11

# On the Canonical Bundle Formula and Adjunction for Generalized Kähler Pairs

Christopher Hacon  Department of Mathematics
University of Utah
155 S 1400 E
Salt Lake City, Utah 84112, USA<EMAIL_ADDRESS>and  Mihai Păun  Institut
für Mathematik
Universität Bayreuth
95440 Bayreuth, Germany<EMAIL_ADDRESS>
###### Abstract.

In this article we prove analogs of Kawamata’s canonical bundle formula,
Kawamata subadjunction and plt/lc inversion of adjunction for generalized
pairs on Kähler varieties. We also show that a conjecture of [BDPP13] in
dimension $n-1$ implies that the cone theorem holds for any $n$-dimensional
Kähler generalized klt pair $(X,B+\boldsymbol{\beta})$.

Christopher Hacon was partially supported by the NSF research grants no:
DMS-1952522, DMS-1840190, DMS-2301374.

Mihai Păun gratefully acknowledges support from the DFG

###### Contents

  1. 1 Preliminaries
    1. 1.1 Boundary and moduli parts
  2. 2 Adjunction of generalized pairs
    1. 2.1 Towards a canonical bundle formula
    2. 2.2 Adjunction
  3. 3 Cone Theorem for generalized klt pairs
  4. 4 Null loci
    1. 4.1 Closed positive currents on normal complex spaces
    2. 4.2 Main results
      1. 4.2.1 Currents with admissible singularities
  5. 5 On subadjunction and the canonical bundle formula
  6. 6 Positivity of the relative adjoint transcendental classes
    1. 6.1 Closed positive currents in twisted relative canonical classes
    2. 6.2 Singularities of the metric
  7. 7 Proof of Theorem 5.2
    1. 7.0.1 Fiber integrals
    2. 7.0.2 Pseudo-effectivity
    3. 7.0.3 Lelong numbers
  8. 8 Proof of Theorem 5.4
  9. 9 Appendix

Generalized pairs have been playing an increasingly prominent role in higher
dimensional birational geometry (see eg. [Birkar21] and references therein).
Their analytic counterparts were introduced in [DHY23] (see Definition 1.1)
where it is shown that the minimal model program for compact Kähler
generalized klt 3-fold pairs holds. Note that even in the projective case,
Definition 1.1 is more general than the usual definition of generalized pairs
since the ”nef” part is only assumed to be a positive (1,1) form (instead of a
nef divisor). This is extremely useful in the Kähler context as, in many
instances, we can replace the use of an arbitrary ample (or big) divisor by a
(modified) Kähler class. In particular, using this extra flexibility, [DHY23]
shows the finiteness of minimal models for compact Kähler generalized klt
3-fold pairs of general type, and that klt Calabi-Yau Kähler 3-folds are
connected by finite sequences of flops. The theory of generalized pairs makes
sense in all dimensions and it is hoped that many results from the projective
minimal model program will carry through to this setting. In this paper we
perform the first steps in this direction. We show that adjunction and
inversion of adjunction hold for generalized pairs (both in the plt and lc
cases), we prove a canonical bundle formula for generalized klt Kähler pairs
and we show that assuming the BDPP conjecture in dimension $n-1$, then the
cone theorem for generalized pairs holds in dimension $n$ (and in particular
it holds unconditionally in dimension $4$). More precisely, we show the
following.

######  Theorem 0.1.

Let $(X,B+\boldsymbol{\beta})$ be a generalized pair and $S$ a component of
$B$ of coefficient 1 with normalization $\nu:S^{\nu}\to S$. Then
$(X,B+\boldsymbol{\beta})$ is generalized lc (resp. generalized plt) on a
neighborhood of $S$ iff $({S^{\nu}},B_{S^{\nu}}+\boldsymbol{\beta}_{S^{\nu}})$
is generalized lc (resp. $S$ is normal and
$({S},B_{S}+\boldsymbol{\beta}_{S})$ is generalized klt).

Next we turn our attention to the following generalization of Kawamata’s
adjunction theorem for generalized pairs (cf. [Kawamata98, Theorem 1]).

######  Theorem 0.2.

Let $(X,B+\boldsymbol{\beta})$ be a generalized log canonical pair such that
$(X,B^{\prime}+\boldsymbol{\beta}^{\prime})$ is a generalized klt pair and
$W\subset X$ is a minimal log canonical center of $(X,B+\boldsymbol{\beta})$.
Then $W$ is normal and
$(K_{X}+B+\boldsymbol{\beta})|_{W}=K_{W}+B_{W}+\boldsymbol{\beta}_{W}$ is a
generalized klt pair.

In order to prove this result, it is necessary to prove the following
canonical bundle formula (cf. Theorem 2.3).

######  Theorem 0.3.

Let $f:X\to Y$ be a projective morphism of compact normal Kähler varieties
such that $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$ and
$(X,B+\boldsymbol{\beta})$ is a generalized klt (or generalized lc) pair. If
$\gamma\in H^{1,1}_{\rm BC}(Y)$ is such that
$[K_{X}+B+\boldsymbol{\beta}_{X}]=f^{*}\gamma$ then
$\gamma=[K_{Y}+B_{Y}+\boldsymbol{\beta}_{Y}]$ where
$(Y,B_{Y}+\boldsymbol{\beta}_{Y})$ is a generalized klt (or generalized lc)
pair.

Note that we expect semistable reduction to hold (unconditionally) for
morphisms of compact analytic varieties, however the necessary references are
not yet available. In the case of projective morphisms, we can deduce
semistable reduction from the algebraic case (see [AK00], [Karu99]). It is
possible that the results of [BdSB23] are already sufficient for our purposes,
but the proof would appear to be more involved and we do not pursue it here.
The proof of this result heavily uses Theorem 6.2 (which is a generalization
of a result of Guenancia [Gue20]), which roughly speaking, states that
$\boldsymbol{\beta}_{Y}$ is pseudo-effective. By [DHP22, Theorem 2.36], to
show that $\boldsymbol{\beta}_{Y}$ is b-nef, it suffices to show that
$\boldsymbol{\beta}_{Y}|_{Z}$ is pseudo-effective for any subvariety $Z\subset
X$. This can be checked by using semistable reduction and applying Theorem
6.2.

Finally, assuming a key conjecture of Boucksom-Demailly-Paun-Peternell
[BDPP13, Conjecture 0.1], we show that the cone theorem for generalized klt
pairs holds in arbitrary dimension (and unconditionally for pseudo-effective
pairs in dimension $\leq 4$). This provides some evidence that the minimal
model program holds in arbitrary dimension for generalized klt pairs. We refer
the reader to [DHY23] for the generalized klt minimal model program in
dimensions $\leq 3$.

######  Conjecture 0.4.

Let $X$ be a compact Kähler manifold. Then the canonical class $K_{X}$ is
pseudoeffective if and only if $X$ is not uniruled (i.e. not covered by
rational curves).

Note that the above conjecture is known to hold in dimension $\leq 3$.
Following ideas from [CH20] and using Theorem 0.3 we then prove the following
result.

######  Theorem 0.5.

Assume that Conjecture 0.4 holds in dimension $n$ (resp. in dimension $n-1$).
Let $X$ be a compact $\mathbb{Q}$-factorial Kähler variety of dimension $n$
such that $(X,B+\boldsymbol{\beta})$ is generalized klt (resp. and
$K_{X}+B+\boldsymbol{\beta}_{X}$ is pseudo-effective), then there are at most
countably many rational curves $\\{\Gamma_{i}\\}_{i\in I}$ such that

$\overline{\rm NA}(X)=\overline{\rm NA}(X)_{K_{X}+B+\boldsymbol{\beta}_{X}\geq
0}+\sum_{i\in I}\mathbb{R}^{+}[\Gamma_{i}],$

where $0<-(K_{X}+B+\boldsymbol{\beta}_{X})\cdot\Gamma_{i}\leq 2n$. Moreover,
if $B+\boldsymbol{\beta}_{X}$ (or $K_{X}+B+\boldsymbol{\beta}_{X}$) is big,
then $I$ is finite.

We now turn to a more detailed description of some of the key results in this
paper. The most important results in this paper concern versions of the
canonical bundle formula (see Theorems 0.3 and 5.2). The typical set up for
the canonical bundle formula is an algebraic fiber space $f:X\to Y$ where say
$X,Y$ are normal projective varieties, $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$,
and a log canonical pair $(X,B)$ such that $K_{X}+B\sim_{\mathbb{Q},Y}0$. We
can then write

$K_{X}+B\sim_{\mathbb{Q}}f^{*}(K_{Y}+B_{Y}+M_{Y})$

where $B_{Y}$ is the boundary part that measures the singularities of $f$, and
the moduli part $M_{Y}$ is a $\mathbb{Q}$-divisor class which measures the
variation of the fibers of $f$. For example if $f$ is an elliptic fbration,
then $M_{Y}=j^{*}\mathcal{O}_{\mathbb{P}^{1}}(\frac{1}{12})$ where $j$ denotes
the $j$ function, and if $X_{P}$ is a smooth fiber of multiplicity $m$ over a
codimension 1 point $P\in Y$ and $B=0$, then the coefficient of $B_{Y}$ along
$P$ is $1-\frac{1}{m}$.

The canonical bundle formula, roughly speaking, states that if the morphism
$f$ is sufficiently well prepared (eg. $B=0$ and all fibers have simple normal
crossings), then $(Y,B_{Y})$ is log canonical and the the moduli part $M_{Y}$
is a nef $\mathbb{Q}$-divisor. In particular $(Y,B_{Y})$ is a generalized log
canonical pair (and in fact this is the key motivation for introducing
generalized pairs [BZ16]).

Results along this line are established in [Kawamata98], see [Kollar07] for a
detailed discussion. The positivity of the nef part is deduced from general
positivity properties of pushforwards of the canonical bundle (see
[Kollar86]). The intuition here is that after performing several reductions,
we can in fact assume that the moduli part coincides with $f_{*}\omega_{X/Y}$.

While the canonical bundle formula has a large number of extremely important
applications (eg. to sub-adjunction [Kawamata98]), it is clear that in order
to run proofs by induction on the dimension, it is important to establish
versions of the canonical bundle formula that work for generalized pairs of
the form $(X,B+M)$. This is achieved in [Fil20]. Note that on projective
varieties, nef classes are limits of ample divisors and hence one hopes the
result for generalized pairs [Fil20] follow as a limit of the result for the
usual pairs [Kawamata98].

In the Kähler context it is more practical to work with generalized pairs
$(X,B+\beta)$ where $\beta\in H^{1,1}_{\rm BC}(X)$ is a nef class (see
Definition 1.1) and $f:X\to Y$ is a holomorphic map of Kähler manifolds such
that $K_{X}+B+\beta=f^{*}\gamma$ for some $\gamma\in H^{1,1}_{\rm BC}(Y)$. One
can then define the boundary $B_{Y}$ and moduli parts $\beta_{Y}\in
H^{1,1}_{\rm BC}(Y)$ just as in the projective case. Unluckily, it does not
follow that $\beta$ is a limit of $\mathbb{Q}$-divisors and hence the
arguments from the projective case do not apply in the Kähler case.

Our strategy is to first prove an analog of the positivity of
$f_{*}\omega_{X/Y}$. We show that if $K_{X}+B+\beta=f^{*}\gamma$ and
$(X,B+\beta)$ is generalized log canonical, then $K_{X/Y}+B+\beta$ is pseudo-
effective (the precise statement is contained in Theorem 2.2, which is an easy
consequence of Theorem 6.2 that generalizes [Gue20]). Once the morphism
$f:X\to Y$ is sufficiently prepared (and $B$ is re-chosen appropriately), we
have $K_{X/Y}+B+\beta=f^{*}\beta_{Y}$ and hence $\beta_{Y}$ is also pseudo-
effective. By [DHP22], it is known that to show that $\beta_{Y}$ is nef, it
suffices to show that $\beta_{Y}|_{W}$ is nef where $W$ is (the normalization
of) any subvariety of $Y$. To verify this we consider $f_{W}:X_{W}\to W$
(where $(\ldots)_{W}$ denotes restriction over $W$) and then apply Theorem 2.2
to the induced pair $(X_{W},B_{W}+\beta_{W})$. For the details of the proof
see Theorem 2.3. Note that for technical reasons we have to assume that $f$ is
a projective morphism, but we expect the result will hold without this
assumption.

Therefore, the technical heart of this paper is Theorem 6.2 (which is the key
ingredient in the proof of Theorem 2.2). To gain some intuition, note that in
the setting of Theorem 6.2 we consider $K_{X/Y}+B+\beta\equiv f^{*}\gamma+L$
where $L$ is a $\mathbb{Q}$-line bundle such that $\kappa(L|_{X_{y}})\geq 0$
for general $y\in Y$. Restricting over open subsets $U\subset Y$ we may
trivialize $\gamma$ and treat $\beta|_{X_{U}}$ as a $\mathbb{Q}$-line bundle
$F_{U}=\mathcal{O}_{X_{U}}(L-K_{X/Y}-B)$. Following [PT18], we construct a
positive current $\Theta_{U}\geq 0$ fiber-wise from the $m$-th root of the
sections of $mL|_{X_{y}}$ (for $m>0$ sufficiently big and divisible). These
currents then glue together to give a positive current $\Theta\equiv
K_{X/Y}+B+\beta$.

Finally, we remark that it is possible to give direct arguments with analytic
techniques to prove stronger versions of the canonical bundle formula Theorem
2.3. In fact Theorem 5.2 shows that if we further assume that $\beta$ contains
a smooth positive representative, then we can conclude the stronger fact that
$\beta_{Y}$ is a closed positive current with zero Lelong numbers.

Acknowledgment. The authors would like to thank J. Cao, O. Das and M. Temkin
for useful communications. Thanks go equally to S. Boucksom and C.-M. Pan for
interesting discussions about Lelong numbers.

##  1\. Preliminaries

Here we recall some definitions and results from [DHY23]. We will say that $S$
is relatively compact if $S\subset S^{\prime}$ is an open subset whose closure
is compact. Similarly $\pi:X\to S$ is a proper morphism to a relatively
compact space $S$ if there is a proper morphism $\pi^{\prime}:X^{\prime}\to
S^{\prime}$ where $S\subset S^{\prime}$ is an open subset whose closure is
compact and $X=\pi^{\prime-1}(S)$.

######  Definition 1.1.

Let $\pi:X\to S$ be a proper morphism of normal Kähler varieties such that $S$
is relatively compact and $X$ is a normal compact Kähler variety,
$\nu:X^{\prime}\to X$ a resolution of singularities, $B^{\prime}$ an
${\mathbb{R}}$-divisor on $X^{\prime}$ with simple normal crossings support,
and $\beta^{\prime}\in H^{1,1}_{\rm BC}(X^{\prime})$, such that

  1. (1)

$B:=\nu_{*}B^{\prime}\geq 0$,

  2. (2)

$[\beta^{\prime}]\in H^{1,1}_{\rm BC}(X^{\prime})$ is nef over $S$, and

  3. (3)

$[K_{X^{\prime}}+B^{\prime}+\beta^{\prime}]=\nu^{*}\gamma$, where $\gamma\in
H^{1,1}_{\rm BC}(X)$.

Then we let $\beta=\nu_{*}\beta^{\prime}$ and we say that
$\nu:(X^{\prime},B^{\prime}+\beta^{\prime})\to(X,B+\beta)$ is a generalized
pair (over $S$). We will often abuse notation and say that $(X/S,B+\beta)$ (or
$(X,B+\beta)$) is a generalized pair (over $S$) and
$\nu:(X^{\prime},B^{\prime}+\beta^{\prime})\to(X,B+\beta)$ is a resolution. If
moreover ${\rm Ex}(\nu)$ is a divisor such that ${\rm Ex}(\nu)+B^{\prime}$ has
simple normal crossings, then we say that $\nu$ is a log resolution and if
${\rm Ex}(\nu)$ supports a relatively ample divisor, then $\nu$ is projective.
We will often assume that $X=S$ and omit $\pi:X\to S$.

######  Remark 1.2.

Note that we can define the corresponding nef b-(1,1) form
$\boldsymbol{\beta}:=\overline{\beta^{\prime}}$ as follows. For any bi-
meromorphic morphism $p:X^{\prime\prime}\to X^{\prime}$ we define
$\boldsymbol{\beta}_{X^{\prime\prime}}=p^{*}\beta^{\prime}$ and for any bi-
meromorphic morphism $q:X^{\prime\prime}\to X^{\prime\prime\prime}$ we let
$\boldsymbol{\beta}_{X^{\prime\prime\prime}}=q_{*}\boldsymbol{\beta}_{X^{\prime\prime}}$.
Using the projection formula, one easily checks that
$q_{*}\boldsymbol{\beta}_{X^{\prime\prime}}$ is well defined (i.e.
$\boldsymbol{\beta}_{X^{\prime\prime\prime}}$ does not depend on the choice of
the common resolution $X^{\prime\prime}$ of $X^{\prime}$ and
$X^{\prime\prime\prime}$) and that for any bi-meromorphic morphism $r:X_{1}\to
X_{2}$ of birational models of $X$, we have
$r_{*}\boldsymbol{\beta}_{X_{1}}=\boldsymbol{\beta}_{X_{2}}$. We say that
$\overline{\beta^{\prime}}$ descends to $X^{\prime}$. Note that for any bi-
meromorphic morphism $p:X^{\prime\prime}\to X^{\prime}$, we also have
$\overline{\beta^{\prime}}=\overline{\beta^{\prime\prime}}$ where
$\beta^{\prime\prime}=\boldsymbol{\beta}_{X^{\prime\prime}}$, and so
$\boldsymbol{\beta}$ also descends to $X^{\prime\prime}$.

Similarly, if $\nu:Y\to X^{\prime}$ is a proper morphism, then write
$K_{Y}+B_{Y}=\nu^{*}(K_{X^{\prime}}+B^{\prime})$. For any proper morphism
$\mu:Y\to Y^{\prime}$ we have $B_{Y^{\prime}}=\mu_{*}B_{Y}$. In this way we
have defined a b-divisor $\mathbf{B}$ (whose trace $\mathbf{B}_{Y}$ on $Y$ is
$B_{Y}$). Since the b-divisor
$\mathbf{K+B}=\overline{K_{X^{\prime}}+B_{X^{\prime}}}$ and the b-$(1,1)$-form
$\beta$$=\overline{\beta_{X^{\prime}}}$ descend to $X^{\prime}$, we say that
the generalized pair $(X,B+\beta)$ descends to $X^{\prime}$.

We will often denote the generalized pair
$\nu:(X^{\prime},B^{\prime}+\beta^{\prime})\to(X,B+\beta)$ by
$(X,B+\boldsymbol{\beta})$ where
$\boldsymbol{\beta}=\overline{\beta^{\prime}}$. Note that then
$\beta^{\prime}=\boldsymbol{\beta}_{X^{\prime}}$ and
$B^{\prime}=\nu^{*}(K_{X}+B+\beta)-(K_{X^{\prime}}+\beta^{\prime})$.

We define the _generalized discrepancies_ $a(P;X,B+\boldsymbol{\beta})=-{\rm
mult}_{P}(\mathbf{B}_{Y})$, where $P$ is a prime divisor on a bimeromorphic
model $Y$ of $X$. We say that $(X,B+\boldsymbol{\beta})$ is _generalized klt_
or generalized Kawamata log terminal (resp. _generalized lc_ or generalized
log canonical) if for any log resolution $\nu:X^{\prime}\to X$, we have
$\lfloor\mathbf{B}_{X^{\prime}}\rfloor\leq 0$, i.e.
$a(P;X,B+\boldsymbol{\beta})>-1$ for all prime divisors $P$ over $X$ (resp.
$a(P;X,B+\boldsymbol{\beta})\geq-1$ for all prime divisors $P$ over $X$). This
can be checked on a single given log resolution. We say that
$(X,B+\boldsymbol{\beta})$ is _generalized dlt_ (divisorially log terminal) if
there is an open subset $U\subset X$ such that
$(U,(B+\boldsymbol{\beta})|_{U})$ is a log resolution (of itself) and $-1\leq
a(P;X,B+\boldsymbol{\beta})\leq 0$ for any prime divisor $P$ on $U$ and
$-1<a(P;X,B+\boldsymbol{\beta})$ for any prime divisor $P$ over $X$ with
center contained in $X\setminus U$.

######  Lemma 1.3.

Let $(X,B+\boldsymbol{\beta})$ be a generalized klt (resp. generalized dlt)
variety. If $K_{X}+B$ is $\mathbb{Q}$-Cartier, then $(X,B)$ is klt (resp.
dlt).

###### Proof.

Let $f:X^{\prime}\to X$ be a log resolution,
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=f^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$,
where $\lfloor B^{\prime}\rfloor\leq 0$, as $(X,B+\boldsymbol{\beta})$ is
generalized klt. Let $K_{X^{\prime}}+B^{\sharp}=f^{*}(K_{X}+B)$, then

$E:=K_{X^{\prime}}+B^{\prime}-f^{*}(K_{X}+B)\equiv
f^{*}\boldsymbol{\beta}_{X}-\boldsymbol{\beta}_{X^{\prime}}$

and so $E$ is exceptional and $-E$ is nef over $X$. By the negativity lemma
$E\geq 0$. But then

$B^{\prime}=E+f^{*}(K_{X}+B)-K_{X^{\prime}}=B^{\sharp}+E$

and so $\lfloor B^{\sharp}\rfloor\leq 0$, i.e. $(X,B)$ is klt. The statement
about dlt singularities follows similarly. ∎

In dimension 2, the situation is particularly simple as shown by the following
lemma.

######  Lemma 1.4.

If $(X,B+\boldsymbol{\beta})$ is a generalized klt, dlt, lc surface, then
$K_{X}+B$ is $\mathbb{R}$-Cartier, $(X,B)$ is klt, dlt, lc and
$\boldsymbol{\beta}_{X}$ is nef.

###### Proof.

See [DHY23]. ∎

The next result shows that, working locally over $X$, generalized klt pairs
behave similarly to the usual klt pairs, and in particular they have rational
singularities.

######  Theorem 1.5.

Let $(X,B+\boldsymbol{\beta})$ be a generalized klt pair, then $X$ has
rational singularities and if we replace $X$ by a relatively compact Stein
open subset, then the following hold:

  1. (1)

there exists a small bimeromorphic morphism $\mu:X^{\sharp}\to X$ such that
$X^{\sharp}$ is $\mathbb{Q}$-factorial,

  2. (2)

if
$K_{X^{\sharp}}+B^{\sharp}+\boldsymbol{\beta}_{X^{\sharp}}=\mu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$,
then $\beta_{X^{\sharp}}\equiv_{X}\Delta^{\sharp}$ so that
$(X^{\sharp},B^{\sharp}+\Delta^{\sharp})$ is klt, and

  3. (3)

if $\Delta=\mu_{*}\Delta^{\sharp}$, then $(X,B+\Delta)$ is klt.

###### Proof.

This follows from [DHY23] but we include a proof for the convenience of the
reader. Note that rational singularities is a local property and hence follows
from (3) and [Fuj22, Theorem 3.12].

(1-2) Let $\nu:X^{\prime}\to X$ be a projective log resolution of
$(X,B+\beta)$ and write
$K_{X^{\prime}}+B^{\prime}+\beta^{\prime}=\nu^{*}(K_{X}+B+\beta)$ so that
$\boldsymbol{\beta}=\overline{\beta^{\prime}}$. Let $E$ be the reduced
exceptional divisor and for $0<\epsilon\ll 1$, let
$B^{*}=(B^{\prime})^{>0}+\epsilon E$ and $F=(B^{\prime})^{<0}+\epsilon E$,
then $K_{X^{\prime}}+B^{*}+\beta^{\prime}=\nu^{*}(K_{X}+B+\beta)+F$ where the
support of $F$ equals the set of all exceptional divisors, and
$(X^{\prime},B^{*}+\beta^{\prime})$ is generalized klt. In particular
$\beta^{\prime}\equiv_{X}F-(K_{X^{\prime}}+B^{*})$ where
$F-(K_{X^{\prime}}+B^{*})$ is an $\mathbb{R}$-divisor, nef over $X$. As $\nu$
is projective and $X$ is Stein, we may assume that $F-(K_{X^{\prime}}+B^{*})$
is big. But then $\beta^{\prime}\equiv_{X}\Delta^{\prime}$, where
$\Delta^{\prime}>0$ is an effective $\mathbb{R}$-divisor such that
$(X^{\prime},B^{*}+\Delta^{\prime})$ is klt. We may therefore run the relative
$K_{X^{\prime}}+B^{*}+\Delta^{\prime}$ mmp ([Fuj22] and [DHP22]) and hence we
may assume that we have a birational map $\psi:X^{\prime}\dasharrow
X^{\sharp}$ such that if $F^{\sharp}=\psi_{*}F$, $B^{\sharp}=\psi_{*}B^{*}$,
$\beta^{\sharp}=\psi_{*}\beta^{\prime}$ and
$\Delta^{\sharp}=\psi_{*}\Delta^{\prime}$, then

$F^{\sharp}\equiv_{X}K_{X^{\sharp}}+B^{\sharp}+\beta^{\sharp}\equiv_{X}K_{X^{\sharp}}+B^{\sharp}+\Delta^{\sharp}$

is nef over $X$ so that $F^{\sharp}=0$ by the negativity lemma. Therefore
$\mu:X^{\sharp}\to X$ is a small bimeromorphic morphism,
$B^{\sharp}=\mu^{-1}_{*}B$ and $X^{\sharp}$ is $\mathbb{Q}$-factorial. Clearly
$(X^{\sharp},B^{\sharp}+\Delta^{\sharp})$ is klt. Note that each step of the
above mmp preserves the numerical equivalence
$\beta^{\sharp}\equiv_{X}\Delta^{\sharp}$, and in particular
$K_{X^{\sharp}}+B^{\sharp}+\beta^{\sharp}=\mu^{*}(K_{X}+B+\beta)$.

(3) By the base point free theorem [Nak87, Theorem 4.8], we have
$K_{X^{\sharp}}+B^{\sharp}+\Delta^{\sharp}\sim_{\mathbb{Q},X}0$ and the claim
follows.∎

The following result is a technical result that is useful in many situations,
especially when proving results by induction on the dimension. It shows that
up to replacing $X$ by a higher model, the locus of non-klt singularities is
contained inside the reduced boundary of a carefully chosen strongly
$\mathbb{Q}$-factorial dlt pair. Recall that a $\mathbb{Q}$-line bundle is a
reflexive rank 1 sheaf such that there exists a positive integer $m$ such that
the reflexive hull $L^{[m]}:=(L^{\otimes m_{0}})^{\vee\vee}$ is a line bundle.
A variety $X$ is strongly $\mathbb{Q}$-factorial if every reflexive rank 1
sheaf is a $\mathbb{Q}$-line bundle.

######  Theorem 1.6 (DLT models).

Let $(X,B+\boldsymbol{\beta})$ be a generalized pair, where $X$ is relatively
compact Stein. Then there exists a projective birational morphism $f^{\rm
m}:X^{\rm m}\to X$ such that $X^{\rm m}$ is strongly $\mathbb{Q}$-factorial,
all exceptional divisors $P$ have discrepancy
$a(X,B+\boldsymbol{\beta},P)\leq-1$ and $(X^{\rm m},{(f^{\rm
m})}^{-1}_{*}B+{\rm Ex}(f^{\rm m}))$ is generalized dlt.

###### Proof.

The proof is similar to the proof in the case of the usual generalized pairs
(see [Fil20, Theorem 3.2]), which in turn is based on ideas of Hacon (see
[KK10]). We include the argument for the convenience of the reader. Recall
that $X$ is strongly $\mathbb{Q}$-factorial if for every reflexive rank 1
sheaf $F$ there exists an integer $m>0$ such that $(F^{\otimes m})^{**}$ is
locally free (see [DH20, Definition 2.2]).

Let $f:X^{\prime}\to X$ be a log resolution of the generalized pair
$(X,B+\boldsymbol{\beta})$ and write
$K_{X^{\prime}}+B^{\prime}+\beta^{\prime}=f^{*}(K_{X}+B+\beta)$ where
$\beta=\boldsymbol{\beta}_{X}$ and
$\beta^{\prime}=\boldsymbol{\beta}_{X^{\prime}}$. We may assume that $f$ is
defined by a sequence of blow ups over centers of codimension $\geq 2$ in $X$,
and hence $f$ is a projective morphism and so we have $C\geq 0$ an
$f$-exceptional divisor such that $-C$ is relatively ample. We write
$B^{\prime}=f^{-1}_{*}\\{B\\}+E^{+}+F-G$ where $E^{+},F,G$ are supported on
the divisors of discrepancy $a\leq-1,\ -1<a<0,\ a>0$ respectively and we let
$E={\rm red}(E^{+})$ be the reduced divisor with the same support as $E$. For
any $0<\epsilon,\mu,\nu<1$, we have

$E+(1+\nu)F-\mu C+\beta^{\prime}=(1-\epsilon\mu)E+(1+\nu)F+\mu(\epsilon
E-C)+\beta^{\prime}.$

Note that $-\mu C+\beta^{\prime}\equiv_{X}-\mu C-(K_{X^{\prime}}+B^{\prime})$
is an ample ${\mathbb{R}}$-divisor (over $X$) and hence for $0<\epsilon\ll 1$,
$\mu(\epsilon E-C)+\beta^{\prime}$ is also numerically equivalent to an ample
${\mathbb{R}}$-divisor (over $X$). Since $X$ is Stein, we may write

$-\mu C+\beta^{\prime}\equiv_{X}H_{1}\qquad{\rm and}\qquad\mu(\epsilon
E-C)+\beta^{\prime}\equiv_{X}H_{2}$

where $B^{\prime}+H_{1}+H_{2}$ has simple normal crossing support and $\lfloor
H_{1}\rfloor=\lfloor H_{2}\rfloor=0$. Let

$\Delta_{\epsilon,\mu,\nu}:=f^{-1}_{*}\\{B\\}+(1-\epsilon\mu)E+(1+\nu)F+H_{2}\equiv_{X}f^{-1}_{*}\\{B\\}+E+(1+\nu)F+H_{1},$

then $(X^{\prime},\Delta_{\epsilon,\mu,\nu})$ is klt for $0<\nu\ll 1$, and by
[DHP22] there is a $\mathbb{Q}$-factorial minimal model
$\psi:X^{\prime}\dasharrow X^{\rm m}_{\epsilon,\mu,\nu}$ over $X$ so that
$K_{X^{\rm m}_{\epsilon,\mu,\nu}}+\Delta_{\epsilon,\mu,\nu}^{\rm m}$ is nef
over $X$. By the equations above, this is also a minimal model for the dlt
pair $(X^{\prime},f^{-1}_{*}\\{B\\}+E+(1+\nu)F+H_{1})$. Let $B^{\rm
m}_{\epsilon,\mu,\nu}=\psi_{*}(f^{-1}_{*}\\{B\\}+E+F)$, then $\Delta^{\rm
m}_{\epsilon,\mu,\nu}=B^{\rm m}_{\epsilon,\mu,\nu}+\psi_{*}(\nu F+H_{1})$, and
$(X^{\rm m}_{\epsilon,\mu,\nu},B^{\rm m}_{\epsilon,\mu,\nu})$ is dlt. Define

$N:=K_{X^{\rm m}_{\epsilon,\mu,\nu}}+B_{\epsilon,\mu,\nu}^{\rm m}+\nu F^{\rm
m}_{\epsilon,\mu,\nu}+H^{\rm m}_{1,\epsilon,\mu,\nu}\equiv_{X}K_{X^{\rm
m}_{\epsilon,\mu,\nu}}+\Delta_{\epsilon,\mu,\nu}^{\rm m}$ $T:=K_{X^{\rm
m}_{\epsilon,\mu,\nu}}+B_{\epsilon,\mu,\nu}^{\rm m}+(E^{+}-E)^{\rm
m}_{\epsilon,\mu,\nu}-G^{\rm m}_{\epsilon,\mu,\nu}+\beta^{\rm
m}_{\epsilon,\mu,\nu}\equiv_{X}0.$

We then have

$T-N\equiv_{X}\mu C^{\rm m}+(E^{+}-E)^{\rm m}_{\epsilon,\mu,\nu}-G^{\rm
m}_{\epsilon,\mu,\nu}-\nu F^{\rm m}_{\epsilon,\mu,\nu}=:D^{\rm
m}_{\epsilon,\mu,\nu},$

where $-D^{\rm m}_{\epsilon,\mu,\nu}$ is nef and the pushforward of $D^{\rm
m}_{\epsilon,\mu,\nu}$ to $X$ is effective. By the negativity lemma, $D^{\rm
m}_{\epsilon,\mu,\nu}\geq 0$. The divisors $C$, $E^{+}-E$, $F$ and $G$ are
independent of ${\epsilon,\mu,\nu}$, thus if $0<\mu\ll\nu\ll 1$, then $G^{\rm
m}_{\epsilon,\mu,\nu}=\nu F^{\rm m}_{\epsilon,\mu,\nu}=0$. Then let $X^{\rm
m}:=X^{\rm m}_{\epsilon,\mu,\nu}$, then $X^{\rm m}$ is strongly
$\mathbb{Q}$-factorial as it is the output of a minimal model program (see
[DH20, Lemma 2.5]) and $(X^{\rm m},{(f^{\rm m})}^{-1}_{*}B+{\rm Ex}(f^{\rm
m}))$ is generalized dlt. ∎

###  1.1. Boundary and moduli parts

Throughout this section we will assume that $\pi:Z\to S$ is a proper morphism
of relatively compact normal analytic varieties, $f:X\to Z$ is a proper
morphism of normal Kähler varieties such that
$f_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$ and $(X/S,B+\boldsymbol{\beta})$ a
generalized pair which is generalized log canonical over an open subset of
$Z$. Recall that by assumption there is a log resolution $\nu:X^{\prime}\to X$
of $(X,B+\boldsymbol{\beta})$, i.e.

  1. (1)

$\boldsymbol{\beta}=\overline{\boldsymbol{\beta}_{X^{\prime}}}$ (that is
$\boldsymbol{\beta}$ descends to $X^{\prime}$),

  2. (2)

$\boldsymbol{\beta}_{X^{\prime}}$ is nef over $S$,

  3. (3)

${\rm Ex}(\nu)$ is a divisor and $\nu^{-1}(B)\cup{\rm Ex}(\nu)$ has simple
normal crossings.

We let
$K_{X^{\prime}}+B_{X^{\prime}}+\boldsymbol{\beta}_{X^{\prime}}=\nu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$
and we say that
$K_{X^{\prime}}+B_{X^{\prime}}+\boldsymbol{\beta}_{X^{\prime}}$ is the log-
crepant pull-back of $K_{X}+B+\boldsymbol{\beta}_{X}$.

######  Definition 1.7.

For any prime divisor $Q$ on $Z$, let

$a_{Q}=a_{Q}(X,B+\boldsymbol{\beta})={\rm
sup}\\{t\in\mathbb{R}|(X,B+tf^{*}Q+\boldsymbol{\beta})\ {\rm is\ glc\ over}\
\eta_{Q}\\}.$

In this definition ”glc” means that there is an analytic open subset
$Z^{0}\subset Z$ intersecting $Q$ such that
$(X,B+a_{Q}f^{*}Q+\boldsymbol{\beta})$ is glc but not gklt over $Z^{0}$. Note
that as $Z$ is normal, $Z_{\rm sing}$ has codimension at least 2 and so $Q$ is
Cartier in codimension 1. Since $(X,B+\boldsymbol{\beta})$ is a generalized
log canonical pair, $f$ is proper, and $Z$ is relatively compact, it is easy
to see that $a_{Q}=1$ for all but finitely many divisors $Q\subset Z$. We can
then define the boundary divisor
$B_{Z}=B(X/Z,B+\boldsymbol{\beta})=\sum(1-a_{Q})Q$.

######  Remark 1.8.

If $\eta:X^{\prime}\to X$ and $\mu:Z^{\prime}\to Z$ are proper bimeromorphic
maps of normal varieties and $f^{\prime}:X^{\prime}\to Z^{\prime}$ is
holomorphic,
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=\eta^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$,
then we say that $(X^{\prime},B^{\prime}+\boldsymbol{\beta})$ is the induced
generalized pair. If $Q^{\prime}=\mu^{-1}_{*}Q$, then it is easy to see that
$a_{Q}(X,B+\boldsymbol{\beta})=a_{Q^{\prime}}(X^{\prime},B^{\prime}+\boldsymbol{\beta})$.
In particular, if $B_{Z^{\prime}}$ is the boundary divisor for
$(X^{\prime},B^{\prime}+\boldsymbol{\beta})$, then
$\mu_{*}B_{Z^{\prime}}=B_{Z}$, i.e. the boundary divisor defined by the above
formula is in fact a b-divisor which we denote by $\mathbf{B}^{Z}$ (so that
$B_{Z}=\mathbf{B}^{Z}_{Z}$ and $B_{Z^{\prime}}=\mathbf{B}^{Z}_{Z^{\prime}}$).

######  Definition 1.9.

If $K_{X}+B+\boldsymbol{\beta}_{X}\equiv f^{*}\gamma$ for some
$\bar{\partial},\partial$ closed form $\gamma$, then we define the moduli part
$\beta_{Z}:=\gamma-(K_{Z}+B_{Z})$ of $(X/Z,B_{X}+\boldsymbol{\beta}_{X})$. If
$K_{Y}+B_{Y}+\boldsymbol{\beta}_{Y}$ is the log-crepant pull-back of
$K_{X}+B+\boldsymbol{\beta}_{X}$ and $\beta_{Z^{\prime}}$ is the moduli part
of $(X^{\prime}/Z^{\prime},B_{X^{\prime}}+\boldsymbol{\beta}_{X^{\prime}})$,
then it is easy to see that $\mu_{*}\beta_{Z^{\prime}}=\beta_{Z}$ and so we
have a b-(1,1) form $\boldsymbol{\beta}^{Z}$ such that
$\beta_{Z}=\boldsymbol{\beta}^{Z}_{Z}$.

######  Definition 1.10.

The pair $(X/Z,B+\boldsymbol{\beta})$ is said to be BP stable (over $Z$) if
$\mathbf{K}+\mathbf{B}=\overline{(K_{Z}+B_{Z})}$ i.e. if $K_{Z}+B_{Z}$ is
${\mathbb{R}}$-Cartier and for any contraction $f^{\prime}:X^{\prime}\to
Z^{\prime}$ which is birationally equivalent to $f$ and such that the induced
maps $\mu:Z^{\prime}\to Z$ and $\eta:X^{\prime}\to X$ are projective
birational morphisms, then
$K_{Z^{\prime}}+\mathbf{B}^{Z}_{Z^{\prime}}=\mu^{*}(K_{Z}+\mathbf{B}^{Z}_{Z})$.
Note that if $K_{X}+B+\boldsymbol{\beta}_{X}\equiv f^{*}\gamma$, then the
moduli part also descends to $Z$ i.e.
$\boldsymbol{\beta}^{Z}=\overline{\beta_{Z}}$

Suppose now that $\eta:X^{\prime}\to X$ is a proper generically finite
morphism. We define the pull-back
$\boldsymbol{\beta}^{\prime}:=\eta^{*}\boldsymbol{\beta}$ as follows. Let
$\nu:Y\to X$ be a log resolution of $(X,B+\boldsymbol{\beta})$ and
$Y^{\prime}$ be a resolution of the normalization of the main component of
$Y\times_{X}X^{\prime}$ so that $\rho:Y^{\prime}\to Y$ is a generically finite
holomorphic map and $\nu^{\prime}:Y^{\prime}\to X^{\prime}$ is a bimeromorphic
map. Then let
$\boldsymbol{\beta}^{\prime}=\overline{({\rho}^{*}\boldsymbol{\beta}_{Y})}$
and $K_{Y^{\prime}}+B_{Y^{\prime}}={\rho}^{*}(K_{Y}+B_{Y})$. Since
$\boldsymbol{\beta}^{\prime}_{Y^{\prime}}={\rho}^{*}\boldsymbol{\beta}_{Y}$ is
nef over $S$, $(Y^{\prime},B_{Y^{\prime}}+\boldsymbol{\beta}^{\prime}/S)$
defines a generalized pair. Now let
$B^{\prime}=\nu^{\prime}_{*}B_{Y^{\prime}}$, then

$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}^{\prime}_{X^{\prime}}=\nu^{\prime}_{*}(\rho^{*}(\nu^{*}(K_{X}+B_{X}+\boldsymbol{\beta}_{X})))=\eta^{*}(K_{X}+B_{X}+\boldsymbol{\beta}_{X})$

by the projection formula. We will say that
$(X^{\prime},B^{\prime}+\boldsymbol{\beta}^{\prime})$ is the log crepant pull-
back of $(X,B+\boldsymbol{\beta})$.

######  Lemma 1.11.

If $\boldsymbol{\beta}$ descends to $X$, then $\boldsymbol{\beta}^{\prime}$
descends to $X^{\prime}$.

###### Proof.

By assumption $\nu^{*}\boldsymbol{\beta}_{X}=\boldsymbol{\beta}_{Y}$. Then

$\nu^{\prime*}(\eta^{*}\boldsymbol{\beta}_{X})=\rho^{*}\nu^{*}\boldsymbol{\beta}_{X}=\rho^{*}\boldsymbol{\beta}_{Y}=\boldsymbol{\beta}^{\prime}_{Y^{\prime}}.$

Thus $\boldsymbol{\beta}^{\prime}=\overline{\eta^{*}\boldsymbol{\beta}_{X}}$,
i.e. $\boldsymbol{\beta}^{\prime}$ descends to $X^{\prime}$. ∎

Given a generically finite map $\mu:Z^{\prime}\to Z$ and $h:X^{\prime}\to
X\times_{Z}Z^{\prime}$ a proper bimeromorphic map from a normal variety to the
main component, then we obtain a base change diagram

(1)
${X^{\prime}}$${X}$${Z^{\prime}}$${Z}$$\scriptstyle{{\eta}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\mu}$

######  Lemma 1.12.

Let $B_{Z}=B(X/Z,B+\boldsymbol{\beta})$ be the boundary divisor for
$f:(X,B+\boldsymbol{\beta})\to Z$ and
$B_{Z^{\prime}}=B(X^{\prime}/Z^{\prime},B_{X^{\prime}}+\boldsymbol{\beta}^{\prime})$
be the boundary divisor for
$f^{\prime}:(X^{\prime},B_{X^{\prime}}+\boldsymbol{\beta}^{\prime})\to
Z^{\prime}$ where $\boldsymbol{\beta}^{\prime}=\eta^{*}\boldsymbol{\beta}$ and
$K_{X^{\prime}}+B_{X^{\prime}}+\boldsymbol{\beta}_{X^{\prime}}$ is the log-
crepant pull-back of $K_{X}+B+\boldsymbol{\beta}_{X}$. If $\mu$ is finite,
then $K_{Z^{\prime}}+B_{Z^{\prime}}=\mu^{*}(K_{Z}+B_{Z})$.

###### Proof.

See [Amb99, Theorem 3.2]. ∎

######  Definition 1.13.

A morphism $f:X\to Z$ is weakly semistable [AK00, Definition 0.1] if

  1. (1)

$X$ and $Z$ admit toroidal structures $U_{X}\subset X$ and $U_{Z}\subset Z$,
with $U_{X}=f^{-1}(U_{Z})$;

  2. (2)

with this structure, the morphism $f$ is toroidal;

  3. (3)

the morphism $f$ is equidimensional;

  4. (4)

all the fibers of the morphism $f$ are reduced; and

  5. (5)

$Z$ is nonsingular.

See [AK00, Section 1] for a discussion on toroidal morphisms. If $(X,B)$ is a
pair, then we say that $(X,B)$ has good horizontal divisors if for every $x\in
X$ there exists a local model
$X_{\sigma}=X_{\sigma^{\prime}}\times\mathbb{A}^{l}$ where the horizontal
divisors (i.e. the divisors dominating $Z$) are exactly the pull-backs of the
coordinate hyperplanes in $\mathbb{A}^{l}$ [Karu99, Definition 9.1].

We recall the following facts.

######  Theorem 1.14.

  1. (1)

Weak semistability is preserved by base change (see [Karu99, Lemma 8.3]).

  2. (2)

Let $f:X\to Z$ be a projective morphism of quasi-projective varieties and
$W\subset X$ a closed subset a proper subscheme, then there exists a diagram

(2) ${U_{X^{\prime}}\subset X^{\prime}}$${X}$${U_{Z^{\prime}}\subset
Z^{\prime}}$${Z}$$\scriptstyle{{\mu_{X}}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\mu_{Z}}$

such that $X^{\prime},Z^{\prime}$ are nonsingular, $\mu_{Z}$ and $\mu_{X}$ are
birational, $f^{\prime}$ is toroidal and $f^{-1}(W)\subset X^{\prime}\setminus
U_{X^{\prime}}$ is a snc divisor [AK00, Theorem 2.1].

  3. (3)

There exists a finite surjective morphism $Z^{\prime\prime}\to Z^{\prime}$
such that denoting by $X^{\prime\prime}$ the normalization of
$X\times_{Z^{\prime}}Z^{\prime\prime}$, we have that
$U_{X^{\prime\prime}}\subset X^{\prime\prime}$ and
$U_{Z^{\prime\prime}}\subset Z^{\prime\prime}$ are toroidal embeddings, the
projection $f^{\prime\prime}:X^{\prime\prime}\to Z^{\prime\prime}$ is an
equidimensional toroidal morphism with reduced fibers, and
$(f^{\prime\prime})^{-1}(U_{Z^{\prime\prime}})=U_{X^{\prime\prime}}$. If
$(X,B)$ is a pair, then we may assume $(X^{\prime\prime},B^{\prime\prime})\to
Z^{\prime\prime}$ is weakly semistable with good horizontal divisors where
$B^{\prime\prime}$ is the support of the inverse image of $B$ and the
$X^{\prime\prime}\to X$ exceptional divisors see [AK00, Proposition 5.1] and
[Karu99, Theorem 9.5].

######  Lemma 1.15.

Suppose that $(X/Z,{\rm Supp}(B))$ is weakly semistable with good horizontal
divisors, $\boldsymbol{\beta}$ descends to $X$ and $(Z,\Sigma)$ is a simple
normal crossings pair such that $f(B^{v})\subset\Sigma$. Then, the
corresponding boundary divisor $B_{Z}$ descends to $Z$, i.e. $(X/Z,B)$ is BP-
stable.

###### Proof.

Since $\boldsymbol{\beta}$ descends to $X$, we can disregard
$\boldsymbol{\beta}$, and the claim follows from the usual argument for pairs,
see eg. [Fil20, 4.13].∎

######  Proposition 1.16.

Let $f:(X,B+\boldsymbol{\beta})\to Z$ be a locally projective morphism of
relatively compact normal analytic varieties such that
$(X,B+\boldsymbol{\beta})$ is a pair which is glc over an open subset of $Z$.
Then $\mathbf{B}^{Z}$ descends to some model $Z^{\prime}$ i.e. if
$f^{\prime}:X^{\prime}\to Z^{\prime}$ is bimeromorphic to $f:X\to Z$ and
$(X^{\prime},B^{\prime}+\boldsymbol{\beta})$ is the induced pair,
$\boldsymbol{\beta}=\overline{\boldsymbol{\beta}_{X^{\prime}}}$ then
$(X^{\prime}/Z^{\prime},B^{\prime})$ is BP-stable.

###### Proof.

We may assume that $\boldsymbol{\beta}$ descends to $X$ and hence we may
disregard $\boldsymbol{\beta}$ in what follows. The question is local over $Z$
and hence we may assume that $f$ is projective and $Z$ is relatively compact
and Stein. By GAGA (see [AT19, Apendix B, C] for the details) we may assume
that $f:X\to Z$ is a projective morphism of quasi-projective varieties.

Let $f^{\prime}:X^{\prime}\to Z^{\prime}$ and
$f^{\prime\prime}:X^{\prime\prime}\to Z^{\prime\prime}$ be the morphisms
defined by Theorem 1.14. We may assume that $\pi_{X}^{-1}({\rm Supp}(B))+{\rm
Ex}(\pi_{X})$ has good horizontal divisors (over $Z^{\prime\prime}$) where
$\pi_{X}:X^{\prime\prime}\to X$. Let
$K_{X^{\prime\prime}}+B^{\prime\prime}=\pi_{X}^{*}(K_{X}+B)$, then
$(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime})$ is BP-stable by Lemma
1.15. We must show that if $\nu_{Z^{\prime}}:Z_{1}^{\prime}\to Z^{\prime}$ is
any birational map, then
$K_{Z^{\prime}_{1}}+B_{Z^{\prime}_{1}}=\nu_{Z^{\prime}}^{*}(K_{Z^{\prime}}+B_{Z^{\prime}})$
where $B_{Z^{\prime}_{1}}$ is the corresponding boundary divisor. Since
$Z^{\prime\prime}\to Z^{\prime}$ is finite, then
$K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}}=\mu_{Z^{\prime}}^{*}(K_{Z^{\prime}}+B_{Z^{\prime}})$
by Lemma 1.12 (where $\mu_{Z^{\prime}}:Z^{\prime\prime}\to Z^{\prime}$ and
$B_{Z^{\prime\prime}}$ is the corresponding boundary divisor). Let
$Z^{\prime\prime}_{1}$ be the normalization of
$Z^{\prime\prime}\times_{Z^{\prime}}Z^{\prime}_{1}$, then
$\mu_{Z^{\prime}_{1}}:Z^{\prime\prime}_{1}\to Z_{1}^{\prime}$ is finite and
hence
$\mu_{Z^{\prime}_{1}}^{*}(K_{Z^{\prime}_{1}}+B_{Z_{1}^{\prime}})=K_{Z^{\prime\prime}_{1}}+B_{Z_{1}^{\prime\prime}}$.
Let $\nu_{Z^{\prime\prime}}:Z_{1}^{\prime\prime}\to Z^{\prime\prime}$. Since
$(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime})$ is BP-stable and
$\mu_{Z^{\prime}}\circ\nu_{Z^{\prime\prime}}=\nu_{Z^{\prime}}\circ\mu_{Z^{\prime}_{1}}$,
then

$\mu_{Z^{\prime}_{1}}^{*}(K_{Z^{\prime}_{1}}+B_{Z_{1}^{\prime}})=K_{Z^{\prime\prime}_{1}}+B_{Z_{1}^{\prime\prime}}=\nu_{Z^{\prime\prime}}^{*}(K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}})=\nu_{Z^{\prime\prime}}^{*}(\mu_{Z^{\prime}}^{*}(K_{Z^{\prime}}+B_{Z^{\prime}}))=\mu_{Z^{\prime}_{1}}^{*}(\nu_{Z^{\prime}}^{*}(K_{Z^{\prime}}+B_{Z^{\prime}})).$

Pushing forward, it follows that
$K_{Z^{\prime}_{1}}+B_{Z_{1}^{\prime}}=\nu_{Z^{\prime}}^{*}(K_{Z^{\prime}}+B_{Z^{\prime}})$.

∎

######  Theorem 1.17.

Let $(X,B)$ be a pair and $f:X\to Z$ be a projective morphism of compact
complex manifolds with connected fibers, then there exists a birational
morphism $Z^{\prime}\to Z$ and a finite morphism $Z^{\prime\prime}\to
Z^{\prime}$ and a morphism $f^{\prime\prime}:X^{\prime\prime}\to
Z^{\prime\prime}$ birational to $X\times_{Z}Z^{\prime\prime}$ such that
$(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime})$ is weakly semistable
with good horizontal divisors where $B^{\prime\prime}$ is the support of the
inverse image of $B$ and the $X^{\prime\prime}\to X$ exceptional divisors.

###### Proof.

Let $\phi:(\mathcal{X},\mathcal{B})\to\mathcal{Z}$ the corresponding component
of the Hilbert scheme. Replacing $\mathcal{Z}$ by a resolution of the image of
$Z$ and replacing $Z$ by a higher model, we may assume that $Z\to\mathcal{Z}$
is surjective, generically finite of smooth manifolds, and
$(X,B)=(\mathcal{X},\mathcal{B})\times_{\mathcal{Z}}Z$ over an open subset of
$Z$. By Theorem 1.14, there are

  1. (1)

birational morphisms $\mu_{\mathcal{Z}}:\mathcal{Z}^{\prime}\to\mathcal{Z}$
and $\mu_{\mathcal{X}}:\mathcal{X}^{\prime}\to\mathcal{X}$ such that the
induced map $\phi^{\prime}:\mathcal{X}^{\prime}\to\mathcal{Z}^{\prime}$ is a
toroidal morphism of smooth varieties,

  2. (2)

a finite surjective morphism
$\mathcal{Z}^{\prime\prime}\to\mathcal{Z}^{\prime}$ such that denoting by
$\mathcal{X}^{\prime\prime}$ the normalization of
$\mathcal{X}^{\prime\prime}\times_{\mathcal{Z}^{\prime}}\mathcal{Z}^{\prime\prime}$
then
$\phi^{\prime\prime}:\mathcal{X}^{\prime\prime}\to\mathcal{Z}^{\prime\prime}$
is an equidimensional toroidal morphism with reduced fibers and
$(\mathcal{X}^{\prime\prime},\mathcal{B}^{\prime\prime})\to\mathcal{Z}^{\prime\prime}$
is weakly semistable with good horizontal divisors where
$\mathcal{B}^{\prime\prime}$ is the support of the inverse image of
$\mathcal{B}$ and the $\mathcal{X}^{\prime\prime}\to\mathcal{X}$ exceptional
divisors.

We let $Z^{\prime}=Z\times_{\mathcal{Z}}\mathcal{Z}^{\prime}$ and
$Z^{\prime\prime}$ be an appropriate resolution of
$Z\times_{\mathcal{Z}}\mathcal{Z}^{\prime\prime}$ (so that the inverse image
of $\mathcal{Z}^{\prime\prime}\setminus U_{\mathcal{Z}^{\prime\prime}}$ is a
divisor with simple normal crossings) and
$X^{\prime\prime}=X\times_{Z}Z^{\prime\prime}$,
$B^{\prime\prime}=B\times_{Z}Z^{\prime\prime}$, then
$f^{\prime\prime}:X^{\prime\prime}\to Z^{\prime\prime}$ is weakly semistable
and $(X^{\prime\prime},B^{\prime\prime})$ has good horizontal divisors where
$B^{\prime\prime}$ is the inverse image of $B$ plus the $X^{\prime\prime}\to
X$ exceptional divisors.∎

######  Remark 1.18.

The morphism $f^{\prime\prime}:X^{\prime\prime}\to Z^{\prime\prime}$ is quasi-
smooth in the following sense (see [Taka22] pages 1736-1737). For every
$x^{\prime\prime}\in X^{\prime\prime}$, there exists an open subset
$x^{\prime\prime}\in U^{\prime\prime}\subset X^{\prime\prime}$ such that
$(U^{\prime\prime},{\rm
Supp}(D^{\prime\prime})|_{U^{\prime\prime}})=(\tilde{U}^{\prime\prime},D_{\tilde{U}^{\prime\prime}})/G$
where $(\tilde{U}^{\prime\prime},D_{\tilde{U}^{\prime\prime}})$ is a log
smooth toric variety and $G$ is a finite abelian group and
$f^{\prime\prime}\circ\pi:(\tilde{U}^{\prime\prime},D_{\tilde{U}^{\prime\prime}})\to
Z^{\prime\prime}$ is toric and flat and $\pi:\tilde{U}^{\prime\prime}\to
U^{\prime\prime}$ is the quotient map. In particular we can pick local
coordinates $(x_{1},\ldots,x_{n+m})$ on $U^{\prime\prime}$ and
$(t_{1},\ldots,t_{m})$ on $Z^{\prime\prime}$ such that
$(f^{\prime\prime}\circ\pi)^{*}(t_{i})=\prod x_{j}^{k_{i,j}})$ where the
$k_{i,j}$ are non-negative integers such that $k_{i,j}\neq 0$ for at most one
$i$ for each index $j$.

##  2\. Adjunction of generalized pairs

In this section we will prove a version of Kawamata’s canonical bundle formula
and of Kawamata’s sub-adjunction for generalized Kähler pairs. We will also
prove plt and log canonical inversion of adjunction in this setting.

###  2.1. Towards a canonical bundle formula

Let $f:X\to Z$ and $Z\to S$ be proper morphisms of normal relatively compact
Kähler varieties such that $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$ and
$(X/S,B+\boldsymbol{\beta})$ is a generalized pair. Suppose that
$K_{X}+B+\boldsymbol{\beta}_{X}=f^{*}\gamma$ for some
$\bar{\partial},\partial$ closed current $\gamma$ on $Z$. Let
$\boldsymbol{\gamma}=\overline{\gamma}$ so that
$\boldsymbol{\gamma}_{Z^{\prime}}=\mu^{*}\gamma$ for any bimeromorphic map
$\mu:Z^{\prime}\to Z$. We define the boundary b-divisor $\mathbf{B}^{Z}$ as
above and for any base change diagram (1), we define $B_{Z^{\prime}}$
accordingly. In particular if $\mu:Z^{\prime}\to Z$ is birational, then
$B_{Z^{\prime}}=\mathbf{B}^{Z}_{Z^{\prime}}$. We also define the moduli part
$\boldsymbol{\beta}^{Z}$ via

$\boldsymbol{\beta}^{Z}=\boldsymbol{\gamma}-(\mathbf{K}+\mathbf{B}^{Z}),\qquad
i.e.\qquad
K_{Z^{\prime}}+\mathbf{B}^{Z}_{Z^{\prime}}+\boldsymbol{\beta}^{Z}_{Z^{\prime}}=\boldsymbol{\gamma}_{Z^{\prime}}.$

One expects, similarly to the algebraic case, that
$\mathbf{B}^{Z},\boldsymbol{\beta}^{Z}$ descend to some model $Z^{\prime}$,
$(Z^{\prime},\mathbf{B}^{Z}_{Z^{\prime}})$ is klt (lc) and that
$\boldsymbol{\beta}^{Z}_{Z^{\prime}}$ is nef. In other words we conjecture the
following.

######  Conjecture 2.1.

Let $f:(X,B+\boldsymbol{\beta})\to Z$ be a generalized klt (lc) pair as above,
then $(Z,B_{Z}+\boldsymbol{\beta}^{Z})$ is a generalized klt (lc) pair.

We will prove this conjecture under additional conditions. As a first step, we
prove the following result which is a generalization of the main result of
[Gue20].

######  Theorem 2.2.

Let $f:X\to Z$ be a surjective projective map with connected fibers between
normal compact Kähler varieties, $(X,B+\boldsymbol{\beta})$ a generalized pair
which is log canonical over an open subset of $Z$, $Z$ is smooth, $\gamma$ a
real (1,1)-class on $Z$, and $L$ a $\mathbb{Q}$-line bundle such that
$h^{0}(X_{z},L^{[m]}|_{X_{z}})\neq 0$ for $m\gg 0$ sufficiently divisible,
$z\in Z$ general. If $K_{X}+B+\boldsymbol{\beta}_{X}=f^{*}\gamma+L$, then
$K_{X/Z}+B+\boldsymbol{\beta}_{X}$ is pseudo-effective.

###### Proof.

Let $\nu:X^{\prime}\to X$ be a resolution and write
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=\nu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})+E$
where $B^{\prime},E\geq 0$ are effective divisors with no common components.
We may assume that $B^{\prime}$ has simple normal crossings support. Since
$E\geq 0$ is $\nu$-exceptional, by the projection formula it suffices to show
that $K_{X^{\prime}/Z}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}$ is pseudo-
effective. Let $B^{\prime}=B^{h}+B^{v}$ where the components of $B^{h}$
dominate $Z$ and the components of $B^{v}$ do not dominate $Z$. Note that
$(X,B^{h})$ is log canonical and it suffices to show that
$K_{X^{\prime}/Z}+B^{h}+\boldsymbol{\beta}_{X^{\prime}}$ is pseudo-effective.
For every $m>0$ we will write

$mF^{m}:=\lfloor B^{h}\rfloor+\\{mB^{h}\\}\qquad{\rm and}\qquad
mB^{m}:=m(B^{h}-F^{m})=\lfloor mB^{h}\rfloor-\lfloor B^{h}\rfloor.$

In particular $(X^{\prime},B^{m})$ is klt and $F^{m}$ has fixed support with
coefficients in $[0,1/m]$. Let $H^{\prime}$ be a relatively ample divisor and
$\omega$ be a Kähler form on $Z$ such that ${f^{\prime}}^{*}\omega+H^{\prime}$
is Kähler. For any $\epsilon>0$ there is an $m>0$ such that
$\beta^{m,\epsilon}:=\boldsymbol{\beta}_{X^{\prime}}+\epsilon({f^{\prime}}^{*}\omega+H^{\prime})+F^{m}$
is Kähler and $L^{\epsilon}:=\nu^{*}L+E-B^{v}+\epsilon H^{\prime}$ is a
$\mathbb{Q}$-line bundle such that
$h^{0}(X^{\prime}_{z},(L^{\epsilon})^{[m]}|_{X^{\prime}_{z}})\neq 0$ for $m\gg
0$ sufficiently divisible, $z\in Z$ general. We may write

$K_{X^{\prime}}+B^{m}+\beta^{m,\epsilon}=f^{\prime*}(\gamma+\epsilon\omega)+L^{\epsilon}.$

By Theorem 6.2, $K_{X^{\prime}/Z}+B^{m}+\beta^{m,\epsilon}$ is pseudo-
effective. Since being pseudo-effective is a closed condition, and

$K_{X^{\prime}/Z}+B^{h}+\boldsymbol{\beta}_{X^{\prime}}=\lim\left(K_{X^{\prime}/Z}+B^{m}+\beta^{m,\epsilon}\right),$

it follows that $K_{X^{\prime}/Z}+B^{h}+\boldsymbol{\beta}_{X^{\prime}}$ is
pseudo-effective. ∎

######  Theorem 2.3.

Let $(X,B+\boldsymbol{\beta})$ be a generalized klt (lc) pair, $f:X\to Z$ a
surjective projective morphism of normal compact Kähler varieties with
connected fibers, such that $K_{X}+B+\boldsymbol{\beta}_{X}=f^{*}\gamma$ for
some $\bar{\partial},\partial$ closed current $\gamma$ on $Z$. Then
$(Z,B_{Z}+\boldsymbol{\beta}^{Z})$ is a generalized klt (lc) pair, i.e.

  1. (1)

$\boldsymbol{\beta}^{Z}$ is b-nef, that is $\boldsymbol{\beta}^{Z}$ descend to
some model $Z^{\prime}$ so that $\boldsymbol{\beta}^{Z}_{Z^{\prime}}$ is nef,
and

  2. (2)

$(Z^{\prime},\mathbf{B}^{Z}_{Z^{\prime}})$ is sub-klt (sub-lc).

The proof of this result is somewhat technical but the strategy that we will
now illustrate is fairly natural. We already know that the moduli part
$\boldsymbol{\beta}^{Z}$ descends to some model, so assume for simplicity that
it descends to $Z$. Similarly assume that $\beta=\boldsymbol{\beta}_{X}$ is
nef and $B$ has simple normal crossings. We must show that
$\beta_{Z}:=\boldsymbol{\beta}^{Z}_{Z}$ is nef. By [DHP22], it suffices to
show that $\beta_{Z}|_{W}$ is pseudo-effective on (the normalization of) any
subvariety $W\subset Z$. When $W=Z$, this follows from Theorem 2.2. If $W\neq
Z$ the situation is more delicate. Assume further that $f$ is weakly
semistable and that $B_{Z}$ is a reduced divisor supported on $Z\setminus
U_{Z}$ (cf. Theorems 1.14, 1.17). If $T$ is any component of $B_{Z}$, then
there is a component $S$ of $B^{=1}$ dominating $T$. By adjunction
$(K_{X}+B+\beta)|_{S}=K_{S}+B_{S}+\beta_{S}=(f|_{S})^{*}(\gamma|_{T})$ and so
$\beta_{T}:=\beta(S/T,B_{S}+\beta_{S})$ is nef by induction on the dimension.
Since $\beta_{Z}|_{T}=\beta_{T}$, then $\beta_{Z}|_{W}$ is nef and hence
pseudo-effective for any $W\subset T$. Finally, suppose that $W\not\subset{\rm
Supp}(B_{Z})$. Let $X_{W}=X\times_{Z}W$, then we expect that
$(X_{W},B_{W}+\beta_{W})$ is a generalized pair with mild singularities and
$K_{X_{W}}+B_{W}+\beta_{W}=(f|_{W})^{*}(\gamma|_{W})$ so that by induction on
the dimension $\beta_{Z}|_{W}=\beta(X_{W}/W,B_{W}+\beta_{W})$ is nef. Of
course there are many technical issues that we will have to address in the
proof.

###### Proof of Theorem 2.3.

By Proposition 1.16, the boundary b-divisor $\mathbf{B}^{Z}$ for
$(X,B+\boldsymbol{\beta}/Z)$ descends to a model $Z^{\prime}$, and hence so
does the moduli part $\boldsymbol{\beta}^{Z}$. It is well known (and easy to
see) that $(Z^{\prime},\mathbf{B}^{Z}_{Z^{\prime}})$ is sub-klt (sub-lc).
Thus, it suffices to show that, after possibly replacing $Z^{\prime}$ by a
higher model, $\boldsymbol{\beta}^{Z}_{Z^{\prime}}$ is nef.

Let $f^{\prime\prime}:X^{\prime\prime}\to Z^{\prime\prime}$ and
$f^{\prime}:X^{\prime}\to Z^{\prime}$ be the bi-meromorphic models of $f:X\to
Z$ defined in Theorem 1.17 and let
$K_{X^{\prime\prime}}+B^{\prime\prime}+\boldsymbol{\beta}_{X^{\prime\prime}}$
and $K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}$ be the log
pull backs of $K_{X}+B+\boldsymbol{\beta}_{X}$. We may assume that
$\boldsymbol{\beta}$ descends to $X^{\prime}$. It suffices to show that
$\beta_{Z^{\prime}}$ is nef where
$\beta_{Z^{\prime}}:=\boldsymbol{\beta}^{Z}_{Z^{\prime}}$ is the moduli part
of $(X^{\prime}/Z^{\prime},B^{\prime}+\boldsymbol{\beta}_{X^{\prime}})$. Let
$B_{Z^{\prime\prime}}$ and
$\beta_{Z^{\prime\prime}}:=\boldsymbol{\beta}^{Z}_{Z^{\prime\prime}}$ be the
boundary and moduli parts of
$(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime}+\boldsymbol{\beta}_{X^{\prime\prime}})$.
Since $\mu_{Z^{\prime}}:Z^{\prime\prime}\to Z^{\prime}$ is finite, then
$\beta_{Z^{\prime\prime}}=\mu_{Z^{\prime}}^{*}\beta_{Z^{\prime}}$ Thus it
suffices to show that $\beta_{Z^{\prime\prime}}$ is nef (see eg. [DHP22, Lemma
2.38]).

Let $B^{\prime\prime}=B^{+}-B^{-}$ where $B^{+},B^{-}$ are effective with no
common components. We may assume that $B_{Z^{\prime\prime}}$ is the reduced
divisor supported on $Z^{\prime\prime}\setminus U_{Z^{\prime\prime}}$, and we
let $F$ be the sum of the components of
$(f^{\prime\prime})^{*}(B_{Z^{\prime\prime}})$ that are not contained in
$\lfloor B^{+}\rfloor$. Pick $0<\delta\ll 1$, $H^{\prime\prime}$ a relatively
ample divisor and let $\psi:X^{\prime\prime}\dasharrow\bar{X}$ be a relative
minimal model for $({X^{\prime\prime}},B^{+}+\delta
F+\boldsymbol{\beta}_{X^{\prime\prime}}+\epsilon H^{\prime\prime})$ where
$0<\epsilon\ll\delta$. We note here that

$K_{X^{\prime\prime}}+B^{+}+\delta F+N_{\epsilon}\equiv
K_{X^{\prime\prime}}+B^{+}+\delta
F+\boldsymbol{\beta}_{X^{\prime\prime}}+\epsilon H^{\prime\prime}\equiv
B^{-}+\delta F,$

where $N_{\epsilon}:=-(K_{X^{\prime\prime}}+B^{\prime\prime})+\epsilon
H^{\prime\prime}\equiv_{Z^{\prime\prime}}\boldsymbol{\beta}_{X^{\prime\prime}}+\epsilon
H^{\prime\prime}$ is a relatively ample $\mathbb{Q}$-divisor and hence the
corresponding minimal model exists by [Fuj22].

We claim that $\psi_{*}(B^{-}+\delta F)=0$. To see this, note that since for
general $z\in Z$, $K_{X_{z}}+B_{z}+\boldsymbol{\beta}_{X}|_{X_{z}}\equiv 0$,
then
$K_{X_{z^{\prime\prime}}}+B^{+}_{z^{\prime\prime}}+\boldsymbol{\beta}_{X^{\prime\prime}}|_{X^{\prime\prime}_{z^{\prime\prime}}}\equiv
B^{-}_{z^{\prime\prime}}$ where $z^{\prime\prime}$ denotes a preimage of $z$
on $Z^{\prime\prime}$. As $B^{-}_{z^{\prime\prime}}$ is exceptional for
$X^{\prime\prime}_{z^{\prime\prime}}\to X_{z}$ and $F_{z^{\prime\prime}}=0$,
then by standard arguments $\psi$ will contract every component of
$B^{-}_{z^{\prime\prime}}$. In particular
$K_{\bar{X}_{z^{\prime\prime}}}+\bar{B}_{z^{\prime\prime}}+\boldsymbol{\beta}_{\bar{X}}|_{\bar{X}_{z^{\prime\prime}}}\equiv
0$ and so
$K_{\bar{X}_{z^{\prime\prime}}}+\bar{B}_{z^{\prime\prime}}+\boldsymbol{\beta}_{\bar{X}}|_{\bar{X}_{z^{\prime\prime}}}+\epsilon\bar{H}|_{\bar{X}_{z^{\prime\prime}}}$
is nef for all $0\leq\epsilon\ll 1$. Since $\bar{B}^{-}+\bar{F}$ is vertical,
up to adding a vertical divisor which is $\equiv_{Z^{\prime\prime}}0$, we may
assume that for any divisor $P$ in the support of
$\bar{f}_{*}(\bar{B}^{-}+\bar{F})$, every divisor $Q$ dominating $P$ has ${\rm
mult}_{Q}(\bar{B}^{-}+\bar{F})\geq 0$ and ${\rm
mult}_{Q}(\bar{B}^{-}+\bar{F})=0$ for one such component. By [Lai11, Lemma
2.9], if ${\rm mult}_{Q}(\bar{B}^{-}+\bar{F})>0$ for one such $Q$, then there
is a component $Q^{\prime}$ dominating $P$ which is contained in
${\mathbf{B}}_{-}(\bar{B}^{-}+\delta\bar{F}/Z^{\prime\prime})$. Since
$\bar{B}^{-}+\delta\bar{F}+\epsilon H^{\prime\prime}$ is nef over
$Z^{\prime\prime}$ (for $0<\epsilon\ll 1$), it follows that no such divisor
exists and hence $\bar{B}^{-}+\bar{F}$ is exceptional (over
$Z^{\prime\prime}$). Since $X^{\prime\prime}\to Z^{\prime\prime}$ is weakly
semistable, $\bar{B}^{-}+\delta\bar{F}=0$ and so
$K_{\bar{X}}+\bar{B}+\boldsymbol{\beta}_{\bar{X}}\equiv\bar{f}^{*}\gamma^{\prime\prime}$.
Note that $\bar{B}\geq\bar{f}^{*}B_{Z^{\prime\prime}}$.

By [DHP22, Theorem 2.36], it suffices to show that for any subvariety
$W\subset Z^{\prime\prime}$ with normalization $W^{\nu}\to W$, then
$\boldsymbol{\beta}_{Z^{\prime\prime}}|_{W^{\nu}}$ is pseudo-effective. Let
$T\subset Z^{\prime\prime}$ be any component of $B_{Z^{\prime\prime}}$. There
exists a component $S$ of $(\bar{B})^{=1}$ such that ${\rm
mult}_{S}(\bar{B})=1$ and $S$ dominates $T$. We replace $S$ by a minimal
stratum of $(\bar{B})^{=1}$ dominating $T$. Then $S$ is normal and by
adjunction
$(K_{\bar{X}}+\bar{B}+\boldsymbol{\beta}_{\bar{X}})|_{S}=K_{S}+B_{S}+\beta_{S}$
where $(S,B_{S}+\beta_{S})$ is generalized dlt,
$(K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}})|_{T}=K_{T}+B_{T}$ where
$(T,B_{T})$ is log canonical, $B_{T}=(B_{Z^{\prime\prime}}-T)|_{T}$. Then

$K_{S}+B_{S}+\beta_{S}=g^{*}(K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}}+\beta_{Z^{\prime\prime}})|_{T}=g^{*}(K_{T}+B_{T}+\beta_{T})$

where $\beta_{S}=\boldsymbol{\beta}_{\bar{X}}|_{S}$, $g=\bar{f}|_{S}$. By
standard arguments (see for example the claim in the proof of Theorem 1.1 of
[FG14]), we may assume that $g$ has connected fibers. Since
$(K_{Z^{\prime\prime}}+B_{Z^{\prime\prime}})|_{T}=K_{T}+B_{T}$, it follows
that $\beta_{Z^{\prime\prime}}|_{T}=\beta_{T}$ is the mobile part of
$(S,B_{S}+\beta_{S}/T)$. By induction on the dimension, $\beta_{T}$ is nef.

Thus, we may now assume that $W$ is not contained in
$Z^{\prime\prime}\setminus U_{Z^{\prime\prime}}$. Replacing $\bar{B}$ by
$\bar{B}-\bar{f}^{*}B_{Z^{\prime\prime}}$ we will also assume that
$B_{Z^{\prime\prime}}=0$. It suffices to show that
$\boldsymbol{\beta}_{Z^{\prime\prime}}|_{\tilde{W}}$ is pseudo-effective where
$\mu_{W}:\tilde{W}\to W$ is a resolution or equivalently that
$(K_{\tilde{X}^{\prime\prime}/\tilde{Z}}+B_{\tilde{X}^{\prime\prime}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}})|_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$
is pseudo-effective (see Theorem 6.1.ii). We may assume that
$\tilde{W}\subset\tilde{Z}$ where $\mu_{Z^{\prime\prime}}:\tilde{Z}\to
Z^{\prime\prime}$ is a resolution. We write
$\tilde{X}=\bar{X}\times_{Z^{\prime\prime}}\tilde{Z}$,
$\tilde{X}^{\prime\prime}=X^{\prime\prime}\times_{Z^{\prime\prime}}\tilde{Z}$
and $\tilde{X}_{\tilde{W}}=\bar{X}\times_{Z^{\prime\prime}}\tilde{W}$,
$\tilde{X}^{\prime\prime}_{\tilde{W}}=X^{\prime\prime}\times_{Z^{\prime\prime}}\tilde{W}$.
We let
$\mu_{\bar{X}}^{*}(K_{\bar{X}/Z^{\prime\prime}}+\bar{B}+\boldsymbol{\beta}_{\bar{X}})=K_{\tilde{X}/\tilde{Z}}+\tilde{B}+\boldsymbol{\beta}_{\tilde{X}}$
and
$K_{\tilde{X}^{\prime\prime}/\tilde{Z}}+\tilde{B}^{\prime\prime}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}}=\tilde{\psi}^{*}(K_{\tilde{X}/\tilde{Z}}+\tilde{B}+\boldsymbol{\beta}_{\tilde{X}})$
where $\mu_{\bar{X}}:{\tilde{X}}\to\bar{X}$ and
$\tilde{\psi}:\tilde{X}^{\prime\prime}\dasharrow\tilde{X}$. By our
construction $\boldsymbol{\beta}_{X^{\prime\prime}}$ is nef over
$Z^{\prime\prime}$ and
$\boldsymbol{\beta}=\overline{\boldsymbol{\beta}_{X^{\prime\prime}}}$. Note
that $\tilde{X}^{\prime\prime}_{\tilde{W}}$ is not contained in the support of
$\tilde{B}^{\prime\prime}$ and that $\tilde{X}^{\prime\prime}_{\tilde{W}}$ and
$\tilde{X}_{\tilde{W}}$ are normal. This last claim follows since
$\tilde{X}^{\prime\prime}_{\tilde{W}}$ and $\tilde{X}_{\tilde{W}}$ are regular
in codimension 1 and all corresponding fibers are S2. We now let

$(K_{\tilde{X}^{\prime\prime}/\tilde{Z}}+B_{\tilde{X}^{\prime\prime}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}})|_{\tilde{X}^{\prime\prime}_{\tilde{W}}}=K_{\tilde{X}^{\prime\prime}_{\tilde{W}}/\tilde{W}}+B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$

where
$K_{\tilde{X}^{\prime\prime}_{\tilde{W}}/\tilde{W}}=K_{\tilde{X}^{\prime\prime}/\tilde{Z}}|_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$
(see [Kollar22, Theorem 2.68]),
$B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}=B_{\tilde{X}^{\prime\prime}}|_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$
and
$\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}}=\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}}|_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$
is nef. Similarly we let
$(K_{\tilde{X}/\tilde{Z}}+B_{\tilde{X}}+\boldsymbol{\beta}_{\tilde{X}})|_{\tilde{X}_{\tilde{W}}}=K_{\tilde{X}_{\tilde{W}}/\tilde{W}}+B_{\tilde{X}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}_{\tilde{W}}}$.
Note that
$K_{\tilde{X}^{\prime\prime}_{\tilde{W}}/\tilde{W}}+B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}}=\psi_{\tilde{W}}^{*}(K_{\tilde{X}_{\tilde{W}}/\tilde{W}}+B_{\tilde{X}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}_{\tilde{W}}})$
and so we have a generalized pair

$\psi_{\tilde{W}}:(\tilde{X}^{\prime\prime}_{\tilde{W}},B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}})\to(\tilde{X}_{\tilde{W}},(\psi_{\tilde{W}})_{*}(B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}})).$

Since $(\psi_{\tilde{W}})_{*}(B_{\tilde{X}^{\prime\prime}_{\tilde{W}}})\geq
B_{\tilde{X}_{\tilde{W}}}$, then
$(\tilde{X}_{\tilde{W}},(\psi_{\tilde{W}})_{*}(B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}}))$
is generalized log canonical over an open subset of $\tilde{W}$. By Theorem
2.2,
$K_{\tilde{X}^{\prime\prime}_{\tilde{W}}/\tilde{W}}+B_{\tilde{X}^{\prime\prime}_{\tilde{W}}}+\boldsymbol{\beta}_{\tilde{X}^{\prime\prime}_{\tilde{W}}}$
is pseudo-effective. This concludes the proof. ∎

######  Remark 2.4.

We believe that if moreover $\boldsymbol{\beta}$ admits a smooth positive
representative, then so does $\boldsymbol{\beta}^{Z}$ (cf. Theorem 5.2).

###  2.2. Adjunction

In what follows, for ease of exposition, we will denote a generalized pair
$(X/Z,B+\boldsymbol{\beta})$ simply by $(X,B+\boldsymbol{\beta})$.

######  Definition 2.5.

Suppose that $(X,B+\boldsymbol{\beta})$ is a generalized pair and $S$ is a
component of $B$ of coefficient 1 with normalization $\nu:S^{\nu}\to S$, then
we define a generalized pair

$K_{S^{\nu}}+B_{S^{\nu}}+\boldsymbol{\beta}^{S^{\nu}}=(K_{X}+B+\boldsymbol{\beta})|_{S^{\nu}}$

as follows. Let $f:X^{\prime}\to X$ be a log resolution of
$(X,B+\boldsymbol{\beta})$ so that $X^{\prime}$ is smooth, $B^{\prime}$ has
simple normal crossings, $\boldsymbol{\beta}_{X^{\prime}}$ is nef,
$\boldsymbol{\beta}$ descends to $X^{\prime}$, and
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}\equiv
f^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$. Let $S^{\prime}=f^{-1}_{*}S$, then
$S^{\prime}$ is smooth and by the usual adjunction for sub-klt pairs, we can
write
$K_{S^{\prime}}+B_{S^{\prime}}=(K_{X^{\prime}}+B^{\prime})|_{S^{\prime}}$
where $B_{S^{\prime}}=(B^{\prime}-S^{\prime})|_{S^{\prime}}$ has simple normal
crossings. We may assume that $\beta^{\prime}$ is smooth and we let
$\boldsymbol{\beta}^{S^{\nu}}=\overline{\beta_{S^{\prime}}}$ where
$\beta_{S^{\prime}}:=\beta^{\prime}|_{S^{\prime}}$ is the induced nef (1,1)
form. Notice that
$[K_{S^{\prime}}+B_{S^{\prime}}+\beta_{S^{\prime}}]=g^{*}(\gamma|_{S^{\nu}})$
where $g:S^{\prime}\to S^{\nu}$ is the induced morphism and
$\gamma=[K_{X}+B+\beta]$. Let
$K_{S^{\nu}}+B_{S^{\nu}}+\beta_{S^{\nu}}:=g_{*}(K_{S^{\prime}}+B_{S^{\prime}}+\beta_{S^{\prime}})$,
then
$g:({S^{\prime}},B_{S^{\prime}}+\beta_{S^{\prime}})\to(S,B_{S^{\nu}}+\beta_{S^{\nu}})$
defines a generalized pair; equivalently
$(S^{\prime},B_{S^{\prime}}+\boldsymbol{\beta}^{S^{\prime}})$ is a generalized
pair.

If $K_{X}+B$ is $\mathbb{R}$-Cartier, then we will write
$K_{X^{\prime}}+B^{\sharp}=f^{*}(K_{X}+B)$ and
$\beta_{X^{\prime}}=f^{*}\beta_{X}-E$ where $E\geq 0$ is exceptional. Note
however that $E_{S^{\prime}}:=E|_{S^{\prime}}$ may not be $g$-exceptional
where $g:S^{\prime}\to S^{\nu}$. If
$K_{S^{\nu}}+B_{S^{\nu}}=(K_{X}+B)|_{S^{\nu}}$ is the usual adjunction, then

$(K_{X}+B+\beta_{X})|_{S^{\nu}}=K_{S^{\nu}}+B_{S^{\nu}}+g_{*}E_{S^{\prime}}+g_{*}\beta_{S^{\prime}}.$

By Lemma 1.4, if $(X,B+\boldsymbol{\beta})$ is generalized lc in codimension
2, then $K_{X}+B$ is $\mathbb{R}$-Cartier in codimension 2 and hence the above
formula can always be used to compute
$(K_{X}+B+\boldsymbol{\beta})|_{S^{\nu}}$.

It is clear that if $(X,B+\boldsymbol{\beta})$ is generalized log canonical
(resp. generalized plt) then $({S^{\nu}},B_{S^{\nu}}+\beta_{S^{\nu}})$ is
generalized log canonical (resp. generalized klt). We now will verify that the
reverse implication also holds. The following statement is often referred to
at plt inversion of adjunction.

######  Theorem 2.6.

Let $(X,B+\boldsymbol{\beta})$ be a generalized pair and $S$ a component of
$B$ of coefficient 1 with normalization $\nu:S^{\nu}\to S$. Then
$(X,B+\boldsymbol{\beta})$ is generalized plt on a neighborhood of $S$ iff $S$
is normal and $({S},B_{S}+\boldsymbol{\beta}^{S})$ is generalized klt.

###### Proof.

Let $f:X^{\prime}\to X$ be a log resolution of the generalized pair
$(X,B+\boldsymbol{\beta})$. We may assume that $f$ is a projective morphism.
We write
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=f^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$
and $S^{\prime}=f^{-1}_{*}S$. We have a short exact sequence

$0\to\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor)\to\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})\to\mathcal{O}_{S^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})\to 0.$

Since, $-\lfloor
B^{\prime}\rfloor\equiv_{X}K_{X^{\prime}}+\\{B^{\prime}\\}+\boldsymbol{\beta}_{X^{\prime}}$
where $(X^{\prime},\\{B^{\prime}\\})$ is klt and
$\boldsymbol{\beta}_{X^{\prime}}$ is nef (and big) over $X$, then
$R^{1}f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor B^{\prime}\rfloor)=0$ and so we
have a surjection

$\phi:f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor B^{\prime}\rfloor+S^{\prime})\to
f_{*}\mathcal{O}_{S^{\prime}}(-\lfloor B^{\prime}\rfloor+S^{\prime}).$

If $(X,B+\boldsymbol{\beta})$ is generalized plt (on a neighborhood of $S$),
then $-\lfloor B^{\prime}\rfloor+S^{\prime}\geq 0$ is effective and
exceptional (over a neighborhood of $S$) and hence $\lfloor
B_{S^{\prime}}\rfloor=(\lfloor B^{\prime}\rfloor-S^{\prime})|_{S^{\prime}}\leq
0$ so that $(S^{\nu},B_{S^{\nu}}+\boldsymbol{\beta}^{S})$ is generalized klt.
We also have that $f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})=\mathcal{O}_{X}$ and so $\phi$ factors through

$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})=\mathcal{O}_{X}\to\mathcal{O}_{S}\subset\nu_{*}\mathcal{O}_{S^{\nu}}=f_{*}\mathcal{O}_{S^{\prime}}\subset
f_{*}\mathcal{O}_{S^{\prime}}(-\lfloor B^{\prime}\rfloor+S^{\prime})$

and therefore $\mathcal{O}_{S}=\nu_{*}\mathcal{O}_{S^{\nu}}$ and $S$ is
normal.

If $(S,B_{S}+\beta_{S})$ is generalized klt (and in particular $S$ is normal),
then $0\leq-\lfloor B_{S^{\prime}}\rfloor=(-\lfloor
B^{\prime}\rfloor+S^{\prime})|_{S^{\prime}}$ and
$f_{*}\mathcal{O}_{S^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})=\mathcal{O}_{S}$, and so we have a surjection
$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})\to\mathcal{O}_{S}$. Since
$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})\subset\mathcal{O}_{X}$, it follows that
$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor
B^{\prime}\rfloor+S^{\prime})=\mathcal{O}_{X}$ on a neighborhood of $S$ and so
$-\lfloor B^{\prime}\rfloor+S^{\prime}\geq 0$ over a neighborhood of $S$, i.e.
$(X,B+\boldsymbol{\beta})$ is generalized plt on a neighborhood of $S$.

∎

The following result is known as log canonical inversion of adjunction. In the
usual pair setting, it was first addressed in [Kawakita07] and refined
in[Hacon14] and [Fil20]. The following proof is based on the ideas of
[Hacon14].

######  Theorem 2.7.

Let $(X,B+\boldsymbol{\beta})$ be a generalized pair and $S$ a component of
$B$ of coefficient 1 with normalization $\nu:S^{\nu}\to S$. Then
$(X,B+\boldsymbol{\beta})$ is generalized lc on a neighborhood of $S$ iff
$({S^{\nu}},B_{S^{\nu}}+\boldsymbol{\beta}^{S^{\nu}})$ is generalized lc.

###### Proof.

Following the arguments above, it is easy to see that if
$(X,B+\boldsymbol{\beta})$ is generalized lc on a neighborhood of $S$, then
$({S^{\nu}},B_{S^{\nu}}+\boldsymbol{\beta}^{S^{\nu}})$ is generalized lc. Thus
it suffices to show that if
$({S^{\nu}},B_{S^{\nu}}+\boldsymbol{\beta}^{S^{\nu}})$ is generalized lc, then
$(X,B+\boldsymbol{\beta})$ is generalized lc on a neighborhood of $S$. The
question is local over $X$ and so we may assume that $X$ is a relatively
compact Stein variety.

Let $\mu:Y\to X$ be a generalized dlt model given by Theorem 1.6 so that $\mu$
is projective, $Y$ is $\mathbb{Q}$-factorial, all exceptional divisors have
discrepancy $a\leq-1$, $(Y,B^{\prime}_{Y}+\boldsymbol{\beta})$ is generalized
dlt, and $B^{\prime}_{Y}:=\mu^{-1}_{*}B+{\rm Ex}(\mu)\leq B_{Y}$ where
$K_{Y}+B_{Y}+\boldsymbol{\beta}_{Y}=\mu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$.
Let $S_{Y}:=\mu^{-1}_{*}S$, $B^{\prime}_{Y}=S_{Y}+\Gamma$ and
$B_{Y}=S_{Y}+\Gamma+\Sigma$. Let $H$ be a sufficiently ample divisor and run
the $K_{Y}+S_{Y}+\Gamma+\boldsymbol{\beta}_{Y}$ mmp with scaling of $H$.

######  Claim 2.8.

There is a sequence of flips and contractions $\phi_{i}:Y_{i}\dasharrow
Y_{i+1}$ and real numbers $s_{0}=1$, $s_{i}\geq s_{i+1}\geq 0$ such that
$K_{Y_{i}}+S_{i}+\Gamma_{i}+\boldsymbol{\beta}_{Y_{i}}+sH_{i}$ is nef over $X$
for $s_{i}\geq s\geq s_{i+1}$. If the minimal model program terminates, then
we may assume that $s_{n+1}=0$, otherwise the we have $\lim s_{i}=0$.

###### Proof.

For any $s>0$, $\boldsymbol{\beta}_{Y}+sH\equiv_{X}-(K_{Y}+B_{Y})+sH$ is an
ample ${\mathbb{R}}$-divisor and hence

$S_{Y}+\Gamma+\boldsymbol{\beta}_{Y}+sH\equiv_{X}\Delta_{s}$

where $(Y,\Delta_{s})$ is klt. By [DHP22] or [Fuj22], for any $0<\epsilon\ll
1$, we can run the $K_{Y}+\Delta_{\epsilon}$ mmp over $X$ with scaling of
$(1-s)H$. The claim now follows easily. ∎

We may assume that for $i\gg i_{0}$, all $\phi_{i}$ are flips.

######  Claim 2.9.

For any $t>0$, there is an ${\mathbb{R}}$-divisor
$\Theta_{t}\equiv_{X}\Gamma+\boldsymbol{\beta}_{Y}+tH$ such that
$(Y,S_{Y}+\Theta_{t})$ is plt.

###### Proof.

Since $H$ is ample and $\boldsymbol{\beta}_{Y}$ is nef, then
$\boldsymbol{\beta}_{Y}+tH\equiv_{X}-(K_{Y}+B_{Y})+tH$ is an ample
${\mathbb{R}}$-divisor and hence

$S_{Y}+\Gamma+\boldsymbol{\beta}_{Y}+tH\equiv_{X}S_{Y}+\Theta_{t}$

where $(Y,S_{Y}+\Theta_{t})$ is plt. ∎

If $t\leq s_{i}$, then $(Y_{i},S_{i}+\Theta_{t,i})$ is plt and in particular
$S_{i}$ is normal. Suppose that $\Sigma_{i}\cap S_{i}\neq\emptyset$, then

$\mu_{i}^{*}(K_{X}+B+\boldsymbol{\beta}_{X})|_{S_{i}}=(K_{Y_{i}}+S_{i}+\Gamma_{i}+\Sigma_{i}+\boldsymbol{\beta}_{Y_{i}})|_{S_{i}}=K_{S_{i}}+{\rm
Diff}_{S_{i}}(\Gamma_{i}+\Sigma_{i})+\boldsymbol{\beta}_{Y_{i}}|_{S_{i}}$

where $K_{S_{i}}+{\rm Diff}_{S_{i}}(\Gamma_{i}+\Sigma_{i})$ is not log
canonical as ${\rm mult}_{P}(\Gamma_{i}+\Sigma_{i})>1$, and this implies that
${\rm mult}_{Q}{\rm Diff}_{S_{i}}(\Gamma_{i}+\Sigma_{i})>1$ where $Q$ is any
component of $S_{i}\cap P$. But then $K_{S_{i}}+{\rm
Diff}_{S_{i}}(\Gamma_{i}+\Sigma_{i})+\boldsymbol{\beta}_{Y_{i}}|_{S_{i}}$ is
not generalized log canonical and so neither is
$({S^{\nu}},B_{S^{\nu}}+\boldsymbol{\beta}^{S^{\nu}})$. Therefore
$\Sigma_{i}\cap S_{i}=\emptyset$ for all $i\geq 0$.

Fix $m>0$ such that $m\Sigma$ is an integral divisor and
$s_{i}>\frac{1}{m}\geq s_{i+1}$. Then

$H_{i}-m\Sigma_{i}-S_{i}\equiv_{X}K_{Y_{i}}+\Theta_{\frac{1}{m},i}+(m-1)(K_{Y_{i}}+S_{i}+\Gamma_{i}+\frac{1}{m}H_{i}+\boldsymbol{\beta}_{Y_{i}})$

where $K_{Y_{i}}+\Theta_{\frac{1}{m},i}$ is klt and
$K_{Y_{i}}+S_{i}+\Gamma_{i}+\frac{1}{m}H_{i}+\boldsymbol{\beta}_{Y_{i}}$ is
nef and big over $X$ so that
$R^{1}(\mu_{i})_{*}\mathcal{O}_{Y_{i}}(H_{i}-m\Sigma_{i}-S_{i})=0$ and hence
we have a surjection

$(\mu_{i})_{*}\mathcal{O}_{Y_{i}}(H_{i}-m\Sigma_{i})\to(\mu_{i})_{*}\mathcal{O}_{S_{i}}(H_{i}-m\Sigma_{i})=(\mu_{i})_{*}\mathcal{O}_{S_{i}}(H_{i}).$

Note that as $Y_{i_{0}}\dasharrow Y_{i}$ is a small bimeromorphic map, then
the sheaves

$(\mu_{i})_{*}\mathcal{O}_{Y_{i}}(H_{i}-m\Sigma_{i})=(\mu_{i_{0}})_{*}\mathcal{O}_{Y_{i_{0}}}(H_{i_{0}}-m\Sigma_{i_{0}})\subset(\mu_{i_{0}})_{*}\mathcal{O}_{Y_{i_{0}}}(H_{i_{0}})$

are contained in
$\mathcal{I}_{V}\cdot(\mu_{i_{0}})_{*}\mathcal{O}_{Y_{i_{0}}}(H_{i_{0}})$ for
$m\gg 0$ where $V=\mu_{i_{0}}(\Sigma_{i_{0}})$ and if $V\cap S\neq\emptyset$,
then this contradicts the above surjection. Therefore
$\mu_{i_{0}}(\Sigma_{i_{0}})\cap S=\emptyset$ and so
$(X,B+\boldsymbol{\beta})$ is generalized log canonical on a neighborhood of
$S$.

∎

###### Proof of Theorem 0.1.

Immediate from Theorems 2.6 and 2.7. ∎

######  Proposition 2.10.

Let $(X,B+\boldsymbol{\beta})$ be a generalized log canonical pair and
$(X,B_{0}+\boldsymbol{\beta}_{0})$ a generalized klt pair. If $V$ a minimal
log canonical center of $(X,B+\boldsymbol{\beta})$, then $V$ is normal.

###### Proof.

The question is local on $X$ and hence we may assume that $X$ is relatively
compact and Stein. By the usual tie breaking arguments, we may assume that
there is a unique log canonical place for $(X,B+\boldsymbol{\beta})$ and that
this place dominates $V$. Thus there is a log resolution of
$(X,B+\boldsymbol{\beta})$, $f:X^{\prime}\to X$ such that
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=f^{*}(K_{X}+B+\beta)$
where $B^{\prime}=S^{\prime}+\Delta^{\prime}$ and $S^{\prime}$ is the unique
LC place for $(X,B+\boldsymbol{\beta})$ so that
$\lfloor\Delta^{\prime}\rfloor\leq 0$ is an exceptional divisor. By the proof
of Theorem 2.6, we have a surjection

$\phi:f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor\Delta^{\prime}\rfloor)\to
f_{*}\mathcal{O}_{S^{\prime}}(-\lfloor\Delta^{\prime}\rfloor).$

Since $-\lfloor\Delta^{\prime}\rfloor$ is effective and exceptional, then
$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor\Delta^{\prime}\rfloor)=\mathcal{O}_{X}$.
Since $S^{\prime}$ is smooth, $S^{\prime}\to V$ factors through the
normalization $\nu:V^{\nu}\to V$ and so $\phi$ factors via
$f_{*}\mathcal{O}_{X^{\prime}}(-\lfloor\Delta^{\prime}\rfloor)=\mathcal{O}_{X}\to\mathcal{O}_{V}$
and the natural inclusions

$\mathcal{O}_{V}\subset\nu_{*}\mathcal{O}_{V^{\nu}}\subset
f_{*}\mathcal{O}_{S^{\prime}}\subset
f_{*}\mathcal{O}_{S^{\prime}}(-\lfloor\Delta^{\prime}\rfloor).$

It follows that $\mathcal{O}_{V}\cong\nu_{*}\mathcal{O}_{V^{\nu}}$ and hence
$V$ is normal. ∎

######  Theorem 2.11.

Let $(X,B+\boldsymbol{\beta})$ be a generalized log canonical pair then ${\rm
nklt}(X,B+\boldsymbol{\beta})$ is seminormal.

###### Proof.

The corresponding result for log canonical pairs is contained in [Amb98]. We
will follow the approach of [Kollar07]. Let $\nu:X^{\prime}\to X$ be a log
resolution of $(X,B+\boldsymbol{\beta})$ and write
$K_{X^{\prime}}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}=\nu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$.
If $B^{\prime}=S-A+\\{B^{\prime}\\}$ where $S,A$ are effective Weil divisors
without common components, then $S$ is seminormal as it is a divisor with
simple normal crossings and hence $f|_{S}:S\to W$ factors through the
seminormilazation $h:W^{\rm sn}\to W$ via $g:S\to W^{\rm sn}$. We have a short
exact sequence

$0\to\mathcal{O}_{X^{\prime}}(A-S)\to\mathcal{O}_{X^{\prime}}(A)\to\mathcal{O}_{S}(A|_{S})\to
0.$

Since
$A-S\equiv_{X}K_{X^{\prime}}+\\{B^{\prime}\\}+\boldsymbol{\beta}_{X^{\prime}}$
and $\boldsymbol{\beta}_{X^{\prime}}$ is nef and big over $X$, it follows that
$R^{1}\nu_{*}\mathcal{O}_{X^{\prime}}(A-S)=0$ and hence
$\nu_{*}\mathcal{O}_{X^{\prime}}(A)\to\nu_{*}\mathcal{O}_{S}(A|_{S})$. Since
$A$ is $\nu$-exceptional, $\nu_{*}\mathcal{O}_{X^{\prime}}(A)=\mathcal{O}_{X}$
and hence $\nu_{*}\mathcal{O}_{S}(A|_{S})=\mathcal{O}_{W}$. But
$\nu_{*}\mathcal{O}_{S}(A|_{S})\supset
h_{*}g_{*}\mathcal{O}_{S}=h_{*}\mathcal{O}_{W^{\rm sn}}$ and so
$h_{*}\mathcal{O}_{W^{\rm sn}}=\mathcal{O}_{W}$ i.e. $h:W^{\rm sn}\to W$ is an
isomorphism. ∎

We will now prove that Theorem 0.2 follows from Theorem 0.3.

###### Proof of Theorem 0.2.

We may assume that $W$ has codimension $\geq 2$. By Proposition 2.10, $W$ is
normal and by a standard tie breaking argument, we may assume that there is a
unique log canonical place for an auxiliary pair
$(X,B^{\sharp}+\boldsymbol{\beta}^{\sharp})$. Let $f:X^{\prime}\to X$ be the
corresponding dlt model (Theorem 1.6), then $f$ has a unique exceptional
divisor $S^{\prime}$ and
$f^{*}(K_{X}+B+\boldsymbol{\beta}_{X})=K_{X^{\prime}}+S^{\prime}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}}$
where $(X^{\prime},S^{\prime}+B^{\prime}+\boldsymbol{\beta}^{\prime})$ is plt.
Note that by the proof of Proposition 2.10, $S^{\prime}\to W$ has connected
fibers. By Theorem 2.6,
$(S^{\prime},B_{S^{\prime}}+\boldsymbol{\beta}_{S^{\prime}})$ is generalized
klt where
$K_{S^{\prime}}+B_{S^{\prime}}+\boldsymbol{\beta}_{S^{\prime}}=(K_{X^{\prime}}+S^{\prime}+B^{\prime}+\boldsymbol{\beta}_{X^{\prime}})|_{S^{\prime}}$.
Let $\nu:S^{\prime}\to W$, then
$K_{S^{\prime}}+B_{S^{\prime}}+\boldsymbol{\beta}_{S^{\prime}}=\nu^{*}((K_{X}+B+\boldsymbol{\beta}_{X})|_{W})$
and so by Theorem 0.3, $(K_{X}+B+\boldsymbol{\beta}_{X})|_{W}\equiv
K_{W}+B_{W}+\boldsymbol{\beta}_{W}$ where $(W,B_{W}+\boldsymbol{\beta}_{W})$
is generalized klt. ∎

######  Remark 2.12.

The above arguments show that if $(X,B+\boldsymbol{\beta})$ and
$(X,B^{\prime}+\boldsymbol{\beta}^{\prime})$ are generalized pairs, $V\subset
X$ is a subvariety and $U\subset X$ an open subset such that
$(U,(B+\boldsymbol{\beta})|_{U})$ is generalized lc,
$(U,(B^{\prime}+\boldsymbol{\beta}^{\prime})|_{U})$ is generalized klt and
$V\cap U$ is a minimal log canonical center of
$(U,(B+\boldsymbol{\beta})|_{U})$, then
$(K_{X}+B+\boldsymbol{\beta})|_{V^{\nu}}=K_{V^{\nu}}+B_{V^{\nu}}+\boldsymbol{\beta}^{V^{\nu}}$
where $V^{\nu}\to V$ is the normalization and
$(V^{\nu},B_{V^{\nu}}+\boldsymbol{\beta}^{V^{\nu}})$ is a generalized pair.

##  3\. Cone Theorem for generalized klt pairs

The results in this section are inspired by [CH20]. They suggest that one of
the main obstructions to the higher dimensional minimal model program for
Kähler varieties is Conjecture 0.4.

######  Proposition 3.1.

Assume Conjecture 0.4 in dimension $\leq n-1$. Let $X$ be a compact
$\mathbb{Q}$-factorial Kähler $n$-fold such that $(X,B+\boldsymbol{\beta})$ is
generalized klt, $K_{X}+B+\boldsymbol{\beta}_{X}$ is pseudo-effective, and
$\omega$ be a Kähler form such that
$\alpha:=[K_{X}+B+\boldsymbol{\beta}_{X}+\omega]$ is nef and big but not
Kähler. Then there is an $\alpha$-trivial rational curve $C$ such that
$0<-(K_{X}+B+\boldsymbol{\beta}_{X})\cdot C=\omega\cdot C\leq
2\operatorname{dim}X$.

###### Proof.

By Proposition 4.21 (see also [CT15, Theorem 1.1] and [Bou04, Theorem 3.17] in
the smooth case) the restricted non-Kähler locus ${E_{nK}^{as}}(\alpha)$
coincides with the null-locus ${\rm Null}(\alpha)$, and there exists a Kähler
current $\eta$ with weak analytic singularities in the class $\alpha$ such
that the Lelong set coincides with ${E_{nK}^{as}}(\alpha)$. Since $\alpha$ is
not Kähler, then ${\rm Null}(\alpha)$ has a positive dimensional component.
Let $Z$ be a maximal dimensional irreducible component of ${\rm Null}(\alpha)$
and $c$ be the log canonical threshold of $(X,B+\boldsymbol{\beta}_{X})$ with
respect to $\eta$ on a neighborhood of general points of $Z$. This means that
if we pick a log resolution $\nu:X^{\prime}\to X$ such that
$K_{X^{\prime}}+B_{X^{\prime}}+\boldsymbol{\beta}_{X^{\prime}}=\nu^{*}(K_{X}+B+\boldsymbol{\beta}_{X})$
where $\boldsymbol{\beta}_{X^{\prime}}$ is nef and
$\nu^{*}\eta=\eta^{\prime}+F$ where $F$ is an effective
${\mathbb{R}}$-divisor, $\eta^{\prime}\geq 0$ and $F+B_{X^{\prime}}$ has
simple normal crossings, then $Z$ is an irreducible component of
$\nu((B_{X^{\prime}}+cF)^{=1})$ and $Z$ is not contained in
$\nu((B_{X^{\prime}}+cF)^{>1})$. If
$\boldsymbol{\eta}=\overline{\eta^{\prime}}$, then $Z$ is a generalized log
canonical center of the generalized pair
$(X,B+c\nu_{*}F+\boldsymbol{\beta}+c\boldsymbol{\eta})$. By Remark 2.12,

$(K_{X}+B+c\nu_{*}F+\boldsymbol{\beta}+c\boldsymbol{\eta})|_{Z^{\nu}}=K_{Z^{\nu}}+B_{Z^{\nu}}+\gamma_{Z^{\nu}}$

where $({Z^{\nu}},B_{Z^{\nu}}+\gamma_{Z^{\nu}})$ is a generalized pair.

By assumption we have $k:=\nu_{\rm
num}(\alpha|_{Z^{\nu}})<\operatorname{dim}Z$ so that
$(\alpha|_{Z^{\nu}})^{k}\not\equiv 0$ and $(\alpha|_{Z^{\nu}})^{k+1}\equiv 0.$
But then

$(K_{Z^{\nu}}+B_{Z^{\nu}}+\gamma_{Z^{\nu}})\cdot\alpha_{Z^{\nu}}^{k}\cdot\omega_{Z^{\nu}}^{\operatorname{dim}Z-k-1}=-\alpha_{Z^{\nu}}^{k}\cdot\omega_{Z^{\nu}}^{\operatorname{dim}Z-k}<0$

where $\alpha_{Z^{\nu}}=\alpha|_{Z^{\nu}}$ and
$\omega_{Z^{\nu}}=\omega|_{Z^{\nu}}$. Since $B_{Z^{\nu}}\geq 0$ and
$\gamma_{Z^{\nu}}$ is pseudo-effective, then $K_{Z^{\nu}}$ is not pseudo-
effective and hence neither is $K_{Z^{\prime}}$ for any resolution
$Z^{\prime}\to Z^{\nu}$.

Consider now the MRC fibration $Z^{\prime}\to Y$, which is non-trivial as
$K_{Z^{\prime}}$ is not pseudo-effective. Passing to a higher model we may
assume that it is a morphism with general fiber $F$. Note that $F$ is
rationally connected and hence $h^{2}(\mathcal{O}_{F})=0$ and so $F$ is
algebraic. Arguing as above, for any $\epsilon>0$ we have

$(K_{Z^{\prime}}+B_{Z^{\prime}}+\gamma_{Z^{\prime}}+(1-\epsilon)\omega_{Z^{\prime}}+t\alpha_{Z^{\prime}})\cdot\alpha_{Z^{\prime}}^{k}\cdot\omega_{Z^{\prime}}^{\operatorname{dim}Z-k-1}=$
$(K_{Z^{\nu}}+B_{Z^{\nu}}+\gamma_{Z^{\nu}}+(1-\epsilon)\omega_{Z^{\nu}})\cdot\alpha_{Z^{\nu}}^{k}\cdot\omega_{Z^{\nu}}^{\operatorname{dim}Z-k-1}=-\epsilon\alpha_{Z^{\nu}}^{k}\cdot\omega_{Z^{\nu}}^{\operatorname{dim}Z-k}<0,$

where $\omega_{Z^{\prime}}=\omega|_{Z^{\prime}}$ and
$\alpha_{Z^{\prime}}=\alpha|_{Z^{\prime}}$. Since $\gamma_{Z^{\prime}}$ is
pseudo-effective and $B_{Z^{\prime}}^{<0}$ is $Z^{\prime}\to Z^{\nu}$
exceptional, it follows that
$(K_{Z^{\prime}}+(1-\epsilon)\omega_{Z^{\prime}}+t\alpha_{Z^{\prime}})\cdot\alpha_{Z^{\prime}}^{k}\cdot\omega_{Z^{\prime}}^{\operatorname{dim}Z-k-1}<0$
and hence
$K_{Z^{\prime}}+(1-\epsilon)\omega_{Z^{\prime}}+t\alpha_{Z^{\prime}}$ is not
pseudo-effective for any $t>0$. Since $Y$ is not uniruled, $K_{Y}$ is pseudo-
effective and hence by [CH20, Theorem 5.2],
$K_{F}+(1-\epsilon)\omega_{F}+t\alpha_{F}$ is also not pseudo-effective for
any $t>0$ and in particular $\alpha_{F}$ is not big. By the cone theorem,
there are finitely many $K_{F}+(1-\epsilon)\omega_{F}$ negative extremal rays,
and there is a non-empty finite collection of $K_{F}+(1-\epsilon)\omega_{F}$
negative extremal rays that are $\alpha$ trivial. Let $\eta:F\to\bar{F}$ be
the induced non-trivial morphism contracting this face. Then
$\alpha_{F}=\eta^{*}\alpha_{\bar{F}}$ where $\alpha_{\bar{F}}$ is ample on
$\bar{F}$. If $\eta$ is birational, then $\alpha_{F}$ is big which is a
contradiction. Thus, $\eta$ is of fiber type and hence $F$ is covered by
$\alpha_{F}$-trivial rational curves $C$. Note that by bend and break, we may
assume that $0<-K_{F}\cdot C\leq 2\operatorname{dim}F$ and hence
$(1-\epsilon)\omega_{F}\cdot C<2\operatorname{dim}F$. But then $Z^{\prime}$ is
covered by $\alpha_{F}$-trivial rational curves and finally $Z$ is covered by
$\alpha$-trivial rational curves $C$ such that $0<\omega\cdot
C<\frac{2}{1-\epsilon}\operatorname{dim}X$. Since these curves belong to
finitely many numerical classes, we may assume that $0<\omega\cdot C\leq
2\operatorname{dim}X$. Finally, we observe that

$0<-(K_{X}+B+\boldsymbol{\beta}_{X})\cdot C=\omega\cdot C\leq
2\operatorname{dim}X.$

∎

######  Corollary 3.2.

Let $(X,B+\boldsymbol{\beta})$ be a compact Kähler $4$-fold generalized klt
pair such that $K_{X}+B$ is pseudo-effective. Then there are at most countably
many rational curves $\\{\Gamma_{i}\\}_{i\in I}$ such that
$-(K_{X}+B+\boldsymbol{\beta}_{X})\cdot\Gamma_{i}\leq 8$ for all $i\in I$ and

$\overline{\rm NA}(X)=\overline{\rm
NA}(X)_{(K_{X}+B+\boldsymbol{\beta}_{X})\geq 0}+\sum_{i\in
I}\mathbb{R}^{+}[\Gamma_{i}].$

###### Proof.

This follows by standard arguments from Proposition 3.1 (see eg the proof of
[DH23, Theorem 1.3]). ∎

##  4\. Null loci

In this section we generalize the main result of [CT15] to the singular
setting. To start with, we recall the notion of Lelong number of a closed
positive current $T\geq 0$ on a normal complex space, since it plays a crucial
role in the formulation of Proposition 4.21 below.

###  4.1. Closed positive currents on normal complex spaces

Let $\Omega\subset\mathbb{C}^{n}$ be the unit ball, and let $\phi$ be a psh
function on $\Omega$. Then

$T:=\sqrt{-1}\partial\overline{\partial}\phi$

defines a closed positive current of $(1,1)$-type on $\Omega$. For example, if
we take $\phi=\log|f|^{2}$, with $f$ holomorphic, then up to a multiple, $T$
is equal to the current of integration along the analytic set $f=0$ (taking
the multiplicities into account). Thus, closed positive $(1,1)$ currents can
be seen as natural generalizations of effective divisors.

It turns out that the function

$r\to\sup_{|z|=r}\phi(z)$

is convex increasing of $\log r$, i.e. if $x:=\log r$, then the function
$\displaystyle x\to\sup_{|z|=e^{x}}\phi(z)$ is convex increasing. It therefore
follows that the limit

$\nu(T,0):=\lim\inf_{z\to 0}\frac{\phi(z)}{\log|z|}$

exists, and it is called the Lelong number of $T$ at $z=0$. In the example
above, this is precisely the multiplicity of the divisor $(f=0)$ at $0$.
Moreover, one can see that the equality

(3) $\nu(T,0)=\sup\\{\nu\geq 0|\phi(z)\leq\nu\log|z|+\mathcal{O}(1)\\}$

holds true.

We collect next a few facts about currents which will be needed later on.

######  Theorem 4.1.

[DeBook] Let $X$ be a complex manifold, and let $T\geq 0$ be a closed positive
current of $(1,1)$-type on $X$. We consider an analytic hypersurface $A\subset
X$, and let $\chi_{A}$ be the characteristic function of $A$. Then the
following assertions hold true.

  1. (1)

The function $a\to\nu(T,a)$ defined on the analytic set $A$ is constant in the
complement of an at most countable union of analytic subsets of $A$, and it
defines the generic Lelong number of $T$ along $A$.

  2. (2)

The currents $\chi_{A}T$ and $\chi_{X\setminus A}T$ obtained by multiplying
$T$ with the characteristic function of $A$ and its complement, respectively
are closed (and of course, positive). Moreover, we have

$\chi_{A}T=\nu(T,A)[A],$

where $\nu(T,A)$ is the generic Lelong number of $T$ along $A$.

Next, consider a surjective map $f:X\to Y$ between two compact complex
manifolds. Given a closed $(1,1)$ current $T$ on the base $Y$, the pull-back
$f^{\star}T$ is a well-defined, closed current on $X$ (this is not necessarily
true for currents of other bi-degrees). The following result clarifies the
connection between the Lelong numbers of $T$ and those of its inverse image
$f^{\star}T$.

######  Theorem 4.2.

[Fav99] Under the assumptions above, there exists a positive constant $C>0$
such that we have

$C\nu(f^{\star}T,x)\leq\nu(T,y)\leq\nu(f^{\star}T,x),$

for all $x\in X$ and $y=f(x)$.

Note that the right-hand side inequality in Theorem 4.2 follows immediately
form the definition (3). Also, this sort of comparison inequalities is far
from true in case of currents obtained by direct images, i.e. if one wishes to
compare the Lelong numbers of a current $\Theta$ on $X$ with those of its
direct image $f_{\star}\Theta$. For example, let $f:X\to Y$ be the blow-up of
a point $y$ of $Y$. We assume that $Y$ is Kähler; then given any Kähler metric
$\omega$ on $X$, the direct image $f_{\star}\omega$ has a positive Lelong
number at $y$.

Still in this context (i.e. $f$ is the blow-up at a point and $T\geq 0$ is a
closed positive current on $Y$), we have the equality

(4) $f^{\star}T=\nu(T,y)[E]+R$

where $E$ is the exceptional divisor of the blow-up $f$, and $R\geq 0$ is a
closed positive current whose generic Lelong number along $E$ is equal to
zero. In particular, we have

(5) $\chi_{E}f^{\star}T=\nu(T,y)[E].$

Let $f:X\to Y$ be a surjective map between compact complex manifolds, and let
$\rho$ be a real, $(1,1)$ cohomology class on the target manifold $Y$. We have
the following well-known remark.

######  Lemma 4.3.

Given any $(1,1)$ closed positive current

$T\in f^{\star}\rho,$

there exists a $(1,1)$ closed positive current $R$ on $Y$ such that

(6) $T=f^{\star}R.$

###### Proof.

The matter is indeed clear: according to our hypothesis there exists an
$L^{1}$ function $\phi$ on $X$ and a smooth representative $a\in\rho$ such
that the following equality

$T=f^{\star}(a)+\sqrt{-1}\partial\overline{\partial}\phi$

holds. Now the restriction $\displaystyle T|_{X_{y}}$ of the current $T$ to
the general fibers $X_{y}$ of $f$ is a well-defined, closed positive current.
On the other hand, we have

$T|_{X_{y}}=\sqrt{-1}\partial\overline{\partial}\phi|_{X_{y}}$

which shows that $\displaystyle\phi|_{X_{y}}$ must be constant on $X_{y}$.
Therefore our assertion follows.∎

The notion of Lelong number of a closed positive current on a normal space
will be needed in order to formulate the main result of this section. We
recall it next.

######  Definition 4.4.

[Dem82]. Let $X$ be a normal complex space, and let $T\geq 0$ be a closed
positive $(1,1)$-current on $X$. We consider a positive function
$\varphi\in\mathcal{C}^{2}(X,{\mathbb{R}}_{+})$, such that $\log\varphi$ is
psh and such that $\operatorname{Supp}(T)\cap(\varphi<R)$ is relatively
compact in $X$ for all $0<R\ll 1$ sufficiently small. The limit

$\nu(T,\varphi):=\lim_{r\to 0}\frac{1}{(2\pi
r)^{2n-2}}\int_{\varphi<r}T\wedge(\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1}$

is called the Lelong number of $T$ with respect to $\varphi$.

Let $y\in X$ be an arbitrary point. If we consider an embedding

(7) $(X,y)\xhookrightarrow{}(\mathbb{C}^{N},0)$

then the coordinate functions $(z_{i})_{i=1,\dots,N}$ on $\mathbb{C}^{N}$
restricted to $X$ induce a generating system $(g_{i})_{i=1,\dots,N}$ of the
maximal ideal of the ring $\mathcal{O}_{X,y}$. The function

(8) $\varphi_{y}:=\sum_{i}|g_{i}|$

is defined on some small open subset $U$ containing $x$, and then the Lelong
number of $T$ at $y$ is defined as follows

(9) $\nu(T,y):=\nu(T|_{U},\varphi_{y}),$

where $T|_{U}$ is the restriction of $T$ to $U$, so the RHS is defined as in
Definition 4.4.

######  Remark 4.5.

It is not immediate that the limit in Definition 4.4 exists, but this is a
consequence of the Jensen formula established in [Dem82], Théorème 3.
Moreover, note that the Lelong number $\nu(T,y)$ is independent of the
embedding (7), as consequence of Théorème 4 in _loc. cit._

Let $n$ be the dimension of $X$. By composing the embedding map (7) with a
generic linear projection on $\mathbb{C}^{n}$, we obtain a proper, finite map

(10) $p:(X,y)\to(\mathbb{C}^{n},0)$

such that $p^{-1}(0)={y}$. The function

(11) $\widetilde{\varphi}_{y}:=\sum|p_{j}|$

(where $p_{j}$ are the components of $p$) verifies the inequalities

$\varphi_{y}^{M}\leq\widetilde{\varphi}_{y}\leq C\varphi_{y}$

locally near $x$, for some positive constants $C$ and $M$. We can assume that
it holds on the open subset $U$. By the comparison theorem for Lelong numbers
(cf. Théorème 4, page 46 in [Dem82]), we have

(12)
$\frac{1}{M}\nu(T,\widetilde{\varphi}_{y})\leq\nu(T,y)\leq\nu(T,\widetilde{\varphi}_{y}).$

######  Remark 4.6.

If $y\in X$ is a regular point, the equality

$\nu(T,y)=\lim\inf_{z\to y}\frac{\varphi_{T}}{\log|z-y|}$

holds, and it provides an alternative definition for the Lelong number of $T$
at $y$, as we have already mentioned. Simple examples ([BEGZ10], Appendix A)
show that as soon as $y\in X_{\rm sing}$, the relation above is no longer
verified in general. However, we always have the inequality
$\displaystyle\nu(T,y)\geq\lim\inf_{z\to y}\frac{\varphi_{T}}{\log|z-y|}$.

In connection with these topics, the following result was obtained very
recently in [P24].

######  Lemma 4.7.

[P24] Let $X$ be a normal complex space, and let $T\geq 0$ be a closed
positive current on $X$. The following equivalence

(13) $\nu(T,y)>0\iff\lim\inf_{z\to y}\frac{\varphi_{T}}{\log|z-y|}>0$

holds true for any point $y\in X$. In other words, the Lelong number of $T$ at
$y$ and the slope of its potential (i.e. the RHS of (13)) are simultaneously
positive or zero.

######  Remark 4.8.

Of course, one expects an inequality of type

(14) $\nu(T,y)\leq C\lim\inf_{z\to y}\frac{\varphi_{T}}{\log|z-y|}$

to be true for some constant $C>0$, uniform on compact subsets of $X$.

To finish this subsection we consider the following the set-up.

  * •

$X$ is a normal, compact Kähler space and $\pi:\widehat{X}\to X$ is
generically finite, such that $\widehat{X}$ is also normal (and Kähler).

  * •

$T=\alpha+\sqrt{-1}\partial\overline{\partial}\varphi$ is a closed current on
$X$, where $\alpha$ is smooth and locally
$\sqrt{-1}\partial\overline{\partial}$-exact, and such that

$T\geq\gamma$

for some smooth locally $\sqrt{-1}\partial\overline{\partial}$-exact
$(1,1)$-form $\gamma$.

  * •

We assume moreover that
$\pi^{\star}T=[E]+\theta+\sqrt{-1}\partial\overline{\partial}\rho,$ where $E$
is effective, $\rho$ is a bounded real function on $\widehat{X}$, and the form
$\theta$ is locally given by the Hessian of a function bounded from above.

Then we claim that the following inequality

(15) $\theta+\sqrt{-1}\partial\overline{\partial}\rho\geq\pi^{\star}\gamma$

holds true (this will be useful in the next sections). This is seen as
follows: we only have to verify (15) locally near a point
$x_{0}\in\operatorname{Supp}(E)$ in the support of the divisor $E$. Let $U$ be
an open subset of $X$ such that $\pi(x_{0})\in U$ and such that $\gamma$
restricted to $U$ is given by the Hessian of the smooth function $f_{\gamma}$.
Assume that there exists an open subset $V$ containing $x_{0}$ such that we
have

$\theta|_{V}=\sqrt{-1}\partial\overline{\partial}f_{\theta}$

where $f_{\theta}$ is bounded from above. It then follows that the function

(16) $f_{\theta}-f_{\gamma}\circ\pi+\rho|_{V_{0}}$

is psh, where $\displaystyle
V_{0}:=\big{(}V\cap\pi^{-1}(U)\big{)}\setminus\operatorname{Supp}E$. On the
other hand, the function in (16) is bounded from above on $V_{0}$. Since
$\widehat{X}$ is normal, it extends as psh function locally near this point,
by result due to Hartogs in the smooth case, see [Dem85], Théorème 1.7 for the
version we need here. This is the analogue of the fact that bounded
holomorphic functions on normal spaces extend.

In conclusion, the $(1,1)$-current obtained by taking the
$\sqrt{-1}\partial\overline{\partial}$ of the function (16) is positive – but
this is simply

$\theta+\sqrt{-1}\partial\overline{\partial}\rho-\pi^{\star}\gamma|_{V},$

and our claim is proved.

###  4.2. Main results

Prior to stating our results we set a few notations and conventions.

######  Definition 4.9.

Given a real $(1,1)$-class $\alpha$ we denote by ${\rm Null}(\alpha)$ the
_null locus_ of $\alpha$, which is given by the union of analytic subsets
$V\subset X$ such that ${\int_{V}\alpha^{\operatorname{dim}V}=0}$.

We next introduce and establish basic properties of a class of closed positive
currents on normal varieties which will play the role of _Kähler currents_ in
[CT15].

####  4.2.1. Currents with admissible singularities

We introduce the following class of singularities.

######  Definition 4.10.

Let $X$ be a normal, compact Kähler space, and let
$\varphi:X\to[-\infty,\infty[$ be a function on $X$. We say that $\varphi$ has
admissible singularities if

$\varphi=\max{(\varphi_{1},\dots,\varphi_{k})}$

where each $\varphi_{j}$ has analytic singularities in the sense of [Dem92]
i.e. it can be locally expressed as

$\gamma\log(\sum|f_{k}|^{2})$

modulo a bounded function. In the expression above, $\gamma\geq 0$ is a real
number and the functions $(f_{k})$ are holomorphic.

A more flexible version of this notion reads as follows.

######  Definition 4.11.

Let $X$ be a normal, compact Kähler space, and let

$T=\alpha+\sqrt{-1}\partial\overline{\partial}\varphi$

be a closed positive current of type $(1,1)$ on $X$, where we denote by
$\alpha$ a smooth, real $(1,1)$-form on $X$, which is locally
$\sqrt{-1}\partial\overline{\partial}$–exact. We say that $T$ has weak
analytic singularities if there exists:

  * •

a biholomorphic map $\pi:\widehat{X}\to X$ such that $\widehat{X}$ is normal,
and

  * •

a closed positive current

$\widehat{T}=\pi^{\star}\alpha+\sqrt{-1}\partial\overline{\partial}\psi\geq 0$

on $\widehat{X}$ such that $\psi$ has admissible singularities and such that
we have $\pi_{\star}\widehat{T}=T.$

######  Remark 4.12.

As consequence of the fact that the current $\widehat{T}$ is assumed to belong
to the class $\pi^{\star}\alpha$, we show that _the function $\psi$ above (in
the second point of Definition 4.11) is constant on every connected component
of positive dimensional fibers of $\pi$_. This can be seen as follows: assume
that the restriction of $\psi$ to a fiber $F$ of $\pi$ is not identically
$-\infty$. Then by Théorème 1.10 in [Dem85] combined with the fact that $\psi$
has admissible singularities we infer that $\psi|_{F}$ is a psh function
defined on a compact analytic space – hence, it must be constant by the
maximum principle. So, there exists a function $\varphi_{1}$ on $X$ such that
$\varphi_{1}\circ\pi=\psi$, and moreover, the difference $\varphi-\varphi_{1}$
is smooth (because it belongs to the kernel of the operator
$\sqrt{-1}\partial\overline{\partial}$).111Since $\psi$ has admissible
singularities, we expect that this should be the case for $\varphi_{1}$ (and
$\varphi$) as well, but it is not clear how such a statement can be proved. It
follows that Definition 4.11 is equivalent to the existence of a birational
map $\pi:\widehat{X}\to X$ together with a current
$\widehat{T}=\pi^{\star}\alpha+\sqrt{-1}\partial\overline{\partial}\psi\geq 0$
on $\widehat{X}$ such that $\pi^{\star}T=\widehat{T}$. In other words, the
”singular analogue” of Lemma 4.3 holds true.

It turns out that this class of currents behaves very well under a few natural
operations which will be needed in the proof of Proposition 4.21 below. In
particular we have the following statement.

######  Lemma 4.13.

Let $X$ and $Y$ be normal compact Kähler spaces, and let $p:Y\to X$ be a
holomorphic map. Let $\beta$ be a smooth, real and closed $(1,1)$–form on $X$.
We have the following assertions.

  1. (a)

Let $T$ be a current with weak analytic singularities on $X$. Then the inverse
image $p^{\star}T$ has weak analytic singularities.

  2. (b)

For $i=1,2$ let $T_{i}:=\beta+\sqrt{-1}\partial\overline{\partial}\varphi_{i}$
be two currents with weak analytic singularities in the class induced by the
smooth form $\beta$. If we define $\varphi:=\max(\varphi_{1},\varphi_{2})$,
then the current $\displaystyle
T:=\beta+\sqrt{-1}\partial\overline{\partial}\varphi$ has weak analytic
singularities.

  3. (c)

Assume moreover that $p$ is birational. If $\Theta\in p^{\star}(\beta)$ is a
closed positive current with weak analytic singularities in the class
$p^{\star}(\beta)$ on $Y$, then the direct image $T:=p_{\star}\Theta$ has weak
analytic singularities.

###### Proof.

Concerning the first point (a), consider the map $\pi_{X}:\widehat{X}\to X$
given in Definition 4.11, so that the inverse image

$\widehat{T}:=\pi_{X}^{\star}T$

is a closed positive current on $\widehat{X}$, whose potential has admissible
singularities. We can construct holomorphic maps

$\pi_{Y}:\widehat{Y}\to Y,\qquad\widehat{p}:\widehat{Y}\to\widehat{X}$

such that the equality $p\circ\pi_{Y}=\pi_{X}\circ\widehat{p}$ holds, the
space $\widehat{Y}$ is normal and $\pi_{Y}$ is birational.

Consider the inverse image $\Theta:=\widehat{p}^{\star}\widehat{T}$. By
Definition 4.10, it is clear that $\Theta$ has admissible singularities. On
the other hand, by Remark 4.12, we may assume that

$\Theta=\widehat{p}^{\star}\big{(}\pi_{X}^{\star}T\big{)}=\pi_{Y}^{\star}(p^{\star}T)$

which shows that the current $p^{\star}T$ has weak analytic singularities.

Point (b) is a direct consequence of (a), so we will not give any further
details.

For assertion (c) we argue as follows. Given that $\Theta$ has weak analytic
singularities, there exist a map $\pi_{Y}:\widehat{Y}\to Y$ and a closed
positive current

$\widehat{\Theta}\in\pi_{Y}^{\star}(p^{\star}\beta)=(p\circ\pi_{Y})^{\star}\beta$

as in Definition 4.11, such that

$\pi_{Y\star}\widehat{\Theta}=\Theta$

and moreover we have
$\widehat{\Theta}=(p\circ\pi_{Y})^{\star}\beta+\sqrt{-1}\partial\overline{\partial}\psi$
for a function $\psi$ with admissible singularities. The map
$p\circ\pi_{Y}:\widehat{Y}\to X$ is birational, and we clearly have

$(p\circ\pi_{Y})_{\star}\widehat{\Theta}=p_{\star}\Theta.$

It therefore follows that the direct image $p_{\star}\Theta$ has weak analytic
singularities. ∎

######  Remark 4.14.

Assume that $X$ is a manifold and that $T\in\alpha$ is a Kähler current with
weak analytic singularities. By the regularisation results in [Dem92], the
class $\alpha$ contains a current with analytic singularities, meaning that
the equality

$\varphi|_{U_{i}}=\gamma_{i}\log\big{(}\sum_{\alpha}|f_{i\alpha}|^{2}\big{)}+\psi_{i}$

holds, where the $f_{i\alpha}$ are holomorphic and $\psi_{i}$ is bounded. We
expect that this still holds in our context, i.e. in case $X$ is a normal,
compact Kähler space.

######  Remark 4.15.

It follows from Lemma 4.13 that in the definition of a current with weak
analytic singularities (4.11) we can assume that the space $\widehat{X}$ is
non-singular.

We prove next another property of currents with weak analytic singularities.
In the proof below, we use the generic notation $"C"$ for a constant that can
change from one line to another.

######  Lemma 4.16.

Let $T$ be a current with weak analytic singularities on a normal compact
Kähler space $X$. Then the set $E_{+}(T):=\bigcup_{c>0}E_{c}(T)$ is a closed,
analytic subset of $X$.

###### Proof.

We are using the notations in the previous Remark 4.12. In particular we
assume that $\widehat{X}$ is non-singular, and moreover the set
$(\varphi=-\infty)$ coincides with the image of $(\psi=-\infty)$ via the map
$\pi$, since we have

(17) $\psi=\varphi\circ\pi$

modulo a bounded quantity. Given that $\psi$ has admissible singularities, it
follows that we have the equality

$(\psi=-\infty)=E_{+}(\widehat{T}),$

and therefore we have $E_{+}(T)\subset\pi\big{(}E_{+}(\widehat{T})\big{)}$,
since $E_{+}(T)\subset(\varphi=-\infty)$ as a consequence of upper-
semicontinuity of $\varphi$.

Actually, more is true, namely the equality

$E_{+}(T)=\pi\big{(}E_{+}(\widehat{T})\big{)}$

holds. To see this, it would be enough to show the existence of a constant
$C>0$ such that we have

(18) $C\nu(T,y)\geq\nu(\pi^{\star}T,x)=\nu(\widehat{T},x)$

where $y\in X$ is an arbitrary point and $x\in\pi^{-1}(y)$. If we admit this
for the moment, the proof of our lemma is complete.

In order to establish inequality (18), we proceed as follows. Consider a local
parametrization $\tau:(X,y)\to(\mathbb{C}^{n},0)$. It is a proper, finite map
such that $\tau^{-1}(0)=y$ (as sets). In this context we have the following
important estimate, cf. [Dem82], Théorème 6

(19) $C\nu(T,y)\geq\nu(\Theta,0),$

which we have already mentioned in (12), where the constant $C$ here is very
explicit (depending on a certain multiplicity associated to the map $\tau$)
and $\Theta:=\tau_{\star}T$ is the direct image of $T$ with respect to the
proper map $\tau$.

By the main result in [Fav99], we have

(20) $C\nu(\Theta,0)\geq\nu\big{(}(\tau\circ\pi)^{\star}\Theta,x\big{)},$

which combined with (19) gives

(21) $C\nu(T,y)\geq\nu\big{(}(\tau\circ\pi)^{\star}\Theta,x\big{)}.$

It would therefore be sufficient to show that the inequality

(22) $\tau^{\star}(\tau_{\star}T)\geq T,$

because then it follows that
$\displaystyle\nu\big{(}(\tau\circ\pi)^{\star}\Theta,x\big{)}\geq\nu(\pi^{\star}T,x)$.

Let $T|_{U}=\sqrt{-1}\partial\overline{\partial}\varphi_{T}$ be the local
expression of the current $T$. Then we have

$\tau_{\star}T=\sqrt{-1}\partial\overline{\partial}\psi$

where $\displaystyle\psi(z):=\sum_{w\in\tau^{-1}(z)}\varphi_{T}(w)$ is the
trace of the local potential $\varphi_{T}$ of $T$. It follows that the
following formula

(23)
$\tau^{\star}(\tau_{\star}T)=\sqrt{-1}\partial\overline{\partial}\psi\circ\tau\geq\sqrt{-1}\partial\overline{\partial}\varphi_{T},$

holds. Indeed, the difference

(24) $\psi\circ\tau(w)-\varphi_{T}(w)=\sum_{x\in F_{w}^{\star}}\varphi_{T}(x)$

is a psh function on $U$, where $\displaystyle F_{w}^{\star}:=\\{x\in
U:\tau(x)=\tau(w),x\neq w\\}$. The argument for this last claim is as follows:
on the unramified locus of $\tau$ things are clear, and on the other hand the
RHS of (24) is uniformly bounded from above. Our proof is finished. ∎

######  Remark 4.17.

The inequality (22) does not holds in general. For example, if instead of
being finite and proper the map $\tau$ is the blow-up of $\mathbb{C}^{2}$ at
$0$, then (22) certainly fails in case $T$ is the current of integration on
the exceptional divisor.

Finally, we introduce the following notion.

######  Definition 4.18.

Let $X$ be a normal compact Kähler space, and let $\alpha$ be a nef and big
real $(1,1)$-class on $X$ (in the Bott-Chern cohomology). The restricted non-
Kähler locus of $\alpha$ is the following set

$E_{nK}^{as}(\alpha):=\bigcap_{T\in\alpha}E_{+}(T)$

where $T$ above is assumed to be a Kähler current with weak analytic
singularities, and $E_{+}(T)\subset X$ is the (analytic) subset of $X$ for
which the Lelong numbers of $T$ are strictly positive.

######  Remark 4.19.

In the case of a non-singular Kähler space $X$, one defines $E_{nK}(\alpha)$
as the intersection of $E_{+}(T)$ for all Kähler currents $T\in\alpha$. Thus,
the difference between $E_{nK}^{as}(\alpha)$ and $E_{nK}(\alpha)$ is that in
the definition of the former we restrict ourselves to currents with weak
analytic singularities. If $X$ is non-singular, then we have
$E_{nK}^{as}(\alpha)=E_{nK}(\alpha)$, thanks to the regularisation results in
[Dem92]. In the general case of a normal space, things are less clear, but we
can at least say that

$E_{nK}(\alpha)\subset E_{nK}^{as}(\alpha)\subset E_{nK}(\alpha)\cup X_{\rm
sing}$

holds true. We actually expect the first inclusion to be an equality.

The following statement will be important in what follows.

######  Corollary 4.20.

Let $X$ be a normal compact complex Kähler space, and let $\alpha$ be a nef
and big $(1,1)$-class. Then $E_{nK}^{as}(\alpha)$ is an analytic subset of
$X$, and there is a Kähler current with weak analytic singularities
$T\in\alpha$ such that $E_{+}(T)=E_{nK}^{as}(\alpha).$

###### Proof.

By (b) of Lemma 4.13, given two Kähler currents with weak analytic
singularities $T_{i}\in\alpha$, we can construct $T\in\alpha$ such that

$E_{+}(T)\subset E_{+}(T_{1})\cap E_{+}(T_{2})$

and moreover $T$ is again a Kähler current with weak analytic singularities.
We can therefore construct a sequence $T_{k}\in\alpha$ of such currents, for
which the following assertions

$E_{+}(T_{m+1})\subset E_{+}(T_{m}),\qquad
E_{nK}^{as}(\alpha)\subset\bigcap_{k}E_{+}(T_{k})$

are true for each $m\geq 1$. Since $X$ is compact and $E_{+}(T_{m})$ are
analytic sets, the corollary follows because by notherian induction there
exists an integer $m_{0}$ such that $E_{+}(T_{m+1})=E_{+}(T_{m})$ for all
$m\geq m_{0}$. ∎

In this context, we have the following statement, which represents the main
result of this section.

######  Theorem 4.21.

Let $X$ be a compact normal Kähler variety, and let $\alpha$ be a smooth
(1,1)-form, which is locally $\sqrt{-1}\partial\overline{\partial}$-exact and
such that the corresponding class is nef and big. Then
$E_{nK}^{as}(\alpha)=\operatorname{Null}(\alpha)$. In particular, the set
$\operatorname{Null}(\alpha)$ is analytic.

######  Remark 4.22.

As we have already mentioned in Remark 4.6, in singular setting the Lelong
number of a positive current can be different from the slope of its potential
at a given point. Therefore one might wonder why the former notion appears in
Proposition 4.21 and not the latter. The explanation is given by Lemma 4.7:
considering slopes instead of Lelong numbers would lead to the same set
$E_{nK}^{as}(\alpha)$.

###### Proof.

The arguments which will follow combine [CT15] (where this statement is
established in case $X$ is a manifold), with additional inputs from [DHP22].

_Step 1: the inclusion $\operatorname{Null}(\alpha)\subset
E^{as}_{nK}(\alpha)$ holds._ Indeed, let $V\subset X$ be an irreducible
component of $\operatorname{Null}(\alpha)$ so that

(25) $\int_{V_{\rm reg}}\alpha^{d}=0,$

where $d$ is the dimension of $V$. If $V\not\subset E_{nK}^{as}(\alpha)$, then
by the definition of this subset, there exists a Kähler current
$\Theta\in\alpha$, whose potentials have weak analytic singularities and whose
Lelong number at a generic point of $V$ is equal to zero. These two properties
show that the restriction $\displaystyle\Theta|_{V}$ is a Kähler current, so
in particular $\alpha|_{V}$ is nef and big. This contradicts the equality
(25).

_Step 2: the set $E^{as}_{nK}(\alpha)$ does not have isolated points._ Let

$T:=\alpha+\sqrt{-1}\partial\overline{\partial}\phi$

be a Kähler current in the class $\alpha$ (in the equality above we abusively
denote by $"\alpha"$ a smooth representative of this class). Assume that the
function $\phi$ has weak analytic singularities and moreover $x\in X$ is an
isolated point in the set $\phi^{-1}(-\infty)$. Then we can remove the pole of
$\phi$ at $x$ as follows (see [Dem92] and the references therein).

Consider $(X,x)\subset U\subset(\mathbb{C}^{N},0)$ a local embedding of $X$,
such that

$\alpha|_{U\cap X}=\sqrt{-1}\partial\overline{\partial}\tau_{x}$

for some smooth function $\tau_{x}$. The sum $\displaystyle\tau_{x}+\phi$ is
the restriction of a function $\psi$ defined on $U$ and whose Hessian is
bigger than a positive multiple of the Euclidean metric on $\mathbb{C}^{N}$.
We now define

$\widetilde{\psi}:=\max(\psi,C_{1}\|Z\|^{2}-C_{2})$

where $C_{1}$ is strictly positive and $C_{2}\gg 0$ is large enough, so that

$\psi|_{\partial(U\cap X)}>(C_{1}\|Z\|^{2}-C_{2})|_{\partial(U\cap X)}$

holds. This is indeed possible, since the restriction of $\psi$ to a small
enough neighborhood of the boundary of $U\cap X$ in $X$ is smooth. We have
denoted by $Z$ the coordinates in $\mathbb{C}^{N}$ and for each open set
$\Omega$ we denote by $\partial(\Omega)$ its boundary.

We then define $\widetilde{\phi}$ the function on $X$ given by
$\widetilde{\psi}|_{U\cap X}-\tau_{x}$ on $U\cap X$ and $\phi$ on $X\setminus
U$. This function has weak analytic singularities, and the current

$\widetilde{T}:=\alpha+\sqrt{-1}\partial\overline{\partial}\widetilde{\phi}$

is greater than a small multiple of the Kähler metric on $X$. Moreover,
$x\not\in\widetilde{\phi}^{-1}(-\infty)$. Therefore, the set
$E^{as}_{nK}(\alpha)$ cannot contain isolated points, all its irreducible
components must have dimension at least one.

_Step 3: the inclusion $E^{as}_{nK}(\alpha)\subset{\rm Null}(\alpha)$ holds._
Let $V\subset E^{as}_{nK}(\alpha)$ be any irreducible component. We have to
show that $\displaystyle\int_{V_{\rm reg}}\alpha^{d}=0$, where $d\geq 1$ is
the dimension of $V$. Assume that this equality does not hold. Then given that
the restriction $\alpha|_{V}$ is nef, the only alternative is

(26) $\int_{V_{\rm reg}}\alpha^{d}>0$

and we show next that (26) leads to a contradiction. This will be done if we
are able to construct a Kähler current in the class $\alpha$, whose potential
has weak analytic singularities and such that it is bounded locally at some
point of $V$.

A first important reduction is that we can assume that $V$ is non-singular.
Indeed, as shown in the proof of [DHP22, Theorem 2.29] there exists a
modification $p:\widehat{X}\to X$ such that the following hold.

  * •

The complex Kähler space $\widehat{X}$ is normal.

  * •

The proper transform $\widehat{V}$ of $V$ is a smooth submanifold of
$\widehat{X}$.

  * •

The map $p$ is an isomorphism locally over an open subset of $V$.

If we are able to construct a Kähler current $\Theta\in p^{\star}\alpha$ with
weak analytic singularities such that $\Theta$ is bounded locally at a very
general point of $\widehat{V}$, then we are done by considering the direct
image $p_{\star}\Theta$, cf. Lemma 4.13, (b). Thus replacing $X$ with
$\widehat{X}$, we can assume from this point on that $V$ is non-singular.

Thanks to inequality (26), it follows that we can construct a Kähler current

$\Theta_{V}:=\alpha|_{V}+\sqrt{-1}\partial\overline{\partial}f$

such that $f:V\to[-\infty,0]$ has _analytic singularities_ , cf. Remark 4.14.

On the other hand, the class $\alpha$ is nef and big on $X$, so there exists a
Kähler current

$\Theta:=\alpha+\sqrt{-1}\partial\overline{\partial}F$

where $F$ has weak analytic singularities along $V\cup Z$, where $Z\subset X$
is an analytic subset of $X$ which does not contain $V$.

The idea is to remove the singularities of $\Theta$ at the generic point of
$V$ (as we did in Step 2) by using $\Theta_{V}$, so that the resulting current
will provide the sought-after contradiction. However, in the actual context
additional complications arise due the singularities of $F$.

_A particular case._ As ”warm-up” for the rest of the proof, we provide here
an argument in case the function $f$ is smooth. By the proof of [Dem90,
Theorem 4] there exists a smooth extension $\widetilde{f}$ defined in an open
subset $U$ of $V$ such that

$\alpha|_{U}+\sqrt{-1}\partial\overline{\partial}\widetilde{f}$

is a Kähler current on $U$. Let now $\psi_{Z}$ be a quasi-psh function on $X$,
with analytic poles along $Z$. If $\delta_{0}>0$ is sufficiently small, then

(27)
$\alpha|_{U}+\sqrt{-1}\partial\overline{\partial}(\widetilde{f}+\delta_{0}\psi_{Z}|_{U})$

is a Kähler current on $U$, and it has poles along the analytic set $Z\cap U$.

Next we will use the hypothesis ”$\alpha$ nef” in order to diminish the order
of poles of $F$. This is necessary, as otherwise we will be unable to glue
$\Theta$ with the form in (27). Indeed, for any positive $\varepsilon>0$ there
exists a smooth function $G_{\varepsilon}$ on $X$ such that

$T_{\varepsilon}:=\alpha+\sqrt{-1}\partial\overline{\partial}G_{\varepsilon}\geq-\varepsilon\omega_{X}$

where $\omega_{X}$ is a reference Kähler metric on $X$. Let $\delta_{1}>0$ be
a strictly positive real number such that
$\displaystyle\Theta\geq\delta_{1}\omega_{X}$. The convex combination

$\Theta_{\varepsilon}:=(1-\frac{\varepsilon}{\delta_{1}})T_{\varepsilon}+\frac{\varepsilon}{\delta_{1}}\Theta$

is a Kähler current in the class $\alpha$. It is easy to check that it is
greater or equal than

$-\varepsilon(1-\frac{\varepsilon}{\delta_{1}})\omega_{X}+\frac{\varepsilon}{\delta_{1}}\delta_{1}\omega_{X}=\frac{\varepsilon^{2}}{\delta_{1}}\omega_{X}.$

On the other hand, the singularities of $\Theta_{\varepsilon}$ are of order
$\mathcal{O}(\varepsilon)$, in other words we can make them as small as we
want/need. Let $\displaystyle
F_{\varepsilon}:=(1-\varepsilon/\delta_{1})G_{\varepsilon}+\varepsilon/\delta_{1}F$
be the potential of $\Theta_{\varepsilon}$. It follows that given a point
$z\in Z\setminus V$, there exists an open subset $U_{z}$ containing $z$ such
that the following inequality

(28) $F_{\varepsilon}>\delta_{0}\psi_{Z}$

holds for all $0<\varepsilon\ll 1$ small enough. This can be seen as
consequence of Hilbert’s Nullstellensatz, as follows.

We recall that $E_{+}(\Theta)$ is equal to $Z\cup V$. Moreover there exists a
birational map $\pi:\widehat{X}\to X$ such that $\pi^{\star}\Theta$ has
admissible singularities, meaning in particular that locally near
$w\in\pi^{-1}(z)$ we have

$F\circ\pi|_{(\widehat{X},w)}=\log(\sum|f_{i}|^{2r_{i}})+b$

where $b$ is bounded, $(f_{i})$ is a set of holomorphic functions and
$r_{i}\geq 0$ are positive real numbers.

In the proof of Lemma 4.16 we showed that $\displaystyle
E_{+}(\Theta)=\pi\big{(}E_{+}(\pi^{\star}\Theta)\big{)}$, which in particular
implies that we have $\displaystyle\pi^{-1}\big{(}E_{+}(\Theta)\big{)}\supset
E_{+}(\pi^{\star}\Theta)$. In fact, the equality

(29) $\pi^{-1}\big{(}E_{+}(\Theta)\big{)}=E_{+}(\pi^{\star}\Theta)$

holds: let $x\in\pi^{-1}\big{(}E_{+}(\Theta)\big{)}$, so that
$\nu\big{(}\Theta,\pi(x)\big{)}>0$. We claim that $\nu(\pi^{\star}\Theta,x)>0$
is strictly positive as well. If this is not the case, the local potential of
$\pi^{\star}\Theta$ is _locally bounded from below_ near $x$. By the analogue
of (17) in our setting here, it follows that the potential of $\Theta$ is
equally locally bounded from below near $\pi(x)$. But this contradicts the
fact that the Lelong number of $\Theta$ at $\pi(x)$ is positive.

Consider next the ideal $\mathcal{I}=(f_{i})$ generated by the functions
appearing in the expression of $F\circ\pi$ above. We remark that the set of
zeros of $\mathcal{I}$ is contained in $(\pi^{-1}(Z),w)$ (this is only true
because we are ”far” from $V$). The function $\psi_{Z}$ is obtained by gluing
functions of type $\log(\sum|h_{j}|^{2})$, where $(h_{j})$ are the local
equations of $Z$ and so there exists an integer $N>0$ such that
$(h_{j}\circ\pi)^{N}\in\mathcal{I}$. In particular we have
$\sum|h_{j}\circ\pi|^{2N}\leq C(\sum|f_{i}|^{2})$ for some positive constant
$C>0$. By arranging the constants, and taking into account the fact that
$\pi^{-1}(z)$ is a compact set, this implies (28).

We fix a value, say $\varepsilon_{0}$, of $\varepsilon$ small enough, such
that (28) holds true for every point of $\partial U\cap Z$: this is possible,
since $\partial U\cap V=\emptyset$. Then we claim that there exists a positive
constant $C>0$ such that the inequality

(30) $\widetilde{f}+\delta_{0}\psi_{Z}<F_{0}+C$

holds true pointwise in a small open subset containing the boundary $\partial
U$ of $U$, where $\displaystyle F_{0}:=F_{\varepsilon_{0}}$.

In order to verify this claim, we note that $\partial U$ is a compact set, and
let $x\in\partial U$ be one of its elements. If $\displaystyle x\in\partial
U\cap Z$, we clearly have $\displaystyle\delta_{0}\psi_{Z}<F_{0}$ locally near
$x$, as consequence of the inequality (28). Moreover, the function
$\widetilde{f}$ is non-singular, so the existence of the constant $"C"$ such
that (30) holds in a small open subset of $x$ is guaranteed. If $\displaystyle
x\not\in\partial U\cap Z$, then things are clear. By compactness, we have
established our claim.

The next claim is that the function

$\widehat{F}:=\max(\widetilde{f}+\delta_{0}\psi_{Z},F_{0}+C)$

is defined on the whole space $X$, is bounded at the generic point of $V$, and
moreover

$\alpha+\sqrt{-1}\partial\overline{\partial}\widehat{F}$

is a Kähler current. Indeed, near the boundary $\partial U$ of the set $U$
where $\widetilde{f}$ is defined the inequality (30) shows that
$\widehat{F}=F_{0}+C$, and thus we define $\widehat{F}=F_{0}+C$ on the
complement of $U$.

_End of the proof._ In general, the function $f$ has no reason to be smooth,
but nevertheless the line of arguments above remains valid, thanks to a very
important remark in [CT15]. The point is that instead of obtaining a smooth
extension $\widetilde{f}$ of $f$ defined on an open subset $U\subset X$ as
above, in the actual context we only get a function with log poles
$\widetilde{f}$ and a ”pinched” neighborhood $U$ of $V\setminus W$, where $W$
is an analytic subset of $V$, such that the analog expression (27) is a Kähler
current. In our context, this can be seen as follows.

Consider finitely many open subsets $(A_{i})$ of $X$, such that each $A_{i}$
is an analytic subset the unit ball of some Euclidean space, and such that
$V\subset\cup A_{i}$. We can assume that the equality

$f|_{A_{i}\cap V}=\delta_{i}\log(\sum_{\alpha}|g_{i\alpha}|^{2})+\tau_{i}$

holds, where $\delta_{i}>0$, the functions $g_{i\alpha}$ defined on $A_{i}\cap
V$ are holomorphic and $\tau_{i}$ are bounded (given that $f$ has analytic
singularities). In particular, the equations $g_{i\alpha}=0$ define a global
analytic set $W$ contained in $V$.

Let $\mathcal{I}_{W}\subset\mathcal{O}_{V}$ be the ideal sheaf of $W$, and let
$\mathcal{J}_{W}\subset\mathcal{O}_{X}$ be its pull-back via the projection
map $\mathcal{O}_{X}\to\mathcal{O}_{V}$. Then there exists a birational map

$\pi:\widehat{X}\to X$

obtained by blowing-up smooth centres contained in $V$, such that the
following hold.

  1. (a)

_The space $\widehat{X}$ is normal and the proper transform $\widehat{V}$ of V
is non-singular. Moreover, the map $\pi$ is biholomorphic near the general
point of $\widehat{V}$._

  2. (b)

_The inverse image of $\mathcal{J}_{W}$ via the restriction of $\pi$ to
$\widehat{V}$ is equal to $\mathcal{O}_{\widehat{V}}(-D)$, where $D$ is an
effective divisor on $\widehat{V}$ whose support is snc._

For the construction of the map $\pi$ we use the same argument as in [DHP22].
We start by constructing a principalization of the ideal
$\mathcal{I}_{W}\subset\mathcal{O}_{V}$: this is achieved by a finite sequence
of blow-ups

$p_{k}:V_{k+1}\to V_{k}$

of smooth centres $\Sigma_{k}\subset V_{k}$, for $k=0,\dots,N-1$ with
$V_{0}:=V$. Next we interpret $\Sigma_{0}\subset V$ as analytic subspace of
$X$, and we blow-up $X$ along $\Sigma_{0}$; let

$\pi_{1}:X_{1}\to X$

be the corresponding map. We get a closed immersion $\displaystyle V_{1}\to
X_{1}$ (whose image is simply the proper transform of $V$), and we repeat this
operation with $\Sigma_{1}$. Notice that the space $X_{N}$ is not necessarily
normal, but the map $\displaystyle X_{N}\to X$ is biholomorphic at the generic
point of $V_{N}$. Since $X$ is normal, it follows that the complement of a
proper analytic subset of $V_{N}$ is contained in the set of normal points of
$X_{N}$. Therefore, the normalization $\nu:\widehat{X}\to X_{N}$ is
biholomorphic near the proper transform $\widehat{V}$ of $V_{N}$ (cf. [GR],
Corollary, page 164). It follows that the map

$\pi:\widehat{X}\to X$

obtained by composing the normalization of $X_{N}$ with the previous sequence
of blow-ups has all the properties we need.

Consider next the pull-back current $(\pi|_{\widehat{V}})^{\star}\Theta_{V}$.
Its singularities are concentrated along the snc divisor $D$ in $\widehat{V}$,
whose support is $\Lambda_{1}+\dots+\Lambda_{N}$. For each $i=1,\dots,N$ we
denote by $\beta_{i}$ an arbitrary, smooth representative of the first Chern
class of $\Lambda_{i}$. By blowing up $\widehat{X}$ along $\Lambda_{i}$, we
can add the following item to the properties of $\pi$ above:

  1. (c)

_There exists a set $(\rho_{i})_{i=1,\dots,N}$ of smooth $(1,1)$-forms on
$\widehat{X}$ such that the equality_

$\rho_{i}|_{\widehat{V}}\equiv\beta_{i}$

_holds for each $i=1,\dots,N$, meaning that the restriction of $\rho_{i}$ to
$\widehat{V}$ belongs to the class $\beta_{i}$._

Indeed, locally analytically we blow up the non-singular set $\Lambda_{i}$ in
$\mathbb{C}^{N}$. The exceptional set of
$\widetilde{\mathbb{C}^{N}}\to\mathbb{C}^{N}$ is a smooth divisor
$F_{i}\to\Lambda_{i}$. Then $E_{i}:=\widetilde{X}\cap F_{i}$ where
$\widetilde{X}\to\widehat{X}$ is the strict transform and even if $E_{i}$ is
neither reduced nor irreducible, it is nevertheless a Cartier divisor. Then
$\mathcal{O}(E_{i})$ is locally free, and so we can endow it with a smooth
metric denoted by $h_{i}$ (this notion is defined precisely as in the usual
case of a line bundle on a manifold). The curvature form corresponding to
$h_{i}$ will be our $\rho_{i}$, for each index $i$. Moreover, notice that
these additional transformations do not affect $\widehat{V}$, since
$\Lambda_{i}$ has codimension one in $\widehat{V}$.

Given the properties (a)–(c) above, we obtain the decomposition

$(\pi|_{\widehat{V}})^{\star}\Theta_{V}=\alpha_{1}+\sum
a_{i}[\Lambda_{i}]+\sqrt{-1}\partial\overline{\partial}\widehat{f}$

where the notations/conventions are as follows:

  * •

The same symbol e.g. $\alpha$ is used to denote a cohomology class and some
fixed representative contained in it (in case we do not intend to emphasize a
particular representative of the said class).

  * •

$\Lambda_{i}$ are the hypersurfaces of $\widehat{V}$ introduced before.

  * •

The smooth $(1,1)$-form $\alpha_{1}$ is defined as

$\alpha_{1}=(\pi|_{\widehat{V}})^{\star}(\alpha|_{V})-\sum
a_{i}\rho_{i}|_{\widehat{V}},$

where $\rho_{i}$ are the forms in (c).

  * •

The function $\widehat{f}$ is bounded.

We note that since $\Theta_{V}$ is a Kähler current, the inequality

$(\pi|_{\widehat{V}})^{\star}\Theta_{V}\geq\delta(\pi|_{\widehat{V}})^{\star}(\omega_{X}|_{V})$

holds, for some $\delta>0$. It follows that we have

$\widehat{\Theta}_{V}:=\alpha_{1}+\sqrt{-1}\partial\overline{\partial}\widehat{f}\geq\delta(\pi|_{\widehat{V}})^{\star}(\omega_{X}|_{V})$

as well, since $\widehat{\Theta}_{V}$ is smooth, and the inequality above
holds in the complement of an analytic set of $\widehat{V}$.

Then we claim that for each $\varepsilon>0$ there exists an open subset
$U_{\varepsilon}\subset\widehat{X}$ containing $\hat{V}$, together with a
smooth function $\widehat{f}_{\varepsilon}:U_{\varepsilon}\to\mathbb{R}$ such
that

(31) $\pi^{\star}(\alpha)-\sum
a_{i}\rho_{i}+\sqrt{-1}\partial\overline{\partial}\widehat{f}_{\varepsilon}\geq\delta\pi^{\star}(\omega_{X})-\varepsilon\omega_{\widehat{X}}$

pointwise on $U_{\varepsilon}$, where $\omega_{\widehat{X}}$ is a fixed Kähler
metric on $\widehat{X}$. Indeed, this is done in two steps: we first apply the
regularisation result in [Dem92] on $\widehat{V}$ in order to convert
$\widehat{f}$ to a smooth function. The price to pay is a loss of positivity,
which can be assumed to be of size
$\displaystyle\frac{\varepsilon}{2}\omega_{\widehat{X}}|_{\widehat{V}}$. Then
by the argument in [DeBook] already used in the particular case above, we
obtain $U_{\varepsilon}$ and $\widehat{f}_{\varepsilon}$.

The inequality (31) above is in particular true if we construct the metric
$\omega_{\widehat{X}}$ as follows:

$\omega_{\widehat{X}}:=\pi^{\star}\omega_{X}-\sum\varepsilon_{i}\rho_{i}$

where $\varepsilon_{i}>0$ are well-chosen real numbers. Then we take
$\displaystyle\varepsilon:=\frac{\delta}{2}$ and if we denote by $U$ and
$\widetilde{f}$ the corresponding set and function, respectively, then all in
all we have

(32)
$\pi^{\star}(\alpha)+\sqrt{-1}\partial\overline{\partial}\big{(}\widetilde{f}+\sum(a_{i}+\frac{\delta}{2}\varepsilon_{i})\log|s_{E_{i}}|_{h_{i}}^{2}\big{)}\geq\frac{\delta}{2}\pi^{\star}(\omega_{X})$

in the sense of currents on $U$.

On the other hand, we also have at our disposal the inverse image current
$\displaystyle\pi^{\star}\Theta\geq\delta_{1}\pi^{\star}(\omega_{X})$ which
has log poles along $\widehat{V}\cup Z$, where $Z\subset\widehat{X}$ is an
analytic set. The procedure we have used in the previous particular case
applies here: indeed, we have only used the fact that $f$ is smooth in order
to construct the open subset $U$. Then adding the function
$\delta_{0}\psi_{Z}$ to it has the effect of diminishing a bit more the lower
bound in (32), but we can afford this since $\delta>0$.

The current obtained after the gluing procedure on $\widehat{X}$ has log poles
and it is non-singular at the generic point of $\widehat{V}$. But as we have
already mentioned, the birational map $\pi$ is a biholomorphism at the general
point of $V$, so the direct image of the said current will be smooth at the
generic point of $V$. Moreover, it has weak analytic singularities by
definition –actually this is the main reason why we have introduced this class
of singularities. ∎

##  5\. On subadjunction and the canonical bundle formula

Let $X$ be a compact Kähler manifold, and consider a real $(1,1)$-class
${\alpha}$ on $X$. We assume that ${\alpha}$ contains a closed positive
current $R\geq 0$ with admissible singularities. This means that we can write

(33) $R=\alpha+\sqrt{-1}\partial\overline{\partial}\phi$

where (abusing notation) $\alpha$ is a smooth representative of the class
$\alpha$ and the function $\phi$ verifies the conditions in Definition 4.10.

We denote by $\pi:\widehat{X}\to X$ a log-resolution of the integral closure
of the ideal generated locally by the functions $(f_{i})$ in Definition 4.10.
The pull back of the current $R$ admits the decomposition

(34) $\pi^{\star}R=[D]+R_{0}$

where we denote by $[D]$ the current of integration along the effective
divisor $D$ and $R_{0}\geq 0$ is a closed, smooth and semi-positive
$(1,1)$-form on $\widehat{X}$. By analogy with the case in which $R$ is
induced by an effective divisor we call such a map $\pi$ a log-resolution of
$R$.

We assume next that there exists a log-resolution of $R$, such that the
following additional requirements are satisfied.

  1. (1)

The support of the $\pi$-exceptional divisor $E$ has simple normal crossings,
and $\pi$ is obtained as composition of blow-ups of smooth centers.

  2. (2)

We have $\displaystyle\pi^{\star}(K_{X}+\alpha)\simeq
K_{\widehat{X}}+\beta+S+\Xi_{1}-\Xi_{2}$, where:

     * •

The notation above means that the relative canonical class $K_{\widehat{X}/X}$
of $\pi$ plus the divisor $S+\Xi_{1}-\Xi_{2}$ coincides with
$\pi^{\star}(\alpha)-\beta$.

     * •

$\beta$ is a nef $(1,1)$-class on $\widehat{X}$.

     * •

$S$ is a smooth hypersurface on $\widehat{X}$ whose image is denoted by
$T:=\pi(S)$.

     * •

The restriction $\pi|_{S}$ admits a decomposition

$\pi|_{S}=\pi_{W}\circ f$

where $\pi_{W}:W\to T$ is a desingularisation of $T$ and $f:S\to W$ is
holomorphic.

     * •

The $\Xi_{i}$’s are effective ${\mathbb{R}}$-divisors on $\widehat{X}$ such
that their supports do not have common components, $S+\Xi_{1}+\Xi_{2}$ is snc,