z,D)$ be a log pair with an invariant $\mathbb{R}$-divisor $D$. Then $\mathcal{D}^{\mathrm{mod}}=0$ by Example (2). Indeed, on one hand, hyperplane sections reduce the determination of the moduli part in prime divisors to a curve case. On the other hand, we can blow up every given prime divisor preserving the assumptions. (4) Let $f\colon X\to Z$ be an unramified double covering of surfaces and $C$ be a nonsingular curve on the base $Z$ which splits on the the covering $X$, that is, $f^*C=C_1+C_2$, where $C_1$, $C_2$ are nonsingular curves on $X$. First consider a log pair $(X/Z,D)$ with a divisor $D=C_1$. Then by Example (2) $D_{\mathrm{div}}=C$, the b-divisor $\mathbb{D}_{\mathrm{div}}=\mathbb{B}(Z,C)$ satisfies BP and stable over $Z$. Remove now a closed point $p\in C_1$ on $X$. Then for the log pair $(X'/Z,D)$ with $X'=X\setminus p$, the map $X'\to Z$ is surjective and surjective for every birational base change. So, the b-divisor $\mathbb{D}_{\mathrm{div}}$ is well-defined but does not satisfies BP. More precisely, $\mathbb{D}_{\mathrm{div}}=\mathbb{B}(Z,0)$ over $f(p)$ and $=\mathbb{B}(Z,C)=\mathbb{B}(Z,0)+\overline{C}$ over the other points of the base $Z$. This is why for log adjunction we usually assume that $f$ is proper. (5) Let $f\colon X\to Z$ be an unramified double covering of surfaces and $C_1$, $C_2$ be nonsingular curves on the base $Z$, which intersect transversally in a single point $p\in Z$ and split on $X$, that is, $f^*C_1=D_1+D_1'$ and $f^*C_2=D_2+D_2'$, where $D_1$, $D_1'$, $D_2$, $D_2'$ are nonsingular curves on $X$. We suppose also that $D_1\cap D_2=D_1'\cap D_2'=\varnothing$. Take a log pair $(X/Z,D)$ with an $\mathbb{R}$-divisor $D=d_1 D_1+d_2 D_2+ D_1'+D_2'$, where $d_1$, $d_2$ are real numbers $<1$ and linearly independent over the rational numbers, that is, the equality $a_1 d_1+a_2 d_2=a$, $a_1,a_2,a\in\mathbb{Q}$, is possible only for $a_1=a_2=a=0$. Then the b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ does not satisfy BP. More precisely, as in Example (4) above the b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ does not satisfy BP over $p\in Z$. Indeed, for the blowup $Z'\to Z$ of the nonsingular point $p$, the multiplicity of $\mathbb{D}_{\mathrm{div}}$ in the exceptional divisor $E$ is $d=\max\{d_1,d_2\}$ and $d<1$ (but $D_{\mathrm{div}}=C_1+C_2$). Denote by $X'=X\otimes_Z Z'$ the corresponding blowup of $X$. On the covering $X'$ of $Z'$, $E$ splits into two exceptional curves of the 1st kind $E_1$, $E_2$, where $E_1$ intersects the proper transform of $D_1$. Then by semiadditivity (2) in General properties [7], Lemma 7.4, the pair $(X/Z,D)$ can be replaced by the log pair $(X'/Z',D')$ with the $\mathbb{R}$-divisor $D'=D+E_1+(1+d_2-d_1)E_2$, where $D$ denotes the proper birational preimage of the $\mathbb{R}$-divisor $D$ on $X'$. We suppose also that $d=d_1> d_2$. The new divisorial part of adjunction as the old one satisfies BP everywhere except for the point $p'=C_1\cap E$, where $C_1$ denotes the proper birational preimage of $C_1$ on $Z'$. The new divisorial part of adjunction stabilizes on same blowups if such exist. The multiplicities $d_1$, $1+d_2-d_1$ of the new pair are linearly independent over the rational numbers. The prove concludes the induction on the number of blowups required to get a stability of BP. The process will newer stops. This contradicts BP. The same argument shows BP for every $\mathbb{Q}$-divisor $D$ (cf. Theorem 3 below). Similarly, one can construct an example with a b-$\mathbb{Q}$-divisor $\mathbb{D}\ne \mathbb{B}(X,D)$ instead of $D$ without BP, the divisorial part of adjunction of which $\mathbb{D}_{\mathrm{div}}$ satisfies BP. (6) Let $\mathfrak{M}_5$ denote the moduli space of stable rational curves with $5$ marked points, $\mathcal{U}_5\to \mathfrak{M}_5$ be the corresponding universally family and $\mathcal{P}_1,\dots,\mathcal{P}_5$ be sections corresponding to the marked points. The family is a smooth three dimensional (nonstandard) conic bundle over a nonsingular surface $\mathfrak{M}_5$. Let $D_1$, $D_1'$ be divisors on $\mathcal{U}_5$ sweeping, respectively, by components $C$, $C'$ of stable curves $C\cup C'$ with points $p_1,p_2,p_3\in C$, $p_4,p_5\in C'$, where $p_i=\mathcal{P}_i\cap(C\cup C')$. Similarly, divisors $D_2'$, $D_2$ on $\mathcal{U}_5$ are sweeping, respectively, by components $C$, $C'$ of stable curves $C\cup C'$ with points $p_1,p_2\in C'$, $p_3,p_4,p_5\in C$. As in Example (5) take a log pair $(\mathcal{U}_5/\mathfrak{M}_5,D)$ with an $\mathbb{R}$-divisor $D=d_1 D_1+d_2 D_2+ D_1'+D_2'$, where $d_1$, $d_2$ are real numbers $<1$ and linearly independent over the rational numbers. Then the divisorial part of adjunction $\mathbb{D}_{\mathrm{div}}$ does not satisfy BP over the (closed) point $p$ of the base $\mathfrak{M}_5$, corresponding to the stable curve $C_1\cup C_2\cup C_3$ with points $p_1,p_2\in C_1$, $p_3\in C_2$, $p_4,p_5\in C_3$. The b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ has the same multiplicities over $p$ as the corresponding b-$\mathbb{R}$-divisor in Example (5). Adding to $D$ sections $\mathcal{P}_i$ with arbitrary multiplicities $\leqslant 1$, one can suppose that $D$ is $\mathbb{R}$-ample over $\mathfrak{M}_5$ with the same b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$. Similarly one can construct an example with a $\mathbb{R}$-divisor $D$ such that $K_{\mathcal{U}_5}+D\sim_\mathbb{R} 0$ over $\mathfrak{M}_5$ and $\mathbb{D}_{\mathrm{div}}$ does not satisfy BP. (According to [7], Theorem 8.1, and Corollary 4 below the horizontal part of $D$ should be not effective, cf. also Example 4 below.) It is sufficient to find such multiplicities $a_1,\dots,a_5\leqslant 1$ that $D:=D+\sum a_i \mathcal{P}_i\equiv 0$ on every irreducible component of the curve over the point $p$. This gives an example over a neighborhood of $p\in\mathfrak{M}_5$. Adding vertical prime divisors with appropriate multiplicities one can find an example over $\mathfrak{M}_5$. More precisely, one can find easily required multiplicities $a_i$, when the multiplicities $d_1$, $d_2$ are close to $1$. In this case one can pick up multiplicities $a_i$ close to $1/2$ for $i\ne 3$ and to $0$ for $i=3$ (actually $a_3<0$). Again for rational $D$ BP holds. A proof of Theorem 3 below in § 4 uses the following fact. Proposition 3 (transitivity of divisorial part of adjunction). Let be a composition of two proper surjective morphisms and $(X/Z,D)$ be a log pair under GLC. Then the pair $(X/Y,D)$ also satisfies GLC. Suppose that the divisorial part of adjunction $\mathbb{D}_Y$ of $(X/Y,D)$ satisfies BP stable over the base $Y$. Then the transitivity holds: $(Y/Z,D_Y)$ is also a log pair under GLC, where $D_Y=(\mathbb{D}_Y)_Y$ the trace of the b-divisor $\mathbb{D}_Y$ on $Y$, and holds, where b-$\mathbb{R}$-divisors $\mathbb{D}_Z$, $(D_Y)_Z$ denote, respectively, divisorial parts of adjunction of pairs $(X/Z,D)$, $(Y/Z,D_Y)$. The usage of the base $Y$ with stability of BP can be omitted. Then, in the transitivity formula, the divisor $D_Y$ should be replaced by the trace $D_{Y'}=(\mathbb{D}_Y)_{Y'}$ on a model $Y'$ of $Y$ over $Z$, over which the stability of BP holds. Proof. If $(X,D)$ is lc over an open subset $U\subset Z$, then the same holds over the open set $g^{-1} U\subset Y$. By surjectivity of $g$ the preimage $g^{-1} U$ is not empty if $U$ is not empty.
The transitivity follows from more precise result – General property (3). It is sufficient to verify the transitivity in each prime b-divisor $W$ of $Z$. By General properties (1), (2) we can suppose that $W$ is a prime divisor on $Z$ and $d_W=\operatorname{mult}_W\mathbb{D}_Z=1$. Then by General property (3), $(X,D)$ is lc but not klt over the generic point of $W$. After blowing up of $X$ and changing of $D$ on its crepant $\mathbb{R}$-divisor, we can suppose that $\operatorname{mult}_V D=1$ for some prime divisor $V$ on $X$ over $W$, that is, $f(V)=W$. We can suppose also that $h(V)$ is a prime divisor on $Y$. Again by General property (3), $d_{f(V)}=\operatorname{mult}_{f(V)} D_Y=1$ and $(Y,D_Y)$ is not klt over the generic point of $W=g(h(V))$. On the other hand, $(Y,D_Y)$ is lc over the generic point of $W$ by Corollary 1 (and the open property of lc). Thus $\operatorname{mult}_W{(D_Y)_Z}=1$ too. $\Box$ In the proof of Theorem 3 we use also the following construction. Example 3 (crepant pull-back). Let $(Z,D_Z)$ be a log pair and $f\colon X\to Z$ be a morphism of normal varieties, separable, finite and surjective over the generic point (separable alteration). Then there exits a natural and unique divisor $D$ on $X$, converting the variety $X$ into a log pair $(X,D)$ crepant to $(Z,D_Z)$, that is, for every canonical divisor $K_Z=(\omega_Z)$ on $Z$, where $\omega_Z$ is a nonzero rational differential form of the top degree on $Z$, holds, where $K=(f^*\omega)$ is a canonical divisor on $X$. In this case by Example 2, (2), $D_{\mathrm{div}}=D_Z$ and $D^{\mathrm{mod}}=D_{\mathrm{mod}}=0$. If the map $f$ is proper and surjective, then $\mathbb{D}_{\mathrm{div}}=\mathbb{B}(Z,D_Z)$ and $\mathcal{D}_{\mathrm{mod}}=0$. In other words, $\mathbb{D}_{\mathrm{div}}$ satisfies BP and stable over $Z$. Corollary 2. (actually, $=$ for appropriate canonical b-divisors of $X$, $Y$, $Z$), where b-$\mathbb{R}$-divisors $\mathcal{M}_{X/Z}$, $\mathcal{M}_{X/Y}$, $\mathcal{M}_{Y/Z}$ are, respectively, the moduli part of adjunction of $(X/Z,D)$, $(X/Y,D)$, $(Y/Z,\mathbb{D}_Y)$. If $\mathbb{D}_Y$ satisfies BP stable over $Y$, then we can replace $(Y/Z,\mathbb{D}_Y)$ by a usual log pair $(Y/Z,D_Y)$. Otherwise, we use an appropriate model $(Y'/Z,D_{Y'})$ of $(Y/Z,\mathbb{D}_Y)$ (cf. Proposition 3 and Example 1, (6)). Proof of Corollary 2.
$$
\begin{equation*}
\begin{aligned} \, \mathcal{M}_{X/Z} &=\mathbb{K}+\mathbb{D}-f^\circ(g^\circ(\mathbb{K}_Z+\mathbb{D}_Z))= \mathbb{K}+\mathbb{D}-f^\circ(\mathbb{K}_Y+\mathbb{D}_Y-\mathcal{M}_{Y/Z}) \\ &=\mathbb{K}+\mathbb{D}-f^\circ(\mathbb{K}_Y+\mathbb{D}_Y)+f^\circ\mathcal{M}_{Y/Z} =\mathcal{M}_{X/Y}+f^\circ\mathcal{M}_{Y/Z}. \end{aligned}
\end{equation*}
\notag
$$
Since the moduli parts are defined up to a linear equivalence we can choose required canonical b-divisors to have $=$ instead of $\sim$. $\Box$ § 4. Main resultsTheorem 3. Let $(X/Z,D)$ be a log pair with a $\mathbb{Q}$-divisor $D$, proper surjective $X/Z$ and under GLC. Then the divisorial part of adjunction $\mathbb{D}_{\mathrm{div}}$ is well-defined and always satisfies BP. Proof. The b-$\mathbb{R}$-divisor $\mathbb{D}_{\mathrm{div}}$ is well-defined by [7], §§ 7.1, 7.2 (cf. also Definition 2 above). Using the birational nature of $\mathbb{D}_{\mathrm{div}}$, namely, General properties (1) and (5), and by Proposition 3 one can reduce the proof to two cases: morphism $f\colon X\to Z$ is finite or is a family of curves.
First we verify BP for the finite morphisms. In this part of the proof, the assumption, that the base field $k$ has the characteristic $0$, is essentially important. For simplicity one can suppose that $k=\mathbb{C}$ is the field of complex numbers. According to Hironaka after an appropriate blowup of the base and replacing $X$ by an induced normal base change one can assume that $Z$ is nonsingular with reduced ($0$ and $1$ are the only multiplicities of) divisor $\Delta$ with only simple normal crossing and such that $f$ is only ramified and $D$ is supported over $\operatorname{Supp}\Delta$. General hyperplane sections and dimensional induction reduce the problem to a verification of BP over a neighborhood of a finite set of isolated closed points $z\in Z$. Over a sufficiently small neighborhood of every such point $z$ in the classical topology, the covering $f$ is a union of simple branches. Again by the transitivity of Proposition 3 it is sufficient to verify BP in the simple mapping case when the fiber $f^{-1} z=X_z$ consists of a single point, and in the case of unramified map $f$. In the first case, up to an analytic isomorphism, the pair $(X/Z,D)$ is toric, that is, $X\to Z$ is a toric finite map with an invariant divisor $D$ on $X$. The invariant divisor on the base $Z$ is $\Delta$. By General properties (2), (3) one can suppose that the divisorial part of adjunction is reduced: $D_{\mathrm{div}}=\Delta$. Then $D$ is also reduced and BP holds over $Z$. (Actually, in this toric situation, the rationality of $D$ does not matter.) The second case, with an unramified map, is a bit harder. Again by the transitivity of Proposition 3 we consider the case of two sheets: $X=X_1\cup X_2$, $X_1\cap X_2=\varnothing$ and the map $X\to Z$ consists of two isomorphisms $X_1,X_2\to Z$. General properties (2), (3) allow to suppose that $D_{\mathrm{div}}=\Delta$. However, this time we know only that the multiplicities of the divisor $D$ are not exceeding $1$ over $\operatorname{Supp}\Delta$ and are equal to $0$ otherwise. If all multiplicities of $D$ over $\operatorname{Supp} \Delta$ are equal to $1$, BP holds. Otherwise there exists a multiplicity $d_i<1$ over a prime divisor of $\operatorname{Supp}\Delta$. By construction and our assumptions every such multiplicity is rational and they form a finite set $\{d_i\}$. Hence there exists a positive integer $m$ such that all numbers $m d_i$ are integral. The following algorithm allows to find a model of $Z$ over which $\mathbb{D}_{\mathrm{div}}$ is stable and BP holds. Put $d=\min\{d_i\}$. Let $D_i$ be a prime divisor on $X$ over $\operatorname{Supp}\Delta$ with the multiplicity $\operatorname{mult}_{D_i}D= d_i=d$. By construction and our assumptions $d<1$. If $D_i$ intersects on $X$ all prime divisors $D_j$ over $\operatorname{Supp}\Delta$ with multiplicities $d_j<1$ and with the image $f(D_j)$ intersecting $f(D_i)$ on $Z$, then BP holds over a neighborhood of $f(D_i)$ (to verify this one can use Example 2, (2)). (Here as usually a prime divisor means closed irreducible subvariety of codimension $1$, but not only its generic point.) Moreover, the multiplicity $d_i$ can be replaced by $1$, not changing $\mathbb{D}_{\mathrm{div}}$. After finitely many steps either there are no prime divisors $D_i$ over $\operatorname{Supp}\Delta$ with the multiplicity $d$, or there is another prime divisor $D_j$ over $\operatorname{Supp}\Delta$ with multiplicity $d_j<1$, with $D_i\cap D_j=\varnothing$, but $f(D_i)\cap f(D_j)\ne \varnothing$. The last intersection is a nonsingular subvariety of the base $Z$ of codimension $2$. In the first case take new $d=\min\{d_i\}$. The algorithm terminates because $d< 1$ and $m d\in \mathbb{Z}$. In the second case blow up the intersection $f(D_i)\cap f(D_j)$ and, respectively, the covering $X$. Let $X'\to Z'$ be the induced unramified covering with blown up divisors $E_1$, $E_2$, $E$ on $X_1'$, $X_2'$, $Z'$, respectively, where $X_1'$, $X_2'$, $Z'$ are blowups of varieties $X_1$, $X_2$, $Z$. To be more precise suppose that $D_i\subset X_1$. Let $D$, $D_i$, $D_j$ denote proper birational transforms of those divisors on $X'$ and $D'$ denote the divisor on $X'$, corresponding to the crepant transform of the pair $(X,D)$. Then $D_i\subset X_1'$, $D_j\subset X_2'$, and
$$
\begin{equation*}
\operatorname{mult}_{E_1} D'=d\leqslant \operatorname{mult}_{E_2}D'=d_j<1.
\end{equation*}
\notag
$$
Again by Example 2, (2), $\operatorname{mult}_{E}D_{\mathrm{div}}'=\max\{d,d_j\}=d_j$. Now we replace the pair $(X/Z,D)$ by $(X'/Z',D'+(1+d-d_j)E_1+E_2)$ and return to the beginning of the algorithm. The validity of BP this does not change. (The new divisor $\Delta'=\Delta+E$.) We contend that the algorithm stops when $\{d_i\}=\varnothing$ that gives BP with stability over the constructed base. Indeed, if $d_j=d$, then a new prime divisor over $\operatorname{Supp}\Delta$ with $d_i<1$ will not appear and one intersection $f(D_i)\cap f(D_j)\ne \varnothing$ disappear. If $d_j>d$, then exactly one new prime divisor $E_1$ over $\operatorname{Supp}\Delta$ will appear with the multiplicity $1>1+d-d_j>d$. Notice that again $m(1+d-d_j)\in \mathbb{Z}$. Hence a new prime divisor $D_j$ with the multiplicity $d$ will not appear and after finitely many steps $D_i$ intersects all prime divisors $D_j$ over $\operatorname{Supp}\Delta$ with multiplicities $d_j<1$ for which $f(D_i)\cap f(D_j)\ne \varnothing$. In this case, as we did it before we increase the multiplicity $d_i=d$ up to $1$.
In conclusion consider the case with the map $f$ being a family of curves. After an appropriate surjective proper base change $Z'\to Z$ with $\dim Z'=\dim Z$ (alteration) one can suppose that the family is maximally good. More precisely, there exists a commutative diagram such that $f'$ is a projective family of (semi)stable curves with the smooth generic fiber and the preimage of horizontal prime divisors of $\operatorname{Supp} D$ is a union of sections $S_i$ of the family $f'$ (see [11], Theorem 2.4). (Sections $S_i$ are prime divisors too.) Maps $X'\to X$ and $Z'\to Z$ are finite generically. Let $D'$ be the crepant pull-back of Example 3. By construction $D'$ is a $\mathbb{Q}$-divisor. Then by the transitivity of Proposition 3 for the composition, $X'\to X\to Z$ gives the same divisorial part of adjunction for the pair $(X'/Z,D')$ as for $(X,D)\to Z$, that is, gives $\mathbb{D}_{\mathrm{div}}$. On the other hand, the transitivity of Proposition 3 for the composition $X'\to Z'\to Z$ and already established BP for the map $Z'\to Z$, BP for $\mathbb{D}_{\mathrm{div}}$ follows from BP for the divisorial part of adjunction of the pair $(X'/Z',D')$. We contend that BP holds for $(X'/Z',D')$ over $Z'$. By [11], Theorem 2.4, (vii)(b) and the Property 2.2.2, $X'\to X$ is unramified generically over $Z$ or $Z'$. The horizontal part of the divisor $D'$ has the same multiplicities in the sections $S_i$ as in the corresponding prime divisors of $X$ in $D$. In our situation GLC means that all $\operatorname{mult}_{S_i} D'\leqslant 1$ for all sections $S_i$. This follows from GLC for $(X/Z,D)$ by construction and [11], Remark 2.5. Again by [11], Remark 2.5 and Theorem 2.4, after an additional blowup of $Z'$ and a normal base change of $f'$, we can suppose that $X'/Z'$ is toroidal with invariant $\operatorname{Supp} D'$. Thus $D_{\mathrm{div}}'$ can be computed by an lc threshold with a vertical divisorial lc center in $\operatorname{Supp} D'$. More precisely, to determine $D_{\mathrm{div}}'$ one can suppose that the base $Z'$ is a curve. By construction all sections $S_i$ and vertical divisors, including vertical prime components of $\operatorname{Supp} D'$, form a divisor with toroidal crossings. Thus in the determination one can reduce all horizontal prime components of $\operatorname{Supp} D'$ in $S_i$, that is, to replace their multiplicities by $1$. Adding locally (in classical or étale topology) (multi)sections $S_i$ with $\operatorname{mult}_{S_i} D'=0$, preserving mutually disjoint and toroidal property of (multi)sections $S_i$, one can suppose that $S=\sum S_i$ is ample over $Z'$ and is in the reduced part of $D'$. (The ample divisor intersects any vertical divisor.) By Example (2) this reduce the determination of $\mathbb{D}_{\mathrm{div}}'$ to that of the divisorial adjunction of the log pair $(S/Z',D'_S)$, where $S=\bigcup S_i\to Z'$ is a toroidal covering and $D'_S$ is the adjunction of $D'$ on $S$ supported in the invariant divisor. Under the base change the crepant divisor change of $D_S'$ will be the adjunction of the crepant change of $D'$ on a modification of $S'$. Thus this case also follows from already known one for finite (toroidal) morphisms.1 $\Box$Theorem 3 and Examples 2, (5), (6), show that, in general, BP is unstable with respect to multiplicities. However, BP holds as a certain good limit, that is, $\mathbb{D}_{\mathrm{div}}$ is always pseudo-BP. A b-$\mathbb{R}$-divisor $\mathbb{D}$ is pseudo-BP if it is a limit of BP b-$\mathbb{R}$-divisors $\mathbb{D}_i$ where the limit is taken componentwise: for every prime b-divisor $P$:
$$
\begin{equation*}
\operatorname{mult}_P\mathbb{D}=\lim_{i\to\infty}\operatorname{mult}_P\mathbb{D}_i.
\end{equation*}
\notag
$$
Equivalently, $\mathcal{D}^{\mathrm{mod}}$ is always pseudo-$\mathbb{R}$-Cartier. Examples 2, (5), (6) also shows that conditions on vertical component multiplicities of $D$ are important. From the birational point of view over $Z$, this should not be so important that shows the next result. Its proof is also a preparation to the proof of our main construction in Proposition-Definition 1. Theorem 4. Let $(X/Z,D)$ be a log pair under GLC. Then there exists a log pair $(X'/Z',D')$ such that (1) morphisms $X\to Z,X'\to Z'$ are birationally equivalent; (2) the pair $(X'/Z',D')$ is crepant to $(X/Z,D)$ over the generic point of $Z$; (3) $(X'/Z',D')$ satisfies GLC; (4) $\mathbb{D}_{\mathrm{div}}'$ satisfies BP. Hence $\mathbb{D}_{\mathrm{div}}$ satisfies BP over the generic point of $Z$. The last property is not surprising (cf. Proposition-Definition 1 below). Proof of Theorem 4. By Proposition 3, Theorem 3 and Stein factorization we can suppose that $X'\to Z'$ is a contraction and to construct rational $\mathbb{D}_{\mathrm{div}}'$ or, moreover, integral.
We can replace $(X/Z,D)$ by a log pair $(X'/Z',D')$ which satisfies (1)–(3); and the crepant property (3) holds everywhere. Moreover, the morphism $f'\colon X'\to Z'$ is toroidal and the divisor $D'$ does not intersect toroidal embedding, that is, it is supported in the invariant divisor. We can use for this a toroidalization of $X\to Z$ with the closed subset $\operatorname{Supp} D$ [12], Theorem 2.1. Recall that by definition the toroidal embedding is nonsingular on $X'$ and on $Z'$. We contend that if we replace all vertical multiplicities of divisor $D'$ outside of the toroidal embedding by $1$, then BP (4) will hold and is stable over $Z'$. Notice for this that all horizontal prime components of $\operatorname{Supp} D'$ are in the invariant divisor and have multiplicities $\leqslant 1$ in $D'$. Hence, in particular, $D_{\mathrm{div}}'=\Delta$ is the complement to the toroidal embedding on $Z'$. Notice also that the last condition is preserved under the modifications of the pair $(X'/Z',D')$ below. We need to verify that $\mathbb{D}_{\mathrm{div}}'$ is BP stable over $Z'$, that is, for every prime exceptional divisor $W$ of $Z'$, the equality $d_W=\operatorname{mult}_W\mathbb{D}_{\mathrm{div}}'=b_W =\operatorname{mult}_W\mathbb{B}(Z',\Delta)$ holds. We can establish this by induction on $1-b_W$. Since $(Z',\Delta)$ is lc, $b_W\leqslant 1$ holds and by construction multiplicities $b_W$ are integers. Indeed, every $b_W$ is the codiscrepancy of toroidal $(Z',\Delta)$ at $W$ and $K_{Z'}+\Delta$ is Cartier. We will use also induction on dimension of $\operatorname{center}_{Z'}W$. General hyperplane sections of the base allow to reduce the dimension of such a center to $0$, that is, to the case with a closed point $\operatorname{center}_{Z'}W$. Suppose first that $b_W=1$ and use induction on the number of blowups, that is, we can suppose that $W$ is the blowup of a closed point on $Z'$. Then there exists a toroidal blowup of $X'$ with a center over $W$. Actually, we can toroidally blow up both varieties $X'$, $Z'$ simultaneously (use subsequent toroidal resolution of $X'$ [12], Proposition 4.4). As above $D_{\mathrm{div}}'=\Delta$ and $d_W=1$. Suppose now that $b_W\leqslant 0$. If $\operatorname{center}_{Z'}W$ is an lc center of $(Z',\Delta)$, then we blow up it as above. Thus we can suppose that $\operatorname{center}_{Z'}W$ is not an lc center. Then we can add a general hyperplane section $H$ through the center and preserve the toroidal property, including the horizontal part of the divisor $D'$. Replace $D'$ by $D'+f'^{*}H$. This increases $b_W$ and completes induction by General property (2). By the crepant assumption (2) and General property (1), $\mathbb{D}_{\mathrm{div}}=\mathbb{D}_{\mathrm{div}}'$ holds over the generic point of the base. This gives the last statement. $\Box$ In general, the behavior of multiplicities $d_W$ is unpredictable, except for general estimations (see General properties (3)–(6)). However, in one important situation, a good behaviour of multiplicities of divisorial part of adjunction is expected (and already known in some cases). See for details [7], Proposition 9.3, (i), and [6], § 6.8. Here we briefly recall only one main result about hyperstandard multiplicities. Corollary 3. Let $\mathfrak R\subset[0,1]$ be a finite subset of rational numbers and $d$ be a natural number. Then there exists a finite subset of rational numbers $\mathfrak R'\subset [0,1]$ such that, for every $0$-pair $(X/Z,B)$ with $X/Z$ of weak Fano type, $\dim X\leqslant d$ and a boundary $B\in \Phi(\mathfrak R)$, holds. In particular, the divisorial part of adjunction $B_{\mathrm{div}}$ is also a boundary and there exists a real number $\varepsilon>0$ such that every nonzero multiplicity of $B_{\mathrm{div}}$ is $\geqslant\varepsilon$. The number $\varepsilon$ depends only on $\mathfrak R$ and $d$. The same follows for any proper $X/Z$ from Index conjecture [6], Conjecture 2. Proof of Corollary 3. The last statement follows from the dcc property of $\Phi(\mathfrak R')$. $\Box$ With the effective b-semiampleness of the moduli part of adjunction $B_{\mathrm{mod}}$ (see [7], (7.13.3), [6], Conjecture 3), this reduces the birational effectiveness of an Iitaka map (see [13], Conjecture 1.1) to the same problem for a big log canonical divisor (see [14], Theorem 1.3; cf. [13], Theorem 1.3). However, Index problem remains the main missing point to complete the proof. Proposition 4. For a log pair $(X/Z,D)$ under GLC, the following properties are equivalent: (1) $(X/Z,D)$ is a log stable pair and $(Z,D_{\mathrm{div}})$ is a log pair; (2) $\mathbb{D}_{\mathrm{div}}$ satisfies BP stable over $Z$, in particular, $(Z,D_{\mathrm{div}})$ is a log pair; and if additionally $X\to Z$ has equidimensional fibers, then (1), (2) are equivalent to (3) $\mathcal{D}^{\mathrm{mod}}$ is $\mathbb{R}$-Cartier over $X$. The log stable property in (1) means that, for every $\mathbb{R}$-Cartier divisor $\Delta$ on $Z$, $(X,D+f^*\Delta)$ is lc if and only if $(Z,D_{\mathrm{div}}+\Delta)$ is lc. The global version of this property is equivalent to the local one over a neighborhood of every point of $Z$. Notice also that, by the log adjunction and the negativity lemma, the b-nef over $Z$ property of $\mathcal{D}^{\mathrm{mod}}$ (cf. Conjecture 1 below) implies that it is enough to assume the if part of the log stability. Proof. (1) $\Rightarrow$ (2). We need to verify that, for every prime exceptional divisor $W$ on $Z$, $d_W=\operatorname{mult}_W\mathbb{D}_{\mathrm{div}}=b_W =\operatorname{mult}_W\mathbb{B}(Z,D_{\mathrm{div}})$ holds. Use a toroidalization $X'\to Z'$ as in the proof of Theorem 4 after a reduction to a contraction $X'\to Z'$. Suppose additionally that the invariant part $\Delta\subset Z'$ contains $W$ and $g^{-1}\operatorname{Supp} D_{\mathrm{div}}$ with all exceptional divisors of a birational projective base change $g\colon Z'\to Z$. Let $D_{Z'}=\mathbb{B}(Z,D_{\mathrm{div}})_{Z'}$ be a codiscrepancy. General property (2) allows to replace $D'$ by $D'+f'^*F'$, where $F'$ is such an $\mathbb{R}$-divisor on $Z'$ supported on $\Delta$ that $\operatorname{mult}_W (D_{Z'}+ F')= 1$ and $(Z',D_{Z'}+F')$ is lc. After an extension of $\Delta$ we can suppose that $F'\sim_\mathbb{R} 0/Z$ and $F'=g^*F$ for an $\mathbb{R}$-Cartier divisor $F$ on $Z$. The pair $(Z,D_{\mathrm{div}}+F)$ is also lc. Thus by the log stability (1) and construction $(X,D+f^*F)$ and its crepant pair $(X',D'+f'^*F')$ are lc. Now make a change $D:=D+f^*F$ and $F:=F':=0$. Then $d_W\leqslant b_W=1$ by General property (3). (In general, $d_W>b_W$ is possible.) Moreover, $d_W=b_W=1$. Indeed, in our construction we can assume that $\mathbb{B}(Z,D_{\mathrm{div}})$ has only one prime b-divisor $W$ over its center in $Z$ with $b_W=1$. Thus if $d_W<1$, then $(X,D)$ is klt over the center. Increasing singularity in the center we get a contradiction with log stability. In other words, the lc property can be replaced by the relative klt property in the definition of the log stability.
(1) $\Leftarrow$ (2). By General property (2) it is sufficient to verify the implication assuming BP stabile over $Z$, that is, $(X,D)$ is lc exactly when $(Z,D_{\mathrm{div}})$ is lc. If $(X,D)$ is lc, then the b-divisor $\mathbb{D}_{\mathrm{div}}$ is lc, that is, it is a subboundary, by General property (3). Hence $(Z,D_{\mathrm{div}})$ is lc by BP stable over $Z$. Conversely, if $(X,D)$ is not lc, then $\mathbb{B}(X,D)$ is not lc, that is, it is not a subboundary. Hence the b-divisor $\mathbb{D}_{\mathrm{div}}$ also is not lc by General property (3) and because a prime b-divisor $W$ on $X$ with the multiplicity $\operatorname{mult}_W\mathbb{B}(X,D)>1$ dominates a prime divisor on a blowup of the base $Z$. Thus $(Z,D_{\mathrm{div}})$ is not lc by BP stable over $Z$. (2) $\Leftrightarrow$ (3) by Proposition 2. $\Box$ Let $(X/Z,D)$ be a log pair the generic fiber of which is a weakly lc pair. Recall that weakly lc means in this situation that $(X_\eta,D_\eta)$ is lc with a boundary $D_\eta$ and $K_{X_\eta}+D_\eta$ is nef where $\eta$ is the generic point of $Z$. Consider the class of weakly lc pairs $(X'/Z',B')$ such that $X'$ is complete, $X'/Z'$ is birationally equivalent to $X/Z$ and $(X_\eta',B_\eta')$ is crepant to $(X_\eta,D_\eta)$. Then the muduli part of adjunction $\mathcal{B}^{\prime \mathrm{mod}}$ for $(X'/Z',B')$ attains its largest value $\mathcal{D}^{\mathrm{mm}}$ on the class of pairs. Pairs $(X'/Z',B')$ with the largest value will be denoted by $(X_m/Z_m,B_m)$ and will be called maximal. That is, $\mathcal{D}^{\mathrm{mm}}=\mathcal{B}^{\mathrm{mod}}_m$, a b-$\mathbb{R}$-divisor of $X_m$ or of a completion $\overline{X}$ of $X$. It is largest modulo linear equivalence on $X$, attained on log pairs naturally related to moduli spaces of the generic fiber, and will be constructed in Proposition-Definition 1 below modulo LMMP. This allows to define a birationally invariant with respect to the base $Z$ moduli part of adjunction $\mathcal{D}^{\mathrm{mm}}=\mathcal{D}^{\mathrm{mm}}(D)=\mathcal{D}^{\mathrm{mm}}(X/Z,D)$ which satisfies certain remarkable properties (e. g., Conjecture 1). However, as usually in mathematics we prefer the adjective maximal instead of largest. Proposition-Definition 1 (maximal log pair). Let $(X/Z,D)$ be a weakly lc pair with a boundary $D$ over the generic point of $Z$. Suppose that LMMP holds in dimensions $\leqslant \dim X$. Then there exists a maximal log pair $(X_m/Z_m,B_m)$, a weakly lc pair birationally equivalent to $(X/Z,D)$ with respect to the base $Z$. The maximal property means the inequality $\mathcal{B}^{\mathrm{mod}}_m\geqslant\mathcal{B}^{\prime \mathrm{mod}}$ modulo linear equivalence for every weakly lc pair $(X'/Z',B')$ with complete $X'$, $X'/Z'$ birationally equivalent to $X/Z$ and $(X_\eta',B_\eta')$ crepant to $(X_\eta,B_\eta$) where $\eta$ is the generic point of $Z$. Thus $\mathcal{D}^{\mathrm{mm}}=\mathcal{B}^{\mathrm{mod}}_m$ is maximal. Moreover, $\mathcal{D}^{\mathrm{mm}}=\mathcal{B}^{\mathrm{mod}}_m$ is a b-nef $\mathbb{R}$-Cartier b-$\mathbb{R}$-divisor. For appropriate canonical divisors on $X$ and $Z$, $\geqslant$ holds literally without the linear equivalence. There are a lot of maximal log pairs $(X_m/Z_m,B_m)$ but $\mathcal{D}^{\mathrm{mm}}$ is unique modulo linear equivalence. Example 4. Let $(X/Z,D)$ be a pair under GLC and generically be a $0$-pair over $Z$. Then it is a maximal log pair exactly when $(X/Z,D)$ is a $0$-pair everywhere over $Z$ with a boundary $D$. The last condition can be omitted keeping the same maximal moduli part of adjunction (cf. [6], Proposition 13, (1)). In this situation $\mathcal{D}^{\mathrm{mm}}=f^*\mathcal{D}_{\mathrm{mod}}$. Thus $\mathcal{D}^{\mathrm{mm}}$ is a generalization of $f^*\mathcal{D}_{\mathrm{mod}}$ with consequent conjectures (cf. Conjecture 1 below). Moduli construction of $(X_m/Z_m,B_m)$ and a proof of the proposition-definition actually need a weaker assumption on LMMP: in $\dim X/Z+1$, and will be explained in [3]. The moduli approach does not use also the b-nef property of $\mathcal{D}^{\mathrm{mm}}$ (cf. Conjecture 1 below). Proof of Proposition-Definition 1. The nef property of $M_X=\mathcal{D}^{\mathrm{mm}}_X$ was established in [1], Theorem 1.1, using the theory of foliations for contractions $X/Z$. The b-nef property of $\mathcal{D}^{\mathrm{mm}}$ follows from stability of $\mathcal{D}^{\mathrm{mm}}$ below. Thus, for contractions $X/Z$, we can suppose the b-nef property. Actually, we need the b-nef property in $\dim X-1$. An alternative approach to the b-nef property will be discussed after Conjecture 1 below.
Fix the birational class of $(X/Z,D)$, that is, the class of pair $(X'/Z',B')$ which are birationally equivalent to $(X/Z,D)$ with respect to the base $Z$. More precisely, we consider the subclass of pairs $(X'/Z',B')$ which are weakly lc with a boundary $B'$ and $(X'/Z',B')$ is crepant to $(X/Z,D)$ generically over $Z$ or $Z'$. Construction of $(X_m/Z_m,B_m)$Suppose first that $X/Z$ is a contraction. By [12], Theorem 2.1, we can suppose that $(X/Z,B)$ is toroidal with a nonsingular projective base $Z$ and with a boundary $B$ such that $\bullet$ $\operatorname{Supp} B$ is in the invariant divisor; $\bullet$ with the same multiplicities as $D$ generically over $Z$, that is, in all those prime divisors nonexceptional on original $X$; and $\bullet$ with multiplicities $1$ in all other invariant prime divisors. By our assumptions we can construct a weakly lc pair $(X_m/Z_m,B_m)$ over $Z\,{=}\,Z_m$. This is a required maximal model. By construction $(X_m/Z_m,B_m)$ belongs to the considered class weakly lc models. Moreover, $\mathbb{B}_{m,\mathrm{div},Z_m}=\Delta_m$ is the invariant divisor on $Z_m$. In general, we do not know the existence of toroidalization. So, we use the following construction. Take a Stein factorization of $X/Z$ where $f$, $g$ are, respectively, a contraction and a finite morphism. For an appropriate model of $X/Z$, we can suppose that $Z$ is nonsingular, $g$ is toroidal and there is a morphism $\varphi\colon Y\to Y_m$ for a maximal log pair $(X_m/Y_m,B_m)$ constructed above such that the divisorial part $(\mathbb{B}_{m,\mathrm{div}})_Y$ is supported in the invariant divisor $D$ of the toroidal finite morphism $g$. We suppose also that $(X/Y,B)$ is isomorphic to $(X_m/Y_m,B_m)$ over $Y\setminus D$ isomorphic to $Y_m\setminus\varphi(D)$. The crepant model $(X/Y,(\mathbb{B}_m)_X)$ of $(X_m/Y_m,B_m)$ is a weakly lc pair if the trace $(\mathbb{B}_m)_X$ is effective. Otherwise, we replace $(\mathbb{B}_m)_X$ by $(\mathbb{B}_m)_X+f^*E$, where $E=D-(\mathbb{B}_{m,\mathrm{div}})_Y$. By construction $E$ is effective, $(\mathbb{B}_{m,\mathrm{div}})_Y+E$ is the invariant divisor $D$. By General property (2), $D$ is the divisorial part of adjunction for $(X/Y,(\mathbb{B}_m)_X+f^*E)$. By construction, the moduli part of adjunction for $(X/Y,(\mathbb{B}_m)_X+f^*E)$ is the same $\mathcal{D}^{\mathrm{mm}}$ and maximal that will be established below (already known for contractions). Then we can reconstruct $(X/Y,(\mathbb{B}_m)_X+f^*E)$ into a weakly lc pair using LMMP over $Y$: take log resolution and replace multiplicities of all exceptional divisors over $D$ by $1$ and by $0$ otherwise, and apply LMMP over $Y$. The maximal property below for contractions warrants that the constructed model is crepant to $(X/Y,(\mathbb{B}_m)_X+f^*E)$ and is maximal over $Y$.Stability of $\mathcal{D}^{\mathrm{mm}}$Since in the last construction the divisorial part of adjunction for $(X/Y,(\mathbb{B}_m)_X+f^*E)$ is an integral invariant divisor $D$, then the stability for $X/Z$ is equivalent to the stability for $X/Y$ by Theorem 3 and Proposition 2. So, we can suppose that $X/Z$ is a contraction. By definition $\mathcal{D}^{\mathrm{mm}}=\mathcal{B}^{\mathrm{mod}}_m$. So, by Proposition 2, it is enough to verify that $\mathbb{B}_{m,\mathrm{div}}$ satisfies BP stable over $Z_m$. Using General properties, dimensional induction and induction on the number of monoidal transformations we can consider only one such transformation in the following situation. Let $P$ be an invariant prime cycle and $Z\to Z_m$ be a monoidal transformation in $P$. The transformation is toroidal and the invariant divisor $\Delta$ on $Z$ is the birational transform of $\Delta_m$ plus the exceptional divisor $E$ (over $P$) of the transformation. To establish required stability it is enough to verify that $\operatorname{mult}_E\mathbb{B}_{m,\mathrm{div}}=1$, that is, $\Delta=\mathbb{B}_{m,\mathrm{div},Z}$. Taking hyperplane sections we can suppose that $P$ is a closed point. We can suppose also that $\Delta_m$ is sufficiently large and $(X_m,B_m)$ itself (not only over $Z_m$) is a weakly lc model. To compute the last divisorial part of adjunction we can take a toroidal resolution of $X_m$ over $Z$ and then construct a model $(X/Z,B)$ as in above Construction using LMMP. By construction $\Delta=\mathbb{B}_{\mathrm{div},Z}$. Thus to verify that $\mathbb{B}_{\mathrm{div},m}=\mathbb{B}_{m,\mathrm{div},Z}$ it is enough (and actually necessary) to verify that $(X,B)$ is a crepant model of $(X_m,B_m)$. (As usually we consider such models over $\mathrm{pt.}$) Indeed, $(Z,\Delta)\to (Z_m,\Delta_m)$ is crepant. On the other hand, the weakly lc pair $(X_m,B_m)$ can be constructed from $(X,B)$ by LMMP and a crepant transformation. If $\Delta$ is sufficiently large the only possible curves negative with respect to $K+B$ are curves $C$ over $E$. These curves are on reduced prime divisors of invariant part $V$. Moreover, by above Construction and adjunction $(C.K+B)=(C.K_V+B_V)=(C.\mathcal{B}_V^{\mathrm{mm}})$ and $\geqslant 0$ by the b-nef property of $\mathcal{B}_V^{\mathrm{mm}}$ for $(V/E,B_V)$ by [1], Theorem 1.1, and we need this property in dimension $\leqslant \dim X-1$. Hence there are no negative curves and $(X,B)$ is crepant to $(X_m,B_m)$. The maximal property of $\mathcal{D}^{\mathrm{mm}}$In particular, $\mathcal{D}^{\mathrm{mm}}$ modulo linear equivalence is independent of construction. Moreover, $\mathcal{D}^{\mathrm{mm}}$ is unique for fixed canonical divisors on $X$ and $Z$. This can be verified directly using General properties. Notice for this especially General property (2) which implies that $\mathcal{D}^{\prime \mathrm{mm}}=\mathcal{D}^{\mathrm{mm}}$ for $D'=D+f^*\Delta$. If $X\to Z$ is a contraction, adding effective divisors we can suppose that $B_{\mathrm{div}}'=B_{m,\mathrm{div}}=\mathbb{B}_{m,\mathrm{div},Z'}$. If additionally $\mathbb{B}'_{m,\mathrm{div}}$ is BP stable over $Z'$, then the required inequality follows from the property that, for a larger divisor, its positive part in the Zariski decomposition is larger too. Perhaps, the same holds even if $\mathbb{B}_{\mathrm{div}}'$ does not satisfies BP. In general, we use the above construction of $(X/Y/Z,B_m)$ with $B_m:=(\mathbb{B}_m)_X+f^*E$, where $\mathbb{B}_{m,\mathrm{div}}=\overline{D}$ is BP stable over $Z$. The maximal moduli part $\mathcal{D}^{\mathrm{mm}}$ over $Z$ is the same as over $Y$:
$$
\begin{equation*}
\begin{aligned} \, \mathcal{D}^{\mathrm{mm}} &=\mathbb{K}+\mathbb{B}_m-(f\circ g)^\circ(\mathbb{K}_Z+\overline{D_Z}) \\ &=\mathbb{K}+\mathbb{B}_m-f^\circ(g^\circ(\mathbb{K}_Z+\overline{D_Z})) =\mathbb{K}+\mathbb{B}_m-f^\circ(\mathbb{K}_Y+\overline{D}), \end{aligned}
\end{equation*}
\notag
$$
where $D_Z$ is the invariant divisor for the toroidal morphism $g$. Notice also that any weakly lc model over $Z'$ is a weakly lc model over $Y'/Z'$ with a birational morphism $Y\to Y'$ for sufficiently high model $Y$ because any curve over $Y'$ is a curve over $Z'$. We do not need to suppose that the divisorial part of adjunction $\mathbb{B}_{\mathrm{div}}'$ of the model $(X'/Y',B')$ is BP stable over $Y$. The required maximal property follows from Corollary 2. Indeed, the corollary
$$
\begin{equation*}
\mathcal{B}'_{X'/Z'}=\mathcal{B}'_{X'/Y'}+\mathcal{M}_{Y/Z},
\end{equation*}
\notag
$$
where $\mathcal{B}'_{X'/Z'}$, $\mathcal{B}'_{X'/Y'}$, $\mathcal{M}_{Y,Z}$ denotes, respectively, the moduli part of adjunction for $(X'/Z',B')$, $(X'/Y',B')$, $(Y/Z,\mathbb{B}_{\mathrm{div}}')$. However, in nonstable situation we consider a moduli part and divisorial part of adjunction as b-divisors. Since $\mathcal{B}'_{X'/Y'}\leqslant \mathcal{D}^{\mathrm{mm}}$, it is enough to verify that $\mathcal{M}_{Y/Z}\leqslant 0$ for appropriate canonical b-divisors of $Y$, $Z$ (cf. Example 2, (2)). By definition, construction and the lc property of $\mathbb{B}_{\mathrm{div}}'$ we can suppose that $\mathbb{B}_{\mathrm{div}}'\leqslant\overline{D}$ and $=\overline{0}$ over $Y\setminus D$ (cf. Theorem 4). By General property (2) after increasing we can suppose that the divisorial part of adjunction $\mathbb{D}'$ for $(Y/Z,\mathbb{B}_{\mathrm{div}}')$ is $\overline{D_Z}$. Since $Y/Z$ is toroidal
$$
\begin{equation*}
\mathcal{M}_{Y/Z}=\mathbb{K}_Y+\mathbb{B}_{\mathrm{div}}'-g^\circ(\mathbb{K}_Z+\overline{D_Z})= \mathbb{K}_Y+\mathbb{B}_{\mathrm{div}}'-\mathbb{K}_Y-\overline{D}= \mathbb{B}_{\mathrm{div}}'-\overline{D}\leqslant 0
\end{equation*}
\notag
$$
holds. This concludes the proof of maximal property. In other words, $\mathcal{D}^{\mathrm{mm}}$ is the positive b-divisor of a relative lc b-divisor. $\Box$ Proof of Theorem 1. Immediate by Proposition-Definition 1. Indeed, by our assumptions $(X/Z,D)$ has a wlc model. $\Box$ Conjecture 1. For every positive $a\in \mathbb{R}$, $\mathcal{D}^{\mathrm{mm}}+a\mathcal{P}$ is b-semiample and effectively b-semiample if the moduli type of irreducible components of generic fiber is bounded where $\mathcal{P}$ is a pull-back b-divisor of canonical polarization for moduli of irreducible components of generic fiber. Moreover, the Iitaka dimension of $\mathcal{D}^{\mathrm{mm}}$ is at least the Kodaira dimension of the generic fiber of $(X/Z,D)$ plus the variation of $(X/Z,D)$. The effective b-semiample property of $\mathcal{D}^{\mathrm{mm}}+a\mathcal{P}$ is meaningful even if $D$ and $a$ are not rational. This can be explained in terms of geography of log models (see Corollary 4 below and [6], Bounded affine span and index of divisor in § 12). A geometrical interpretation of the conjecture: the corresponding contraction gives a family of log canonical models of components of fibers. So, in addition to a rational morphism from $Y$ in the Stein factorization $X\to Y\to Z$ to the coarse moduli of irreducible components of fibers, we have a rational morphism of $X$ to the family with log canonical models in fibers whereas $\mathcal{P}$ is ample on the base, $\mathcal{D}^{\mathrm{mm}}$ is nef on the family and ample on its fibers. In particular, the coarse moduli can be extended to a fibration with corresponding log canonical models. Notice that the conjecture implies the Kodaira additivity in the strong form (with variation). The semiampleness of $\mathcal{D}^{\mathrm{mm}}$ does not hold in general according to [15], § 3. The conjecture implies also that $\mathcal{D}^{\mathrm{mm}}$ is b-nef and even NQC (see Corollary 4 below). This property naturally related to the Viehweg positivity for the direct image of a relative dualizing sheaf and the polarization $\mathcal{P}$. We already used the b-nef property in the proof of Proposition-Definition 1. The author thinks that this property follows from its special klt case with a $1$-dimensional base by [16], Theorem 0.1, and adjunction. Finally, we expect the following strong form of stability of $\mathcal{D}^{\mathrm{mm}}$ with respect to horizontal part of $D$ which is supposed to be a boundary. Corollary 4 (geography of log adjunction). Under the assumptions of Theorem 1 let $X/Z$ be a dominant morphism and $S=\{S_i\}$ be a finite set of horizontal over $Z$ prime b-divisors of $X$. Then there exists a closed rational convex polyhedron $\mathfrak N_S$ in the cube and its finite decomposition into rational convex polyhedrons such that $D\in \mathfrak B_S$ belons to $\mathfrak N_S$ if and only if $(X/Z,D)$ has a nonnegative log Kodaira dimension over the generic point of $Z$ and the maximal moduli part $\mathcal{D}^{\mathrm{mm}}$ of $(X/Z,D)$ is well-defined. Moreover, $\mathcal{D}^{\mathrm{mm}}$ is a continuous piecewise $\mathbb{Q}$-linear function of $D\,{\in}\, \mathfrak N_S$, $\mathbb{Q}$-linear on every polyhedron $\mathfrak P$ of the decomposition of $\mathfrak N_S$: for every two divisors $D_1,D_2\in\mathfrak P$ and any two real numbers $w_1,w_2\in[0,1]$ with $w_1+w_2=1$, Since the geography is rational, every $\mathcal{D}^{\mathrm{mm}}$ satisfies NQC. Since $\mathfrak P$ is rational, it is enough to verify Conjecture 1 for $\mathbb{Q}$-divisors $D$. Proof of Corollary 4. As in the proof of Proposition-Definition 1 we can suppose that $(X/Y,\Delta)$ is a toroidal pair with projective $Y$, the invariant divisor $\Delta$ over $Y$ where $X\to Y\to Z$ is a Stein factorization and $S$ is in $\Delta$. Actually, we extend $S$ and suppose that $S$ is the horizontal part of $\Delta$. Respectively, we extend every $D\in \mathfrak B_S$ by the invariant horizontal divisors with multiplicity $1$. Let $V=\{V_j\}$ be the set of vertical over $Y$ or $Z$ prime invariant divisors of $X$. Then
$$
\begin{equation*}
\mathfrak N_S=\mathfrak N_{S+V}\cap (\mathfrak B_S\times V)
\end{equation*}
\notag
$$
with the polyhedral decomposition induced by the wlc geography of $\mathfrak N_{S+V}$ for pair $(X/Y,S+V)$; we identify $V$ and $S$ with reduced divisors $S=\sum S_i$, $V=\sum V_i$. Recall that the wlc geography is a polyhedral decomposition of the convex polyhedron $\mathfrak N_{S+V}$ of the cube
$$
\begin{equation*}
\mathfrak{B}_{S+V}=(\oplus [0,1]S_i)\oplus(\oplus [0,1]V_j),
\end{equation*}
\notag
$$
where an $\mathbb{R}$-divisor $D\in\mathfrak{B}_{S+V}$ belongs to $\mathfrak N_{S+V}$ if and only if $(X/Z,D)$ has a wlc model. The polyhedral decomposition is given by the wlc equivalence $\sim_{\mathrm{wlc}}$ on $\mathfrak N_{S+V}$: for $B,B'\in\mathfrak N_{S+V}$, $B\sim_{\mathrm{wlc}}B'$ if pairs $(X/Y,B)$, $(X/Y,B')$ have the same wlc models, that is, $(X'/Y,B^{\mathrm{log}}_{X'})$ is a wlc model of $(X/Y,B)$ if and only if $(X'/Y,B^{\prime \mathrm{log}}_{X'})$ is a wlc model of $(X/Y,B')$ (see [17], § 3, for details). This gives a polyhedral decomposition of $\mathfrak N_{S+V}$ and $\mathfrak N_S$ by [17], Theorem 3.4.
By the proof of Proposition-Definition 1, for every $D\in \mathfrak N_S$, $(X/Y,D+V)$ has a maximal model $(X_m/Y,D_m+V_m)$ with $\mathbb{Q}$-factorial $X_m$ that is a wlc model of $(X/Y,D+V)$ too and by definition
$$
\begin{equation*}
\mathcal{D}^{\mathrm{mm}}=\mathcal{D}^{\mathrm{mod}}_m =\mathbb{K}+\mathbb{D}_m+\overline{V_m}-f^\circ (\mathbb{K}_Y+\mathbb{D}_{\mathrm{div}})= \overline{K_{X_m}+D_m+V_m}-f^\circ(\overline{K_Y+\Delta_Y}),
\end{equation*}
\notag
$$
where $D_{\mathrm{div},Y}=\Delta_Y$ is the invariant divisor on $Y$ for toroidal morphism $X\to Y$. As in the proof of Proposition-Definition 1 $\mathbb{D}_{\mathrm{div}}=\overline{\Delta_Y}$ holds. Since $K_{X_m},V_m$ and $K_Y+\Delta_Y$ are constant divisors on $X_m$ and $Y$, respectively, and the model $X_m/Y$ is the same for $D\in \mathfrak P$ of the decomposition, then the $\mathbb{Q}$-linear property of $\mathcal{D}^{\mathrm{mm}}$ on $\mathfrak P$ follows from the $\mathbb{Q}$-linear property of $D_m$ on $\mathfrak P$, the image of $D$ on $X_m$. $\Box$
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\Bibitem{Sho23} |
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