# A High-Q Vertical Light Emission From Active Parity-Time Symmetric Gratings

Tahere Hemati  School of Electrical and Computer Engineering, University of
Oklahoma, Norman OK 73019, United States  Yi Zou<EMAIL_ADDRESS>School of Information Science and Technology, ShanghaiTech University,
Shanghai 201201, China  Binbin Weng<EMAIL_ADDRESS>School of Electrical
and Computer Engineering, University of Oklahoma, Norman OK 73019, United
States

###### Abstract

This work presents a theoretical investigation of an active photonic grating
of the parity-time (PT) symmetric architecture. The analytical study of the
free-space mode propagation from the grating structure indicates the unique
bifurcation property due to the PT-symmetry modulation. It is shown that both
the gain/loss contrast and the lattice constant parameters are critical
factors to modulate the active photonic system in between the PT-symmetry to
the symmetry-broken phases. Furthermore, numerical simulations via the
Rigorous Coupled-Wave Analysis (RCWA) method discover the existence of a
unique Spectral Singularity (SS) phenomenon in this PT grating structure,
which corresponds to a non-trivial single-mode and near-zero bandwidth
photonic resonant emission. Also, the guiding procedure for fulfilling SS
modes is found to be related to the unique formation of the scattering matrix
applied in the PT-symmetric photonic gratings. This theoretical work takes a
fresh look into the active PT-symmetric photonic gratings focusing on the
discovery of new free-space emission modes rather than the commonly studied
unidirectional properties, which could contribute to the development of novel
low-threshold and super-coherent surface-emitting devices.

††preprint: APS/123-QED

##  I Introduction

Photonic gratings as the first optical components operating at subwavelength
region can be considered the aspiration of modern nano-photonics concepts such
as photonic crystals and metamaterials [1]. The main purpose of all these
optical components is to control the dynamic of light, especially on a
nanometer scale [2]. Today, the applications of gratings (e.g., diffraction)
are expanded from telecommunications [3] and astronomy [4] to chemistry [5]
and biosensing [6]. The recent progress in nano-fabrication technologies
enabled us to create highly precise grating structures and revealed the
potential of these optical components in practice. Thus, in the recent decade,
special attention has been attracted to this field.

The unique properties of photonic gratings are associated with the spatial
modulation of the Refractive Index (RI). By optimizing grating parameters such
as the period, thickness and fill factor, we can modulate the amplitude and
phase of the incident wave and design the gratings for various purposes,
including optical filters [7], reflectors [8, 9], absorbers [10], resonators
[11, 12], and so on.

More recently, by drawing the fundamental concepts in quantum physics (i.e.,
Parity-Time (PT) symmetry) to the realm of optics and photonics, novel
applications, such as unidirectional emission [13] and asymmetric diffraction
[14], appeared from periodical structures, including photonic gratings. The
simplest definition of a PT-symmetric system is a non-Hermitian system whose
Hamiltonian remains invariant under spatial reflection ($p$ $\xrightarrow{}$
$-p$, $x$ $\xrightarrow{}$ $-x$) and time-reversal ($p$ $\xrightarrow{}$ $-p$,
$x$ $\xrightarrow{}$ $x$, $i$ $\xrightarrow{}$ $-i$) operators, where $p$,
$x$, and $i$ are momentum, location, and imaginary unite, respectively [15].
It is indicated that to satisfy this condition, the related potential should
be $\hat{V}(x)=\hat{V}^{*}(-x)$. The equivalent condition for a optical system
is ${n}(x)={n}^{*}(-x)$ [16].

In practice, two identical coupled waveguides, one showing loss and the other
one indicating the same amount of gain can be the simplest example of an
optical PT-symmetric system [16]. In this system, by increasing the gain(loss)
coupling contrast between these two waveguides, the system transits from a PT-
symmetric regime where all eigenvalues are real into a PT-broken regime where
the eigenvalues are complex.

There is a transient point between these two regimes named exceptional point,
where both eigenvalues and eigenstates coalesce [17]. The uniqueness of a PT-
symmetric system lies in indicating real eigenvalues, despite the non-
Hermiticity and non-orthogonality of its eigenstates [18], which lead to
extraordinary phenomena. Indeed, in PT-symmetric systems, we take advantage of
the beneficial role of loss, while in trivial systems, loss always has a
destructive effect, and it should be eliminated.

Scholars gradually found that using more complicated optical platforms such as
periodical structures to realize the PT-symmetry concept can introduce new
non-trivial outcomes, including loss-induced transparency [19], non-reciprocal
light propagation [20], unidirectional invisibility [21, 13], and directional
emission [22]. Among different kinds of periodical structures, only a few
works have focused on grating structures as a platform for exploring PT-
symmetric effects [23, 24, 25, 26], and the accomplished works mostly
investigate asymmetric diffraction [23] and unidirectional properties [27].

Herein, we instead investigate an “active” type of the PT-symmetric gratings
and study the possible new non-trivial phenomena besides the asymmetric
diffraction effect. This research will theoretically and numerically study the
emission properties of diffracted modes from an active grating by Helmholtz
equation, driving the related Hamilton and indicating symmetry, symmetry-
broken and exceptional point regions. Moreover, the properties of resonating
modes induced by PT-symmetric diffraction grating will be discussed in detail.

##  II Theory and analytical solutions

As just mentioned, the guiding procedure in gratings lies in the spatial RI
modulation. On the other hand, to realize the PT-symmetry, the RI modulation
needs to satisfy ${n}(x)={n}^{*}(-x)$. Therefore, the real part of RI
modulation has to be a symmetric function, and its imaginary part should be an
asymmetric function. Thus the RI modulation can be defined as:

$n(x)=n_{0}+n\text{cos}(2\pi x/a)+i\gamma\text{sin}(2\pi x/a)$ (1)

where $a$ is the period, $n_{0}$ is the effective refractive index, and $n$
and $\gamma$ are the real and imaginary parts of the refractive index
modulation, respectively.

Figure 1: (a) The real and imaginary parts of a PT-symmetric RI modulation in
the x-direction. (b) The real RI modulation in x- and z-direction.

Fig. 1(a) shows the real (solid blue line) and imaginary (dashed red line)
parts of the PT-symmetric refractive index distribution. The positive part of
the imaginary RI indicates gain, and the negative part shows the induced loss.
Fig. 1(b) displays the real part of the RI modulation in two dimensions where
the RI solely changes in the x-direction, and the incident light is
perpendicular to the periodicity. $A_{-}$, $A_{0}$, and $A_{+}$ show the
amplitude of the $-1$, $0$, and $+1$ orders of diffracted grating modes,
respectively.

Here, we study the wave propagation of an incident light entering along the
z-axis where the wave $E(x,z)$ obeys the Helmholtz equation [28, 29]:

$\nabla^{2}E+k_{0}^{2}n^{2}(x)E=0$ (2)

in which $k_{0}=2\pi/\lambda$. Since the incident light is normal to the
periodicity, both x and z components are effective. The general solution for
$E(x,z)$ can be derived from Bloch’s wave equation.

$E(x,z)=u(x)e^{i\vec{\beta}.\vec{r}}$ (3)

where $u(x)$ is a periodic function in the x-direction

$u(x)=\sum_{m=-\infty}^{\infty}A_{m}e^{-i2\pi mx/a}$ (4)

and in a 2D plane

$e^{i\vec{\beta}.\vec{r}}=e^{i\beta_{x}x}e^{i\beta_{z}z}$ (5)

By substituting Eq. (4) and Eq. (5) into Eq. (3):

$E(x,z)=\sum_{m=-\infty}^{\infty}A_{m}e^{i\beta_{z}z}e^{i(\beta_{x}-2\pi
m/a)x}$ (6)

Considering the normal light incident, the wavenumber in x-direction equals
$2\pi m/a$, and consequently, the dispersion relation leads to:

$\beta_{z}=\sqrt{k_{0}^{2}n_{0}^{2}-({2\pi m/a)}^{2}}$ (7)

If we consider the grating in a diffracted regime where $a>>\lambda$,
$\beta_{z}$ will be reduced to $\beta_{z}=k_{0}n_{0}$. Thus, Eq. (6) can be
rephrased as:

$E(x,z)=\sum_{m=-\infty}^{\infty}A_{m}e^{-i[mKx-k_{0}n_{0}z]}$ (8)

where $A_{m}$ is the amplitude of the $m_{th}$ diffracted mode, and
$K=2\pi/a$.

To analytically solve Eq. (2), in addition to $E(x,z)$ we need to find $n(x)$.
Through Eq. (1), supposing $n$ and $\gamma$ are significantly smaller than
$n_{0}$, the RI modulation can be rephrased as:

$n(x)=\sqrt{n_{0}^{2}+n_{0}[c^{+}e^{iKx}+c^{-}e^{-iKx}}]$ (9)

where $c^{+}=n+\gamma$ and $c^{-}=n-\gamma$. Now, by substituting Eq. (8) and
Eq. (9) into equation Eq. (2), and considering the Raman-Nath approximation
the wave coupled equation (Eq. (10)) is obtained [30, 24, 31].

$\frac{\partial A_{m}}{\partial
z}+\frac{im^{2}K^{2}A_{m}}{2k_{0}n_{0}}-\frac{ik_{0}}{2}[c^{-}A_{m+1}+c^{+}A_{m-1}]=0$
(10)

In contrast with Bragg gratings, in which only two first orders ($m=0,1$) at
Bragg angle show non-zero amplitude, higher orders can indicate non-zero
amplitude in thin gratings. However, in this active PT grating work with a
normal incident light input, the amplitude of higher orders ($m=\pm 2$) are
set to be significantly small [30]. Therefore, neglecting higher orders, which
give us a reasonable approximation of the exact solution and provide a clear
insight into the PT-symmetry concept [24]. Hence, Eq. (10) reduced to a set of
coupled equations (Eq. (11)), which can also be written in a matrix form with
Hamiltonian $H$, shown in Eq. (12).

$\begin{split}\frac{\partial A_{0}}{\partial
z}-\frac{ik_{0}}{2}[c^{-}A_{+1}+c^{+}A_{-1}]=0\\\ \frac{\partial
A_{+1}}{\partial
z}+\frac{iK^{2}A_{+1}}{2k_{0}n_{0}}-\frac{ic^{+}k_{0}A_{0}}{2}=0\\\
\frac{\partial A_{-1}}{\partial
z}+\frac{iK^{2}A_{-1}}{2k_{0}n_{0}}-\frac{ic^{-}k_{0}A_{0}}{2}=0\end{split}$
(11)

$H=\begin{bmatrix}\sigma&-k_{0}c^{+}/2&0\\\ -k_{0}c^{-}/2&0&-k_{0}c^{+}/2\\\
0&-k_{0}c^{-}/2&\sigma\end{bmatrix}$ (12)

where $\sigma=K^{2}/2k_{0}n_{0}$. The eigenvalues of this Hamiltonian, or in
other words, the amplitude of the diffracted modes, are:

$\displaystyle\lambda_{1}=\sigma$ (13)
$\displaystyle\lambda_{2}=1/2[\sigma-\sqrt{4k_{0}^{2}(n^{2}-\gamma^{2})+\sigma^{2}}]$
(14)
$\displaystyle\lambda_{3}=1/2[\sigma+\sqrt{4k_{0}^{2}(n^{2}-\gamma^{2})+\sigma^{2}}]$
(15)

While $\lambda_{1}$ is independent of the real and imaginary parts of the
refractive index modulation, $\lambda_{2}$ and $\lambda_{3}$ are dependent on
the real and imaginary parts. Thus, once Eq. (16) is satisfied, the system
will be located at its exceptional point, and $\lambda_{2}$ and $\lambda_{3}$
and their related eigenstates coalesce.

$\gamma=\sqrt{n^{2}+\frac{\sigma^{2}}{4k_{0}^{2}}}$ (16)

Therefore, once the imaginary part of the refractive index modulation is
$\mid\gamma\mid$ $>$ $\sqrt{n^{2}+\frac{\sigma^{2}}{4k_{0}^{2}}}$, the system
enters into the symmetry-broken phase, indicating imaginary eigenvalues. Fig.
2(a) and Fig. 2(b) show the real and imaginary parts of eigenvalues according
to the imaginary part of the refractive index ($\gamma$), where the dashed
blue line indicates $\lambda_{2}$ and the solid red line displays
$\lambda_{3}$. As we can see, in the symmetry region, both eigenvalues show a
pure real value. However, immediately after the exceptional point, shown here
by the dashed gray line, two eigenvalues bifurcates so that one of them
possesses a negative imaginary part ($\lambda_{3}$) and the other one
indicates a positive imaginary part ($\lambda_{2}$). It can be interpreted
that one mode is trapped mostly in gain and the other one captured in loss and
suppressed. The bifurcation is the most remarkable property of PT-symmetric
systems [32, 33]. This unique property is used in designing single-mode lasers
[34, 35], where undesired competing modes are captured in the loss and
suppressed, and only confined modes in the gain are amplified.

Figure 2:  (a) The real part and, (b) The imaginary part of eigenvalues
according to the $\gamma$ for a system with the periodicity of 2 $\mu m$. The
exceptional point is shown by the dashed gray line. Figure 3:  (a) The real
and, (b) The imaginary parts of eigenvalues according to the period for a PT-
symmetric grating. The exceptional point is shown by the dashed gray line.

In gratings that incident light is parallel to the periodicity (along
x-direction in Fig. 1(b)), only the ratio between real and imaginary parts of
the refractive index ($n$ and $\gamma$) determines the phase of the system
[21]. However, in our designed grating, where the incident light is
perpendicular to periodicity, the period plays an essential role.

Figs. 3(a) and 3(b) show the real and imaginary parts of eigenvalues according
to different period selections, respectively. This figure indicates that
increasing the period will drive the system into the symmetry-broken phase.
The dashed gray line shows the location of the exceptional point. As reflected
from the Eq. (16) in addition to real and imaginary parts of RI modulation,
period and the effective refractive index can alter the system phase from
symmetry to symmetry-broken phase. In other words, more design freedom exists
in this unique active photonic grating platform.

This section analytically solved the Helmholtz equation for the dominant
grating modes considering the Raman-Nath approximation. We showed the impact
of the imaginary part of RI modulation and period to shift the system from a
PT-symmetry phase to the symmetry-broken stage. In the following section, we
will design a thin grating that follows the theory described above, and report
the unique PT-symmetric photonic modes solved numerically by the RCWA method.
Moreover, the related electric field and amplitude distribution over the
grating will be displayed, which will extend the understanding of physical
mechanisms leading to extraordinary phenomena observed in the active PT-
symmetric photonic grating structures.

##  III Numerical simulations

Fig. 4 displays a 3D schematic view of the designed diffraction grating, where
yellow and red parts show the gain and loss ($\mid$$\gamma$$\mid$),
respectively. In this numeral study, we adopt common material parameters from
the III-V semiconductors, InGaAs just as an example, to design and study the
PT-symmetric grating effects. It is noted that conclusions to be made can also
be applied to other materials systems in other spectral domains.

Figure 4:  A 3D schematic view of the PT-symmetric diffraction gratings.

One of the most well-known criteria of PT-symmetric systems is the spatial
bifurcation of two modes in the symmetry-broken region. As presented in Fig.
2, in the symmetry-broken phase, two modes with the same amplitude (same real
part of eigenvalues) show the bifurcation, where one is captured mostly in the
gain other trapped mainly in the loss. Thus, one mode is amplified while the
other one is suppressed. This phenomenon is the primary mechanism behind the
single-mode lasing in most of the PT-symmetric lasers. We expect that the
numerical simulation demonstrates the same mechanism.

Fig. 5 shows the spatial electric field distribution over a grating with the
period of 1 $\mu m$ and the thickness of 821 nm. The substrate and gain/loss
areas are demonstrated in this figure, where G stands for gain and L stands
for loss. The grating parameters were selected in a way that satisfies the
Raman-Nath approximation criteria (Eq. (17)) that was implemented in the
theoretical section.

$\begin{cases}Q^{{}^{\prime}}\zeta<1\\\ Q^{{}^{\prime}}=\frac{2\pi\lambda
d}{n_{0}\Lambda^{2}cos\theta}\\\ \zeta=\frac{\pi nd}{2\lambda
n_{0}}\end{cases}$ (17)

In Eq. (17), $\lambda$ is the free-space wavelength, $d$ is the grating
thickness, $n_{0}$ is the effective refractive index, $\Lambda$ is the grating
period, $n$ is the real part of the RI modulation, and $\theta$ is the
incident angle.

Figure 5:  The electric field distribution over the PT-symmetric grating where
(a) $\gamma=0.2$, associated with the symmetry phase. (b) $\gamma=0.22$
associated with the symmetry-broken phase (amplified mode). (c) $\gamma=0.22$
associated with the symmetry-broken phase (suppressed mode).

Fig. 5(a) shows the symmetric distribution of the electric field over the gain
and loss areas so that no mode experiences net amplification or suppression.
This pattern can be associated with the PT-symmetry phase, where both
eigenvalues are purely real. By increasing gain/loss ($\gamma$) from 0.2 to
0.22, the system is shifted to the symmetry-broken phase, where one mode is
captured in the gain and experiences the amplification. Fig. 5(b) displays
this mechanism clearly. However, the other mode is trapped in the loss and is
suppressed (Fig. 5(c)).

After indicating the bifurcation over the PT-symmetric diffraction grating, we
found out that the system can show a single-mode, almost zero-bandwidth (0.04
nm) mode through optimizing gain/loss contrast ($\gamma$). To better
understand this non-trivial mode, we compared the diffraction efficiency of
the transmitted mode in the logarithm form, where $\gamma=0.18$, $\gamma=0.2$,
and $\gamma=0.22$. Fig. 6(a) illustrates this comparison, where the solid blue
line is related to the system with $\gamma=0.22$, the dashed red line is
associated with $\gamma=0.2$, and the black dotted line is associated with
$\gamma=0.18$. The inset shows field amplitude distribution over a unit cell
in a grating with $\gamma=0.2$ and $\gamma=0.22$. The magnitude of amplitude
is displayed by the colored bar. Furthermore, Fig. 6(b), Fig. 6(c), and Fig.
6(d) show the 2D transmitted emission coefficient map over wavelengths, where
$\gamma$ changes from 0 to 0.18, 0.2, and 0.22, respectively. As we can see,
the broadness of the mode decreases by increasing $\gamma$, which is matched
with the full-width half maximum of the spectrum displayed in Fig. 6(a)

As Fig. 6(a) shows, the transmitted mode when $\gamma=0.22$ is $10^{6}$
greater than when $\gamma=0.2$. The primary reason for this huge difference
can be related to the intensity of the confined field in the gain area. The
inset illustrates that the amplitude of the confined mode in the gain area for
the grating with $\gamma=0.22$ is almost $10^{3}$ greater than the amplitude
of the trapped mode in the gain for the same grating but with $\gamma=0.2$.
The physical mechanism for this extraordinary phenomenon has roots in the
scattering theory in PT-symmetric systems and the concept of spectral
singularity [36, 37, 38], which will be discussed in detail in the following
paragraphs.

Figure 6:  (a) The logarithm form of the transmitted spectra over wavelengths
from the grating with $\gamma=0.18$ (the black dotted line), $\gamma=0.2$ (the
red dashed line), and $\gamma=0.22$ (the solid blue line). The amplitude
distribution of the spectrum with $\gamma=0.2$ and $\gamma=0.22$ is displayed
over one grating period in inset. The transmitted emission coefficient map
over wavelengths where $\gamma$ changes from 0 to (b) 0.18, (c) 0.2, and (d)
0.22.

The properties of gratings as scattering systems can be defined by the
Scattering matrix (S-matrix). This matrix relates the input field to the
output field, passing through a scattering system. Eq. (18) shows the S-matrix
($S_{nm}$) that associates the incoming mode amplitude ($\phi_{m}$) to the
outgoing mode amplitude ($\eta_{n}$).

${\sum_{n}\ S_{nm}(\omega)\phi_{m}=\eta_{n}}$ (18)

The required condition to define an S-matrix is the unitary condition
($\mid$det S$\mid$=1). For passive structures with real frequencies, this
unitary condition is only satisfied by having unimodular S-matrix eigenvalues
($\mid$$s_{\pm}$$\mid$=1) [39]. Applying PT operator on S-matrix meets the
unitary condition. However, in contrast with passive structures, the only way
to meet the unitary condition in PT-symmetric systems is not having unimodular
eigenvalues. In these systems, the unitary condition can be satisfied by pairs
reciprocal moduli as well ($s_{\pm}$=$1/s^{*}_{\mp}$) [40]. The former
possibility to satisfy the unitary condition is related to the PT-symmetry
(unimodular eigenvalues), and the latter way (reciprocal eigenvalues) is
associated with the PT-symmetry-broken.

In the PT-symmetry-broken phase, there are points at which poles and zeros of
the S-matrix intersect on the real axis [41, 42]. This intersection leads to
approaching $s_{n}$ to zero and consequently $1/s^{*}_{n}=\infty$, while their
product is still unity. In fact, these singular points, named spectral
singularity, correspond to the giant amplitude enhancement of the transmitted
emission modes for an optimized value of gain/loss.

Figure 7: The Q factor of the resonance mode in the logarithm form with
different introduced gain/loss ($\gamma$) to the grating.

SSs are a family member of resonant modes [43, 44]. However, the main feature
for SSs that distinguishes them from trivial resonances is eliminating the
imaginary part of the resonant frequencies [36]. The imaginary part of the
resonant frequencies is responsible for evanescent energy or, in other words,
the bandwidth of the resonant peaks, the lower evanescent energy, narrower
bandwidth. The uniqueness of SSs lies in being purely real, which in theory
leads to zero-bandwidth resonances [36]. It should be noted that SSs are
different from the Bound States In Continuum (BICs); however, both show zero-
bandwidth resonances [45].

Fig. 7 indicates the Q factor of the resonant mode (1.463 $\mu m$) with
different introduced gain/loss ($\gamma$) to the grating. A sharp enhancement
is observed when $\gamma$ touches 0.22. This figure shows that only at a
specific gain/loss value, here $\gamma=0.22$, SS can occur, and higher or
lower values cannot satisfy the SS condition. This is matched with the theory
that claims only and only at the SS point, $s_{n}$ goes to zero, while
$1/s^{*}_{n}$ goes to $\infty$, and the giant transmitted emission enhancement
is related to $1/s^{*}_{n}$, which its amplitude approaches infinity. This
transmitted emission coefficient map indicates that by this unique active PT
grating design, $\gamma=0.22$ leads to appearing one SS at 1.46 $\mu m$.

##  IV Conclusion

In this work, we report a new active granting platform having the optical gain
and loss to realize non-trivial PT-symmetry modulation effects. Herein, by
solving the Helmholtz equation in the Raman-Nath approximation, the
bifurcation leading to confining the amplified mode in the gain region was
indicated. We showed that in the PT photonic grating structure, in addition to
the gain/loss parameter, the period selection can also drive the system from
PT-symmetry to the symmetry-broken phase. Moreover, a zero-bandwidth photonic
emission mode (i.e., at 1.463 $\mu m$) can be stimulated and numerically
demonstrated through the RCWA simulation. It is suggested that, by optimizing
the optical gain/loss contrast, this non-trivial high Q-factor resonance mode
is the result of the photonic system’s special spectral singularity effect,
where one of the scattering states approaches zero while its reciprocal state
approaches infinity. Considering the functionality of gratings in a broad
range of applications, we anticipate introducing active PT-symmetric gratings
will offer new engineering strategies for developing new low-threshold and
super-coherent surface-emitting lasers, and thus, advancing modern on-chip
photonic applications.

##  V ACKNOWLEDGMENTS

The authors would like to thank Dr. Mykola Kulishov for the insightful
discussions. B.W. acknowledges the partial supports from OU’s Big Idea
Challenge award and the Oklahoma Center for the Advancement of Science and
Technology’s Research Grant #AR21-052. Y. Z. acknowledges the partial supports
from the Natural Science Foundation of Shanghai (21ZR1443100) and the National
Natural Science Foundation of China (61705099).

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# A note on lc-trivial fibrations

Kenta Hashizume  Department of Mathematics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>
(Date: 2022/06/08, version 0.01)

###### Abstract.

For every lc-trivial fibration $(X,\Delta)\to Z$ from an lc pair, we prove
that after a base change, there exists a positive integer $n$, depending only
on the dimension of $X$, the Cartier index of $K_{X}+\Delta$, and the
sufficiently general fibers of $X\to Z$, such that $n(K_{X}+\Delta)$ is
linearly equivalent to the pullback of a Cartier divisor.

###### Key words and phrases:

lc-trivial fibration, Cartier index, effective base point freeness

###### 2010 Mathematics Subject Classification:

primary: 14E30, secondary: 14D06, 14J40

###### Contents

  1. 1 Introduction
  2. 2 Definitions
  3. 3 Proofs of main results
  4. 4 Lc-trivial fibration with log big moduli part

##  1\. Introduction

Throughout this note, we will work over the complex number field.

In this note, we prove the following lemma and discuss its application.

######  Lemma 1.1 (Main result).

For every $d$, $m\in\mathbb{Z}_{>0}$ and $v\in\mathbb{R}_{>0}$, there exists
$n\in\mathbb{Z}_{>0}$, depending only on $d$, $m$, and $v$, satisfying the
following. Let $(X,\Delta)$ be a projective lc pair, let
$\pi\colon(X,\Delta)\to Z$ be an lc-trivial fibration with the sufficiently
general fiber $F$, and let $A\geq 0$ be a Weil divisor on $X$ such that

  * •

${\rm dim}X=d$,

  * •

$mf^{*}(K_{X}+\Delta)$ is Cartier for some resolution $f\colon X^{\prime}\to
X$ of $X$,

  * •

$A|_{\pi^{-1}(U)}$ is $\mathbb{Q}$-Cartier and ample over $U$ for some open
subset $U\subset Z$,

  * •

$(F,\Delta|_{F}+tA|_{F})$ is an lc pair for some real number $t>0$, and

  * •

${\rm vol}(A|_{F})=v$.

Then there is a generalized lc pair $(Z,\Delta_{Z},\boldsymbol{\rm M})$
defined with the canonical bundle formula such that

  * •

$n(K_{X}+\Delta)\sim n\pi^{*}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$,

  * •

$n\boldsymbol{\rm M}$ is b-Cartier, and

  * •

$n\phi^{*}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$ is Cartier for some
resolution $\phi\colon Z^{\prime}\to Z$ of $Z$.

Furthermore, if there is a klt pair on $Z$, then
$n(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$ is Cartier.

Lc-trivial fibrations are one of special kinds of fibrations that naturally
appear in the birational geometry. For example, every fibration induced by the
log canonical divisor of a good minimal model is an lc-trivial fibration,
every fibration from a Calabi–Yau manifold is also an lc-trivial fibration,
and we can define the structure of an lc-trivial fibration for any Mori fiber
space. Because of the strong condition of lc-trivial fibrations, it has been
revealed that the geometries of the total variety, the general fiber, and the
base variety of lc-trivial fibration are closely related to each other
([Ka98], [A04], [A05], [FG12], [FG14], [B16], [B18], [B21]). Especially, a
relation between the total variety and the base variety called the canonical
bundle formula, developed by Kawamata ([Ka98]) and Ambro ([A04], [A05]), plays
a fundamental role in the recent development of the birational geometry.

Lemma 1.1 looks similar to [B21, Lemma 7.4] by Birkar. In both lemmas, we fix
the dimension of the lc pair and the set of the coefficients of the boundary
divisor, and we assume a kind of boundedness condition on the general fibers
of the lc-trivial fibration. In Lemma 1.1, the linear equivalence between
$n(K_{X}+\Delta)$ and $n\pi^{*}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$ and
the b-Cartier property of $n\boldsymbol{\rm M}$ are not new because the two
properties have already been proved in [B21, Lemma 7.4]. The new part of Lemma
1.1 is that we can get the b-Cartier property of
$n(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$ as a b-divisor if we further fix
the b-Cartier index of $K_{X}+\Delta$ as a b-divisor. Birkar’s result [B21,
Lemma 7.4] plays a crucial role for the boundedness of the base varieties of
the Iitaka fibrations (see [BiHa22, Theorem 1.3] by Birkar–Hacon), whereas
Lemma 1.1 is useful to study effective base point freeness. The following
result is an application of Lemma 1.1.

######  Theorem 1.2.

For every $d$, $m\in\mathbb{Z}_{>0}$, and $v\in\mathbb{R}_{>0}$, there exists
$n\in\mathbb{Z}_{>0}$, depending only on $d$, $m$, and $v$, satisfying the
following. Let $(X,\Delta)$ be a projective klt pair such that
$e(K_{X}+\Delta)$ is semi-ample for an $e\in\\{1,-1\\}$, let $\pi\colon X\to
Z$ be the contraction induced by $e(K_{X}+\Delta)$, and let $A\geq 0$ be a
$\mathbb{Q}$-Cartier Weil divisor on $X$ such that

  * •

${\rm dim}X=d$,

  * •

$m(K_{X}+\Delta)$ is Cartier, and

  * •

${\rm vol}(A|_{F})=v$, where $F$ is a sufficiently general fiber of $\pi$.

Then $ne(K_{X}+\Delta)$ is base point free and the linear system
$|ne(K_{X}+\Delta)|$ defines $\pi$.

Theorem 1.2 is a partial generalization of Kollár’s effective base point
freeness [Ko93]. We should mention the effective base point freeness by
boundedness results of klt pairs. In [J22], Jiao proved the boundedness of
$\epsilon$-lc pairs up to crepant birational maps under fixed dimension, fixed
Iitaka volume, and a condition of the general fibers of the Iitaka fibrations
that is similar to the condition in Lemma 1.1. Though we do not discuss in
this note, other statement of effective base point freeness for semi-ample log
canonical divisors should follow from [J22] if we fix the Iitaka volumes of
log canonical divisors instead of the Cartier indices.

######  Acknowledgments.

The author was partially supported by JSPS KAKENHI Grant Number JP22K13887.
The author is grateful to Doctor Masafumi Hattori for fruitful discussions
which improved Theorem 1.2 in the first draft. The author thanks Professor
Osamu Fujino for informing him of the paper [F17].

##  2\. Definitions

We will freely use the notations and the definitions in [KM98] and [BCHM10]
for singularities of pairs except that $a(D,X,\Delta)$ denotes the log
discrepancy of $D$ with respect to $(X,\Delta)$ in this note. We will use the
definition of generalized pairs in [BZ16] and [B19], however, we adopt the
notations of generalized pairs in [Ha22] for the clear distinction between the
boundary part and the nef part of generalized pairs.

######  Definition 2.1 (Generalized lc pair).

A generalized pair $(X,\Delta,\boldsymbol{\rm M})/Z$ consists of

  * •

a projective morphism $X\to Z$ from a normal variety to a variety,

  * •

an effective $\mathbb{R}$-divisor $\Delta$ on $X$, and

  * •

a b-nef$/Z$ $\mathbb{R}$-b-Cartier b-divisor $\boldsymbol{\rm M}$ on $X$

such that $K_{X}+\Delta+\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier. When
$\boldsymbol{\rm M}=0$, the generalized pair $(X,\Delta,\boldsymbol{\rm M})/Z$
coincides with a usual pair. When $Z$ is a point, we simply denote
$(X,\Delta,\boldsymbol{\rm M})$.

Let $(X,\Delta,\boldsymbol{\rm M})/Z$ be a generalized pair, and let $f\colon
Y\to X$ be a projective birational morphism from a normal variety $Y$. Then
there is an $\mathbb{R}$-divisor $\Gamma$ on $Y$ such that

$K_{Y}+\Gamma+\boldsymbol{\rm M}_{Y}=f^{*}(K_{X}+\Delta+\boldsymbol{\rm
M}_{X}).$

We say that a generalized pair $(X,\Delta,\boldsymbol{\rm M})/Z$ is a
generalized lc pair if ${\rm coeff}_{P}(\Gamma)\geq-1$ for every projective
birational morphism $Y\to X$ and every prime divisor $P$ on $Y$.

######  Definition 2.2 (Lc-trivial fibration for lc pair).

In this note, an lc-trivial fibration, often denoted by $(X,\Delta)\to Z$,
means a pair of a projective lc pair $(X,\Delta)$ and a contraction $X\to Z$
of normal projective varieties such that $\Delta$ is a $\mathbb{Q}$-divisor
and $K_{X}+\Delta\sim_{\mathbb{Q},Z}0$.

In general, lc-trivial fibrations are defined for contractions and sub-pairs
that are not necessarily sub-lc ([A04], [A05], [FG12], [FG14], [FL19],
[Hu20]). However, in this note, we adopt the above convention for simplicity
of the definition. In particular, we always assume that the lc pairs of lc-
trivial fibrations have $\mathbb{Q}$-boundary divisors.

######  Definition 2.3 (Ambro model and log smooth Ambro model).

Let $(X,\Delta)\to Z$ be an lc-trivial fibration, and let $\boldsymbol{\rm B}$
and $\boldsymbol{\rm M}$ be the discriminant b-divisor and the moduli
b-divisor of the canonical bundle formula, respectively.

We say that a normal variety $Z^{\prime}$ with a projective birational
morphism $Z^{\prime}\to Z$ is an Ambro model if $\boldsymbol{\rm M}$ descends
to $Z^{\prime}$ and $\boldsymbol{\rm M}_{Z^{\prime}}$ is nef. If an Ambro
model $Z^{\prime}\to Z$ satisfies the log smoothness of $(Z^{\prime},{\rm
Supp}\boldsymbol{\rm B}_{Z^{\prime}})$, we say that $Z^{\prime}\to Z$ is a log
smooth Ambro model.

##  3\. Proofs of main results

######  Lemma 3.1.

Let $(X,\Delta)$ is a klt pair, and let $D$ be a $\mathbb{Q}$-Cartier divisor
on $X$. If there is a projective birational morphism $f\colon X^{\prime}\to X$
from a normal variety $X^{\prime}$ such that $f^{*}D$ is Cartier, then $D$ is
Cartier.

###### Proof.

Because the Cartier property can be checked locally, we may assume that $X$ is
affine. We may write

$K_{X^{\prime}}+\Delta^{\prime}=f^{*}(K_{X}+\Delta)+E^{\prime},$

where $\Delta^{\prime}\geq 0$ and $E^{\prime}\geq 0$ have no common
components. By replacing $f$ with a log resolution of $(X,\Delta)$, we may
assume that $(X^{\prime},\Delta^{\prime})$ is a $\mathbb{Q}$-factorial klt
pair.

By running a $(K_{X^{\prime}}+\Delta^{\prime})$-MMP over $X^{\prime}$ with
scaling of an ample divisor, we get a birational contraction $\phi\colon
X^{\prime}\dashrightarrow X^{\prime\prime}$ over $X$ such that $E^{\prime}$ is
contracted by $\phi$. By the cone and contraction theorem [KM98, Theorem 3.7],
it follows that $\phi_{*}f^{*}D$ is Cartier. Thus, replacing $X^{\prime}$ by
$X^{\prime\prime}$, we may assume $E^{\prime}=0$. In particular, we may assume
that $X^{\prime}$ is of Fano type over $X$.

By the base point free theorem [KM98, Theorem 3.24], for every integer $m\gg
0$ it follows that $mf^{*}D$ is linearly equivalent to the pullback of a
Cartier divisor on $X$. Then $mD$ is Cartier for all $m\gg 0$, so $D$ is
Cartier. ∎

###### Proof of Lemma 1.1.

Since $mf^{*}(K_{X}+\Delta)$ is Cartier for some resolution $f\colon
X^{\prime}\to X$ of $X$, we see that $m\Delta$ is a Weil divisor. By [B21,
Lemma 7.4], there exists a positive integer $n^{\prime}$, depending only on
$d$, $m$, and $v$, such that a generalized lc pair
$(Z,\Delta_{Z},\boldsymbol{\rm M})$ defined with the canonical bundle formula
satisfies

  * •

$n^{\prime}(K_{X}+\Delta)\sim
n^{\prime}\pi^{*}(K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z})$, and

  * •

$n^{\prime}\boldsymbol{\rm M}$ is b-Cartier.

By applying [B20, Theorem 1.7] to $(F,\Delta|_{F})$ and $A|_{F}$, we may find
a positive real number $t_{0}$, depending only on $d$, $m$, and $v$, such that
$(F,\Delta|_{F}+t_{0}A|_{F})$ is lc. By shrinking $U$ and removing the
vertical part of $A$ with respect to $X\to Z$ if necessary, we may assume that
$\bigl{(}\pi^{-1}(U),(\Delta+t_{0}A)|_{\pi^{-1}(U)}\bigr{)}$ is lc and all
components of $A$ dominate $Z$.

We put $D=K_{Z}+\Delta_{Z}+\boldsymbol{\rm M}_{Z}$. By Lemma 3.1, it is
sufficient to prove the existence of a positive integer $n^{\prime\prime}$
depending on $d$, $m$, $v$, $n^{\prime}$, and $t_{0}$ such that
$n^{\prime\prime}\phi^{*}D$ is Cartier for some resolution $\phi\colon
Z^{\prime}\to Z$ of $Z$. Indeed, supposing the existence of such
$n^{\prime\prime}$, then it is easy to see that
$n:=mn^{\prime}n^{\prime\prime}$ is the desired positive integer. From now on,
we will prove the existence of such $n^{\prime\prime}$ as above. We will
follow [B21, Proof of Lemma 7.4].

######  Step 1.

In this step, we construct a birational modification $X^{\prime}\to
Z^{\prime}$ of $X\to Z$ and $\mathbb{Q}$-divisors $\Delta^{\prime}$ and
$A^{\prime}$ on $X^{\prime}$.

Let $h\colon Z^{\prime}\to Z$ be a log resolution of $(Z,{\rm Supp}D)$. Let
$\Sigma^{\prime}$ be a divisor on $Z^{\prime}$ whose support contains ${\rm
Supp}h^{*}D\cup{\rm Ex}(h)$. Shrinking $U$ and adding some prime divisors to
$\Sigma^{\prime}$ if necessary, we may assume that
$Z^{\prime}\setminus\Sigma^{\prime}=h^{-1}(U)$ and the image of the vertical
part of $\Delta$ maps into $Z\setminus U$. Then, $h$ is an isomorphism over
$U$. Let $f\colon X^{\prime}\to X$ be a resolution of $X$ such that
$mf^{*}(K_{X}+\Delta)$ is Cartier. Replacing $f$ if necessary, we may assume
that the induced map $\pi^{\prime}\colon X^{\prime}\dashrightarrow Z^{\prime}$
is a morphism and $\bigl{(}X^{\prime},{\rm
Supp}(f^{-1}_{*}(\Delta+A)+\pi^{\prime*}\Sigma^{\prime})\cup{\rm
Ex}(f)\bigr{)}$ is log smooth. We define a $\mathbb{Q}$-divisor
$\Delta^{\prime}$ on $X^{\prime}$ as follows: For any prime divisor $P$ on
$X^{\prime}$, we define

${\rm coeff}_{P}(\Delta^{\prime}):=\left\\{\begin{array}[]{lll}1&&\text{($P$
is $f$-exceptional or $\pi^{\prime}(P)\subset\Sigma^{\prime}$)}\\\ {\rm
coeff}_{P}(f^{-1}_{*}\Delta)&&\text{(otherwise)}\end{array}\right.$

We put $V=\pi^{-1}(U)$,
$V^{\prime}=\pi^{\prime-1}(Z^{\prime}\setminus\Sigma^{\prime})$, and
$f_{V^{\prime}}=f|_{V^{\prime}}$. Since
$Z^{\prime}\setminus\Sigma^{\prime}=h^{-1}(U)$, we have
$V^{\prime}=f^{-1}(V)$. We have the following diagrams.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi^{\prime}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{V}}$$\textstyle{Z}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z^{\prime}}$$\scriptstyle{h}$$\textstyle{U}$$\textstyle{Z^{\prime}\setminus\Sigma^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

We put $A^{\prime}=f^{-1}_{*}A$. Since all components of $A$ dominate $Z$ and
$\bigl{(}V,(\Delta+t_{0}A)|_{V}\bigr{)}$ is lc,
$\Delta^{\prime}+t_{0}A^{\prime}$ is a boundary $\mathbb{R}$-divisor. Then
$(X^{\prime},\Delta^{\prime}+t_{0}A^{\prime})$ is lc. We may write

$\begin{split}K_{X^{\prime}}+\Delta^{\prime}=f^{*}(K_{X}+\Delta)+E^{\prime}+\Xi^{\prime}\end{split}$

where $E^{\prime}$ is an effective $f$-exceptional $\mathbb{Q}$-divisor whose
components intersect $V^{\prime}$ and $\Xi^{\prime}$ is an
$\mathbb{Q}$-divisor whose support is mapped into $\Sigma^{\prime}$ by
$\pi^{\prime}$. We may write

$f_{V^{\prime}}^{*}(A|_{V})=A^{\prime}|_{V^{\prime}}+E_{0}$

for some $f_{V^{\prime}}$-exceptional $\mathbb{Q}$-divisor $E_{0}\geq 0$ on
$V^{\prime}$. From them, we have

($1$)
$(K_{X^{\prime}}+\Delta^{\prime}+t_{0}A^{\prime})|_{V^{\prime}}=f_{V^{\prime}}^{*}\bigl{(}(K_{X}+\Delta+t_{0}A)|_{V}\bigr{)}+E^{\prime}|_{V^{\prime}}-t_{0}E_{0}$

and $E^{\prime}|_{V^{\prime}}-t_{0}E_{0}$ is $f|_{V^{\prime}}$-exceptional.
Since $\bigl{(}V,(\Delta+t_{0}A)|_{V}\bigr{)}$ is lc, the definition of
$\Delta^{\prime}$ implies that every $f_{V}$-exceptional prime divisor $P$ on
$V^{\prime}$ satisfies

${\rm
coeff}_{P}(E^{\prime}|_{V^{\prime}}-t_{0}E_{0})=a\bigl{(}P,V,(\Delta+t_{0}A)|_{V}\bigr{)}\geq
0.$

This implies that $E^{\prime}|_{V^{\prime}}-t_{0}E_{0}$ is effective.

######  Step 2.

In this step, we prove that $(X^{\prime},\Delta^{\prime}+t_{0}A^{\prime})$ has
a good minimal model over $Z^{\prime}$.

Since $\bigl{(}V,(\Delta+t_{0}A)|_{V}\bigr{)}$ is an lc pair and
$K_{V}+\Delta|_{V}+t_{0}A|_{V}\sim_{\mathbb{Q},U}t_{0}A|_{V}$ is ample over
$U$, the relation ($1$) in Step 1 shows that
$\bigl{(}V^{\prime},(\Delta^{\prime}+t_{0}A^{\prime})|_{V^{\prime}}\bigr{)}$
has a good minimal model over $Z^{\prime}\setminus\Sigma^{\prime}$. By
construction of $\Delta^{\prime}$, we can find an effective
$\mathbb{Q}$-divisor $T^{\prime}$ on $X^{\prime}$, which is a multiple of
$\pi^{\prime*}\Sigma^{\prime}$ by a positive rational number, such that
$\Delta^{\prime}-T^{\prime}\geq 0$. Then the pair
$(X^{\prime},\Delta^{\prime}-T^{\prime}+t_{0}A^{\prime})$ is lc and all lc
centers of the pair intersect
$\pi^{\prime-1}(Z^{\prime}\setminus\Sigma^{\prime})$. By [Ha19, Theorem 1.2],
the lc pair $(X^{\prime},\Delta^{\prime}-T^{\prime}+t_{0}A^{\prime})$ has a
good minimal model over $Z^{\prime}$. Moreover, the relation
$T^{\prime}\sim_{\mathbb{Q},Z^{\prime}}0$ shows

$K_{X^{\prime}}+\Delta^{\prime}-T^{\prime}+t_{0}A^{\prime}\sim_{\mathbb{Q},Z^{\prime}}K_{X^{\prime}}+\Delta^{\prime}+t_{0}A^{\prime}.$

From these facts, it follows that
$(X^{\prime},\Delta^{\prime}+t_{0}A^{\prime})$ has a good minimal model over
$Z^{\prime}$

######  Step 3.

In this step, we define some varieties and $\mathbb{R}$-divisors.

By running a $(K_{X^{\prime}}+\Delta^{\prime}+t_{0}A^{\prime})$-MMP over
$Z^{\prime}$, we get a birational contraction

$(X^{\prime},\Delta^{\prime}+t_{0}A^{\prime})\dashrightarrow(X^{\prime\prime},\Delta^{\prime\prime}+t_{0}A^{\prime\prime})$

over $Z^{\prime}$ to a good minimal model
$(X^{\prime\prime},\Delta^{\prime\prime}+t_{0}A^{\prime\prime})$. Let $g\colon
X^{\prime\prime}\to Y$ be the contraction over $Z^{\prime}$ induced by
$K_{X^{\prime\prime}}+\Delta^{\prime\prime}+t_{0}A^{\prime\prime}$, and we put
$\Gamma=g_{*}(\Delta^{\prime\prime}+t_{0}A^{\prime\prime})$ and we denote
$Y\to Z^{\prime}$ by $\pi_{Y}$. We have the following diagram

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi^{\prime}}$$\textstyle{X^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime\prime}}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Y}}$$\textstyle{Z}$$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$

such that $K_{Y}+\Gamma$ is ample over $Z^{\prime}$ and
$K_{X^{\prime\prime}}+\Delta^{\prime\prime}+t_{0}A^{\prime\prime}=g^{*}(K_{Y}+\Gamma)$.
Let $G$ be a sufficiently general fiber of $\pi_{Y}$. By construction, we have

${\rm vol}\bigl{(}(K_{Y}+\Gamma)|_{G}\bigr{)}={\rm
vol}\bigl{(}(K_{X}+\Delta+t_{0}A)|_{F}\bigr{)}=t_{0}^{{\rm dim}F}{\rm
vol}(A|_{F}).$

Because ${\rm vol}(A|_{F})=v$ and $t_{0}\leq 1$, which follows from the facts
that $A$ is a Weil divisor, we see that ${\rm
vol}\bigl{(}(K_{Y}+\Gamma)|_{G}\bigr{)}\leq v$.

######  Step 4.

In this step, we prove that for any generic point $\eta$ of $\Sigma^{\prime}$,
the multiplicity of the fiber of $\pi_{Y}$ over $\eta$ has an upper bound
depending only on $d$, $m$, $t_{0}$ and $v$.

We may assume that $Z^{\prime}$ is a curve by cutting with hyperplane
sections. We fix an arbitrary generic point $\eta$ of $\Sigma^{\prime}$. Then
we may write

$\pi_{Y}^{*}\eta=\sum_{i}\mu_{i}G_{i},$

where $G_{i}$ are prime divisors on $Y$. For each $i$, let $G_{i}^{\nu}$ be
the normalization of $G_{i}$. For every $i$, the inequality ${\rm
vol}\bigl{(}(K_{Y}+\Gamma)|_{G}\bigr{)}\leq v$ implies

$\begin{split}v\geq&G\cdot(K_{Y}+\Gamma)^{{\rm dim}G}\\\
\geq&\mu_{i}G_{i}\cdot(K_{Y}+\Gamma)^{{\rm dim}G_{i}}\\\
=&\mu_{i}\cdot((K_{Y}+\Gamma)|_{G_{i}^{\nu}})^{{\rm dim}G_{i}^{\nu}}\\\
=&\mu_{i}\cdot{\rm
vol}\bigl{(}(K_{Y}+\Gamma)|_{G_{i}^{\nu}}\bigr{)}>0.\end{split}$

By the definition of $\Delta^{\prime}$ in Step 1, it follows that
$\lfloor\Delta^{\prime}\rfloor$ contains all components of
$\pi^{\prime*}\Sigma^{\prime}$. Hence $G_{i}$ is a component of
$\lfloor\Gamma\rfloor$. Since $m(K_{X}+\Delta)$ is Cartier, which is the
hypothesis of Lemma 1.1, the definition of $\Delta^{\prime}$ in Step 1 shows
that $m\Delta^{\prime}$ is a Weil divisor on $X^{\prime}$. Hence, the
coefficients of $\Gamma$ belong to $\frac{1}{m}\mathbb{Z}_{\geq 0}\cup
t_{0}\mathbb{Z}_{\geq 0}$. By applying divisorial adjunction to $(Y,\Gamma)$
and $G_{i}$ and applying the DCC for volumes [HMX14, Theorem 1.3], the volume
${\rm vol}\bigl{(}(K_{Y}+\Gamma)|_{G_{i}^{\nu}}\bigr{)}$ is bounded from below
by a positive real number depending only on $d$, $m$, and $t_{0}$. Thus,
$\mu_{i}$ has an upper bound depending only on $d$, $m$, $t_{0}$ and $v$.

In this way, we see that the multiplicity of the fiber of $\pi_{Y}\colon Y\to
Z^{\prime}$ over any generic point of $\Sigma^{\prime}$ has an upper bound
depending only on $d$, $m$, $t_{0}$ and $v$.

######  Step 5.

In this step, we prove that $n^{\prime\prime}h^{*}D$ is Cartier for some
positive integer $n^{\prime\prime}$ that depends only on $d$, $m$, $t_{0}$,
$v$, and $n^{\prime}$, where $n^{\prime}$ is the positive integer defined at
the start of this proof.

We define a $\mathbb{Q}$-divisor $\Theta^{\prime}$ on $X^{\prime}$ by
$K_{X^{\prime}}+\Theta^{\prime}=f^{*}(K_{X}+\Delta)$. Let $\Theta_{Y}$ be the
birational transform of $\Theta^{\prime}$ on $Y$. By the hypothesis of Lemma
1.1 that $mf^{*}(K_{X}+\Delta)$ is Cartier, $m(K_{Y}+\Theta_{Y})$ is a Weil
divisor on $Y$. Then there is a rational function $\sigma$ on $Y$ such that

($2$) $mn^{\prime}(K_{Y}+\Theta_{Y})+{\rm
div}(\sigma)=mn^{\prime}\pi_{Y}^{*}h^{*}D$

as $\mathbb{Q}$-divisors. Moreover, the left hand side is a Weil divisor. We
can write

$h^{*}D=\sum_{j}a_{j}D_{j}$

where $D_{j}$ are prime divisors on $Z^{\prime}$. By Step 4, the multiplicity
of the fiber of $\pi_{Y}$ over any generic point of $\Sigma^{\prime}$ has an
upper bound, which we denote $\beta$, depending only on $d$, $m$, $t_{0}$ and
$v$. This shows that $\pi_{Y}^{*}D_{j}$ has a component $Q_{j}$ such that
${\rm coeff}_{Q_{j}}(\pi_{Y}^{*}D_{j})\leq\beta$, and ($2$) implies
$mn^{\prime}a_{j}\cdot{\rm coeff}_{Q_{j}}(\pi_{Y}^{*}D_{j})\in\mathbb{Z}$. In
particular, $mn^{\prime}\lfloor\beta\rfloor!a_{j}\in\mathbb{Z}$.

We define $n^{\prime\prime}:=mn^{\prime}\lfloor\beta\rfloor!$. Then
$n^{\prime\prime}$ depends only on $d$, $m$, $t_{0}$, $v$, and
$n^{\prime\prime}$. Moreover, $n^{\prime\prime}h^{*}D$ is a Weil divisor. In
particular, $n^{\prime\prime}h^{*}D$ is Cartier.

By defining $n:=mn^{\prime}n^{\prime\prime}$ and $\phi:=h\colon Z^{\prime}\to
Z$, we complete the proof. ∎

###### Proof of Theorem 1.2.

Let $\pi\colon(X,\Delta)\to Z$ be the contraction as in Theorem 1.2. We pick
$t>0$ such that $(X,\Delta+tA)$ is klt. Since $A$ is big over $Z$, it follows
from [BCHM10] that $(X,\Delta+tA)$ has a good minimal model over $Z$. In
particular, $(X,\Delta+tA)$ has a log canonical model over $Z$, i.e., there is
a birational contraction

$(X,\Delta+tA)\dashrightarrow(X^{\prime},\Delta^{\prime}+tA^{\prime})$

over $Z$ such that $K_{X^{\prime}}+\Delta^{\prime}+tA^{\prime}$ is ample over
$Z$ and $a(P,X,\Delta+tA)\leq a(P,X^{\prime},\Delta^{\prime}+tA^{\prime})$ for
every prime divisor $P$ over $X$. By using the negativity lemma and Lemma 3.1
after taking a common resolution of $X\dashrightarrow X^{\prime}$, we see that
$e(K_{X^{\prime}}+\Delta^{\prime})$ is semi-ample, $X^{\prime}\to Z$ is the
contraction induced by $e(K_{X^{\prime}}+\Delta^{\prime})$, and
$m(K_{X^{\prime}}+\Delta^{\prime})$ is Cartier. Furthermore, ${\rm
vol}(A^{\prime}|_{F^{\prime}})=v$, where $F^{\prime}$ is a sufficiently
general fiber of $X^{\prime}\to Z$, and Theorem 1.2 holds for $(X,\Delta)\to
Z$ and $A$ if and only if Theorem 1.2 holds for
$(X^{\prime},\Delta^{\prime})\to Z$ and $A^{\prime}$. Therefore, replacing
$(X,\Delta)$ and $A$ by $(X^{\prime},\Delta^{\prime})$ and $A^{\prime}$
respectively, we may assume that $A$ is ample over $Z$.

Because there is a klt pair on $Z$, by Lemma 1.1, there exists
$n^{\prime}\in\mathbb{Z}_{>0}$ depending only on $d$, $m$, and $v$ such that
$n^{\prime}(K_{X}+\Delta)\sim\pi^{*}D$ for some Cartier divisor $D$ on $Z$.
Then $eD$ is ample by construction of $\pi\colon X\to Z$. By [A05, Theorem
4.1], there is a klt pair $(Z,B)$ such that $K_{Z}+B\sim_{\mathbb{Q}}D$. By
[Ko93, Theorem 1.1] and [F17, Lemma 7.1], there exists
$n^{\prime\prime}\in\mathbb{Z}_{>0}$, depending only on ${\rm dim}Z$, such
that $2n^{\prime\prime}eD$ is very ample. By replacing $n^{\prime\prime}$, we
may assume that $n^{\prime\prime}$ depends only on $d$. Hence
$n:=2n^{\prime}n^{\prime\prime}$ is the desired positive integer. ∎

##  4\. Lc-trivial fibration with log big moduli part

In this section, we study lc-trivial fibrations whose moduli parts satisfy the
log bigness on certain log smooth Ambro models.

######  Definition 4.1 (Lc-trivial fibration with log big moduli part).

Let $(X,\Delta)\to Z$ be an lc-trivial fibration. Let $\boldsymbol{\rm B}$ and
$\boldsymbol{\rm M}$ be the discriminant b-divisor and the moduli b-divisor of
the canonical bundle formula, respectively. We say that $(X,\Delta)\to Z$ is
an lc-trivial fibration with log big moduli part if there exists a log smooth
Ambro model $Z^{\prime}\to Z$ such that $\boldsymbol{\rm M}_{Z^{\prime}}$ is
log big with respect to $(Z^{\prime},\boldsymbol{\rm B}_{Z^{\prime}})$.

The motivation of the topic comes from the following result, that is a
consequence of the argument by Floris–Lazić [FL19] (see also [Hu20, Theorem
1.2] by Hu).

######  Theorem 4.2 (cf. [FL19]).

Let $(X,\Delta)$ be a projective lc pair, and let $(X,\Delta)\to Z$ be an lc-
trivial fibration. Let $\boldsymbol{\rm B}$ and $\boldsymbol{\rm M}$ be the
discriminant b-divisor and the moduli b-divisor of the canonical bundle
formula, respectively. Then there is a log smooth Ambro model $Z^{\prime}\to
Z$ such that $\boldsymbol{\rm M}_{Z^{\prime}}$ is log abundant with respect to
$(Z^{\prime},\boldsymbol{\rm B}_{Z^{\prime}})$, i.e., for every stratum
$T^{\prime}$ of $(Z^{\prime},\boldsymbol{\rm B}_{Z^{\prime}})$, there exist a
birational morphism $h\colon W\to T^{\prime}$ from a normal projective variety
$W$, a surjective morphism $\psi\colon W\to V$ to a normal projective variety
$V$, and a nef and big $\mathbb{Q}$-divisor $N$ on $V$ such that
$h^{*}(\boldsymbol{\rm
M}_{Z^{\prime}}|_{T^{\prime}})\sim_{\mathbb{Q}}\psi^{*}N$.

Note that $Z^{\prime}$ is not uniquely determined.

Lc-trivial fibrations with log big moduli parts appear as a case where there
is a log smooth Ambro model $Z^{\prime}\to Z$ such that the map $\psi\circ
h^{-1}\colon T^{\prime}\dashrightarrow V$ in Theorem 4.2 is birational for
every stratum $T^{\prime}$ of $(Z^{\prime},\boldsymbol{\rm B}_{Z^{\prime}})$.

By combining Lemma 1.1 and a result in [Ha22], we will prove an effective base
point free theorem for lc-trivial fibrations with log big moduli parts. From
now on, for every lc-trivial fibration $(X,\Delta)\to Z$, we denote the
discriminant b-divisor and the moduli b-divisor of the canonical bundle
formula by $\boldsymbol{\rm B}$ and $\boldsymbol{\rm M}$, respectively.

######  Theorem 4.3.

For every $d$, $m\in\mathbb{Z}_{>0}$, and $v\in\mathbb{R}_{>0}$, there exists
$n\in\mathbb{Z}_{>0}$, depending only on $d$, $m$, and $v$, satisfying the
following. Let $(X,\Delta)$ be a projective lc pair, let
$\pi\colon(X,\Delta)\to Z$ be an lc-trivial fibration with log big moduli
part, let $F$ be the sufficiently general fiber of $\pi$, and let $A$ be a
Weil divisor on $X$ such that

  * •

${\rm dim}X=d$,

  * •

$m(K_{X}+\Delta)$ is nef and Cartier,

  * •

$A|_{\pi^{-1}(U)}$ is $\mathbb{Q}$-Cartier and ample over $U$ for some open
subset $U\subset Z$,

  * •

$(F,\Delta|_{F}+tA|_{F})$ is an lc pair for some real number $t>0$, and

  * •

${\rm vol}(A|_{F})=v$.

Then $n(K_{X}+\Delta)$ is base point free.

###### Proof.

By Lemma 1.1, There is $n^{\prime}$ that depends on $d$, $m$, and $v$ such
that a generalized lc pair $(Z,\boldsymbol{\rm B}_{Z},\boldsymbol{\rm M})$
defined by the canonical bundle formula satisfies

  * •

$n^{\prime}(K_{X}+\Delta)\sim n^{\prime}\pi^{*}(K_{Z}+\boldsymbol{\rm
B}_{Z}+\boldsymbol{\rm M}_{Z})$,

  * •

$n^{\prime}\boldsymbol{\rm M}$ is b-Cartier, and

  * •

$n^{\prime}\phi^{*}(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M}_{Z})$ is
Cartier for some resolution $\phi\colon Z^{\prime}\to Z$ of $Z$.

Put $D=(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M}_{Z})$. By Lemma 3.1,
for any resolution $\psi\colon Z^{\prime\prime}\to Z$, the Cartier property of
$n^{\prime}\phi^{*}D$ implies that $n^{\prime}\psi^{*}D$ is Cartier. Since
$(X,\Delta)\to Z$ is an lc-trivial fibration with log big moduli part,
replacing $Z^{\prime}$ if necessary we may assume that $\boldsymbol{\rm
M}_{Z^{\prime}}$ is log big with respect to $(Z^{\prime},\boldsymbol{\rm
B}_{Z^{\prime}})$. Then

$n^{\prime}(K_{Z^{\prime}}+\boldsymbol{\rm B}_{Z^{\prime}}+\boldsymbol{\rm
M}_{Z^{\prime}})=n^{\prime}\phi^{*}D$

is Cartier.

Let $B^{\prime}$ be the effective part of $\boldsymbol{\rm B}_{Z^{\prime}}$.
Then

$K_{Z^{\prime}}+B^{\prime}+\boldsymbol{\rm
M}_{Z^{\prime}}=\phi^{*}(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm
M}_{Z})+E^{\prime},$

where $E^{\prime}$ is the negative part of $\boldsymbol{\rm B}_{Z^{\prime}}$.
By running a $(K_{Z^{\prime}}+B^{\prime}+\boldsymbol{\rm M}_{Z^{\prime}})$-MMP
over $Z$, we get a birational contraction

$(Z^{\prime},B^{\prime},\boldsymbol{\rm
M})\dashrightarrow(W,B^{\prime}_{W},\boldsymbol{\rm M})$

over $Z$ such that $E^{\prime}$ is contracted by the MMP. Then
$B^{\prime}_{W}=\boldsymbol{\rm B}_{W}$. Moreover, the divisor
$n^{\prime}(K_{Z^{\prime}}+\boldsymbol{\rm B}_{Z^{\prime}}+\boldsymbol{\rm
M}_{Z^{\prime}})$ is trivial over the extremal contractions of the MMP over
$Z$. Thus, $n^{\prime}(K_{W}+\boldsymbol{\rm B}_{W}+\boldsymbol{\rm M}_{W})$
is Cartier and the divisor is the pullback of $K_{Z}+\boldsymbol{\rm
B}_{Z}+\boldsymbol{\rm M}_{Z}$ to $W$. By [Ha22, Theorem 5.8], there exists
$n$, depending only on ${\rm dim}Z$ and $n^{\prime}$, such that
$n(K_{W}+\boldsymbol{\rm B}_{W}+\boldsymbol{\rm M}_{W})$ is base point free.
By replacing $n$, we may assume that $n$ depends only on $d$, $m$, and $v$,
and we may further assume that $mn^{\prime}$ divides $n$.

Let $f\colon X^{\prime}\to X$ be a resolution of $X$ such that the induced map
$\pi^{\prime}\colon X^{\prime}\dashrightarrow W$ is a morphism. Then
$\pi^{\prime}$ is a contraction and

$f^{*}\bigl{(}n(K_{X}+\Delta)\bigr{)}\sim\pi^{\prime*}\bigl{(}n(K_{W}+\boldsymbol{\rm
B}_{W}+\boldsymbol{\rm M}_{W})\bigr{)}.$

Since $n(K_{X}+\Delta)$ is Cartier, we see that $n(K_{X}+\Delta)$ is base
point free. ∎

######  Theorem 4.4.

For every $d$, $m\in\mathbb{Z}_{>0}$, and $v\in\mathbb{R}_{>0}$, there exists
$n\in\mathbb{Z}_{>0}$, depending only on $d$, $m$, and $v$, satisfying the
following. Let $(X,\Delta)$ be a projective lc pair, let
$\pi\colon(X,\Delta)\to Z$ be an lc-trivial fibration with log big moduli
part, let $F$ be the sufficiently general fiber of $\pi$, and let $A$ be a
Weil divisor on $X$ such that

  * •

${\rm dim}X=d$,

  * •

$-m(K_{X}+\Delta)$ is nef and Cartier,

  * •

$A|_{\pi^{-1}(U)}$ is $\mathbb{Q}$-Cartier and ample over $U$ for some open
subset $U\subset Z$,

  * •

$(F,\Delta|_{F}+tA|_{F})$ is an lc pair for some real number $t>0$, and

  * •

${\rm vol}(A|_{F})=v$.

Then $-n(K_{X}+\Delta)$ is base point free. In particular, $-(K_{X}+\Delta)$
is semi-ample.

###### Proof.

By Lemma 1.1, There is $n^{\prime}$ that depends on $d$, $m$, and $v$ such
that a generalized lc pair $(Z,\boldsymbol{\rm B}_{Z},\boldsymbol{\rm M})$
defined by the canonical bundle formula satisfies

  * •

$n^{\prime}(K_{X}+\Delta)\sim n^{\prime}\pi^{*}(K_{Z}+\boldsymbol{\rm
B}_{Z}+\boldsymbol{\rm M}_{Z})$,

  * •

$n^{\prime}\boldsymbol{\rm M}$ is b-Cartier, and

  * •

$n^{\prime}\phi^{*}(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M}_{Z})$ is
Cartier for some resolution $\phi\colon Z^{\prime}\to Z$ of $Z$.

Put $D=(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M}_{Z})$. As in the proof
of Theorem 4.3, replacing $Z^{\prime}$ if necessary, we may assume that
$\boldsymbol{\rm M}_{Z^{\prime}}$ is log big with respect to
$(Z^{\prime},\boldsymbol{\rm B}_{Z^{\prime}})$. Then

$n^{\prime}(K_{Z^{\prime}}+\boldsymbol{\rm B}_{Z^{\prime}}+\boldsymbol{\rm
M}_{Z^{\prime}}-2\phi^{*}D)=-n^{\prime}\phi^{*}D$

is Cartier. Then the proof of Theorem 4.3 works by replacing $\boldsymbol{\rm
M}$ with $\boldsymbol{\rm M}-2\overline{D}$. ∎

Finally, we prove the minimal model theory for lc pairs admiting an lc-trivial
fibration with log big moduli part. Note that we do not need Lemma 1.1 for the
proof.

######  Theorem 4.5.

Let $(X,\Delta)$ be a projective lc pair admitting an lc-trivial fibration
with log big moduli part. Then $(X,\Delta)$ has a good minimal model or a Mori
fiber space.

###### Proof.

We may assume that $K_{X}+\Delta$ is pseudo-effective. Let $(X,\Delta)\to Z$
be an lc-trivial fibration with log big moduli part, and let $\phi\colon
Z^{\prime}\to Z$ be a log smooth Ambro model such that $\boldsymbol{\rm
M}_{Z^{\prime}}$ is log big with respect to $(Z^{\prime},\boldsymbol{\rm
B}_{Z^{\prime}})$. We may write

$K_{Z^{\prime}}+B^{\prime}+\boldsymbol{\rm
M}_{Z^{\prime}}=\phi^{*}(K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm
M}_{Z})+E^{\prime}$

where $B^{\prime}\geq 0$ and $E^{\prime}\geq 0$ have no common components. By
[Ha22, Theorem 1.1], the generalized lc pair
$(Z^{\prime},B^{\prime},\boldsymbol{\rm M})$ has a good minimal model. In
particular, $K_{Z}+\boldsymbol{\rm B}_{Z}+\boldsymbol{\rm M}_{Z}$ birationally
has a Nakayama–Zariski decomposition with semi-ample positive part. By
applying [Na04, III, 5.17 Corollary] to a suitable base change of $X\to Z$, we
see that $K_{X}+\Delta$ birationally has a Nakayama–Zariski decomposition with
semi-ample positive part. By [Ha20, Theorem 2.23] (see also [BiHu14, Theorem
1.1]), we see that $(X,\Delta)$ has a good minimal model. ∎

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