# Magnet-Free Nonreciprocal Phase-Shifter Based on Time Modulation

Sajjad Taravati and George V. Eleftheriades
The Edward S. Rogers Sr. Department of Electrical and Computer Engineering,
University of Toronto, Toronto, Ontario M5S 3H7, Canada
Email<EMAIL_ADDRESS>
###### Abstract

Recently, nonreciprocal phase shifters have attracted a surge of interest
thanks to the advent of nonreciprocal electromagnetic systems, such as
nonreciprocal metasurfaces, nonreciprocal-beam antennas, and invisibility
cloaks. To overcome the limitations associated with conventional technologies
for realizing nonreciprocal phase shifters and gyrators, here we propose a
low-noise, lightweight, low-profile, and linear magnetless nonreciprocal phase
shifter formed by two temporal loops. The proposed temporal apparatus operates
based on the generation of time-harmonic signals and destructive and
constructive interferences for the undesired and desired time harmonics,
respectively, at different locations of the structure. An external time-
harmonic modulation signal injects an effective electronic angular momentum to
the system to control the phase and frequency of the two loops. Such a
temporal nonreciprocal phase shifter offers low insertion loss, and a large
return loss (input matching) of greater than 28.1 dB. Additionally, this
nonreciprocal phase shifter possesses a reconfigurable architecture and can be
directly embedded in integrated circuit (IC) technology to create high power
handling and linear IC-based nonreciprocal phase shifters.

††preprint: AIP/123-QED

Phase shifters are key elements of wireless communication systems, biomedical
systems, radars, optical wave processors, and many other applications.
However, conventional passive and reciprocal phase shifters are restricted by
the Lorentz reciprocity and have limited applications in the modern
communication and electromagnetic systems. Recently, several exotic
electromagnetic systems have been proposed by taking advantage of
nonreciprocal phase shifting, including nonreciprocal metasurfaces and
nonreciprocal-beam antennas. Conventionally, nonreciprocal phase shifters are
created using magneto-optical structures Okamura, Negami, and Yamamoto (1985);
Hung and Anjan (2002); Zhou  _et al._ (2012), ferrite-based magnetic
structures Adhikari  _et al._ (2013) and transistor-loaded transmission lines
Ayasli (1989a); Ciccognani  _et al._ (2012); Palomba  _et al._ (2018); Tanaka,
Shimomura, and Ohtake (1965a); Smith (1988); Ayasli (1989b); Kalialakis  _et
al._ (2000); Carchon and Nanwelaers (2000); Taravati and Eleftheriades
(2021a); Sajjad Taravati (2021); Karimian  _et al._ (2020). However, ferrite-
based magnetic structures are practically realized using bulky, costly, heavy
architectures, and are not available at high frequencies. On the other hand,
transistor-based nonmagnetic nonreciprocity and magneto-optical and electro-
optical modulators substitute the absence of cumbersome magnetic bias for
other tangible drawbacks, such as poor noise performance and strong
nonlinearities in the case of transistors, or the complexity and large size in
the case of optical modulators.

Recently, nonmagnetic time-modulation-based nonreciprocity has spurred a surge
of research activity to eliminate the issues associated with conventional
nonreciprocal devices Taravati and Kishk (2020); Camacho, Edwards, and Engheta
(2020); Darvish and Kishk (2022); Elnaggar and Milford (2021); Kumar, Dash,
and Sarkar (2022); Oudich  _et al._ (2022). Space-time-modulated structures
take advantage of one-way progressive wave coupling to create nonreciprocal
devices and isolators Taravati (2017); Li  _et al._ (2019); Taravati (2018a);
Taravati and Eleftheriades (2019, 2021b); Wan  _et al._ (2021). Other
applications of space-time modulation include parametric wave amplification
Tien and Suhl (1958); Tien (1958); Zhu  _et al._ (2020), pure frequency
generation Taravati (2018b); Taravati and Eleftheriades (2021c); Bui  _et al._
(2022); Gaxiola-Luna and Halevi (2021), nonreciprocal filters Wu  _et al._
(2020), metasurface-based wave processors Zang  _et al._ (2019); Taravati and
Kishk (2019a); Taravati and Eleftheriades (2022a); Khorrami  _et al._ (2022);
Taravati and Eleftheriades (2022b); Bai  _et al._ (2022); Fathnan  _et al._
(2022); Zhang  _et al._ (2022); Barati Sedeh, Salary, and Mosallaei (2022);
Ghanekar  _et al._ (2022), antennas Zang, Alvarez-Melcon, and Gomez-Diaz
(2019); Do  _et al._ (2022); Zang  _et al._ (2022); Do  _et al._ (2022),
unidirectional beam splitting Taravati and Kishk (2019b), and full-transceiver
schemes Taravati and Eleftheriades (2020a).

Phase-tailored time modulation is shown to be an excellent alternative
approach to realize unique functionalities and devices Zang, Alvarez-Melcon,
and Gomez-Diaz (2019); Taravati and Eleftheriades (2020b, 2022c, 2022d) in a
compact fashion. Such a technique avoids the long profile that is typically
required for the realization of space-time modulation Tien and Suhl (1958);
Taravati (2017). This letter proposes a nonmagnetic nonreciprocal phase
shifter comprising two phase-tailored time-modulated loops. As a result, a
lightweight and low-profile device can be realized thanks to the fact that
only four time-modulated varactors are required. Due to the unique and optimal
architecture of the proposed loop-based temporal nonreciprocal phase shifter,
low insertion loss of less than 1.2 dB is achieved between its two ports, by
imposing specified constructive and destructive interferences for the ports of
the nonreciprocal phase shifter. Additionally, the undesired time harmonics
are strongly suppressed. The nonreciprocal phase shifter exhibits highly
linear response corresponding to the 1 dB compression point of +31.1 dBm,
broad-band operation where a nonreciprocal phase shift with $4^{\circ}$
differential phase shift deviation is achieved across a 28$\%$ fractional
bandwidth.

Figure 1: Forward and backward frequency-phase transitions in a time-modulated
transmission line. (a) A phased time-modulated transmission line. (b)
Dispersion diagram showing forward transition from the $n=0$ time harmonic at
$\omega_{0}$ to the $n=+1$ time harmonic at $\omega_{0}+\Omega$ resulting in
phase alteration of $+\phi$, and backward transition from $\omega_{0}+\Omega$
to $\omega_{0}$ with phase alteration of $-\phi$.

Fig. 1(a) shows a transmission line loaded with a phased time-modulated
varactor. Thus, the time-varying capacitance of the transmission line reads
$C_{\text{eq}}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi)$, where
$C_{\text{av}}$ represents the average capacitance of the time-varying
varactor. The total capacitance of the transmission line reads
$C_{\text{tot}}(t)=C_{0}C_{\text{eq}}(t)$, where $C_{0}$ is the capacitance
per unit length of the transmission line. By proper engineering of the
dispersion of the transmission line, the output voltage acquires a frequency
transition (up/down-conversion) accompanied by a phase transition. Such
frequency and phase transitions are shown in the dispersion diagram in Fig.
1(b), where a frequency up-conversion (from $\omega_{0}$ to
$\omega_{0}+\Omega$) is associated with a phase addition of $\phi$.

The dispersion bands of the structure are tailored in a way that only the
fundamental and the first higher-order time-harmonics are excited, and all
undesired higher-order time-harmonics are suppressed. Hence, the space-time
varying voltage between the two conductors of the transmission line is defined
based on the superposition of the $n=0$ and $n=-1$ space-time harmonics
fields, i.e.,

$V_{\text{S}}(z,t)=a_{0}(z)e^{-i\left(k_{0}z-\omega_{0}t\right)}+a_{1}(z)e^{-i\left(k_{0}z-(\omega_{0}+\Omega)t\right)}.$
(1)

Since the transmission line is modulated in time, and not in space, we
consider an identical wave number $k_{0}$ for both the fundamental and the
higher-order harmonics. As shown in Fig. 1(b), as a result of the time-
modulated sinusoidally periodic capacitance, the dispersion diagram of the
time-modulated transmission line is periodic with respect to the $\omega$ axis
Jorge R Zurita-Sánchez and Cervantes-González (2009), that is
$k(\omega_{0}+\Omega)=k(\omega_{0})=k_{0}$. Such a time-modulated periodicity
introduces vertical electromagnetic transitions in the dispersion diagram, as
demonstrated in Fig. 1(b). The unknown spatially variant amplitudes $a_{0}(z)$
and $a_{1}(z)$ should be determined through satisfying both the Telegrapher’s
equations and the initial conditions at $z=0$. The up-conversion assumes the
initial conditions of $v_{0}^{\text{up-c}}(0)=a_{0}$ and
$v_{1}^{\text{up-c}}(0)=0$, which yields

$v_{0}^{\text{up-c}}(z)=a_{0}\cos\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right),$ (2a)
$v_{1}^{\text{up-c}}(z)=a_{0}\dfrac{k_{1}}{k_{0}}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right)e^{+i\phi}.$ (2b)

Equations (2a) and (2b) show that the change of the frequency and phase of the
up-converted signal corresponds to the frequency and phase of the time-
modulated modulation signal, $\Omega$ and $+\phi$. In addition, the maximum
amplitude of the up-converted signal $a_{1}^{\text{up-c}}(z)$ occurs at
$\delta k_{1}/C_{\text{av}}=2\pi$. This reveals that, one can control the
amplitude of the up-converted signal by changing the modulation amplitude
$\delta$. Following the same procedure, we can determine the input and output
voltages for the down-conversion.

The down-conversion assumes the initial conditions of $v_{0}(0)=0$ and
$v_{1}^{\text{dn-c}}(0)=a_{1}=\text{Max}[v_{1}^{\text{dn-c}}(z)]=a_{0}k_{1}/k_{0}$,
which gives

$v_{1}^{\text{dn-c}}(z)=a_{1}\cos\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right),$ (3a) and
$v_{0}^{\text{dn-c}}(z)=a_{0}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right)e^{-i\phi}.$ (3b)

Equations (3a) and (3b) show that the change of the frequency and phase of the
down-converted signal corresponds to the frequency and phase of the time-
modulated modulation signal, $\Omega$ and $-\phi$.

Figure 2 shows a schematic representation of the dispersion diagram for
asymmetric frequency-phase transitions in a time-modulated transmission line.
Here, a frequency up-conversion from $\omega_{0}$ to $\omega_{0}+\Omega$ is
accompanied by a positive additive phase, i.e., $+\phi_{1}$ in the forward
transmission and $+\phi_{2}$ in the backward transmission. However, a
frequency down-conversion from $\omega_{0}+\Omega$ to $\omega_{0}$ is
accompanied by a negative additive phase, i.e., $-\phi_{2}$ in the forward
transmission and $-\phi_{1}$ in the backward transmission.

Figure 2: Schematic representation of forward and backward frequency-phase
transitions in two cascaded time-modulated transmission lines.

The time-varying nonreciprocal phase shifting architecture in Fig. 2 assumes
only two time harmonics are supported by the transmission line, while all the
undesired time harmonics are not excited. In reality, however, this is not
true, and all the time harmonic side-bands are generated which degrades the
performance of nonreciprocal phase shifting. To overcome this issue, here we
propose a new architecture that is constituted of four temporal unit cells in
the form of two loops. Such an architecture guarantees propagation and
transmission of desired time harmonics and eliminates the undesired time
harmonic side-bands, leading to an excellent nonreciprocal phase shifting
performance. Figure 3 shows the proposed phase-engineered loop-based time-
modulated nonreciprocal phase shifter, formed by two time-varying loops, each
of which is composed of two phase-engineered time-modulated transmission
lines. In comparison with the simple nonreciprocal phase shifting platform in
Fig. 2, the loop-based architecture in Fig. 3 introduces three junctions
between its top and bottom arms that are utilized for governing the time
harmonics, that is, forming constructive interferences at the desired time
harmonics and destructive interferences at the undesired time harmonics, as
will be explained later.

The left time-modulated loop is composed of two time-modulated transmission
lines, where the capacitance of the transmission lines is modulated in time
introducing a $\pi$ phase difference between the bottom and top time-modulated
cells. The right time-modulated loop is composed of two time-modulated
transmission lines, where the capacitance of the transmission lines is
modulated in time introducing a $\pi$ phase difference between the bottom and
top time-modulated cells and $\phi$ phase difference with the left loop, that
is,

$C_{1,\text{b}}(t)=C_{\text{av}}+\delta\cos(\Omega t),$ (4a)
$C_{1,\text{t}}(t)=C_{\text{av}}+\delta\cos(\Omega t+\pi),$ (4b)
$C_{2,\text{b}}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi),$ (4c)
$C_{2,\text{t}}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi+\pi).$ (4d)

In Eq. (4), $C_{\text{av}}$ represents the average capacitance of the
transmission line loaded with time-modulated capacitance, $\delta$ is the
strength of the time modulation, $\Omega$ is the modulation frequency, and
$\phi$ represents the modulation phase. Next, by proper design of the time
modulation of the two loops, a desired nonreciprocal phase shift is achieved.
These two time-modulated loops introduce constructive interferences for the
desired time harmonics and destructive interferences for the undesired time
harmonics. Two band-pass filters (BPF) at $\omega_{0}$ are used at the two
ports of the nonreciprocal phase shifter to further suppress and eliminate the
undesired side-band time harmonics.

Figure 3: Architecture of the nonreciprocal phase shifter formed by two phase-
tailored time-modulated loops.

Figure 4: Appropriate constructive and destructive signal interferences in the
two loops are achieved through engineered frequency and phase transitions in
the proposed time-modulated nonreciprocal phase shifter. (a) Forward signal
transmission introducing a phase alteration of $-\phi$ at the same frequency
$\omega_{0}$. (b) Backward signal transmission introducing a phase alteration
of $+\phi$ at the same frequency $\omega_{0}$.

Figures 4 and 4 illustrate the signal transmission through the two arms of the
nonreciprocal phase shifter under forward and backward signal injections,
respectively. Two power splitters/combiners possessing two quarter-wavelength
arms are placed at the two sides of the structure to govern the desired
constructive and destructive interferences. In addition, a half-wavelength
transmission line is placed at the middle of the structure to separate the two
loops from each other and provide desired constructive and destructive
interferences for different time harmonics. We aim to achieve a two-way
complete signal transmission, but with opposite phase shifts for the forward
and backward transmissions. This is accomplished by proper engineering of the
forward (Figures 4) and backward ( 4) frequency and phase transitions in the
proposed time-modulated nonreciprocal phase shifter.

Fig. 4 shows the forward signal injection and transmission. Here, the input
signal is fed to the common port of the left power splitter and propagates
through the two arms of the left loop. These two signals make a transition to
the first higher order time harmonic $\omega_{+1}$ but with different phases,
that is, $+\pi$ and $0$ phase shifts in the upper and lower arms,
respectively. The half-wavelength interconnector at the middle of the
structure makes a constructive interference at $\omega_{+1}$, and therefore,
the generated $\omega_{+1}$ time harmonic excites the right side loop. This
leads to the generation of the $\omega_{0}$ harmonic by the right side loop,
possessing identical phase shift of $-\phi$ in the upper and lower arms of the
output of the right side loop. As a consequence, since the power splitter is
formed by equal quarter-wavelength long arms, the signals add up at the common
port of the right side power divider. This results in a transmission at
$\omega_{0}$ with the phase shift of $-\phi$. The higher order time harmonics
$\omega_{+1}=\omega_{0}+\Omega$ and $\omega_{-1}=\omega_{0}-\Omega$ acquire a
$\pi$ phase shift difference at the two arms of the output (right) power
divider, and hence, they form a complete null at the output of the structure,
and therefore are not be transmitted.

Figure 4 illustrates the backward signal transmission, where the input signal
is injected to the common port of the right side power divider and propagates
through the two arms of the right loop. Here, the input signal is fed to the
common port of the right power splitter and splits into two signals at two
output ports of the power splitter. Next, these two signals transition to the
higher order time harmonic $\omega_{+1}$ but with $+\phi+\pi$ and $+\phi$
phase shifts in the upper and lower arms, respectively. Similar to the forward
signal transmission case, the half-wavelength transmission in the middle
interconnector makes a constructive interference at $\omega_{+1}$ and a
destructive interference at $\omega_{0}$. Hence, the generated $\omega_{+1}$
time harmonic reaches to the left side loop. This leads to the generation of
the $\omega_{0}$ harmonic by the left side loop, with identical phase shift of
$+\phi$ in the upper and lower arms of the output of the right side loop.
Hence, since the power divider is formed by equal quarter-wavelength long
arms, the signals add up at the common port of the left side power combiner.
This results in a transmission at $\omega_{0}$ with the phase shift of $+\phi$
which is the opposite of the phase shift of the forward direction. Since the
higher order time harmonics $\omega_{+1}$ and $\omega_{-1}$ acquire a $\pi$
phase shift difference at the two arms of the output power divider, they form
a complete null at the output of the left power combiner. The difference
between the phase shift of the forward and backward transmissions arises due
to the asymmetric frequency-phase transitions in the proposed phase-tailored
time-modulated cells.

Figure 5 shows a proof of concept fabricated time-modulated nonreciprocal
phase shifter using four phase-shifted time-varying varactors. Here, the two
quarter-wavelength transmission lines on top of the figure form the two fixed
reciprocal $90^{\circ}$ phase shifters. The nonreciprocal time-modulated phase
shifter is constituted of the following components. Four varactors are used to
create a phase-shifted time modulation. Two transmission-line-based power
splitters feed the four time-modulated transmission lines, and the half-
wavelength transmission line in the middle of the nonreciprocal time-modulated
phase shifter provides constructive interference for the desired time
harmonics and destructive interference for the undesired time harmonics.
Furthermore, four fixed inductors denoted by $L_{\text{chk}}=390$ nH prevent
leakage of the main microwave signal to the modulation line and phase
shifters, eight fixed capacitances denoted by $C_{\text{cp}}=10$ pF decouple
the dc bias and low frequency modulation signals from the incident and
transmitted high frequency incident microwave signals.

Figure 5: Practical realization of the time-modulated nonreciprocal phase
shifter by four time-modulated varactors.

Figures 6 and 6 show the top and bottom views of the fabricated nonreciprocal
phase shifter, respectively. The nonreciprocal phase shifter is designed at
the center frequency of 3.5 GHz with the modulation frequency of $\Omega=220$
MHz and modulation signal power of +15 dBm. We have utilized a RT6010
substrate with permittivity $\epsilon_{\text{r}}=10.2$, thickness $h=1.27$ mm
and $\tan\delta=0.0023$. In addition, four SMV2019-079LF Skyworks Solutions
Inc. varactors are placed on top of the substrate and are grounded through
four via holes. Ports 1 and 2 are the main high frequency ports of the
nonreciprocal phase shifter (support 3.5 GHz), while two other ports at the
bottom of the structure, that is, Mod. 1 input and Mod. 2 input, provide the
low frequency modulation signal injection with different phase shifts. As
shown in Fig. 6, the two modulation signals, with same amplitudes and
frequencies but different phases, are injected to the structure through the
two 50 Ohm coplanar waveguide (CPW) transmission lines at the back of the
structure. Two ZFBT-6GQ+ minicircuits bias-tees are used to concurrently feed
the modulation signals and DC bias to the varactors.

Figure 6: Proof of concept fabricated prototype at microwave frequencies. (a)
Top view. (b) Bottom view.

Figures 7 plots the experimental results for wave transmission between the two
ports of the nonreciprocal phase shifter. It may be seen from these results
that an insertion loss of less than 1.2 dB between the two ports is achieved,
whereas the undesired time harmonics are strongly suppressed, i.e., they are
31 dB lower than the main signals. Figure 7 plots the return loss at the two
ports of the nonreciprocal phase shifter, that is, $P_{11}$ and $P_{22}$,
where $P_{11}<-28.1$ dBm ($\text{RL}_{11}>28.1$ dB) and $P_{22}<-34.5$ dBm
($\text{RL}_{22}>34.5$ dB).

Figures 8 plots the experimental results for differential phase shift, that
is, $\Delta\phi=\phi_{\text{backward}}-\phi_{\text{forward}}=2\phi$, where the
input frequency $\omega_{0}$ is swept from 3 to 4 GHz to show the broad-band
response of the temporal nonreciprocal phase shifter. This results show that
the proposed temporal nonreciprocal phase shift offers a controllable
nonreciprocal phase shift in a broad frequency band.

Such a temporal nonreciprocal phase shifter can be built in a more compact
fashion, and may be integrated into IC technology for high frequencies thanks
to the availability of variable capacitors at high frequencies. In addition,
in contrast to transistor-based nonreciprocal phase shifters Tanaka,
Shimomura, and Ohtake (1965b) where nonlinear and low power handling
transistors are placed in series with the incident signal, our nonreciprocal
phase shifter is endowed by high power rating as the varactors are placed in
parallel to the incident signal. In our experiment, we applied up to +33 dBm
input signal, and the power rating is expected to exceed $+47$ dBm. Having
leveraged the linear properties of the time modulation, the proposed time-
modulated nonreciprocal phase shifter exhibits an outstanding linear response,
where the P1dB is +31.1 dBm.

Figure 7: Experimental results for $\omega_{0}=3.5$ GHz. (a) Power
transmission from port 1 to port 2, $P_{21}$, and from port 2 to port 1,
$P_{12}$. (b) Return loss (input matching) at the two ports of the
nonreciprocal phase shifter. Figure 8: Experimental results for differential
phase shift $\Delta\phi=\phi_{\text{backward}}-\phi_{\text{forward}}=2\phi$.
Here, the input frequency $\omega_{0}$ is swept from 3 to 4 GHz to show the
broad-band response of the temporal nonreciprocal phase shifter.

In conclusion, a magnetless temporal nonreciprocal phase shifter is proposed.
The experimental results demonstrate a low insertion loss of less than 1.2 dB
which can be further improved. Furthermore, the unwanted time harmonics are 30
dB lower than the main signal. Furthermore, the fabricated nonreciprocal phase
shifter exhibits a highly linear response, with the P1dB of +30.4 dBm. In
addition, the proposed device can be redesigned for integration into IC
technology to achieve much smaller footprint and versatile operation.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts of interest to declare.

DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

ACKNOWLEDGEMENTS

This work is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC).

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# Semiorthogonal indecomposability of minimal irregular surfaces

Shinnosuke Okawa  Department of Mathematics, Graduate School of Science, Osaka
University, Machikaneyama 1–1, Toyonaka, Osaka, 560–0043, Japan.
<EMAIL_ADDRESS>
###### Abstract.

We prove a relative version of the fact that semiorthogonal decompositions of
the bounded derived category of coherent sheaves are strongly constrained by
the base locus of the canonical linear system. As an application we prove that
the derived category of minimal surfaces $X$ with
$H^{1}(X,\mathcal{O}_{X})\neq 0$ are semiorthogonally indecomposable.

###### Contents

  1. 1 Introduction
  2. 2 Basics of relative semiorthogonal decomposition
  3. 3 Relative semiorthogonal indecomposability
    1. 3.1 Relative base locus
    2. 3.2 The relative version of [2015arXiv150800682K]
    3. 3.3 Applications to semiorthogonal indecomposability
  4. 4 Semiorthogonal indecomposability of irregular minimal surfaces
  5. 5 Alternative proof via relative Hochschild homology

##  1\. Introduction

It is impossible to stress enough the significance of the relative point of
view for the minimal model theory. Even if one is only interested in the
birational geometry of a variety $X$, in many places it is useful to choose an
appropriate morphism $f\colon X\to Y$ and consider the geometry of $X$
relative to $f$.

This paper is devoted to bringing this insight into the study of
semiorthogonal decomposition (SOD) of the bounded derived category of coherent
sheaves and demonstrating its usefulness by showing the semiorthogonal
indecomposability of minimal surfaces with positive irregularity.

A guiding principal for the study of SODs of the bounded derived category
$\mathbf{D}(X)=\mathbf{D}^{\mathrm{b}}\operatorname{coh}X$ of coherent sheaves
on a smooth projective variety $X$ is the _DK hypothesis_. It asserts that a
step of a minimal model program $X\dasharrow Y$ (or more generally a
$K$-inequality $K_{X}>K_{Y}$) induces an SOD of $\mathbf{D}(X)$ by
“$\mathbf{D}(Y)$” and some other components; since $Y$ could be singular even
if $X$ is nonsingular, part of the hypothesis is to define the category
“$\mathbf{D}(Y)$” suitably (see, say, [MR3838122] and references therein).
Note that the hypothesis implies that if $\mathbf{D}(X)$ is indecomposable,
then $X$ should be minimal.

On the other hand, the other implication

$\displaystyle\mathrm{SOD}\Rightarrow\mathrm{MMP}$ (1.1)

does not always hold in dimensions at least $2$:

######  Example 1.1.

Suppose that $H^{i}(X,\mathcal{O}_{X})=0$ for $i>0$. Then any invertible sheaf
on $X$ is an exceptional object and hence decomposes $\mathbf{D}(X)$. There
are several examples of minimal surfaces of Kodaira dimensions $1$ or $2$ of
this sort. They are rich sources of (quasi-)phantom categories (see, say,
[2015arXiv150800682K, MR3728631, MR3090263] and references therein).

Nevertheless, known results seem to suggest the following bold hypothesis.

######  Hypothesis 1.2.

For almost all smooth projective variety $X$ which is minimal, the derived
category $\mathbf{D}(X)$ admits no nontrivial semiorthogonal decomposition.

Making the phrase ‘almost all’ in Hypothesis 1.2 as precise as possible is
part of the problem. Note that it is a special case of the following even
bolder hypothesis. One can tell from experience, however, that Hypothesis 1.2
is the essential part (see [2020arXiv200607643P, Proposition 5.7]).

######  Hypothesis 1.3.

For almost all smooth projective variety $X$ which is _not necessarily_
minimal, any SOD of the derived category $\mathbf{D}(X)$ comes from MMPs
starting with $X$ through the DK hypothesis up to autoequivalences of
$\mathbf{D}(X)$. Namely, for almost all $X$ the implication (1.1) holds (up to
autoequivalences).

Hypothesis 1.3 is completely settled for curves in [MR2838062]. See
[2015arXiv150800682K] and the rest of this introduction for what is known in
dimensions $\geq 2$.

In this paper we add a new technique for investigating SODs of
$\mathbf{D}(X)$, which is based on the _relative point of view_. Let $X$ be a
nonsingular variety and $f\colon X\to Y$ be a projective morphism to another
variety (or more generally an algebraic space). As we mentioned in the
beginning of this introduction, it is quite important to consider the
birational geometry of $X$ over $Y$.

We claim that the notion of _$f$ -linear semiorthogonal decompositions_
(Definition 2.4) is the categorical counterpart of birational geometry over
$Y$. The correspondence is summarized in Figure 1.1 below.

| geometry | category
---|---|---
absolute | MMP of $X$ | SOD of $\mathbf{D}(X)$
relative | MMP of $X$ over $Y$ |  $f$-linear SOD of $\mathbf{D}(X)$
Figure 1.1. Dictionary

This, for example, is supported by the following result due to Pirozhkov,
which plays an important role in this paper.

######  Theorem 1.4 ($=$[2020arXiv201112743P, Theorem 4.5], simplified for
brevity).

Let $f\colon X\to Y$ be a morphism of projective varieties such that $X$ is
nonsingular. Suppose that $Y$ admits a finite morphism to an abelian variety.
Then any SOD of $\mathbf{D}(X)$ is $f$-linear.

######  Remark 1.5.

Theorem 1.4 is comparable to the fact that, in the same situation, any MMP
starting with $X$ is over $Y$ since there is no rational curve on an abelian
variety.

In [2015arXiv150800682K, Theorem 1.2] Kawatani and the author proved that
semiorthogonal decompositions of a smooth projective variety $X$ is strongly
constrained by the base locus of the canonical linear system. A main result of
this paper is that the straightforward generalization to the relative settings
holds as follows.

######  Theorem 1.6 ($=$Theorem 3.10).

Let $f\colon X\to Y$ be a proper morphism of finite dimensional noetherian
algebraic spaces such that $X$ is a regular connected scheme. Let

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle$ (1.2)

be an $f$-linear semiorthogonal decomposition. Then one of the followings
holds, where $\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ is the relative
canonical base locus of $X$ (see Definition 3.1).

  1. (A)

For any closed point $x\not\in\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ the
sheaf $\mathbf{k}(x)$ is contained in $\mathcal{A}$.

  2. (B)

For any closed point $x\not\in\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ the
sheaf $\mathbf{k}(x)$ is contained in $\mathcal{B}$.

Moreover, when Item A (resp. Item B) holds, then the support of any object of
$\mathcal{B}$ (resp. $\mathcal{A}$) is contained in the closed subset
$\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$.

The case where $Y=\operatorname{Spec}\mathbf{k}$ of Theorem 1.6 is exactly the
same as [2015arXiv150800682K, Theorem 1.2], where the relative canonical base
locus is nothing but the usual canonical base locus.

Note that Theorem 1.6 is only about those semiorthogonal decompositions of $X$
which are $f$-linear. However, if we are in the situation of Theorem 1.4, then
any semiorthogonal decomposition of $X$ is automatically $f$-linear. Since the
relative canonical base locus is smaller than the usual canonical base locus
by Lemma 3.5, and in some cases they do differ, we obtain stronger constrains
from Theorem 1.6 than [2015arXiv150800682K, Theorem 1.2]. See Example 3.12 for
the example which motivated this work.

When the variety of interest is of positive irregularity, then we obtain for
free a nontrivial morphism to an abelian variety; i.e., the Albanese morphism.
With this in mind, as an application of Theorem 1.6, we prove the following

######  Theorem 1.7 ($=$Theorem 4.1).

Let $X$ be a minimal surface with $H^{1}\left(X,\mathcal{O}_{X}\right)\neq 0$.
Then $\mathbf{D}(X)$ admits no non-trivial semiorthogonal decomposition.

The proof goes as follows (see Section 4 for details). The assumptions of
Theorem 1.7 imply that $X$ admits a morphism to a variety of positive
dimension such that any semiorthogonal decomposition of $\mathbf{D}(X)$ is
linear with respect to the morphism. Hence, as we explained just before
Theorem 1.7, it amounts to understanding the relative canonical base loci of
such morphisms. Thankfully, for surfaces Kazuhiro Konno established structure
theorems for the relative canonical base loci which are strong enough for our
purpose. Thus we can conclude the proof.

The strategy itself, however, applies to varieties of higher dimensions of
positive irregularity as well. Though we leave it for future, it would be
interesting to use the same strategy to obtain semiorthogonal
indecomposability in dimensions $\geq 3$.

In view of Theorem 1.7 and the other results for surfaces established in
[2015arXiv150800682K], it seems fair to make the following conjecture. This is
Hypothesis 1.2 made precise in dimension $2$.

######  Conjecture 1.8.

A minimal surface $X$ is not semiorthogonally indecomposable if and only if
$\mathcal{O}_{X}$ is acyclic; namely, $H^{i}\left(X,\mathcal{O}_{X}\right)=0$
for $i=1,2$.

In view of Example 1.1, it seems natural to generalize Conjecture 1.8 to the
following hypothesis, which partly refines Hypothesis 1.3 in dimension $2$.

######  Hypothesis 1.9.

Let $X$ be a surface of non-negative Kodaira dimension. Then any
semiorthogonal decomposition of $\mathbf{D}(X)$, if any, is induced by
exceptional collections.

Corollary 1.10 below summarizes the remaining cases of Conjecture 1.8.

######  Corollary 1.10 ($=$Corollary 4.7).

Conjecture 1.8 is true except possibly for those $X$ which satisfy all of the
following conditions.

  * •

$X$ is a minimal surface of general type.

  * •

$H^{2}(X,\mathcal{O}_{X})\neq 0$ ($\iff H^{0}(X,\omega_{X})\neq 0$).

  * •

$H^{1}(X,\mathcal{O}_{X})=0$.

  * •

There exists a 1-dimensional component of
$\operatorname{\mathsf{Bs}}\left(\omega_{X}\right)$ which can not be
contracted to a point in the category of algebraic spaces.

In order to tackle $X$ as in Corollary 1.10, we need a better understanding of
the (absolute) canonical base locus $\operatorname{\mathsf{Bs}}(\omega_{X})$.
Unfortunately this does not seem to have been worked out in the literature, as
it is mentioned as a very interesting question in the end of [MR2654305,
Introduction].

### Relations to other works

This paper is a continuation of the series of works [MR2838062,
2015arXiv150800682K], which aims at Hypothesis 1.3. This paper came as a
result of an attempt to interpret from a natural point of view the work
[2021arXiv211006795C, Theorem 1.5], which identifies the paracanonical base
locus as the relative base locus of the canonical line bundle with respect to
the Albanese morphism, where the importance of the former in the context of
semiorthogonal decompositions was first discovered in [2021arXiv210709564L,
Theorem 1.4]. See Example 3.12 for details.

Theorem 4.1 is proved for some examples of surfaces of general type in
[2021arXiv210709564L, Theorem 4.16], and Caucci informed the author that they
also obtained Theorem 4.1 for surfaces of general type of maximal Albanese
dimension (private communication).

In Section 5 we give an alternative proof of Theorem 3.10 using the Hochschild
homology, which is a relative version of what is discussed in
[2020arXiv200607643P, Section 5]. As is the case for the results of
[2020arXiv200607643P, Section 5], actually we get slightly finer information
than Theorem 3.10. Though we do not pursuit it in this paper, it is
interesting to ask if we can obtain results about Hypothesis 1.3 analogous to
[2020arXiv200607643P, Proposition 5.7] from it.

During the preparation of this paper, an independent work
[2023arXiv230106818L] appeared on the arXiv. Their main result
[2023arXiv230106818L, Theorem 2.3] is more general than Theorem 1.6, in that
they do not assume that $X$ is nonsingular. See Remark 3.11 for detailed
comparison.

### Acknowledgements

The author is indebted to Ana Cristina López Martín and Fernando Sancho de
Salas for their various comments on this paper, especially as to dualizing
complexes, and valuable exchanges, Alexander Kuznetsov for various useful
comments, and Federico Caucci for many comments and corrections. He also
thanks Kotaro Kawatani, Kazuhiro Konno, and Xun Lin for useful discussions.

The author was partially supported by JSPS Grants-in-Aid for Scientific
Research (18H01120, 19KK0348, 20H01797, 20H01794, 21H04994).

##  2\. Basics of relative semiorthogonal decomposition

The aim of this section is to generalize the rudiments of semiorthogonal
decompositions to the relative setting as much as possible.

######  Definition 2.1.

Let $X$ be a regular noetherian scheme of finite dimension. We let
$\mathbf{D}(X)$ denote the bounded derived category of coherent sheaves on
$X$.

By the theorem of Serre, the natural embedding
${\mathrm{perf}}(X)\hookrightarrow\mathbf{D}(X)$ is full for $X$ as in
Definition 2.1.

We use the Grothendieck duality in an essential way:

######  Theorem 2.2 ($=$[MR0222093, Chapter VII Corollary 3.4 (c)]).

Let

$\displaystyle f\colon X\to Y$ (2.1)

be a proper morphism of finite dimensional noetherian schemes. Then there is a
functor

$\displaystyle
f^{!}\colon\mathbf{D}^{+}_{\operatorname{coh}}(Y)\to\mathbf{D}^{+}_{\operatorname{coh}}(X)$
(2.2)

which is right adjoint to the derived pushforward functor

$\displaystyle
f_{\ast}\colon\mathbf{D}^{+}_{\operatorname{Qcoh}}(X)\to\mathbf{D}^{+}_{\operatorname{Qcoh}}(Y)$
(2.3)

in the sense that there exists a bi-functorial isomorphism in
$\mathbf{D}^{+}_{\operatorname{Qcoh}}(Y)$ as follows for any
$\mathcal{F}\in\mathbf{D}^{-}_{\operatorname{Qcoh}}(X)$ and
$\mathcal{G}\in\mathbf{D}^{+}_{\operatorname{coh}}(Y)$.

$\displaystyle
f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(\mathcal{F},f^{!}\mathcal{G})\to\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{Y}(f_{\ast}\mathcal{F},\mathcal{G})$
(2.4)

The following special case of Theorem 2.2 is particularly relevant to us (see
Definitions 3.8 and 3.9 for the notion of dualizing complexes).

######  Corollary 2.3.

In the situation of Theorem 2.2, there is a natural isomorphism as follows for
any $F,G\in{\mathrm{perf}}(X)$, where
$\omega^{\bullet}_{X},\omega^{\bullet}_{Y}$ are dualizing complexes of $X$ and
$Y$, respectively, satisfying
$f^{!}\omega^{\bullet}_{Y}\simeq\omega^{\bullet}_{X}$.

$\displaystyle
f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(G,F\otimes\omega^{\bullet}_{X})\simeq\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{Y}(f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(F,G),\omega^{\bullet}_{Y})$
(2.5)

###### Proof.

Apply Theorem 2.2 to $\mathcal{F}=F^{\vee}\otimes G$ and
$\mathcal{G}=\omega^{\bullet}_{Y}$. ∎

######  Definition 2.4.

Let $f\colon X\to Y$ be a projective morphism of noetherian algebraic spaces
such that $X$ is regular and of finite dimension. An _$f$ -linear
semiorthogonal decomposition_ (alternatively _semiorthogonal decomposition of
$\mathbf{D}(X)$ relative to $f$ _ or _semiorthogonal decomposition of
$\mathbf{D}(X)$ over $Y$ _) of $\mathbf{D}(X)$ (of length $2$) is a pair of
(full) triangulated subcategories
$\mathcal{A},\mathcal{B}\subseteq\mathbf{D}(X)$ such that

  * •

$\mathbf{D}(X)$ coincides with the smallest triangulated subcategory
containing both $\mathcal{A}$ and $\mathcal{B}$, and

  * •

The pair is semiorthogonal over $Y$, in the sense that

$\displaystyle f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(b,a)=0$
(2.6)

for any $a\in\mathcal{A}$ and $b\in\mathcal{B}$.

We use the shorthand notation

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle_{f}$ (2.7)

to indicate the situation.

An $f$-linear semiorthogonal decomposition for the structure morphism $f\colon
X\to\operatorname{Spec}\mathbb{Z}$ is simply called _semiorthogonal
decomposition_ and denoted by

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle.$ (2.8)

######  Remark 2.5.

In the situation of Definition 2.4, let $g\colon Y\to Z$ be a projective
morphism of noetherian algebraic spaces and put $h=g\circ f$. Then an
$f$-linear semiorthogonal decomposition of $X$ is automatically an $h$-linear
semiorthogonal decomposition. In particular, any $f$-linear semiorthogonal
decomposition is a semiorthogonal decomposition.

######  Definition 2.6.

In the situation of Definition 2.4, an $f$-linear semiorthogonal decomposition
(2.7) is said to be an  _orthogonal decomposition_ if it also satisfies

$\displaystyle\mathbf{D}(X)=\langle\mathcal{B},\mathcal{A}\rangle_{f}.$ (2.9)

We indicate the situation by

$\displaystyle\mathbf{D}(X)=\mathcal{A}\perp_{f}\mathcal{B}.$ (2.10)

######  Definition 2.7.

In the situation Definition 2.4 we say that _$X$ is semiorthogonally
indecomposable over $Y$ (SI over $Y$)_ or _$f$ is semiorthogonally
indecomposable ($f$ is SI)_ if any $f$-linear semiorthogonal decomposition of
$X$ is an orthogonal decomposition. We say $X$ is semiorthogonally
indecomposable (SI) if it is so over $\operatorname{Spec}\mathbb{Z}$.

######  Remark 2.8.

If $X$ is connected, then $X$ being SI implies that any semiorthogonal
decomposition of $\mathbf{D}(X)$ is trivial; i.e., there is no semiorthogonal
decomposition of $\mathbf{D}(X)$ other than $\langle 0,\mathbf{D}(X)\rangle$
and $\langle\mathbf{D}(X),0\rangle$.

As we pointed out in Remark 2.5, $f$-linear semiorthogonal decompositions are
semiorthogonal decompositions satisfying an additional condition. Lemma 2.9
below, which is a slight generalization of [MR2801403, Lemma 2.7],
characterizes the condition in a slightly different way. This guarantees that
Definition 2.4 is equivalent to the original definition of $f$-linearity given
in [MR2238172, MR2801403].

######  Lemma 2.9.

Suppose that $Y$ is a quasi-separated and quasi-compact scheme. Then a
semiorthogonal decomposition
$\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle$ is $f$-linear if and
only if $\mathcal{A}$ (resp. $\mathcal{B}$) is stable under the action of
${\mathrm{perf}}(Y)$, in the sense that $a\otimes f^{\ast}y\in\mathcal{A}$ for
all $a\in\mathcal{A}$ and $y\in{\mathrm{perf}}(Y)$ (resp. $b\otimes
f^{\ast}y\in\mathcal{B}$ for all $b\in\mathcal{B}$ and
$y\in{\mathrm{perf}}(Y)$).

###### Proof.

If the semiorthogonal decomposition is $f$-linear, then

$\displaystyle
f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(b,a)\otimes y=0$ (2.11)

for all $a\in\mathcal{A},b\in\mathcal{B},y\in{\mathrm{perf}}(Y)$. Then
$\mathbb{R}\Gamma(Y,?)$ of the left hand side is isomorphic to
$\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}(b,a\otimes f^{\ast}y)$. Thus we
have confirmed that $a\otimes f^{\ast}y\in\mathcal{B}^{\perp}=\mathcal{A}$.
Conversely the vanishing of

$\displaystyle\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}(b,a\otimes
f^{\ast}y)\simeq\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{Y}(y^{\vee},f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(b,a))$
(2.12)

for all $y\in{\mathrm{perf}}(Y)$ implies the vanishing of
$f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(b,a)$, since
${(-)}^{\vee}$ is a (contravariant) autoequivalence of ${\mathrm{perf}}(Y)$
and ${\mathrm{perf}}(Y)$ generates $\mathbf{D}^{b}\operatorname{coh}Y$
([Bondal-van_den_Bergh, Theorem 3.1.1]). ∎

######  Definition 2.10 ($=$ the original version of [2020arXiv201112743P,
Definition 1.3]).

A noetherian scheme $Y$ is said to be _stably semiorthogonally indecomposable_
(_SSI_ for short) if any proper morphism $f\colon X\to Y$ from a regular
scheme $X$, any semiorthogonal decomposition of $\mathbf{D}(X)$ is $f$-linear.

In [2020arXiv201112743P] Pirozhkov proved that a scheme which admits a finite
morphism to an abelian variety is SSI. This paper heavily relies on this
result.

######  Remark 2.11.

In fact [2020arXiv201112743P, Definition 1.3] introduces a notion which is
stronger than SSI. It is called _NSSI_ (noncommutatively SSI), and Pirozhkov
proved that the aforementioned class of schemes belong to this category.
However, the notion of SSI is sufficient for the purpose of this paper.

######  Remark 2.12.

As shown in [2020arXiv201112743P, Lemma 2.3], if $Y$ is a smooth proper
variety and SSI, then it is SI.

######  Example 2.13.

A noetherian affine scheme $Y=\operatorname{Spec}R$ is SSI since
$R\in{\mathrm{perf}}(Y)$ is a classical generator (in the sense of [Bondal-
van_den_Bergh]).

The following remark is obvious but plays a central role in this paper.

######  Remark 2.14.

Let $f\colon X\to Y$ be as in Definition 2.4. Suppose that $Y$ is SSI and $f$
is SI. Then $X$ is SI.

##  3\. Relative semiorthogonal indecomposability

###  3.1. Relative base locus

######  Definition 3.1.

Let $f\colon X\to Y$ be a morphism of algebraic spaces, and let $L$ be an
invertible sheaf on $X$. The _base locus of $L$ relative to $f$_ or the
_(relative) base locus of $L$ over $Y$_ is the closed subspace
$\operatorname{\mathsf{Bs}}_{f}(L)\hookrightarrow X$ whose structure sheaf is
defined as follows.

$\displaystyle
f^{\ast}f_{\ast}L\otimes_{\mathcal{O}_{X}}L^{\vee}\to\mathcal{O}_{X}\to\mathcal{O}_{\operatorname{\mathsf{Bs}}_{f}(L)}\to
0$ (3.1)

We call $\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ the _(relative) canonical
base locus of $f$_ or the _canonical base locus of $X$ over $Y$ (relative to
$f$)_.

######  Remark 3.2.

If $Y$ is affine in Definition 3.1, then the first map of (3.1) is canonically
isomorphic to the map $H^{0}(X,L)\otimes L^{-1}\to\mathcal{O}_{X}$ and hence
$\operatorname{\mathsf{Bs}}_{f}(L)$ is nothing but the usual base locus
$\operatorname{\mathsf{Bs}}(L)$.

######  Lemma 3.3.

In the situation of Definition 3.1, the following equality of closed subspaces
of $X$ holds for any invertible sheaf $M$ on $Y$.

$\displaystyle\operatorname{\mathsf{Bs}}_{f}(L)=\operatorname{\mathsf{Bs}}_{f}(L\otimes
f^{\ast}M)$ (3.2)

###### Proof.

The projection formula

$\displaystyle f_{\ast}\left(L\otimes f^{\ast}M\right)\simeq f_{\ast}L\otimes
M$ (3.3)

implies

$\displaystyle f^{\ast}f_{\ast}\left(L\otimes
f^{\ast}M\right)\otimes\left(L\otimes
f^{\ast}M\right)^{-1}\stackrel{{\scriptstyle{\eqref{equation:projection
formula}}}}{{\simeq}}f^{\ast}f_{\ast}L\otimes f^{\ast}M\otimes\left(L\otimes
f^{\ast}M\right)^{-1}\simeq f^{\ast}f_{\ast}L\otimes L^{-1}.$ (3.4)

The assertion immediately follows from this. ∎

######  Lemma 3.4.

In the situation of Definition 3.1, let $y\colon Y^{\prime}\to Y$ be a flat
morphism and consider the following Cartesian diagram.

${X^{\prime}}$${X}$${Y^{\prime}}$${Y}$$\scriptstyle{{\widetilde{y}}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{y}$
(3.5)

Then

$\displaystyle\operatorname{\mathsf{Bs}}_{f^{\prime}}({\widetilde{y}}^{\ast}L)={\widetilde{y}}^{-1}\left(\operatorname{\mathsf{Bs}}_{f}(L)\right).$
(3.6)

###### Proof.

It follows from the flatness of $y$ that the base change map

$\displaystyle y^{\ast}f_{\ast}L\to f^{\prime}_{\ast}{\widetilde{y}}^{\ast}L$
(3.7)

is an isomorphism ([Hartshorne, Chapter III Proposition 9.3]). The assertion
follows from this. ∎

######  Lemma 3.5.

In the situation of Definition 3.1, suppose we have another morphism of
algebraic spaces $g\colon Y\to Z$. Then there is a canonical inclusion of
closed subspaces of $X$ as follows.

$\displaystyle\operatorname{\mathsf{Bs}}_{g\circ
f}(L)\hookrightarrow\operatorname{\mathsf{Bs}}_{f}(L)$ (3.8)

Moreover, (3.8) is an equality if $g$ is an affine morphism.

###### Proof.

Note that there is a following commutative diagram as follows, where each
arrow is the counit of the adjoint pair of functors next to it.

${(g\circ f)^{\ast}(g\circ
f)_{\ast}L}$${f^{\ast}g^{\ast}g_{\ast}f_{\ast}L}$${f^{\ast}f_{\ast}L}$${L}$$\scriptstyle{g^{\ast}\dashv
g_{\ast}}$$\scriptstyle{\left(g\circ f\right)^{\ast}\dashv\left(g\circ
f\right)_{\ast}}$$\scriptstyle{f^{\ast}\dashv f_{\ast}}$ (3.9)

The first assertion immediately follow from this. The second assertion follows
from that the horizontal arrow is an epimorphism if $g$ is affine. ∎

######  Example 3.6.

Since $\operatorname{\mathsf{Bs}}_{\operatorname{id}}(L)=\emptyset$ for the
obvious reason, Lemma 3.5 implies that
$\operatorname{\mathsf{Bs}}_{g}(L)=\emptyset$ for any affine (in particular
finite) morphism $g$.

In usual situation of birational geometry, there is an alternative description
of the relative base locus as follows.

######  Lemma 3.7.

In the situation of Definition 3.1, suppose further that $f$ is projective and
that $Y$ is a quasi-projective variety. Then there is an alternative
description for the relative base locus as follows.

$\displaystyle\mathcal{O}_{\operatorname{\mathsf{Bs}}_{f}(L)}\simeq\operatorname{coker}\left(\bigoplus_{\begin{subarray}{c}M\in\operatorname{Pic}Y\end{subarray}}H^{0}(X,L\otimes
f^{\ast}M)\otimes_{\mathbf{k}}{\left(L\otimes
f^{\ast}M\right)}^{-1}\to\mathcal{O}_{X}\right)$ (3.10)

Moreover, we have the following equality for any sufficiently ample invertible
sheaf $M$ on $Y$.

$\displaystyle\operatorname{\mathsf{Bs}}_{f}(L)=\operatorname{\mathsf{Bs}}(L\otimes
f^{\ast}M)$ (3.11)

###### Proof.

To ease notation, let $B\hookrightarrow X$ be the closed subscheme of $X$
defined by the right hand side of (3.10) and let $g\colon
Y\to\operatorname{Spec}\mathbf{k}$ be the structure morphism of $Y$. There is
a following commutative diagram for any invertible sheaf
$M\in\operatorname{Pic}Y$.

${f^{\ast}f_{\ast}L\otimes L^{-1}}$${f^{\ast}f_{\ast}\left(L\otimes
f^{\ast}M\right)\otimes\left(L\otimes
f^{\ast}M\right)^{-1}}$${\mathcal{O}_{X}}$${H^{0}(X,L\otimes
f^{\ast}M)\otimes_{\mathbf{k}}\left(L\otimes
f^{\ast}M\right)^{-1}}$$\scriptstyle{\simeq}$$\scriptstyle{\eqref{equation:projection
formula}}$$\scriptstyle{g^{\ast}\dashv g_{\ast}}$ (3.12)

Thus we obtain the following commutative diagram, which implies that there is
a natural morphism from the right hand side to the left hand side of (3.10).

${f^{\ast}f_{\ast}L\otimes
L^{-1}}$${\bigoplus_{\begin{subarray}{c}M\in\operatorname{Pic}Y\end{subarray}}H^{0}(X,L\otimes
f^{\ast}M)\otimes_{\mathbf{k}}\left(L\otimes
f^{\ast}M\right)^{-1}}$${\mathcal{O}_{X}}$ (3.13)

On the other hand, let $M$ be sufficiently ample so that $f_{\ast}L\otimes M$
is a globally generated coherent sheaf on $Y$. Then we obtain the epimorphism

$\displaystyle H^{0}(Y,f_{\ast}L\otimes
M)\otimes_{\mathbf{k}}M^{-1}\twoheadrightarrow f_{\ast}L,$ (3.14)

from which we further obtain the epimorphism

$\displaystyle H^{0}(Y,f_{\ast}L\otimes M)\otimes_{\mathbf{k}}{\left(L\otimes
f^{\ast}M\right)}^{-1}\twoheadrightarrow f^{\ast}f_{\ast}L\otimes L^{-1}.$
(3.15)

Thus we have confirmed the surjectivity of the vertical arrow of (3.13), from
which the assertion immediately follows. ∎

###  3.2. The relative version of [2015arXiv150800682K]

In this section we establish the relative version of [2015arXiv150800682K,
Theorem 3.1], which constrains semiorthogonal decompositions in terms of the
base locus of the canonical linear system. We prove that the relative base
locus of the canonical bundle constrains relative semiorthogonal
decompositions.

Let us first clarify what we mean by dualizing complexes and their relative
base loci.

######  Definition 3.8.

Let $f\colon X\to Y$ be a proper morphism of finite dimensional connected
noetherian schemes such that $X$ is regular and $Y$ admits a dualizing complex
$\omega_{Y}^{\bullet}\in\mathbf{D}^{\mathrm{b}}\operatorname{coh}Y$ (see, say,
[stacks-project, Tag 0A87] for the definition of dualizing complexes). Choose
$\omega_{Y}^{\bullet}$ such that

$\displaystyle\omega_{X}^{\bullet}\coloneqq f^{!}\omega_{Y}^{\bullet}$ (3.16)

is an invertible sheaf ([stacks-project, Tag 0AWV]). Then we call
$\omega_{X}\coloneqq\mathcal{H}^{0}\left(\omega_{X}^{\bullet}\right)$ _the_
dualizing sheaf of $X$. We let $\operatorname{\mathsf{Bs}}_{f}\omega_{X}$
denote the relative base locus of $\omega_{X}$.

######  Remark 3.9.

Though the dualizing sheaf $\omega_{X}$ defined in Definition 3.8 is unique
only up to tensoring by invertible sheaves on $Y$ (see [stacks-project, Tag
0ATP]), the relative base locus
$\operatorname{\mathsf{Bs}}_{f}\omega_{X}\hookrightarrow X$ is well-defined by
Lemma 3.3.

If $X$ is smooth and proper over a field $\mathbf{k}$, then we can choose

$\displaystyle\omega_{Y}^{\bullet}\coloneqq\left(Y\to\operatorname{Spec}\mathbf{k}\right)^{!}\mathbf{k}[-\dim
X]$ (3.17)

so that $\omega_{X}\simeq\Omega^{\dim X}_{X/\mathbf{k}}$. If moreover
$\mathbf{k}$ is of characteristic $0$, so that the resolution of singularity
is guaranteed, then we can drop the properness assumption (the author owes the
following argument to Alexander Kuznetsov). Indeed, we can compactify $f$ into
a morphism between proper schemes over $\mathbf{k}$ such that the domain of
the morphism is smooth over $\mathbf{k}$. Then we can reduce the problem to
the case where $X$ is smooth and proper over $\mathbf{k}$ by the base-change
theorem for upper shriek [stacks-project, Tag 0E9U] since open immersions are
flat.

Finally, note that the relative dualizing complex
$\omega^{\bullet}_{X/\mathbf{k}}$ is different from a dualizing complex
$\omega^{\bullet}_{X}$ unless $X$ is proper over $\mathbf{k}$.

######  Theorem 3.10.

Let $f\colon X\to Y$ be a proper morphism of finite dimensional noetherian
algebraic spaces such that $X$ is a regular connected scheme. Let $\omega_{X}$
be the dualizing sheaf of $X$ as in Definition 3.8 (see also Remark 3.9) and

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle$ (3.18)

be an $f$-linear semiorthogonal decomposition. Then one of the followings
holds.

  1. (A)

For any closed point $x\not\in\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ the
sheaf $\mathbf{k}(x)$ is contained in $\mathcal{A}$.

  2. (B)

For any closed point $x\not\in\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ the
sheaf $\mathbf{k}(x)$ is contained in $\mathcal{B}$.

Moreover, when Item A (resp. Item B) holds, then the support of any object of
$\mathcal{B}$ (resp. $\mathcal{A}$) is contained in the closed subset
$\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$.

###### Proof.

We follow the strategy of the proof of [2015arXiv150800682K, Theorem 3.1].
Take a closed point $x\not\in\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$. We
first prove that either $\mathbf{k}(x)\in\mathcal{A}$ or
$\mathbf{k}(x)\in\mathcal{B}$ holds.

Take an étale morphism

$\displaystyle y\colon Y^{\prime}\to Y$ (3.19)

from a noetherian affine scheme $Y^{\prime}$ and consider the Cartesian
diagram as in (3.5). Choose $y$ so that

  * •

$f(x)\in\operatorname{Im}(y)$

  * •

$|y^{-1}\left(f(x)\right)|=1$.

Let $x^{\prime}\in X^{\prime}$ be the unique point of $X^{\prime}$ such that
${\widetilde{y}}(x^{\prime})=x$.

The flatness of $y$, the isomorphism
${\widetilde{y}}^{\ast}\omega_{X}\simeq\omega_{X^{\prime}}$ (by the base-
change theorem [stacks-project, Tag 0E9U] as $y$ is étale), and Lemma 3.4
imply that
$x^{\prime}\not\in\operatorname{\mathsf{Bs}}_{f^{\prime}}(\omega_{X^{\prime}})\subseteq
X^{\prime}$. Then by Remark 3.2, there is a section $s\in
H^{0}(X^{\prime},\omega_{X^{\prime}})$ such that $s(x)\neq 0$.

Consider the decomposition of $\mathbf{k}(x)$ with respect to the $f$-linear
semiorthogonal decomposition (3.18):

$\displaystyle
b\xrightarrow[]{\xi}\mathbf{k}(x)\xrightarrow[]{\eta}a\xrightarrow[]{\zeta}b[1],$
(3.20)

where $a\in\mathcal{A}$ and $b\in\mathcal{B}$. Apply the functor
${\widetilde{y}}^{\ast}$ to (3.20) to obtain the following distinguished
triangle, where
$b^{\prime}\coloneqq{\widetilde{y}}^{\ast}b,\xi^{\prime}\coloneqq{\widetilde{y}}^{\ast}\xi$
and so on.

$\displaystyle
b^{\prime}\xrightarrow[]{\xi^{\prime}}{\widetilde{y}}^{\ast}\mathbf{k}(x)=\mathbf{k}(x^{\prime})\xrightarrow[]{\eta^{\prime}}a^{\prime}\xrightarrow[]{\zeta^{\prime}}b^{\prime}[1],$
(3.21)

Then it follows that

$\displaystyle\left[a^{\prime}\xrightarrow[]{\zeta^{\prime}}b^{\prime}[1]\xrightarrow[]{\otimes
s}b^{\prime}\otimes\omega_{X^{\prime}}[1]\right]$
$\displaystyle\in\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X^{\prime}}\left(a^{\prime},b^{\prime}\otimes\omega_{X^{\prime}}[1]\right)$
(3.22) $\displaystyle\stackrel{{\scriptstyle\textrm{ $Y^{\prime}$ is affine
}}}{{=}}f^{\prime}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X^{\prime}}\left(a^{\prime},b^{\prime}\otimes\omega_{X^{\prime}}[1]\right)$
(3.23)
$\displaystyle\stackrel{{\scriptstyle\text{\lx@cref{creftype~refnum}{corollary:Grothendieck
duality}}}}{{\simeq}}{\left(f^{\prime}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X^{\prime}}\left(b^{\prime}\otimes\omega_{X^{\prime}}[1],a^{\prime}\otimes\omega_{X^{\prime}}^{\bullet}\right)\right)}^{\vee}$
(3.24) $\displaystyle\stackrel{{\scriptstyle\textrm{
\lx@cref{creftype~refnum}{definition:dualizing complex}
}}}{{\simeq}}{\left(f^{\prime}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X^{\prime}}\left(b^{\prime},a^{\prime}\right)\right)}^{\vee}[1],$
(3.25)

where

$\displaystyle{\left(?\right)}^{\vee}\coloneqq\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{Y^{\prime}}(?,\omega_{Y^{\prime}}^{\bullet})\colon{\mathbf{D}(Y^{\prime})}^{\mathrm{op}}\xrightarrow{\sim}\mathbf{D}(Y^{\prime}).$
(3.26)

Then (3.25) $=0$ as

$\displaystyle
f^{\prime}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X^{\prime}}\left(b^{\prime},a^{\prime}\right)$
$\displaystyle=f^{\prime}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X^{\prime}}\left({\widetilde{y}}^{\ast}b,{\widetilde{y}}^{\ast}a\right)$
(3.27) $\displaystyle\simeq
f^{\prime}_{\ast}{\widetilde{y}}^{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(b,a\right)$
(3.28)
$\displaystyle\stackrel{{\scriptstyle\textrm{\cite[cite]{[\@@bibref{}{Hartshorne}{}{},
Chapter~{}III~{}Proposition~{}9.3]}}}}{{\simeq}}y^{\ast}f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(b,a\right)$
(3.29) $\displaystyle=y^{\ast}0$ (3.30) $\displaystyle=0.$ (3.31)

On the other hand, multiplication by the section $s$ is an isomorphism on its
non-vanishing locus $U\coloneqq X^{\prime}_{s}\subseteq X^{\prime}$. Thus we
conclude that

$\displaystyle\zeta|_{U}=0.$ (3.32)

It follows from this and (3.20) that either
$(a^{\prime}|_{U},b^{\prime}|_{U})\simeq(0,\mathbf{k}(x))$ or
$(\mathbf{k}(x),0)$. This implies that either $a$ or $b$ is $0$ on an open
neighborhood of $x$ in $X$. If the former is the case, then it implies that
$\eta=0$, so that

$\displaystyle b\simeq\mathbf{k}(x)\oplus a[-1].$ (3.33)

As $\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}\left(b,a[-1]\right)=0$ by the
semiorthogonality, this implies that $a=0$ and $b\simeq\mathbf{k}(x)$.
Similarly, if the latter is the case, then it follows that $b=0$ and
$a\simeq\mathbf{k}(x)$.

Next, the connectedness of $X$ implies that it is impossible to have two
closed points $x_{1}\neq x_{2}\in X$ such that
$\mathbf{k}(x_{1})\in\mathcal{A}$ and $\mathbf{k}(x_{2})\in\mathcal{B}$.
Indeed, if this happens, then it follows that the support of objects of
$\mathcal{A}$ and $\mathcal{B}$ are closed subsets of $X$ which do not contain
the generic point of $X$. On the other hand, there is a triangle

$\displaystyle b\to\mathcal{O}_{X}\to a\to b[1]$ (3.34)

obtained from (3.18). Then

$\displaystyle
X=\operatorname{Supp}(\mathcal{O}_{X})\subseteq\operatorname{Supp}(a)\cup\operatorname{Supp}(b)\neq
X,$ (3.35)

a contradiction. Thus we obtain the first assertion.

Now assume that the case Item A holds, and take an arbitrary object
$b\in\mathcal{B}$. Suppose for a contradiction that there exits a closed point
$x\in\operatorname{Supp}(b)\setminus\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$.
Let

$\displaystyle j\colon\left\\{x\right\\}\hookrightarrow X$ (3.36)

be the embedding of the point. The derived version of the Nakayama’s lemma
(see, say, [2002.03303v1, Lemma 2.3]) implies that $j^{\ast}b\neq 0$, which in
turn implies that there exits a non-trivial morphism

$\displaystyle b\to j_{\ast}j^{\ast}b.$ (3.37)

Hence we see that

$\displaystyle
f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(b,j_{\ast}j^{\ast}b\right)\neq
0.$ (3.38)

On the other hand, the assumption Item A implies $\mathbf{k}(x)\in\mathcal{A}$
and hence $j_{\ast}j^{\ast}b\in\mathcal{A}$. Thus we obtain a contradiction.

Finally, if the case Item B holds, then we can similarly prove the assertion
by contradiction by using Corollary 2.3. ∎

######  Remark 3.11.

Theorem 3.10 is also shown in [2023arXiv230106818L, Theorem 2.3] in a greater
generality. Here we briefly compare this paper with [2023arXiv230106818L].

First of all, the definition of the relative canonical base locus given in
[2023arXiv230106818L, Definition 1.2] (_base locus of $f$_ in their
terminology and denoted by $\operatorname{\mathsf{Bs}}|f|$) is different from
the one we gave in Definition 3.1. They coincide when $Y$ is the spectrum of a
field (see the end of [2023arXiv230106818L, Section 1.2]). They also
introduced a slightly modified version
$\operatorname{\mathsf{Bs}}^{\prime}|f|$ with the following properties (see
[2023arXiv230106818L, Example 1.3 (2)] for details).

  * •

$\operatorname{\mathsf{Bs}}^{\prime}|f|\subseteq\operatorname{\mathsf{Bs}}|f|$,
and they coincide if $f$ is of finite homological dimension.

  * •

$\operatorname{\mathsf{Bs}}^{\prime}|f|$ is local on the base; i.e., for any
open embedding $U\hookrightarrow Y$ one has
$\operatorname{\mathsf{Bs}}^{\prime}|f|\cap
f^{-1}(U)=\operatorname{\mathsf{Bs}}^{\prime}|f|_{U}|$.

  * •

The main results of [2023arXiv230106818L] are valid if one replaces
$\operatorname{\mathsf{Bs}}|f|$ with $\operatorname{\mathsf{Bs}}^{\prime}|f|$.

Since $\operatorname{\mathsf{Bs}}_{f}\omega_{X}$ is also local on the base,
and the inclusion

$\displaystyle\operatorname{\mathsf{Bs}}^{\prime}|f|\subseteq\operatorname{\mathsf{Bs}}_{f}\omega_{X}$
(3.39)

is obvious when $Y$ is affine, it follows that the inclusion (3.39) holds true
in general. Hence [2023arXiv230106818L, Theorem 2.3] implies Theorem 3.10. It
is not clear to the author whether (3.39) is always an equality or not.

Though our base locus is (at least a priori) bigger than theirs, it is simpler
and more natural for birational geometry; these features are exploited in
Section 4. On the other hand [2023arXiv230106818L, Definition 1.2] works for
singular $X$ as well, for which the dualizing complex is not necessarily a
shift of an invertible sheaf. They prove [2023arXiv230106818L, Theorem 2.3] by
introducing a well-tailored spanning class (see also [MR4395364] for the case
of Cohen-Macaulay varieties). It is not clear how to generalize the arguments
of this paper to singular $X$.

From [2023arXiv230106818L, Theorem 2.3] they obtain a couple of
indecomposability results, which, for (proper morphisms from) regular schemes,
we can also obtain from Theorem 3.10.

###  3.3. Applications to semiorthogonal indecomposability

Example 3.12 below motivated this work.

######  Example 3.12.

The paracanonical base locus $\operatorname{\mathsf{PBs}}(X)$ of $X$ is
defined as follows ([2021arXiv210709564L, Corollary 3.5]).

$\displaystyle\mathcal{O}_{\operatorname{\mathsf{PBs}}(X)}\simeq\operatorname{coker}\left(\bigoplus_{\begin{subarray}{c}M^{\prime}\in\operatorname{Pic}^{0}X\end{subarray}}H^{0}(X,\omega_{X}\otimes
M^{\prime})\otimes_{\mathbf{k}}{\left(\omega_{X}\otimes
M^{\prime}\right)}^{-1}\to\mathcal{O}_{X}\right)$ (3.40)

Put $f=\operatorname{alb}_{X}\colon X\to Y=\operatorname{Alb}_{X}$, the
albanese morphism of $X$. Then the natural isomorphism
$\operatorname{Pic}^{0}(Y)=\operatorname{Pic}^{0}(\operatorname{Alb}_{X})\simeq\operatorname{Pic}^{0}(X)$
and Lemma 3.7 imply the obvious quotient map
$\mathcal{O}_{\operatorname{\mathsf{PBs}}(X)}\twoheadrightarrow\mathcal{O}_{\operatorname{\mathsf{Bs}}_{f}(\omega_{X})}$;
i.e., a closed immersion
$\operatorname{\mathsf{Bs}}_{f}(\omega_{X})\hookrightarrow\operatorname{\mathsf{PBs}}(X)$.
It is shown in [2021arXiv211006795C, Theorem 1.5] that this actually is an
isomorphism.

In [2021arXiv210709564L, Theorem 4.5] it is shown that the paracanonical base
locus is empty for the $k$-th symmetric product of a smooth projective curve
of genus $g\geq 2$ for $k\leq g-1$. On the contrary, the canonical base locus
of such a variety is not necessarily empty and hence we can not conclude the
semiorthogonal indecomposability by using [2015arXiv150800682K, Theorem 1.2].

Xun Lin proved in [2021arXiv210709564L, Theorem 1.4] that one can replace the
canonical base locus in [2015arXiv150800682K, Theorem 1.2] with the
paracanonical base locus, by bootstrapping the arguments of
[2015arXiv150800682K]. From our point of view, the same result is obtained by
applying Theorem 3.10 to the Albanese morphism and then by Theorem 1.4. In any
case, this implies the semiorthogonal indecomposability of aforementioned
varieties by the aforementioned emptiness.

It is worth noting that the technique of [2021arXiv210709564L] applies to
singular $X$ as well, whereas $\operatorname{Pic}^{0}(X)$ for such $X$ may not
be an abelian variety and hence the arguments of this paper do not apply. In
this sense the former is more flexible than the latter. In fact, in the
author’s appendix to [2023arXiv230106818L], he uses the technique to prove
that semistable curves are semiorthogonally indecomposable.

Even if the relative canonical base locus is not empty, in good cases we can
still conclude the relative semiorthogonal indecomposability.

######  Corollary 3.13.

In the situation of Theorem 3.10, suppose that for each connected component of
$\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$ there exists a Zariski open
neighborhood in $X$ on which $\omega_{X}$ is isomorphic to the pullback of an
invertible sheaf on $Y$ (i.e., $f$-trivial). Then $X$ admits no $f$-linear
semiorthogonal decomposition.

###### Proof.

It is enough to show that any $f$-linear semiorthogonal decomposition
$\langle\mathcal{A},\mathcal{B}\rangle$ is an orthogonal decomposition. Take
$a\in\mathcal{A}$ and $b\in\mathcal{B}$. Suppose that Theorem 3.10 Item A
holds, so that $a$ is supported in
$\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$. Then it follows from the
assumption of Corollary 3.13 and the $f$-linearity that

$\displaystyle a\otimes\omega_{X}^{\bullet}\in\mathcal{A}.$ (3.41)

Hence

$\displaystyle
f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(a,b)\stackrel{{\scriptstyle\text{~{}\lx@cref{creftype~refnum}{corollary:Grothendieck
duality}
}}}{{\simeq}}{f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(b,a\otimes\omega_{X}^{\bullet}\right)}^{\vee}=0.$
(3.42)

The last equality follows from the vanishing
$f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(b,a\otimes\omega_{X}^{\bullet})=0$,
which follows from (3.41). Thus we have confirmed that the decomposition is
actually orthogonal. If we assume Theorem 3.10 Item B instead, then similarly
it follows that $b\otimes\omega_{X}^{-1}\in\mathcal{B}$ and one can confirm
$f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}(a,b)=0$ by similar
arguments as above. ∎

######  Corollary 3.14.

Let $f\colon X\to Y$ be a morphism of smooth projective varieties such that
$Y$ is SSI. If moreover

$\displaystyle|\operatorname{\mathsf{Bs}}_{f}(\omega_{X})|<\infty,$ (3.43)

then $X$ admits no semiorthogonal decomposition.

###### Proof.

Immediately follows from Corollary 3.13 and Remark 2.14. ∎

######  Lemma 3.15.

In the situation of Theorem 3.10, suppose further that there exists a closed
subset $Z\hookrightarrow X$ such that

$\displaystyle\dim f\left(Z\right)\leq
0\left(\iff\left|f\left(Z\right)\right|<\infty\right).$ (3.44)

Then any SOD

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle$ (3.45)

such that

$\displaystyle\operatorname{Supp}\left(a\right)\subseteq Z$ (3.46)

for any $a\in\mathcal{A}$ is $f$-linear.

###### Proof.

The assertion is trivial if $Y$ is a scheme, but we find the assertion for
algebraic spaces not completely trivial. As we could not find a proof in the
literature, we include a proof below.

Take objects $a\in\mathcal{A}$ and $b\in\mathcal{B}$. By [MR2434186, Lemma
7.40], for some $m>0$ there exists a bounded complex of coherent sheaves
${\overline{a}}$ on $mZ$ such that $a\simeq\iota_{\ast}{\overline{a}}$, where
$\iota\colon mZ\hookrightarrow X$ is the immersion morphism. Then we have

$\displaystyle
f_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{X}\left(b,a\right)\simeq
f_{\ast}\iota_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{mZ}\left(\iota^{\ast}b,{\overline{a}}\right).$
(3.47)

On the other hand, by the assumption there exists a closed immersion from a
$0$-dimensional algebraic space $j\colon W\hookrightarrow Y$ such that
$f\circ\iota$ decomposes into a morphism $g\colon mZ\to W$ and $j$, so that

$\displaystyle
f_{\ast}\iota_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{mZ}\left(\iota^{\ast}b,{\overline{a}}\right)\simeq
j_{\ast}g_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{mZ}\left(\iota^{\ast}b,{\overline{a}}\right).$
(3.48)

As is known well, since $\dim W=0$, $W$ is indeed an affine scheme. Hence

$\displaystyle
g_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{mZ}\left(\iota^{\ast}b,{\overline{a}}\right)=\operatorname{Hom}_{mZ}\left(\iota^{\ast}b,{\overline{a}}\right)\simeq\operatorname{Hom}_{X}\left(b,\iota_{\ast}{\overline{a}}=a\right)=0.$
(3.49)

This concludes the proof. ∎

##  4\. Semiorthogonal indecomposability of irregular minimal surfaces

Throughout Section 4 we work over an algebraically closed field $\mathbf{k}$
of an arbitrary characteristic.

######  Theorem 4.1.

Let $X$ be a minimal surface with $H^{1}\left(X,\mathcal{O}_{X}\right)\neq 0$.
Then $\mathbf{D}(X)$ admits no non-trivial semiorthogonal decompositions.

In the proof of Theorem 4.1, we use the following results on relative
canonical base locus due to Kazuhiro Konno.

######  Theorem 4.2 ($=$[MR2481686, Theorem 2.4]).

Let $f\colon X\to Y$ be a birational projective morphism of normal algebraic
spaces of dimension $2$ such that $X$ is a nonsingular scheme and $\omega_{X}$
is $f$-nef. Then there exits a factorization

${X}$${X^{\prime}}$${Y}$$\scriptstyle{g}$$\scriptstyle{f}$ (4.1)

such that $X^{\prime}$ has only rational singularities and

$\displaystyle\left|\operatorname{\mathsf{Bs}}_{f}\omega_{X}\setminus\mathsf{Exc}(g)\right|<\infty.$
(4.2)

######  Theorem 4.3.

In the setting of Theorem 4.2, assume further that $Y$ has only rational
singularities. Then $\operatorname{\mathsf{Bs}}_{f}\omega_{X}=\emptyset$.

###### Proof.

See the paragraph before [MR2481686, Proposition 2.7]. ∎

######  Theorem 4.4 ($=$[MR2654305, Theorem 3.1]).

Let

$\displaystyle h\colon X\to C$ (4.3)

be a projective morphism of schemes of connected fibers where $X$ is a
nonsingular surface, $C$ is a nonsingular curve, and $\omega_{X}$ is $h$-nef.
Suppose further that the generic fiber of $h$ is of genus $\geq 2$. Let
$F=X_{c}\hookrightarrow X$ be the schematic fiber of a closed point $c\in C$.
Then $\operatorname{\mathsf{Bs}}\omega_{F}$ supports at most exceptional sets
of rational singularities or weak elliptic singularities.

######  Lemma 4.5.

In the situation of Theorem 4.4, we have

$\displaystyle
F\cap\operatorname{\mathsf{Bs}}_{h}\left(\omega_{h}\right)\simeq\operatorname{\mathsf{Bs}}\omega_{F}.$
(4.4)

###### Proof.

Immediately follows from the following isomorphisms.

$\displaystyle\left(h_{\ast}\omega_{h}\right)|_{c}\simeq
H^{0}\left(F,\omega_{h}|_{F}\right)\simeq H^{0}\left(F,\omega_{F}\right),$
(4.5)

where the first isomorphism follows from the cohomology and base change
[Hartshorne, Chapter III Theorem 12.11] and the relative duality. ∎

######  Corollary 4.6.

In the situation of Theorem 4.4, there exists a factorization

${X}$${Y}$${C}$$\scriptstyle{f}$$\scriptstyle{h}$ (4.6)

such that $Y$ is a normal algebraic space, $f$ is a birational projective
morphism, and

$\displaystyle\left|\operatorname{\mathsf{Bs}}_{h}\omega_{X}\setminus\mathsf{Exc}(f)\right|<\infty$
(4.7)

###### Proof.

Note first that

$\displaystyle\operatorname{\mathsf{Bs}}_{h}\omega_{X}=\operatorname{\mathsf{Bs}}_{h}\omega_{h}$
(4.8)

by Lemma 3.3. Then Lemma 4.5 immediately implies the following equality.

$\displaystyle\operatorname{\mathsf{Bs}}_{h}\omega_{h}=\bigcup_{F\colon\mathrm{singular}}\operatorname{\mathsf{Bs}}\omega_{F}$
(4.9)

Then by Theorem 4.4, there is a birational contraction $f\colon X\to Y$ to an
algebraic fiber space satisfying the assertions of Corollary 4.6. ∎

###### Proof of Theorem 4.1.

By the assumption, we have
$\dim\operatorname{alb}_{X}\left(X\right)\in\left\\{1,2\right\\}$.

Suppose first that $\dim\operatorname{alb}_{X}\left(X\right)=2$. Take the
Stein factorization of the Albanese morphism of $X$ as follows.

${X}$${\operatorname{Alb}_{X}}$${Y}$$\scriptstyle{\operatorname{alb}_{X}}$$\scriptstyle{f}$
(4.10)

The assumption implies $\dim Y=2$, and $Y$ is SSI by Theorem 1.4. Hence it is
enough to show that $f$ is SI. Now we can apply Theorem 4.2 to the morphism
$f$ of (4.10) to obtain the factorization as in (4.1).

Take an arbitrary $f$-linear semiorthogonal decomposition of $\mathbf{D}(X)$.
Then it follows from Theorems 4.2, 3.15 and 3.10 that the semiorthogonal
decomposition under consideration is indeed $g$-linear. Then by applying
Theorem 4.3 to the morphism $g$ and Theorem 3.10 again, we conclude that the
semiorthogonal decomposition is trivial.

Suppose next that $\dim\operatorname{alb}_{X}\left(X\right)=1$. Take the Stein
factorization of the Albanese morphism of $X$ as follows.

${X}$${\operatorname{Alb}_{X}}$${C}$$\scriptstyle{\operatorname{alb}_{X}}$$\scriptstyle{h}$
(4.11)

As in the previous case, the assumption implies that $\dim C=1$ and $C$ is SSI
by Theorem 1.4, so it is enough to show that $h$ is SI. Now we can apply
Corollary 4.6 to the morphism $h$ of (4.11) to obtain the factorization as in
(4.6).

Take an arbitrary $h$-linear semiorthogonal decomposition of $\mathbf{D}(X)$.
Assume first that the general fiber of $h$ is of genus $\geq 2$. Then it
follows from Corollaries 4.6, 3.15 and 3.10 that the semiorthogonal
decomposition under consideration is indeed $f$-linear. Then by applying the
arguments for the case where $\dim\operatorname{alb}_{X}\left(X\right)=2$,
which is given above, to the morphism $f$, we can conclude that the
semiorthogonal decomposition is indeed trivial.

Finally, suppose that the general fiber of $h$ is of genus $1$. Then it
follows from the canonical bundle formula for elliptic fibrations (see, say,
[2015arXiv150800682K, Section 4 (4.1)]) that there exists an open dense subset
$V\hookrightarrow C$ such that

$\displaystyle
f\left(\operatorname{\mathsf{Bs}}_{f}(\omega_{f})\right)\subseteq C\setminus
V.$ (4.12)

Hence one of the components of the semiorthogonal decomposition is supported
in $f^{-1}\left(C\setminus V\right)$ by Theorem 3.10. Then the proof of
[2015arXiv150800682K, Theorem 4.2] literally applies to our situation. Thus we
conclude the proof. ∎

To conclude, let us summarize the remaining cases of Conjecture 1.8 in
dimension $2$:

######  Corollary 4.7.

Conjecture 1.8 is true except possibly for those $X$ which satisfy all of the
following conditions.

  * •

$X$ is a minimal surface of general type.

  * •

$H^{2}(X,\mathcal{O}_{X})\neq 0$ ($\iff H^{0}(X,\omega_{X})\neq 0$).

  * •

$H^{1}(X,\mathcal{O}_{X})=0$.

  * •

There exists a 1-dimensional component of
$\operatorname{\mathsf{Bs}}\left(\omega_{X}\right)$ which can not be
contracted to a point in the category of algebraic spaces.

###### Proof.

The case where $X$ is of Kodaira dimension $0$ is established in
[2015arXiv150800682K]. The case where $X$ is of Kodaira dimension $1$ is also
established in [2015arXiv150800682K], except the case where $\dim
H^{1}(X,\mathcal{O}_{X})=1$ and $\dim H^{2}(X,\mathcal{O}_{X})=0$. This case
is partially settled in [2021arXiv210709564L, Theorem 4.14], and fully by
Theorem 1.7.

Hence it remains to consider the case when $X$ is of general type with $\dim
H^{1}(X,\mathcal{O}_{X})=0$. We can also assume $H^{2}(X,\mathcal{O}_{X})\neq
0$, since otherwise line bundles on $X$ are exceptional objects and hence we
can conclude that $X$ is not semiorthogonally indecomposable. If any
1-dimensional components of $\operatorname{\mathsf{Bs}}(\omega_{X})$ is
contractible to a point in the category of algebraic spaces, then we can show
the indecomposability (this is done in [2015arXiv150800682K], but we can also
prove it by Lemma 3.15). Hence we obtain the assertion. ∎

##  5\. Alternative proof via relative Hochschild homology

Here we give an alternative proof of Theorem 3.10 using the relative
Hochschild homology, under the extra assumption that $f$ is smooth. Indeed,
the result is finer than Theorem 3.10. This is the relative version of the
absolute case (i.e., the case $Y=\operatorname{Spec}\mathbf{k}$) which is
worked out in [2020arXiv200607643P, Section 5].

In what follows we freely use the notation in the following diagram.

${X}$${X\times_{Y}X}$${X}$${X}$${Y}$$\scriptstyle{\operatorname{id}_{X}}$$\scriptstyle{\Delta_{f}}$$\scriptstyle{\operatorname{id}_{X}}$$\scriptstyle{\mathrm{pr}_{2}}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{f^{2}}$$\scriptstyle{f}$$\scriptstyle{f}$
(5.1)

In this section we discuss an $f$-linear semiorthogonal decomposition

$\displaystyle\mathbf{D}(X)=\langle\mathcal{A},\mathcal{B}\rangle.$ (5.2)

Then we obtain the base change

$\displaystyle{\mathrm{perf}}(X\times_{Y}X)=\langle\mathrm{pr}_{1}^{\ast}\mathcal{A},\mathrm{pr}_{1}^{\ast}\mathcal{B}\rangle$
(5.3)

which is automatically $\mathrm{pr}_{2}$-linear (see [MR2801403, Section
5.1]). Let us define the objects
$K_{\mathcal{A}},K_{\mathcal{B}}\in{\mathrm{perf}}(X\times_{Y}X)$ by the
following distinguished triangle which is obtained from the semiorthogonal
decomposition (5.3). Note that $K_{\mathcal{A}},K_{\mathcal{B}}$ are the
kernels of the projection functors to the subcategories $\mathcal{A}$ and
$\mathcal{B}$, respectively (see [kuznetsov2009hochschild, Theorem 3.7] for
details).

$\displaystyle K_{\mathcal{B}}\to\Delta_{f\ast}\mathcal{O}_{X}\to
K_{\mathcal{A}}\to K_{\mathcal{B}}[1]$ (5.4)

######  Definition 5.1.

In the situation of Theorem 3.10, the _relative Hochschild homology_ of $X$
over $Y$, $\mathcal{A}$, and $\mathcal{B}$ is defined as follows,
respectively.

$\displaystyle\operatorname{HH}_{\ast}\left(X/Y\right)$
$\displaystyle\coloneqq
f^{2}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X\times_{Y}X}\left(\Delta_{f\ast}\mathcal{O}_{X},\Delta_{f\ast}\omega_{X}[\dim
X]\right)$ (5.5)
$\displaystyle\operatorname{HH}_{\ast}\left(\mathcal{A}\right)$
$\displaystyle\coloneqq
f^{2}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X\times_{Y}X}\left(K_{\mathcal{A}},K_{\mathcal{A}}\circ\Delta_{f\ast}\omega_{X}[\dim
X]\right)$ (5.6)
$\displaystyle\operatorname{HH}_{\ast}\left(\mathcal{B}\right)$
$\displaystyle\coloneqq
f^{2}_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X\times_{Y}X}\left(K_{\mathcal{B}},K_{\mathcal{B}}\circ\Delta_{f\ast}\omega_{X}[\dim
X]\right)$ (5.7)

Then one can naturally define a morphism of $\mathbb{Z}$-graded coherent
sheaves on $Y$

$\displaystyle\gamma_{\mathcal{A}}\oplus\gamma_{\mathcal{B}}\colon\operatorname{HH}_{\ast}\left(X/Y\right)\to\operatorname{HH}_{\ast}\left(\mathcal{A}\right)\oplus\operatorname{HH}_{\ast}\left(\mathcal{B}\right)$
(5.8)

as follows.

Choose $m\in\mathbb{Z}$. To define the morphism

$\displaystyle\gamma_{\mathcal{A},m}\colon\operatorname{HH}_{m}\left(X/Y\right)\to\operatorname{HH}_{m}\left(\mathcal{A}\right)$
(5.9)

by using the Yoneda lemma, take an arbitrary object
$\mathcal{E}\in{\mathrm{perf}}(Y)$. All we have to do is to construct a map

$\displaystyle\operatorname{Hom}_{{\mathrm{perf}}(Y)}\left(\mathcal{E},\operatorname{HH}_{m}\left(X/Y\right)\right)\to\operatorname{Hom}_{{\mathrm{perf}}(Y)}\left(\mathcal{E},\operatorname{HH}_{m}\left(\mathcal{A}\right)\right)$
(5.10)

which is functorial in $\mathcal{E}$. By the standard adjunctions it is
equivalent to giving a map as follows.

$\displaystyle\operatorname{Hom}_{{\mathrm{perf}}(X\times_{Y}X)}\left(f^{2\ast}\mathcal{E}\otimes\Delta_{f\ast}\mathcal{O}_{X},\Delta_{f\ast}\omega_{X}[\dim
X+m]\right)$ (5.11)
$\displaystyle\to\operatorname{Hom}_{{\mathrm{perf}}(X\times_{Y}X)}\left(f^{2\ast}\mathcal{E}\otimes
K_{\mathcal{A}},K_{\mathcal{A}}\circ\Delta_{\ast}\omega_{X}[\dim X+m]\right)$
(5.12)

This is achieved by the same arguments of [2020arXiv200607643P, Section 5.1],
so we omit the details. The morphism $\gamma_{\mathcal{B},m}$ is also obtained
from the same arguments. Finally, by adjusting the proof of
[kuznetsov2009hochschild, Proposition 5.5] to the relative setting, we obtain
the following

######  Proposition 5.2.

The morphism (5.8) is an isomorphism.

###### Proof.

See [kuznetsov2009hochschild, Proposition 5.5, Theorem 7.3, Corollary 3.10]
and their proofs for the absolute case. We can adjust them to the relative
setting. ∎

######  Remark 5.3.

More abstract proof is given (in the absolute setting) in
[2021arXiv211210312L, Theorem 4.7]. We opted to follow the concrete approach
of [kuznetsov2009hochschild] in this paper.

As in the absolute case, we have the following assertion as to the
decomposition (5.8).

######  Lemma 5.4.

Either of the following assertions holds.

  1. (A)

$\operatorname{HH}_{-\dim X}\left(\mathcal{A}\right)=0$

  2. (B)

$\operatorname{HH}_{-\dim X}\left(\mathcal{B}\right)=0$

Moreover, when the case Item A holds, then for any $a\in\mathcal{A}$ the
canonical map

$\displaystyle
f^{\ast}f_{\ast}\omega_{X}\otimes_{\mathcal{O}_{X}}a\to\omega_{X}\otimes_{\mathcal{O}_{X}}a$
(5.13)

is $0$. Hence for any étale morphism from an affine scheme as in (3.19) and a
section $s\in H^{0}\left(X^{\prime},\omega_{X^{\prime}}\right)$ (see (3.5) for
the definition of $X^{\prime}$) the multiplication map

$\displaystyle a^{\prime}\xrightarrow[]{\otimes
s}a^{\prime}\otimes\omega_{X^{\prime}}$ (5.14)

is $0$. In particular, it holds that
$\operatorname{Supp}(a)\subset\operatorname{\mathsf{Bs}}_{f}(\omega_{X})$. If
the case Item B holds, then the same assertions hold for objects of
$\mathcal{B}$.

###### Proof.

Without loss of generality, we can assume that $f$ is surjective and hence $Y$
is also connected.

Suppose that $\operatorname{HH}_{-\dim X}\left(\mathcal{B}\right)\neq 0$. Let
$X^{\prime}$ be the base change of $X$ as in (3.5) by an étale morphism from
an affine scheme as in (3.19) such that

$\displaystyle\operatorname{HH}_{-\dim
X}\left(\mathcal{B}^{\prime}\right)\simeq y^{\ast}\operatorname{HH}_{-\dim
X}\left(\mathcal{B}\right)\neq 0$ (5.15)

From the isomorphism (5.8) in degree $-\dim X$ we obtain a section $0\neq s\in
H^{0}\left(X^{\prime},\omega_{X^{\prime}}\right)$ such that
$\gamma_{\mathcal{A}^{\prime}}(s)=0$. This implies that the objects of
$\mathcal{A}^{\prime}$ are supported in the vanishing locus of $s$, which
means that there is a proper closed subset $Z_{\mathcal{A}}\subsetneq X$ in
which objects of $\mathcal{A}$ are supported. Likewise, if
$\operatorname{HH}_{-\dim X}\left(\mathcal{A}\right)\neq 0$, then it follows
that there is a proper closed subset $Z_{\mathcal{B}}\subsetneq X$ in which
objects of $\mathcal{B}$ are supported. Then the skyscraper sheaves (if any)
of the closed points of the locus

$\displaystyle X\setminus\left(Z_{\mathcal{A}}\cup Z_{\mathcal{B}}\right)$
(5.16)

should be orthogonal to both $\mathcal{A}$ and $\mathcal{B}$, which is a
contradiction. As we assume that $X$ is connected, it follows that either
$Z_{\mathcal{A}}$ or $Z_{\mathcal{B}}$ is empty.

To see the rest of the assertions, note

$\displaystyle
f_{\ast}\omega_{X}\stackrel{{\scriptstyle\gamma_{\mathcal{A},-\dim
X}}}{{\to}}f^{2}_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{X\times_{Y}X}\left(K_{\mathcal{A}},K_{\mathcal{A}}\circ\Delta_{\ast}\omega_{X}\right)$
(5.17) $\displaystyle\iff$ $\displaystyle f^{2\ast}f_{\ast}\omega_{X}\otimes
K_{\mathcal{A}}\to K_{\mathcal{A}}\circ\Delta_{\ast}\omega_{X}$ (5.18)

Think of (5.18) as a morphism of kernels. Take $a\in\mathcal{A}$ and evaluate
at $\omega_{X}^{-1}\otimes a$, to obtain the first row of the following
commutative diagram (see the corresponding diagram in [2020arXiv200607643P,
Proof of Lemma 5.4]), where $\Phi_{\bullet}$ is our notation for the integral
transformation by the kernel $\bullet$. The left vertical arrow is the adjoint
unit

$\displaystyle\operatorname{id}_{\mathbf{D}(X)}\to\Phi_{K_{\mathcal{A}}}$
(5.19)

tensored with the identity of $f^{2\ast}f_{\ast}\omega_{X}$.

${f^{2\ast}f_{\ast}\omega_{X}\otimes\omega_{X}^{-1}\otimes
a}$${a}$${f^{2\ast}f_{\ast}\omega_{X}\otimes\Phi_{K_{\mathcal{A}}}\left(\omega_{X}^{-1}\otimes
a\right)}$${a}$$\scriptstyle{\eqref{equation:action of omega on
a}}$$\scriptstyle{\operatorname{id}_{a}}$$\scriptstyle{\eqref{equation:map of
kernels}\left(\omega_{X}^{-1}\otimes a\right)}$ (5.20)

The bottom arrow is trivial as we assumed $\gamma_{\mathcal{A}}=0$. Thus we
see ${\eqref{equation:action of omega on a}}=0$. The rest is straightforward.
∎