$(\widehat{X},S+\Xi_{1})$ is plt and any hypersurface $Y$ contained in the
support of $\Xi_{2}$ is $\pi$-exceptional.

We let

(35) $\gamma:=\pi_{W}^{\star}(K_{X}+\alpha)|_{T}-K_{W},$

and we write

(36) $f^{\star}(\gamma)+\Xi_{2}|_{S}\simeq K_{S/W}+(\beta+\Xi_{1})|_{S}.$

Note that the pair $(\widehat{X},\Xi:=\\{\Xi_{1}-\Xi_{2}\\})$ is klt and that
$T$ is the unique center of log canonical singularities for the generalized
pair $(\hat{X},S+\Xi_{1}-\Xi_{2}+\beta)$. In particular $T$ is normal.

  1. 3.

The class $\beta$ contains a smooth, positive representative.

  2. 4.

The coefficients of the divisors $\Xi_{i}$ are rational.

A first result we establish here is the following.

######  Theorem 5.1.

Assume that the requirements 1-4 above are satisfied. Then the class
$K_{S/W}+(\beta+\Xi_{1})|_{S}$ contains a closed positive current $\Theta\geq
0$ such that:

  * (1)

For each general fiber $S_{w}=f^{-1}(w)$, the restriction
$\displaystyle\Theta|_{S_{w}}$ is induced by the space of sections of the line
bundle associated to $m\Xi_{2}|_{S_{w}}$, for $m$ large and sufficiently
divisible.

  * (2)

Consider the divisor $\Xi_{2}^{\prime}\leq\Xi_{2}|_{S}$ obtained by discarding
the components of $\Xi_{2}|_{S}$ whose image is contained in the singular
subset of $T$. Then, $\Theta\geq[\Xi_{2}^{\prime}]$ – that is to say, the
current $\Theta$ is singular along the divisor $\Xi_{2}^{\prime}$.

The following results can be seen as ”transcendental” versions of the
canonical bundle formula. They can be used to refine Theorem 5.1, but they are
of independent interest as they apply to a variety of other contexts.

Let $f:S\to W$ be a surjective map of compact Kähler manifolds. Let $P:=\sum
P_{i}$ and $Q=\sum Q_{j}$ be two reduced, snc divisors on $S$ and $W$,
respectively such that moreover $f^{-1}Q\subset P$. We decompose the divisor
$P$

$P=P^{h}+P^{v}$

into $f$-horizontal and $f$-vertical parts, and we assume moreover that the
restriction of $f$ to the support of $P^{h}$ is relatively snc on the
complement of the support of $Q$, and moreover
$f(\operatorname{Supp}P^{v})=Q$.

Let $B=\sum d_{i}P_{i}$ be a $\mathbb{Q}$-divisor on $S$, and let $\beta$ be a
$(1,1)$-class such that the following requirements are satisfied.

  1. (a)

The pair $(S,B)$ is sub-klt.

  2. (b)

The morphism $\mathcal{O}_{W}\to f_{\star}\mathcal{O}_{S}(\lceil-B\rceil)$ is
surjective at general points of $W$.

  3. (c)

We have $K_{S}+B+\beta\simeq f^{\star}\gamma$ and moreover $\beta$ contains a
smooth positive representative.

  4. ($\star$)

For any point $z_{0}\in S$ and $w_{0}=f(z_{0})\in W$ there exist local
coordinates $(x_{1},\dots,x_{n+m})$ on $S$ centred at $z_{0}$ and
$(t_{1},\dots,t_{m})$ on $W$ centred at $w_{0}$ such that $\displaystyle
t_{i}\circ f(x)=\prod x_{j}^{k_{ij}}$ where the $k_{ij}$ are non-negative
integers such that $k_{ij}\neq 0$ for at most one $i$ for each index $j$.

Then the following result holds true – the case $\beta=0$ corresponds to the
original result of Y. Kawamata in [Kawamata98].

######  Theorem 5.2.

Assume that conditions (a), (b), (c) as well as $(\star)$ hold. Then the class
$\\{\gamma\\}$ can be decomposed as $K_{W}+B_{W}+\beta_{W}$ where $B_{W}$ is
the discriminant divisor and $\beta_{W}$ is a cohomology class containing a
closed positive current with zero Lelong numbers. In particular, $\beta_{W}$
is a nef class.

######  Remark 5.3.

Note that the hypothesis (a), (b), (c) are very natural, identical to the set-
up in [Kawamata98]. We expect Theorem 5.2 to hold without the additional
hypothesis $(\star)$, but there are serious technical difficulties to
overcome.

One could ask the same type of questions in a more flexible and natural
context, in which $\beta$ is only assumed to be nef (so that we start with a
nef class on $S$ and the ”output” is a nef class on $W$). It turns out that
the situation is a bit more complicated –the reason being that a perturbation
of $\beta$ could destroy the first hypothesis in (c)–, and in order to treat
the nef case we consider the following assumptions.

  1. (d)

There exists a Kähler metric $\omega$ on $S$ such that

$\omega=f^{\star}g+\theta,$

where $g$ is a Kähler metric on $W$ and $\theta$ is a _rational_ $(1,1)$-form
on $S$. Therefore, for any coordinate subset $\Omega\subset W$ biholomorphic
to a ball we have a $\mathbb{Q}$-line bundle $A_{\Omega}$ on
$V:=f^{-1}(\Omega)$ whose curvature equals $\omega|_{V}$.

  2. (b’)

For any fixed, sufficiently big and divisible integer $m_{0}>0$, there is an
integer $k_{0}$ such that the natural inclusion
$f_{\star}\mathcal{O}_{V}(m_{0}A_{\Omega}+k_{0}\lceil-B\rceil)\subset
f_{\star}\mathcal{O}_{V}(m_{0}A_{\Omega}+k\lceil-B\rceil)$ is an isomorphism
for all $k\geq k_{0}$ (i.e. the local sections vanish along
$(k-k_{0})\lceil-B\rceil$), where $A_{\Omega}$ is defined above.

  3. (c’)

We have $K_{S}+B+\beta\simeq f^{\star}\gamma$ and $\beta$ is a nef class.

######  Theorem 5.4.

Assume that conditions (a), (b’), (c’), (d), as well as $(\star)$ hold. Then
the class $\\{\gamma\\}$ can be decomposed as $K_{W}+B_{W}+\beta_{W}$ where
$B_{W}$ is the discriminant divisor and $\beta_{W}$ is a nef class.

######  Remark 5.5.

The proof of Theorem 5.1 will show that the hypothesis (b’) is quite natural,
in the sense that if the map $f:S\to W$ is induced by a birational
transformation $\pi$, then these hypothesis hold true.

The content of the following sections is organized as follows. There are two
techniques of constructing closed positive currents in twisted relative
classes of a map between compact Kähler manifolds. One can either use
fiberwise holomorphic sections (normalized in a canonical manner), or
fiberwise Kähler-Einstein metrics, cf. [Gue20] and the references therein.
Here we will use the former, since the latter is not sufficiently general to
be implemented in our context.

Indeed, given a holomorphic surjective map $f:S\to W$ between two Kähler
manifolds and a Hermitian line bundle $(L,h_{L})\to X$, the spaces

$H^{0}\big{(}S_{w},(K_{S_{w}}+L|_{S_{w}})\otimes\mathcal{I}(h_{L}|_{S_{w}})\big{)}$

of $L^{2}$ sections (for $w\in W$ general) can be ”pieced together” in order
to construct a metric on $K_{S/W}+L$, which is semi-positively curved e.g. in
case the curvature current $\sqrt{-1}\Theta(L,h_{L})\geq 0$ is positive, cf.
[BP08]. The same is true in the pluricanonical case, i.e. we can construct a
positively curved metric on $mK_{S/W}+L$, by replacing the $L^{2}$
normalization with an $L^{\frac{2}{m}}$ condition. As a result, the rational
class $\displaystyle K_{S/W}+\frac{1}{m}L$ contains a closed positive current,
whose restriction to the general fiber of $f$ is induced by the subspace of
sections of

$H^{0}\big{(}S_{w},mK_{S_{w}}+L|_{S_{w}})$

which satisfy an $L^{\frac{2}{m}}$ integrability condition.

In section 6.1 we show that the same holds true if we replace $\frac{1}{m}L$
with a $(1,1)$-class $\alpha$, provided that the we can still define the space
above, i.e. the restriction $\alpha|_{S_{w}}$ of our class to the fibers of
$f$ is rational. This will settle the first part of Theorem 5.1. The
singularities of the current constructed are analyzed by using techniques
borrowed from extension of pluricanonical forms.

Concerning Theorem 5.2, recall that any nef class is pseudo-effective, but in
general the two cones are different. Nevertheless, if a $(1,1)$ class contains
a closed positive current whose Lelong numbers at each point of the ambient
space are equal to zero, then the class in question is psef, cf. [Dem92]. The
nefness of the moduli part (in our notations, the class $\beta_{W}$) in the
canonical bundle formula was established by S. Takayama in [Taka22] along
these lines. Here we will adopt the same strategy –i.e., we will conclude by
showing that $\beta_{W}$ contains a closed positive current with zero Lelong
numbers–, and by the same token, simplify a little the arguments in _loc.
cit._

##  6\. Positivity of the relative adjoint transcendental classes

We begin this section by recalling the following results.

######  Theorem 6.1.

Let $f:S\to W$ be a surjective map between two compact complex manifolds. Let
$\alpha$ be a real class of type $(1,1)$ on $W$. Then

  1. (1)

${\alpha}$ is nef if and only if ${f^{\star}\alpha}$ is nef.

  2. (2)

${\alpha}$ is nef if and only if $\alpha|_{Z}$ is pseudo-effective for all
irreducible proper subvarieties $Z\subset W$ and ${f^{\star}\alpha}$ is
pseudo-effective.

###### Proof.

(1) follows immediately from [DHP22, Lemma 2.38].

We will now show that (2) also follows from the proofs of [DHP22, Theorem
2.36, Lemma 2.38]. By [DHP22, Theorem 2.36], we know that if $Z^{\nu}\to Z$ is
the normalization of an irreducible proper subvariety of $W$, then
$\alpha|_{Z^{\nu}}$ is nef. By [DHP22, Corollary 2.39], it suffices to show
that if $\omega$ is Kähler on $W$ and $\operatorname{dim}W=d$, then
$\int_{W}\alpha^{k}\wedge\omega^{d-k}\geq 0$ for $0<k\leq d$. Suppose that
$\alpha$ is not nef, then we let $t>0$ be the nef threshold so that
$\alpha+t\omega$ is nef but not Kähler. Clearly $(\alpha+t\omega)|_{Z^{\nu}}$
is Kähler and so by [DHP22, Corollary 2.39]
$\int_{W}(\alpha+t\omega)^{k}\wedge\omega^{d-k}=0$ for some $0<k\leq d$. Let
$F$ be a general fiber of $S\to W$, $\eta$ a Kähler class on $S$ and
$\lambda=\int_{F}\eta^{n-d}$ where $n=\operatorname{dim}S$. Then

$\lambda\cdot\int_{W}(\alpha+t\omega)^{k}\wedge\omega^{d-k}=\int_{S}f^{*}((\alpha+t\omega)^{k}\wedge\omega^{d-k})\wedge\eta^{n-d}\geq\int_{S}f^{*}(t^{k}\omega^{d})\wedge\eta^{n-d}=\lambda
t^{k}\cdot\int_{W}\omega^{d}$

which is impossible as the LHS equals 0 and the RHS is strictly positive. Thus
$t=0$ and $\alpha$ is nef.

Another way of establishing the point (2) is by using [DHP22, Corollary 2.32]
: this shows that $\alpha$ contains a closed positive current, i.e. it is
pseudo-effective. The conclusion follows by using [DHP22, Theorem 2.36]. ∎

Therefore, in order to show that the class $\beta_{W}$ in Theorem 5.2 is nef,
it is sufficient to show that this is true for its pull back via $f$. This
will follow from the main results in the next two subsections. In the first
one we collect a few results about the construction of closed positive
currents.

###  6.1. Closed positive currents in twisted relative canonical classes

To start with, we introduce the following set of notations, _which will only
be used in this subsection_.

  1. (1)

$f:X\to Y$ is a surjective map with connected fibers between compact Kähler
manifolds.

  2. (2)

$D=\sum a_{i}D_{i}$ is an effective, snc divisor with rational coefficients
$0<a_{1}<1$ and $L$ is a $\mathbb{Q}$-line bundle on $X$. Thus there exists a
positive integer $m_{0}$ such that the reflexive hull
$L^{[m_{0}]}:=(L^{\otimes m_{0}})^{\vee\vee}$ is a genuine line bundle, and a
metric on $L$ will simply be given by a collection of functions
$\displaystyle\Big{(}\frac{\varphi_{i}}{m_{0}}\Big{)}_{i\in I}$, where
$(\varphi_{i})_{i\in I}$ are the weights of a metric on $L^{[m_{0}]}$. By
abuse of notation we will often denote $L^{[m_{0}]}$ by $m_{0}L$.

  3. (3)

$\alpha$ and $\gamma$ are real $(1,1)$–classes on $X$ and $Y$, respectively.
Moreover, $\alpha$ contains a smooth positive representative denoted by
$\theta$.

  4. (4)

The class $f^{\star}\gamma-\alpha$ coincides with the first Chern class of
$K_{X/Y}+D-L$.

In this context we prove the next statement.

######  Theorem 6.2.

Assume that conditions (1), (2), (3) and (4) hold and that for some
sufficiently big and divisible integer $m>0$ we have $\displaystyle
H^{0}(X_{y},mL|_{X_{y}})\neq 0$ for general $y\in Y$. Then the class

$K_{X/Y}+D+\alpha$

contains a closed positive current $\Theta\geq 0$ whose restriction to the
general fiber of $f$ is (well-defined and) induced by the sections of $mL$
restricted to the fibers of $f$.

###### Proof.

To begin with, we remark that if $D=0$ and if some multiple of $\alpha$
belongs to $H^{2}(X,\mathbb{Z})$, then the matter is clear. Indeed, in this
case we can choose a $\mathbb{Q}$-line bundle $F$ on $X$ whose Chern class is
$\alpha$ and such that

$(K_{X/Y}+D+F)|_{X_{y}}\simeq L|_{X_{y}}$

for all general $y\in Y$. The results proved in [PT18] show that the current
$\Theta$ constructed fiber-wise by the $m^{\rm th}$ root of the sections of
$mL|_{X_{y}}$ is positive.

Even though $\alpha$ may not be a rational class, hypothesis (4) implies that
this is the case locally over $Y$. This will allow us to conclude via an
approach similar to the one in [PT18], [CH20] (with slight modifications). The
details are as follows.

We denote by $\gamma$ any closed, real $(1,1)$-form contained in the class
$\gamma$ given by hypothesis (3) -and apologize for the abuse of notation. As
consequence of the hypothesis (4) above, the $(1,1)$-form $\mu$ defined by the
equality

(37) $\mu:=\theta-f^{\star}(\gamma)$

is closed, real, and its corresponding class is _rational_.

Let $h$ be a metric on $L-(K_{X/Y}+D)$ whose corresponding curvature form
equals $\mu$ (here we are using the convention in (2) above). We consider a
finite open cover $(U_{i})_{i\in I}$ of $Y$ such that

(38) $\gamma|_{U_{i}}=dd^{c}\tau_{i}$

for some smooth real function $\tau_{i}$ defined on $U_{i}$.

For each index $i$ we endow the restriction

$(L-(K_{X/Y}+D))|_{f^{-1}(U_{i})}$

with the metric $h_{i}:=e^{-\tau_{i}\circ f}h$. The equality

(39) $\theta|_{f^{-1}(U_{i})}=dd^{c}(f^{\star}\tau_{i})+\mu|_{f^{-1}(U_{i})}$

shows that the curvature corresponding to the metric $h_{i}$ is equal to
$\theta|_{f^{-1}(U_{i})}$.

All in all, we can define a Hermitian $\mathbb{Q}$-line bundle $(F_{i},h_{i})$
on the inverse image $f^{-1}(U_{i})$ such that the following hold.

  1. (a)

For each $i\in I$ the following holds

(40) $(K_{X/Y}+D)|_{f^{-1}(U_{i})}+F_{i}\simeq L|_{f^{-1}(U_{i})}$

especially the sections of $\displaystyle mL|_{X_{y}}$ correspond to sections
of $m(K_{X/Y}+D+F_{i}|_{X_{y}})$ for each general point $y\in U_{i}$, where
$X_{y}:=f^{-1}(y)$.

  2. (b)

We describe here more precisely the metric $h_{i}$. Let $(V_{j})_{j\in J}$ be
an open covering of $X$, such that the restriction of the bundles
$K_{X},f^{\star}K_{Y},mD,mL$ to each $V_{j}$ is trivial. Recall that we have
fixed a metric $h$ on $L-K_{X/Y}-D$, and denote by $\rho_{j}$ its weight on
the set $V_{j}$. Then the weight of the metric $h_{i}$ on the set $V_{j}\cap
f^{-1}(U_{i})$ is

$\varphi_{ij}:=\rho_{j}|_{V_{j}\cap f^{-1}(U_{i})}+\tau_{i}\circ f|_{V_{j}\cap
f^{-1}(U_{i})}.$

We stress the fact that the only ”non-global” part of the metric $h_{i}$
corresponds to the pull-back of $\tau_{i}$.

  3. (c)

It follows that we have

(41) $\sqrt{-1}\Theta(F_{i},h_{i})=\theta|_{f^{-1}(U_{i})}.$

and remark that even if $(F_{i},h_{i})$ is only locally defined (with respect
to the base $Y$), the corresponding curvature is a global form on $X$.

Relation (40) allows us to define a metric $h_{X/Y,i}$ on
$(K_{X/Y}+D)|_{f^{-1}(U_{i})}+F_{i}$ whose corresponding curvature is
positive. This was done in [PT18], and we recall next the construction. Let
$x_{0}\in f^{-1}(U_{i})$ be an arbitrary point. We fix coordinates $(t_{k})$
and $(z_{l})$ on $U_{i}$ and near $x_{0}$, respectively. Assume that $f$ is
smooth over $Y\setminus\Sigma$. For each $y\in U_{i}\setminus\Sigma_{i}$ and
$\displaystyle\xi\in V_{m,y}$ let

(42)
$\|\xi\|^{\frac{2}{m}}_{y,i}:=\int_{X_{y}}|\xi|^{\frac{2}{m}}e^{-\varphi_{D}-\varphi_{i}}$

be the $L^{2/m}$-seminorm on the space of sections

$V_{m,y}:=H^{0}\big{(}X_{y},mL_{y}\big{)}=H^{0}\big{(}X_{y},m(K_{X_{y}}+D_{y}+F_{i,y})\big{)}$

where the notations are explained below. The subscript $(\ldots)_{y}$ denotes
restriction to the fiber $X_{y}$ and $m$ is assumed to be sufficiently
divisible so that all divisors in question are Cartier. In (42) the symbol
$e^{-\varphi_{i}}$ means that we are using the metric $h_{i}$ (cf. (b) above)
on the bundle $F_{i}$. The section $\xi$ in (42) is interpreted as a twisted
pluricanonical form, so that the quantity under the integral is a
$(n,n)$–form.

Then the weight of the metric $h_{X/Y,i}$ at the point $x_{0}$ is equal to

(43)
$e^{\varphi_{X/Y,i}(x_{0})}:=\sup_{\|\xi\|_{y_{0},i}=1}\left|\xi_{0}(x_{0})\right|^{\frac{2}{m}}$

where $y_{0}:=f(x_{0})$ and $\xi_{0}$ is given by the equality

$\xi\wedge f^{\star}(dt^{\otimes m})=\xi_{0}dz^{\otimes m}$

written locally near $x_{0}$.

We have given such a detailed description of the metric $h_{X/Y,i}$ because
thanks to it, it is easy to deduce its dependence on the index $i$. Indeed, we
assume that we choose the same coordinates $t$ and $z$ on $U_{i}\cap U_{k}$
and near $x_{0}\in V_{j}$, respectively, where it is understood that
$f(x_{0})\in U_{i}\cap U_{k}$. Notice that the space of holomorphic sections
$V_{m,y}$ involved in the definition of the relative metric is independent of
$i$, but this may be not the case for the semi-norm (42). By (b) of (40), we
can write

(44)
$\int_{X_{y}}|\xi|^{\frac{2}{m}}e^{-\varphi_{D}-\varphi_{i}}=e^{-\tau_{i}(y)}\int_{X_{y}}|\xi|^{\frac{2}{m}}e^{-\varphi_{D}-\rho_{j}},$

where the notation is indicating the weight $\varphi_{D}+\rho_{j}$ we are
using on the set $V_{j}\cap X_{y}$. This is a consequence of the definition of
the metric $h_{i}$ in (b). Moreover, we remark that the second factor of the
product on the RHS of (44) is independent on the index ”$i$”.

Thus, by (42) we infer that the equality

(45)
$\|\xi\|^{\frac{2}{m}}_{y,i}=e^{\tau_{k}(y)-\tau_{i}(y)}\|\xi\|^{\frac{2}{m}}_{y,k}$

holds for any point $y\in U_{i}\cap U_{k}$ and $\xi\in V_{m,y}$.

Moreover, we can assume that the difference

(46) $\tau_{k}-\tau_{i}=\Re(\tau_{ik})$

is the real part of some holomorphic function $\tau_{ik}$ defined on the
intersection $U_{i}\cap U_{k}$, since their respective Hessian forms coincide
by (39).

It follows that

(47)
$\sup_{\|\xi\|_{y_{0},i}=1}\left|\xi_{0}(x_{0})\right|=\sup_{\|\xi\|_{y_{0},k}=e^{-\Re(\tau_{ik}(y_{0}))}}\left|\xi_{0}(x_{0})\right|=e^{-m\Re(\tau_{ik}(y_{0}))}\sup_{\|\xi\|_{y_{0},k}=1}\left|\xi_{0}(x_{0})\right|.$

Finally, we get

(48)
${\varphi_{X/Y,i}(x_{0})}={\varphi_{X/Y,k}(x_{0})-\frac{1}{2}\Re(\tau_{ik}(y_{0}))}$

and since the point $x_{0}$ was arbitrary and $y_{0}=f(x_{0})$ it follows that
we have

(49) ${\varphi_{X/Y,i}}={\varphi_{X/Y,k}-\frac{1}{2}\Re(\tau_{ik}\circ f)}$

locally near a fixed point on $f^{-1}(U_{i}\cap U_{k})$. In particular we
obtain the equality

(50) $dd^{c}\varphi_{X/Y,i}=dd^{c}\varphi_{X/Y,k},$

on the overlapping $U_{i}$’s.

In conclusion, (50) shows that the curvature currents we construct locally on
the base agree on the intersection of the corresponding sets, and the
construction of $\Theta$ is finished, since the positivity of this current was
already established in [PT18, Theorem 4.2.7]. ∎

###  6.2. Singularities of the metric

In order to prove Theorem 5.1 we can apply Theorem 6.2 for the following data:
$X:=S,Y:=W,D=\Xi_{1}$ and $L=\Xi_{2}$, together with $\alpha:=\beta$. The
output is a current

$\Theta\in c_{1}(K_{S/W})+(\beta+\Xi_{1})|_{S}$

with the properties stated in the point (1) of Theorem 5.1. The assertion (2)
will be established along the following lines.

###### Proof of Theorem 5.1, (2).

To begin with, we recall an important class of manifolds on which $L^{2}$
methods can be applied.

######  Definition 6.3.

A manifold/complex space $X$ is called weakly pseudo-convex if it admits a
smooth, plurisubharmonic exhaustion function $\psi$, so that the closure of
the sets $(\psi<C)\Subset X$ are compactly contained in $X$, for any constant
$C$.

Obviously, compact holomorphic manifolds have this property, but this is
equally the case for any complex space which admits a proper map into a Stein
manifold. In particular, consider the map $f:S\to W$ given in Theorem 5.1; for
any Stein open subset $U\subset W$ the inverse image $f^{-1}(U)\subset S$ is
an example of weakly pseudo-convex manifold which will be important in what
follows.

Consider next the blow-up map $\pi:\widehat{X}\to X$ introduced at the
beginning of Section 5, and denote by $E=\sum E_{j}$ the corresponding
exceptional divisor. We define the following form

(51) $\omega_{\widehat{X}}:=\pi^{\star}\omega_{X}+\sum a_{i}\theta_{i}$

where $\omega_{X}$ is a Kähler metric on the base $X$, the coefficients
$a_{i}$ are positive rational numbers and the forms $\theta_{i}$ belong to the
Chern class of $\mathcal{O}_{\hat{X}}(-E_{i})$. By an appropriate choice of
the coefficients $a_{i}$, we can assume that
$\displaystyle\omega_{\widehat{X}}>0$ – so we have a Kähler metric on
$\widehat{X}$ for which the only ”transcendental” part is pulled-back from the
base $X$.

Let $w_{0}\in W$ be an arbitrary point, and let $\Omega\subset X$ be a Stein
co-ordinate subset which contains the image $\iota_{W}(w_{0})$, cf. diagram
(52) below.

(52)
${S}$${\widehat{X}}$${W}$${X}$$\scriptstyle{\iota_{S}}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\scriptstyle{\iota_{W}}$

Consider the following sets

(53) $\widehat{X}_{\Omega}:=\pi^{-1}(\Omega),\qquad
U:=\iota_{W}^{-1}(\Omega),\qquad S_{U}:=f^{-1}(U)$

contained in $\hat{X}$, $W$, and $S$, respectively. We have the following
statement.

######  Lemma 6.4.

There exist Hermitian line bundles $(A_{\Omega},h)\to\widehat{X}_{\Omega}$ and
$(A_{U},h)\to U$ on $\widehat{X}_{\Omega}$ and $U$, respectively such that the
corresponding curvature forms are multiple of Kähler forms, i.e.

$\sqrt{-1}\Theta(A_{\Omega},h)=N\omega_{\widehat{X}}|_{\widehat{X}_{\Omega}},\qquad\sqrt{-1}\Theta(A_{U},h)=N\omega_{W}|_{U}$

where $N$ is positive and sfficiently divisible.

###### Proof.

This follows by standard arguments. Since the restriction
$\displaystyle\omega_{X}|_{\Omega}$ is $dd^{c}$-exact, it can be interpreted
as trivial bundle over $\Omega$ endowed with a non-trivial metric whose
corresponding curvature form is $\displaystyle\omega_{X}|_{\Omega}$. The rest
follows as consequence of (51) – in particular we only need the positive
integer $N$ in order to clear the denominators of the coefficients $a_{i}$. A
similar argument applies for $\omega_{W}$. ∎

######  Remark 6.5.

Note that we may assume $\iota_{W}:W\to X$ is given by a finite sequence of
blow ups whose centers are contained in the singular locus $T_{\rm sing}$ of
$T$. The metric $\omega_{W}$ can be obtained by the same formula as in (51),
so that the corresponding exceptional divisors $E_{i}\subset W$ map into the
singular locus of the centre $X$.

After these preparations, we proceed with the second part of Theorem 5.1. Let
$\Omega\subset X$ be an open subset as above. By the same procedure as in the
proof of Theorem 6.2, we can construct a $\mathbb{Q}$-line bundle
$(F_{\Omega},h_{F})\to\widehat{X}_{\Omega}$ such that the following relations
hold

(54)
$K_{\widehat{X}}+S+\Xi_{1}+F_{\Omega}\simeq\Xi_{2},\qquad\sqrt{-1}\Theta(F_{\Omega},h_{F})=\beta$

on $\widehat{X}_{\Omega}\subset\widehat{X}$.

On the other hand, let $\rho$ be any non-singular $(1,1)$-form on $W$, such
that $\rho\in c_{1}(K_{W})$. We can assume that the constant $N$ in Lemma 6.4
is large enough, so that the following inequality

(55) $\rho+N\omega_{W}\geq 0$

holds point-wise on $W$. We then consider the closed positive current

(56) $\Theta+f^{\star}(\rho+N\omega_{W})\geq 0$

which belongs to the class $\displaystyle
K_{S}+(\Xi_{1}+\beta)|_{S}+Nf^{\star}(\omega_{W})$.

Next, given the expression of the metric $\omega_{W}$ combined with Remark
6.5, there exist integers $k_{i}$ and divisors $E_{i}$ such that the current

(57) $\widehat{\Theta}:=\Theta+f^{\star}(\rho+N\omega_{W})+\sum
k_{i}[E_{i}|_{S}]$

has the following properties

  * •

It belongs to the cohomology class
$\displaystyle(K_{X}+S+\Xi_{1}+\beta+N\pi^{\star}(\omega_{X}))|_{S}$

  * •

The divisors $E_{i}$ appearing in (57) project into $T_{\rm sing}$.

When restricted to the set

$S_{U}=S\cap\pi^{-1}\Omega$

(cf. (53) for the notations), the class $\beta$ corresponds to the line bundle
$\displaystyle F_{\Omega}|_{S}$, so by abuse of notation we can write

(58) $\widehat{\Theta}|_{S_{U}}\simeq K_{X}+S+\Xi_{1}+F_{\Omega}|_{S_{U}}$

by which we mean that the bundle on the RHS admits a singular metric
$h_{\theta}=e^{-\varphi_{\theta}}$ defined on $S_{U}$ whose curvature form is
precisely the restriction $\displaystyle\widehat{\Theta}|_{S_{U}}$.

For each $k\geq 1$ sufficiently divisible we consider the line bundle

$L_{k}:=(k(K_{X}+S+\Xi_{1}+F_{\Omega})+A_{\Omega})|_{S_{U}}$

and the corresponding Hilbert space of homolorphic sections

(59) $\mathcal{H}_{k}:=\\{s\in
H^{0}(S_{U},L_{k})/\int_{S_{U}}|s|^{2}e^{-k\varphi_{\theta}-\varphi_{A}}dV<\infty\\}$

in the multiplier ideal induced by the current $k\widehat{\Theta}|_{S_{U}}$.
Then we recall the following result, basically proved in [Dem09, Section 13]
and references therein.

######  Theorem 6.6.

Let $\widehat{\Theta}_{k}\geq 0$ be the closed positive current on $S_{U}$
given by a family of orthonormal sections of $\mathcal{H}_{k}$. Then we have

$\nu(\widehat{\Theta},x)=\lim_{k}\frac{1}{k}\nu({\widehat{\Theta}_{k},x})$

where $x\in S_{U}$ is an arbitrary point and where we denote by
$\nu(\widehat{\Theta},x)$ the Lelong number of $\widehat{\Theta}$ at $x$.

######  Remark 6.7.

We note that in _loc. cit._ the result above is established in the setting of
bounded pseudo-convex subsets in $\mathbb{C}^{n}$, but the proof applies in
the context of Theorem 6.6, so we will not reproduce it here. As a matter of
fact, it is at this point that the pseudo-convexity of the set $S_{U}$ (cf.
Definition 6.3) is very important.

In other words, in order to evaluate the singularities of $\widehat{\Theta}$
it would suffice to have a uniform lower bound for the vanishing orders of the
sections $s\in\mathcal{H}_{k}$ as $k\to\infty$. To this end, we recall that as
a consequence of the results e.g. in [BP10] the following local version of the
_invariance of plurigenera_ holds true.

######  Theorem 6.8.

In the above set-up, any holomorphic section $s$ of the bundle $L_{k}$ extends
to $\widehat{X}_{\Omega}$ as section $\widetilde{s}$ of
$k(K_{X}+S+\Xi_{1}|_{\widehat{X}_{\Omega}}+F_{\Omega})+A_{\Omega}.$

We offer next a few explanations about 6.8 in the very particular case in
which we have to extend a section $s$ of the bundle $\displaystyle
k(K_{X_{\Omega}}+S+L)|_{S_{U}}$, where $(L,h_{L})$ is a semi-positively curved
line bundle, such that $h_{L}$ is non-singular and it is defined over
$X_{\Omega^{\prime}}$ for some $\Omega\Subset\Omega^{\prime}$. As we have seen
above, we have an ample line bundle $A_{\Omega}$ over $X_{\Omega}$, and thus,
in order to construct the extension of $s$ we need the following.

  * •

_A local version of the Ohsawa-Takegoshi extension theorem_. The statement we
need is available, cf. [DemOT].

  * •

_A finite family of holomorphic sections for the bundles_

$\displaystyle(k+r)(K_{X_{\Omega}}+S+L)+C(k)A_{\Omega}|_{X_{\Omega}}$

_for $r=0,\dots,k$ such that for each $r$, their common set of zeroes is
empty._ This is easy to see, despite of the fact that $S_{U}$ and $X_{\Omega}$
are not compact: the point is that all the bundles/metrics extend over
$X_{\Omega^{\prime}}$ and we construct our sections by a quick compactness
argument.

These two points granted, one follows the usual algorithm, see e.g.[BP10] and
the references therein.

However, in our case there is an additional level of difficulty, induced by
the presence of the $\mathbb{Q}$-divisor $\Xi_{1}$. This can also be treated
by the known techniques (i.e. work by Hacon-McKernan, Ein-Popa …), given the
fact that $S+\Xi_{1}+\Xi_{2}$ is snc and the relation (54) (and therefore the
condition (7) in _loc. cit._ is automatically satisfied).

Now, by relation(54), the extension $\widetilde{s}$ can be seen as section of
the bundle $\displaystyle k\Xi_{2}|_{\widehat{X}_{\Omega}}+A_{\Omega}$. Given
that the support of $\Xi_{2}$ is $\pi$-contractible and that the set
$\widehat{X}_{\Omega}$ is the inverse image of $\Omega$ by the map $\pi$, it
follows that the vanishing order of $\widetilde{s}$ along $\Xi_{2}$ is at
least $k-k_{0}$ (we ”loose” a fixed amount $m_{0}$ because of the ample bundle
$A_{\Omega}$).

In conclusion, it follows that we have

(60) $\widehat{\Theta}_{k}\geq(k-k_{0})[\Xi_{2}]|_{S_{U}}$

and the proof is finished by using Theorem 6.6. ∎

##  7\. Proof of Theorem 5.2

The main steps of the proof of our version of the canonical bundle formula –
Theorem 5.2 – are as follows. Let $\Xi:=\\{B\\}$ be the fractional part of $B$
and we write $B=\Xi+\lfloor B\rfloor=\Xi-\lceil-B\rceil$ as difference of two
effective $\mathbb{Q}$-divisors. We assume that the discriminant divisor
$B_{W}$ is equal to zero (we can do this without altering any of our
hypothesis). We then have the numerical identity

$K_{S/W}+\Xi+\beta\simeq f^{\star}\beta_{W}+\lceil-B\rceil.$

Next, we apply the methods already used in the proof of Theorem 5.1 in order
to construct a closed positive current $\Theta\geq 0$ in the class
corresponding to the LHS of the relation above. The said current is proved to
be singular along $\lceil-B\rceil$: this follows as consequence of the
hypothesis (b) (which replaces the fact that the map $f:S\to W$ might not be
induced by a log-resolution $\pi$).

The heart of the matter is to show the (highly non-trivial) fact that the
Lelong numbers of the difference

$\Theta-\lceil-B\rceil$

are equal to zero. To this end we adapt the method used in [Taka22] in our
context.

We start with a general discussion –and a simple result– concerning fiber
integrals.

####  7.0.1. Fiber integrals

Let $p:X\to Y$ be a proper, surjective holomorphic map, where $X$ is a
$(n+m)$-dimensional Kähler manifold and $Y$ is the unit disk in
$\mathbb{C}^{m}$. We denote by $Y_{0}\subset Y$ the set of regular values of
$p$. Let $t=(t_{1},\dots,t_{m})$ be coordinates on $\mathbb{C}^{m}$ induced by
a fixed base. Consider a $\mathbb{Q}$-line bundle $(L,h_{L})$ on the total
space $X$, endowed with a metric $h_{L}$ eventually singular, but whose
curvature is semi-positive. Let $s\in H^{0}\big{(}X,k(K_{X}+L)\big{)}$ be a
pluricanonical form with values in $kL$, where $k$ is a positive, sufficiently
divisible integer so that $kL$ is a line bundle. For each $y\in Y_{0}$ let
$\displaystyle s_{y}\in H^{0}\big{(}X_{y},k(K_{X_{y}}+L_{y})\big{)}$ be the
induced form on $X_{y}$, in the sense that

(61) $s|_{X_{y}}=s_{y}\wedge p^{\star}(dt)^{\otimes k}.$

In this setting we show that the following holds true.

######  Lemma 7.1.

We assume moreover that there exists a section $\sigma$ of a line bundle
$\Lambda$ such that the quotient $\displaystyle\frac{s}{\sigma}$ is a
holomorphic section of $k(K_{X}+L)-\Lambda$. There exists a positive constant
$C_{0}>0$ independent of $s$ such that the inequality

(62) $\int_{X_{y}}|s_{y}|^{\frac{2}{k}}e^{-\varphi_{L}}\geq
C_{0}\sup_{X_{y}}\left|\frac{s}{\sigma}\right|^{\frac{2}{k}}$

holds for any $y\in Y_{0}$ such that $|y|<\frac{1}{2}$. The norm on the RHS is
with respect to a fixed, smooth metric on $k(K_{X}+L)-\Lambda$, and an upper
bound for the constant $C_{0}$ can be obtained from the proof that follows.

###### Proof.

Let $z_{0}\in X_{y}$ be a point such that

$\sup_{X_{y}}\left|\frac{s}{\sigma}\right|^{\frac{2}{k}}=\left|\frac{s}{\sigma}\right|^{\frac{2}{k}}(z_{0})$

and let $y=f(z_{0})$ be its image. We take the local coordinates
$z=(z_{1},\dots,z_{n+m})$ and $t=(t_{1},\dots,t_{m})$ centred at $z_{0}$ and
$y$ respectively. The $t$-coordinates are defined on some open set
$\Omega\subset Y$, and the $z$-coordinates are defined on $V\subset
f^{-1}(\Omega)$ biholomorphic to the unit ball in $\mathbb{C}^{n+m}$. Let
$\omega$ be an arbitrary Kähler metric on $X$.

Corresponding to this data we define the function
$\psi:V\to\mathbb{R}\cup\\{-\infty\\}$ as follows

(63) $\omega^{n}\wedge f^{\star}(\sqrt{-1}dt\wedge
d\overline{t})=e^{\psi}\sqrt{-1}dz\wedge d\overline{z}$

where we use the notations

$\sqrt{-1}dt\wedge d\overline{t}:=\prod_{i=1}^{m}\sqrt{-1}dt_{i}\wedge
d\overline{t}_{i},\qquad\sqrt{-1}dz\wedge
d\overline{z}:=\prod_{i=1}^{m+n}\sqrt{-1}dz_{i}\wedge d\overline{z}_{i}.$

Therefore, the restriction of the form $\displaystyle e^{-\psi}\omega^{n}$ to
the fiber $X_{y}\cap V$ is equal to the measure sometimes denoted with
$\displaystyle\left|\frac{dz}{f^{\star}dt}\right|^{2}$ on $X_{y}$.

We assume that the bundles $L$ and $\Lambda$ are trivial when restricted to
$V$, and let $u\in\mathcal{O}(V)$ be the local holomorphic function
corresponding to the section $s|_{V}$. Then we clearly have the inequality

(64) $\int_{X_{y}\cap
V}|u|^{\frac{2}{k}}e^{-\varphi_{L}-\psi}\omega^{n}\leq\int_{X_{y}}|s_{y}|^{\frac{2}{k}}e^{-\varphi_{L}}.$

On the other hand, by the $L^{\frac{2}{k}}$-version of the Ohsawa-Takegoshi
theorem established in [PT18], Proposition 1.2 there exists a function
$U\in\mathcal{O}(V)$ such that

(65) $U|_{V\cap X_{y}}=u|_{V\cap
X_{y}},\qquad\int_{V}|U|^{\frac{2}{k}}e^{-\varphi_{L}}d\lambda\leq C_{\rm
univ}\int_{X_{y}\cap V}|u|^{\frac{2}{k}}e^{-\varphi_{L}-\psi}\omega^{n},$

where $C_{\rm univ}$ is a numerical constant.

Let now $\sigma_{V}\in\mathcal{O}(V)$ be the local holomorphic function
induced by the section $\sigma$ and let $N\gg 0$ be a large enough integer so
that the integral

(66) $\int_{V}\frac{d\lambda}{|\sigma_{V}|^{\frac{2}{N}}}<\infty$

is convergent. The first part of (65) combined with the fact that $z_{0}$ is
the maximum point and the mean-value inequality gives

(67) $\sup_{X_{y}}\left|\frac{s}{\sigma}\right|^{\frac{2}{k}}\leq
C\left|\frac{U}{\sigma_{V}}(z_{0})\right|^{\frac{1}{kN}}\leq
C\int_{V}\left|\frac{U}{\sigma_{V}}\right|^{\frac{1}{kN}}d\lambda$

where the first constant is due to the fixed metric on $k(K_{X}+L)-\Lambda$
and it follows -thanks to Hölder inequality- that

(68) $\sup_{X_{y}}\left|\frac{s}{\sigma}\right|^{\frac{2}{k}}\leq
C_{0}\int_{V}|U|^{\frac{2}{k}}e^{-\varphi_{L}}d\lambda,$

where $C_{0}$ depends on (67), and an upper bound for $\varphi_{L}$. This
inequality, combined with the estimate in (65) completes the proof of Lemma
7.1. ∎

####  7.0.2. Pseudo-effectivity

We remark that we can assume $B_{W}=0$, by simply replacing $B$ with
$B-f^{\star}(B_{W})$ and noticing that under the transversality hypothesis in
our statement, the new pair $(S,B)$ is sub-klt and moreover the hypothesis (b)
still holds.

Under the assumption that $B_{W}=0$, it follows from the hypothesis (c) of
Theorem 5.2 that we have

(69) $K_{S/W}+B+\beta\simeq f^{\star}\beta_{W}.$

Since the pair $(S,B)$ is sub-klt, we can write $B=\\{B\\}+\lfloor
B\rfloor:=\Xi-\lceil-B\rceil$ and therefore we obtain

(70) $K_{S/W}+\Xi+\beta\simeq f^{\star}\beta_{W}+\lceil-B\rceil,$

where $(S,\Xi)$ is klt and $\lceil-B\rceil$ is effective, with integer
coefficients. We apply Theorem 6.2 to the following data:
$X:=S,Y:=W,\alpha:=\beta,\gamma:=\beta_{W},L:=\mathcal{O}_{S}(\lceil-B\rceil)$
and finally $D:=\Xi$. It follows that there is a closed positive current
$\Theta\geq 0$ in the class (70), induced by the sections of
$\displaystyle\mathcal{O}_{S}(\lceil-B\rceil)|_{S_{w}}$ for $w\in W$ general.

We then formulate our next assertion:

######  Claim 7.2.

The inequality

$\Theta\geq\lceil-B\rceil$

holds in the sense of currents on $S$, where the RHS is interpreted as current
of integration on the divisor $\lceil-B\rceil$.

###### Proof of the Claim.

We start with a little comment: if a hypersurface $Y\subset S$ belongs to the
support of the divisor $\lceil-B\rceil$ is such that $f(Y)=W$ (i.e. $Y$ is
horizontal with respect to the map $f$), then the hypothesis (c) together with
the construction of $\Theta$ show immediately that $\Theta\geq\mu[Y]$, where
$\mu$ is the multiplicity of $\lceil-B\rceil$ along $Y$. However, things are
less clear for the vertical part of the support of $\lceil-B\rceil$, since we
only have the explicit expression of $\Theta$ over general points of $W$. It
is at this point that the techniques from the subsection 7.0.1 come into play.
The argument which follows has its origins in [BP08], as well as in [CH20].
The reason why we review it here is to show that it can be easily adapted to
the pluricanonical case, needed a bit later.

Let $w\in W$ be any regular value of the map $f$, and let $w\in\Omega\subset
W$ be a coordinate set of $W$, biholomorphic with the unit ball in
$\mathbb{C}^{m}$. Recall from the proof of Theorem 6.2 that there exists a
Hermitian line bundle $(F,h_{F})$ defined over $f^{-1}(\Omega)$, whose
associated curvature form is equal to $\beta$, and such that restricting to
$f^{-1}(\Omega)$ we have

(71) $K_{S/W}+\Xi+F\simeq\lceil-B\rceil.$

Next, let $u$ be any holomorphic section of $\displaystyle
K_{S_{w}}+(\Xi+F)|_{S_{w}}$, such that

(72) $\int_{S_{w}}|u|^{2}e^{-\varphi_{\Xi}-\varphi_{F}}=1.$

As recalled in 7.0.1, there exists a section $U$ of $K_{S}+\Xi+F$ such that

(73) $U|_{S_{w}}=u\wedge
f^{\star}(dt),\qquad\int_{f^{-1}(\Omega)}|U|^{2}e^{-\varphi_{\Xi}-\varphi_{F}}\leq
C_{0}.$

Since the canonical bundle of $W$ is trivial when restricted to $\Omega$, we
can interpret $U$ as a section of $K_{S/W}+\Xi+F$, which is the same as
$\displaystyle\lceil-B\rceil|_{f^{-1}(\Omega)}$ thanks to $\eqref{new30}$. In
particular, by (b) the quotient

$\tau:=\frac{U}{s_{\lceil-B\rceil}}$

becomes a _holomorphic_ function on $f^{-1}(\Omega)$, where $\displaystyle
s_{\lceil-B\rceil}$ is the canonical section of $\mathcal{O}(\lceil-B\rceil)$.

By Lemma 7.1 we infer the following inequality

(74) $\sup_{S_{w}}|\tau|\leq C,$

–because of the normalisation (72)– where the constant $C$ in (74) is
independent of $u$.

As consequence of (72) combined with the definition of the relative metric we
obtain

(75) $\varphi_{S/W}\leq C+\log|s_{\lceil-B\rceil}|^{2},$

from which our claim follows. ∎

####  7.0.3. Lelong numbers

Next we show that the Lelong numbers of the closed positive current

(76) $T:=\Theta-\lceil-B\rceil$

are equal to zero. To this end we will use an important result due to S.
Takayama. Actually we will ”extract” from the proof in [Taka22] the result
below (which will be useful for the proof of Theorem 5.4 as well). To begin
with, we recall the construction of a natural metric on $K_{S/W}$, cf. [MP12].

Let $z_{0}\in S$ be an arbitrary point, and $t_{0}=f(z_{0})$ be its image. We
take coordinates $z=(z_{1},\dots,z_{n+m})$ and $t=(t_{1},\dots,t_{m})$ centred
at $z_{0}$ and $t_{0}$ respectively. The $t$-coordinates are defined on some
open set $\Omega\subset W$, and the $z$-coordinates are defined on $V\subset
f^{-1}(\Omega)$. Let $\omega$ be an arbitrary Kähler metric on $S$.
Corresponding to this data we define the function $\psi$ as follows

(77) $\omega^{n}\wedge f^{\star}(\sqrt{-1}dt\wedge
d\overline{t})=e^{\psi}\sqrt{-1}dz\wedge d\overline{z}$

where we use the notations

$\sqrt{-1}dt\wedge d\overline{t}:=\prod_{i=1}^{m}\sqrt{-1}dt_{i}\wedge
d\overline{t}_{i},\qquad\sqrt{-1}dz\wedge
d\overline{z}:=\prod_{i=1}^{m+n}\sqrt{-1}dz_{i}\wedge d\overline{z}_{i}.$

We now consider a covering of $S$ and $W$ with coordinates sets as above.
Given the equality (77), the resulting functions $e^{-\psi}$ define a metric
$h$ on the relative canonical bundle $K_{S/W}$, which is general is singular.
Let $h_{0}=e^{-\psi_{0}}$ be an arbitrary, smooth metric on $K_{S/W}$. The
difference of the weights corresponding to the two metrics

$\psi_{f}:=\psi-\psi_{0}$

is a global function on $S$.

For each regular value $t\in W$ of $f$ we define the function

(78) $F(w):=\int_{S_{w}}e^{-\psi_{f}-\phi_{B}}\omega^{n}$

where $\phi_{B}:=\log|s_{B}|^{2}$ is the log of the norm of the canonical
section of the divisor $B$ (which we recall, is not necessarily effective).

We remark that so far the hypothesis $(\star)$ was not used in our arguments.
It comes into play through the following important result, established in
[Taka22]. Although it is not stated in this form explicitly, it is a direct
consequence of the proof of Theorem 3.1 in _loc. cit._

######  Theorem 7.3.

[Taka22] Assume that the hypothesis of Theorem 5.2 are satisfied, as well as
the following.

  * •

The divisor $B_{W}=0$ is zero.

  * •

given any point $z_{0}\in S$ and $w_{0}=f(z_{0})\in W$ there exist local
coordinates $(x_{1},\dots,x_{n+m})$ on $S$ centred at $z_{0}$ and
$(t_{1},\dots,t_{m})$ on $W$ centred at $w_{0}$ such that $\displaystyle
t_{i}\circ f=\prod x_{j}^{k_{ij}}$ where the $k_{ij}$ are non-negative
integers such that $k_{ij}\neq 0$ for at most one $i$ for each index $j$.

Then for any point $w_{0}\in W$ the following inequality holds

$F(w)\leq C\prod_{j}\log\frac{1}{|w_{j}|}$

where $C>0$ is a positive constant and $w$ are coordinates centred at any
$w_{0}$.

######  Remark 7.4.

For the comfort of the readers, we will provide a complete proof of Theorem
7.3 in the Appendix of this article.

######  Remark 7.5.

We note that the inequality in Theorem 7.3 holds for any $w$ such that
$\mu_{W}(w)$ belongs to the set of regular values of $f$, and the constant $C$
is uniform.

Consider $z_{0}\in S$ and $w_{0}:=f(z_{0})$, together with the corresponding
local coordinates chosen as in Theorem 7.3. By the definition of the relative
metric we have

(79)
$e^{\varphi_{S/W}(z)}\geq\frac{|f_{\lceil-B\rceil}|^{2}}{\int_{S_{w}}|s_{\lceil-B\rceil,w}|^{2}e^{-\varphi_{\Xi}-\varphi_{F}}}$

where we denote by $s_{\lceil-B\rceil,w}$ the restriction of the section
$s_{\lceil-B\rceil}$ to the fiber $S_{w}$, so that all in all the expression
$\displaystyle|s_{\lceil-B\rceil,w}|^{2}e^{-\varphi_{\Xi}-\varphi_{F}}$ is a
volume form on $S_{w}$.

Moreover, given the definition of the divisors $\Xi$ and $\lceil-B\rceil$, we
have

(80) $|s_{\lceil-B\rceil,w}|^{2}e^{-\varphi_{\Xi}-\varphi_{F}}\leq
Ce^{-\psi_{f}-\phi_{B}}\omega^{n}|_{S_{w}}$

for some constant $C>0$ (remark that the proximity of $w$ to the singular loci
of $f$ is luckily irrelevant for the uniformity of $C$).

Then we have the following inequality for the potential $\varphi_{T}$ of the
current $T$ introduced in (76)

(81) $e^{\varphi_{T}(z)}\geq\frac{C}{F(w)}$

where $w=f(z)$. The second bullet in Theorem 7.3 together with the upper bound
for the function $F$ provided by this result show that

(82) $\nu(T,z_{0})=0,$

and therefore Theorem 5.2 is completely proved, modulo the regularisation
theorem in [Dem92] (a class containing a closed positive current whose Lelong
numbers are equal to zero is nef)

######  Remark 7.6.

In general, a nef cohomology class does not necessarily contain a closed
positive current with zero Lelong numbers. Therefore, the property we are
establishing in the proof of Theorem 5.2 is stronger than neffness. Moreover,
we construct the current $T$ is a very explicit manner, so in principle it
should be possible to further analyze its singularities.

######  Remark 7.7.

We expect that a more general form of Theorem 5.2 holds true, namely one
should obtain a version of this result in the absence of the hypothesis
$\star$. This promises to be a difficult problem (given the arguments invoked
to prove it in [Taka22]).

##  8\. Proof of Theorem 5.4

The main steps of the proof that follows are the same as in the previous
subsection. To begin with, recall that by hypothesis (d) we have a Kähler
metric $\omega$ on $S$ such that

(83) $\omega=f^{\star}g+\theta,$

where $g$ is a Kähler metric on $W$ and $\theta$ is a _rational_ $(1,1)$-form
on $S$.

Consider a positive integer $m_{0}$ divisible enough such that $m_{0}\theta\in
H^{2}(S,\mathbb{Z})$. For each $k\geq 1$ the class

(84) $\beta_{k}:=\beta+\frac{m_{0}}{k}\omega$

contains a positive representative, since $\beta$ is nef. Therefore, we can
write

(85) $K_{S/W}+B+\beta_{k}=f^{\star}(\gamma_{k})+A_{k}$

where

$\gamma_{k}:=\gamma+\frac{m_{0}}{k}g,\qquad\frac{m_{0}}{k}\theta\in
c_{1}(A_{k})$

i.e. $A_{k}$ is a $\mathbb{Q}$-bundle such that $kA_{k}$ becomes a holomorphic
line bundle which admits a metric $h_{k}$ whose curvature is precisely
$m_{0}\theta$.

We therefore find ourselves in the framework of Theorem 6.2: we obtain a
closed positive current $\Theta_{k}$ belonging to the class $\displaystyle
K_{S/W}+\Xi+\beta_{k}$, constructed by using the global sections of

$k(A_{k}+\lceil-B\rceil)|_{S_{w}}$

for $w\in W$ generic.

Moreover, by hypothesis (b’) together with the arguments in sub-section 7.0.1
and the Claim 7.2, we infer that we have

(86) $\Theta_{k}\geq(1-\delta_{k})\lceil-B\rceil$

where $\delta_{k}\to 0$ as $k\to\infty$.

Finally, we analyse next the the singularities of the closed positive current

(87) $T_{k}:=\Theta_{k}-(1-\delta_{k})\lceil-B\rceil,\qquad T_{k}\in
K_{S/W}+B+\beta_{k}+\delta_{k}\lceil-B\rceil.$

To this end, we first observe that for each co-ordinate ball $\Omega\subset W$
the restriction

(88) $kA_{k}|_{V}$

admits a metric whose curvature is equal to $m_{0}\omega|_{V}$, where
$V:=f^{-1}(\Omega)$. We can assume that $m_{0}$ is large enough, so that the
bundle in (88) is generated by its global sections.

As in the proof of Theorem 5.2 assume that the morphism $f$ satisfies the
additional hypothesis in the statement of Theorem 7.3. We consider $z_{0}\in
S$ such that $w_{0}:=f(z_{0})\in\Omega$, and let $x$ and $t$ be coordinates
having the second bullet property in Theorem 7.3. Let $u$ be a holomorphic
section of the bundle $kA_{k}|_{V}$, such that $z_{0}\not\in(u=0)$. The
product

$\rho:=u\otimes s_{\lceil-B\rceil}^{\otimes k}$

can be interpreted as section of $\displaystyle k(K_{S/W}+\Xi+F_{k}|_{V})$
(notations as in Section 6) and by the definition of the $L^{2/k}$-metric we
get

(89)
$e^{\varphi_{S/W}(z)}\geq\frac{|f_{\rho}(z)|^{\frac{2}{k}}}{\int_{S_{w}}|\rho_{w}|^{\frac{2}{k}}e^{-\varphi_{\Xi}-\varphi_{F}}}$

for any point $z$ near $z_{0}$ and $w:=f(z)$. In (88) we denote by $f_{\rho}$
the local holomorphic function corresponding to the section $\rho$.

Inequality (80) still applies, so we infer that

(90) $\nu(T_{k},z_{0})\leq\delta_{k}\nu(\lceil-B\rceil,z_{0})$

given the definition of $\rho$, provided that we assume beforehand that
$B_{W}=0$ so that we can use Theorem 7.3. Now the quantity $\delta_{k}$ is
independent of the point $z_{0}$, and it follows that

(91) $\sup_{z\in Z}\nu(T_{k},z)\leq\delta_{k}C,$

where $C$ is the maximum multiplicity of the divisor $\lceil-B\rceil$ at
points of $S$.

We next use [Dem92] in order to obtain a smooth representative
$\widetilde{T}_{k}\in K_{S/W}+B+\beta_{k}+\delta_{k}\lceil-B\rceil$ such that

$\widetilde{T}_{k}\geq-C_{S}\delta_{k}\omega$

where the constant $C_{S}$ only depends on the geometry of $(S,\omega)$. Since
$k$ was arbitrary, it follows that the class

$K_{S/W}+B+\beta=f^{\star}\gamma$

is nef, hence the same is true for $\gamma$ by Theorem 6.1.

##  9\. Appendix

The main result of this subsection is a direct argument for Theorem 7.3. We
first fix a few notations:

  * •

$U\subset S$ is an open subset of $S$ small enough so that we have the
coordinates $x=(x_{1},\dots,x_{n+m})$ on $U$ with the property that

$f_{1}(x)=\prod_{i=1}^{l_{1}}x_{i}^{a_{i}},\quad
f_{2}(x)=\prod_{i=l_{1}+1}^{l_{2}}x_{i}^{a_{i}},\quad\dots,\quad
f_{m}(x)=\prod_{i=l_{m-1}+1}^{l_{m}}x_{i}^{a_{i}}$

where $a_{i}\geq 1$ and $0=l_{0}<l_{1}<\dots<l_{m}\leq n+m$ and
$f_{i}:=t_{i}\circ f$ for $i=1,\dots,m$.

  * •

For every multi-index $I:=(i_{1},\dots,i_{m})$ such that $i_{k}\in
J_{k}:=\\{l_{k}+1,\dots l_{k+1}\\}$ we define a form of by-degree $(n,n)$
through the formula

$\sqrt{-1}dx_{I}\wedge d\overline{x}_{I}:=\Big{(}\prod_{k=1}^{m}\prod_{i\in
J_{k},i\neq i_{k}}\sqrt{-1}dx_{i}\wedge
d\overline{x}_{i}\Big{)}\wedge\prod_{i=l_{m}+1}^{n+m}\sqrt{-1}dx_{i}\wedge
d\overline{x}_{i}$

  * •

$\omega:=\sum_{i}\sqrt{-1}dx_{i}\wedge d\overline{x}_{i}$ is the local version
of the reference Kähler metric. We set

$\omega_{l}:=\sum_{I}\sqrt{-1}dx_{I}\wedge d\overline{x}_{I}$

and up to a constant, we see that we have

$\omega_{l}=\big{(}\sum_{i=1}^{l_{1}}\sqrt{-1}dx_{i}\wedge
d\overline{x}_{i}\big{)}^{l_{1}-1}\wedge\dots\wedge\big{(}\sum_{i=l_{m-1}+1}^{l_{m}}\sqrt{-1}dx_{i}\wedge
d\overline{x}_{i}\big{)}^{l_{m}-1}\wedge\bigwedge_{i=l_{m}+1}^{n+m}\sqrt{-1}dx_{i}\wedge
d\overline{x}_{i}$

We now proceed to the evaluation of the function $\psi$ defined in (77). The
first remark is that

(92)
$\frac{df_{i}}{f_{i}}=\sum_{j=l_{i-1}+1}^{l_{i}}a_{i}\frac{dx_{i}}{x_{i}}$

and therefore a few simple calculations that we skip show that we have

(93)
$e^{\psi}\simeq\big{(}\prod_{i=1}^{m}|f_{i}|^{2}\big{)}\prod_{k=1}^{m}\sum_{i=l_{k-1}+1}^{l_{k}}\frac{1}{|x_{i}|^{2}}$

where the symbol $"\simeq"$ in (93) means that the quotient of the two
functions is two-sided bounded away from zero.

The hypothesis $B_{W}=0$ shows that the inequality

(94) $e^{-\psi-\phi_{B}}\leq
C\prod_{k=1}^{m}\frac{1}{|X_{k}|^{2d_{k}-2}}\prod_{j\in
J}\frac{1}{|x_{j}|^{2-2\beta_{j}}}=:\Lambda(x)$

holds, where $\displaystyle d_{k}:=l_{k}-l_{k-1}$ and
$\displaystyle|X_{k}|^{2}:=\sum_{i=l_{k-1}+1}^{l_{k}}|x_{i}|^{2}$. Moreover,
we have $J\subset\\{l_{m}+1,\dots,m+n\\}$ and $\beta_{j}>0$ for each $j$.

On the other hand, by the Poincaré-Lelong formula, the local version of the
quantity we have to analyse equals

(95)
$F_{U}(t):=\int_{U}\theta(x)e^{-\psi-\phi_{B}}\bigwedge_{i=1}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}\wedge\omega^{n}$

where $\theta$ is a truncation function defined as follows

$\theta(x):=\theta(|X^{\prime}|^{2})\prod_{i=1}^{m}\theta(|X_{i}|^{2})$

and $\displaystyle|X^{\prime}|^{2}:=\sum_{i=l_{m}+1}^{n+m}|x_{i}|^{2}$.

Now, given the expression of the functions $f_{i}$, we infer that the integral
(95) is bounded by the following expression

(96)
$\int_{U}\theta(x)\Lambda(x)\omega_{l}\wedge\bigwedge_{i=1}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}$

up to a constant independent of $t$, where we recall that the function
$\Lambda$ was defined in (94). Indeed, the equality

$\bigwedge_{i=1}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}\wedge\omega^{n}=\omega_{l}\wedge\bigwedge_{i=1}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}$

holds modulo a constant, because $f_{i}$ only depends on the variables
$\displaystyle x_{l_{i-1}+1},\dots,x_{l_{i}}$.

In order to evaluate (96) we use integration by parts: this expression is the
same as

(97)
$\int_{U}\log|t_{1}-f_{1}(x)|^{2}dd^{c}\big{(}\theta\Lambda\big{)}\wedge\omega_{l}\wedge\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}$

and we will consider first the term containing
$\displaystyle\theta(x)dd^{c}\Lambda$.

Since $\displaystyle dx^{\alpha}\wedge
d\overline{x}^{\beta}\wedge\omega_{l}\wedge\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}=0$
if $\max(\alpha,\beta)\geq l_{1}+1$, we see that the integral

$dd^{c}\Lambda\wedge\omega_{l}\wedge\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}$

is equal to

(98) $\small\prod_{k=2}^{m}\frac{1}{|X_{k}|^{2d_{k}-2}}\prod_{j\in
J}\frac{1}{|x_{j}|^{2-2\beta_{j}}}dd^{c}\frac{1}{|X_{1}|^{2d_{1}-2}}\wedge\omega_{l}\wedge\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}.$

We recall that we have the equality

$dd^{c}\frac{1}{|X_{1}|^{2d_{1}-2}}\wedge dd^{c}|X_{1}|^{2}=\delta_{0},$

the Dirac distribution at the origin in $\mathbb{C}^{l_{1}}$, so that we have

(99) $\int_{U}\log|t_{1}-f_{1}(x)|^{2}\theta
dd^{c}\big{(}\Lambda\big{)}\wedge\omega_{l}\wedge\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}=$
$\log\frac{1}{|t_{1}|^{2}}\int_{U^{\prime}}\theta_{1}\Lambda_{1}\bigwedge_{i=2}^{m}dd^{c}\log|t_{i}-f_{i}(x)|^{2}\wedge\omega_{l^{\prime}}$

where $U^{\prime}\subset\mathbb{C}^{n+m-l_{1}}$ is the unit ball,
$x^{\prime}=(x_{l_{1}+1},\dots,x_{n+m})$ and

$\theta_{1}(x^{\prime}):=\theta(|X^{\prime}|^{2})\prod_{i=2}^{m}\theta(|X_{i}|^{2}),\quad
l^{\prime}=(l_{2},\dots,l_{m}),\qquad\Lambda_{1}:=\prod_{k=2}^{m}\frac{1}{|X_{k}|^{2d_{k}-2}}\prod_{j\in
J}\frac{1}{|x_{j}|^{2-2\beta_{j}}}.$

A quick argument by induction gives the expected estimate for the RHS of (99).

The remaining terms involve the differential of $\displaystyle
X_{1}\to\theta(|X_{1}|^{2})$, on the support of which
$\displaystyle\frac{1}{|X_{1}|^{2d_{1}-2}}$ is smooth.

In conclusion, after the first integration by parts we get

(100)
$\int_{U}\theta_{1}(X_{1})\psi_{1}(X_{1})\log|t_{1}-f_{1}(x)|^{2}\theta_{1}(x)\Lambda_{1}(X)\bigwedge_{i=1}^{l_{1}}dd^{c}|X_{1}|^{2}\wedge\omega_{l^{\prime}}\wedge\bigwedge_{j=2}^{m}dd^{c}\log|t_{j}-f_{j}(x)|^{2}$

modulo the terms (99).

We now repeat this procedure, integrating by parts using the factor
$\displaystyle dd^{c}\log|t_{2}-f_{2}(x)|^{2}$. Notice that, because of the
form $\displaystyle\bigwedge_{i=1}^{l_{1}}dd^{c}|X_{1}|^{2}$ the derivatives
of the function

$\theta_{1}(X_{1})\psi_{1}(X_{1})\log|t_{1}-f_{1}(x)|^{2}$

don’t come into play.

Thus, after a finite number of steps the last term we still have to deal with
equals

(101)
$\int_{U}\theta(x)\psi(x)\prod_{i=1}^{m}\log|t_{i}-f_{i}(x)|^{2}\frac{d\lambda}{\prod_{j\in
J}{|x_{j}|^{2-2\beta_{j}}}}$

where $\psi$ is a truncation function. Since $\beta_{j}>0$, the integral (101)
is uniformly bounded as soon as we fix a bound for $|t|$. Collecting all the
terms, Theorem 7.3 is proved.

######  Remark 9.1.

We consider the function

$f:\mathbb{C}^{3}\to\mathbb{C}^{2},\qquad f(z)=(z_{1}z_{2},z_{1}z_{3}).$

Let $\theta$ be a truncation function which equals 1 near the origin of
$\mathbb{C}^{3}$. A simple calculation shows that we have

$\int_{\mathbb{C}^{3}}\theta e^{-\psi}dd^{c}\log|t_{1}-f_{1}(x)|^{2}\wedge
dd^{c}\log|t_{2}-f_{2}(x)|^{2}\wedge\omega\simeq\frac{1}{|t_{1}|^{2}+|t_{2}|^{2}}$

therefore the hypothesis $(\star)$ is crucial.

######  Remark 9.2.

Let $f:S\to W$ be a morphism such that all the hypothesis of Theorem 5.2
except perhaps for $(\star)$ are satisfied. Then the techniques developed in
our article show that the current $T$ in (76) can only have positive Lelong
numbers along an analytic subset of $X$ which projects in codimension two. The
reason is that one can construct a subset $W_{0}\subset W$ whose codimension
is at least two, and such that the morphism $f$ satisfies $(\star)$ in the
complement of $W_{0}$. It follows that if $\operatorname{dim}W=2$, Theorem 5.2
holds true for morphisms which only satisfy the assumptions (a), (b), (c),
thanks to the following general fact.

######  Theorem 9.3.

[Dem92] Let $X$ be a compact complex manifold, and let $T$ be a $(1,1)$–closed
positive current on $X$. If the level sets

$E_{c}(T):=\\{x\in X:\nu(T,x)\geq c\\}$

have dimension zero for any $c>0$, then the cohomology class of $T$ is nef.

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