# Generation of imprinted strain gradients for spintronics

G. Masciocchi<EMAIL_ADDRESS>Institute of Physics, Johannes Gutenberg
University Mainz, Staudingerweg 7, 55099 Mainz, Germany  Sensitec GmbH,
Walter-Hallstein-Straße 24, 55130 Mainz, Germany  M. Fattouhi  Universidad de
Salamanca, Department of Applied Physics, E-37008 Salamanca, Spain  E.
Spetzler  Institute for Materials Science, Kiel University, Kaiserstraße 2,
24143 Kiel, Germany  M.-A. Syskaki  Institute of Physics, Johannes Gutenberg
University Mainz, Staudingerweg 7, 55099 Mainz, Germany  Singulus Technologies
AG, Hanauer Landstrasse 107, 63796 Kahl am Main, Germany  R. Lehndorff
Allegro MicroSystems Germany GmbH, Vangerowstraße 18/1, 69115 Heidelberg,
Germany  E. Martinez  Universidad de Salamanca, Department of Applied Physics,
E-37008 Salamanca, Spain  J. McCord  Institute for Materials Science, Kiel
University, Kaiserstraße 2, 24143 Kiel, Germany  Kiel Nano, Surface and
Interface Science (KiNSIS), Kaiserstraße 2, 24143 Kiel, Germany  L. Lopez-Diaz
Universidad de Salamanca, Department of Applied Physics, E-37008 Salamanca,
Spain  A. Kehlberger  Sensitec GmbH, Walter-Hallstein-Straße 24, 55130 Mainz,
Germany  M. Kläui<EMAIL_ADDRESS>Institute of Physics, Johannes
Gutenberg University Mainz, Staudingerweg 7, 55099 Mainz, Germany

###### Abstract

In this work, we propose and evaluate an inexpensive and CMOS-compatible
method to locally apply strain on a Si/SiOx substrate. Due to high growth
temperatures and different thermal expansion coefficients, a SiN passivation
layer exerts a compressive stress when deposited on a commercial silicon
wafer. Removing selected areas of the passivation layer alters the strain on
the micrometer range, leading to changes in the local magnetic anisotropy of a
magnetic material through magnetoelastic interactions. Using Kerr microscopy,
we experimentally demonstrate how the magnetoelastic energy landscape, created
by a pair of openings, in a magnetic nanowire enables the creation of pinning
sites for in-plane vortex walls that propagate in a magnetic racetrack. We
report substantial pinning fields up to 15 mT for device-relevant
ferromagnetic materials with positive magnetostriction. We support our
experimental results with finite element simulations for the induced strain,
micromagnetic simulations and 1D model calculations using the realistic strain
profile to identify the depinning mechanism. All the observations above are
due to the magnetoelastic energy contribution in the system, which creates
local energy minima for the domain wall at the desired location. By
controlling domain walls with strain, we realize the prototype of a true
power-on magnetic sensor that can measure discrete magnetic fields or Oersted
currents. This utilizes a technology that does not require piezoelectric
substrates or high-resolution lithography, thus enabling wafer-level
production.

One of the promising "Beyond COMS" technologies are nanomagnetic and
spintronic devices due to their non-volatile nature, high operating speed,
extremely low power consumption, and well explored routes to read and write
dataManipatruni, Nikonov, and Young (2018). One example is nanomagnetic
tracks, where information (stored in domain walls - DWs) is propagated and
manipulated by dipolar interaction along soft ferromagnetic nanowiresOno  _et
al._ (1999); Atkinson  _et al._ (2003). The manipulation of DWs has quite a
long history and a turning point in this research area was the demonstration
of a current-controlled magnetic DW shift register Hayashi  _et al._ (2008);
Kumar  _et al._ (2022) (racetrack memory). Since then, more work has been done
on the development of DW-based memoriesFranken, Swagten, and Koopmans (2012),
logic devicesLuo  _et al._ (2020) and sensorsZhang  _et al._ (2018); Diegel
_et al._ (2009); Khan  _et al._ (2021); Borie  _et al._ (2017) or neuromorphic
computingAlamdar  _et al._ (2021); Yue  _et al._ (2019). However, feasibility
of the fabrication process and compatibility with existing CMOS devices must
be ensured before full technological realization is achieved.

One of the key challenges with these devices is the control of DWsBoulle,
Malinowski, and Kläui (2011), typically realized using geometric constraints
(notches) Omari  _et al._ (2019); Shiu  _et al._ (2021); Al Bahri  _et al._
(2019) or the local manipulation of the magnetic anisotropy through strain
Fattouhi  _et al._ (2022); Masciocchi  _et al._ (2021) using
magnetostrictive/piezoelectric systems Lei  _et al._ (2013); Zhou  _et al._
(2020); Hu  _et al._ (2016); Dean  _et al._ (2011). However, these approaches
are not attractive for most sensor manufacturers due to high cost and
complexity. Respectively, because high-resolution notches and presence of the
multiferroic stack would require significant investments in tools for high-
resolution lithography and layer deposition. Also, the presence of voltages
for piezoelectric actuation via metallic contacts increases design complexity
and area usage. It is moreover difficult to realize an arbitrary shape of
strain and strain gradients down to the micrometer range with piezoelectric
substrates because it is technologically nontrivial to confine the electric
fieldsBarra  _et al._ (2021).

An alternative method of transferring strain to a thin filmChu  _et al._
(2009); Doherty  _et al._ (2020), is the use of capping layers Martin  _et
al._ (2009) widely used in the semiconductor and photovoltaic industries
because they provide protection from hostile environments.

In this work, we propose and experimentally demonstrate a low-cost and CMOS-
compatible method to induce local strain on a Si/SiOx substrate by removing
selected regions of the passivation layer. Arbitrary strain magnitudes and
strain gradients can be realized by simply choosing the design of the removed
part. The magnitude and profile of the strain are determined by combining
anisotropy and stress measurements with finite elements simulations. We
experimentally demonstrate, using Kerr microscopy, that this local strain
allows for domain wall pinning in a racetrack element. This is verified by
micromagnetic simulations and 1D model calculations. Finally, to show the
technological relevance of this method, we propose and verify a non-volatile
magnetic peak-field sensor based on this technology.

Figure 1:  Racetracks structuring (a), SiN layer deposition (uniform strain
generated) (b), and locally relieved strain after RIE (c). (d) section of the
layers used in this work. (e) SEM image of a pair of openings in the SiN in
the proximity of a magnetic nanowire.

Samples of AlOx(10 nm)/HfOx(10)/Co70Fe30(30) and AlOx(10
nm)/HfOx(10)/Co40Fe40B20(30) were prepared by DC magnetron sputtering using a
Singulus Rotaris system on a SiOx(1.5 µm)/Si(625 µm) substrate. Using optical
lithography and etching, nanowires were fabricated with a variable width -
between 800 and 500 nm - and a length of 70 µm. A reservoir at the left end
allows for DWs injection at lower fields. After the first lithography step,
the wafer was covered with a 1 µm thick SiN layer using plasma-enhanced
chemical vapor deposition (PECVD) at a temperature of 250∘C. The residual
stress on the wafer is quantified using a standard wafer bow measurement. A
second optical lithography step is used in combination with reactive ion
etching (RIE) to remove selected areas (up to $20\times 20$ µm2) of the SiN
layer (openings) without damaging the magnetic layer, as shown in Figs.1 (a) -
(d) while the wafer surface is still largely covered. The hysteresis loops of
the thin films are measured using a BH-looper with a setup to measure
magnetostriction. The magneto-optic Kerr effect (MOKE) was used to image the
magnetization state in the devicesMcCord (2015); Soldatov and Schäfer (2017).

To understand the origin of the intrinsic stress in our system, one should
consider the coefficients of thermal expansion of a film and a substrate,
along with the high temperature during deposition. If the thermal expansion
coefficients are different, thermal stresses arise when the whole stack cools
down to room temperature after deposition Ohring (1992). Relaxation of this
stress leads to a deformation, i.e. bending, of the wafer (Figs. 1(a)-(b))
allowing for the residual stress to be estimatedMarks, Hassan, and Cheong
(2014). The measured in-plane (compressive) stress is planar and in our case
has a magnitude of $-495(5)$ MPa. To create a non-uniform stress on the
substrate surface, selected areas of the SiN are completely removed, creating
some openings in the passivation layer (Fig. 1 (c)). The etching is monitored
to stop the process at the AlOx/SiN interface, as shown in Fig. 1 (d), so that
the integrity of the magnetic layer is preserved. An example of the final
device is shown in a scanning electron microscopy image (SEM) in Fig. 1 (e)
for a pair of square apertures $10\times 10$ µm2 in size. With a suitable
lithography mask, arbitrary shapes, sizes, and spacing of the apertures can be
realized with sub µm resolution. In the example presented here, the openings
are spaced 1 µm apart and the 800 nm wide magnetic track under the SiN layer
shows no signs of damage caused by the etching process.

Figure 2:  (a) - (b) effective strain ($\epsilon_{xx}-\epsilon_{yy}$) and and
strain gradient ($\frac{d}{dx}\left(\epsilon_{xx}-\epsilon_{yy}\right)$) for,
respectively, a square and a triangular pair of opening obtained from FEM
simulations. The $x$ axis refers to the red dashed line. (c) - (d) surface
strain $\epsilon_{xx}-\epsilon_{yy}$ obtained with FEM simulations. (e) - (f)
in-plane hysteresis loops obtained with Kerr microscopy on a full film of
Co70Fe30 (30 nm) for, respectively, a square and a triangular pair of opening.
The contrast was measured before the opening (diamond) and between them (star)
according to the marker position. The magnetic field was applied along the
direction $\Phi=0^{\circ}$.

To determine the magnitude of stress relieved, finite-element-method (FEM)
simulations were performed using the COMSOL Multiphysics® Structural Mechanics
ModuleCOMSOL AB, Stockholm, Sweden (2023). More details about FEM simulations
can be found in Section S1 of the supplementary material. To have a well
defined strained region in racetrack type devices, it is convenient to
consider a pair of openings - to be realized at each side of a magnetic
nanowire. Figs. 2 (a) - (d) contain the computed values of the surface strain
$\epsilon_{xx}-\epsilon_{yy}$ at the interface between SiN/SiOx for two
different opening geometries. As shown in Figs. 2 (c) and (d), the effective
surface strain $\epsilon_{xx}-\epsilon_{yy}$ is close to zero at a distance
greater than 20 µm from the etched areas and becomes non-uniform in their
proximity. The geometry of the opening determines the strain profile. This can
be seen in Figs. 2 (a) and (b) where the effective strain is plotted along the
dashed line running between the two openings shown in Figs. 2 (c) and (d),
respectively. For a square pair of openings (Fig. 2 (a)), the effective
uniaxial strain profile $\epsilon_{xx}-\epsilon_{yy}$ is mostly flat and
confined between them with strain gradient maxima (minima) at the exit
(entrance) of the strained area. The strain reaches values of
$\epsilon_{xx}-\epsilon_{yy}\simeq 0.2\%$. For a diamond shaped pair of
openings, the strain is again confined between the openings but its magnitude
increases almost linearly towards the center. This time the strain gradient is
mostly constant.

To experimentally confirm the magnitude and sign of this local strain, we
measured the magnetization curves of a unpatterned film of AlOx/HfOx/Co70Fe30
(30 nm) underneath the patterned SiN. The hysteresis loops were measured with
field applied along $\Phi=0^{\circ}$ at different locations on the sample,
selecting a region of interest of $5\times 5$ µm2 size within the field of
view of the Kerr microscope. Full angular dependence of the anisotropy is
reported in section S3 of the supplementary information. Since Co70Fe30 has
considerable magnetostriction ($\lambda_{s}\simeq 80\times 10^{-6}$), the
strain acting on the film is coupled to the magnetization via the
magnetoelastic effect, as expressed in the anisotropy energyFinizio  _et al._
(2014)

$K_{ME}=\frac{3}{2}\lambda_{s}Y\left(\epsilon_{xx}-\epsilon_{yy}\right),$ (1)

where $Y$ is the Young’s modulus and $\lambda_{s}$ is the saturation
magnetostriction. As done in previous works Masciocchi  _et al._ (2021, 2022)
the in-plane magnetoelastic anisotropy $K_{ME}$ can be locally estimated.
Measuring hysteresis loops, where an in-plane field is applied along two
perpendicular directions, can give us a direct measurement of the local
strainMcCord (2004); Urs  _et al._ (2014); Thórarinsdóttir  _et al._ (2022).

In Figs. 2 (e) - (f) hysteresis loops of an unpatterned film, this time of
SiN/AlOx/HfOx/Co70Fe30(30 nm), are shown. The openings geometry is the one of
Figs. 2 (c) - (d), respectively. Looking at Fig. 2 (e) we can compare the
magnetization curve before (diamond) and between (star) the square openings.
The anisotropy field increases, due to (uniaxial) magnetoelastic anisotropy.
As Co70Fe30 has a positive magnetostriction, the increase in anisotropy field
($K_{ME}\simeq 8.9(2)$ kJ/m3) is caused by a negative (compressive)
$\epsilon_{xx}-\epsilon_{yy}$ strain, in agreement with our FEM simulation.
Using Eq. 1 and the values of magnetoelastic anisotropy difference we can
estimate the strain to be $\epsilon_{xx}-\epsilon_{yy}\simeq-0.05(1)\%$ for a
square opening of this size. The same measurement can be performed for a
diamond-shaped pair of openings and is reported in Fig. 2 (f). The calculated
maximum strain difference for this case is
$\epsilon_{xx}-\epsilon_{yy}\simeq-0.02(1)\%$.

The strain, created by removing specific areas of the SiN layer could be used
as a mechanism to move, change direction, or stop a DW, a feature often needed
in the device implementationMasciocchi  _et al._ (2021); Diegel  _et al._
(2009). Typical ways to do so relies on the modification of the DW energy
making it a spatially variable quantity. In analogy with the conventional
field-driven case, the magnetoelastic field can be considered as a force that
pushes the DW along the direction of decreasing energy, i.e., increasing
compressive strain if $\lambda_{s}>0$ for the in-plane-strain-gradient case.
This force is proportional to the local gradient of the spatially variable
quantity Wen  _et al._ (2020); Fattouhi  _et al._ (2022, 2021), and its effect
is essentially that of an effective (magnetoelastic) field

$B_{ME}=-\frac{1}{M_{s}}\frac{du_{ME}}{dx},$ (2)

where $u_{ME}$ is the magnetoelastic DW energy per unit area.

Figure 3:  MOKE images showing a DW (a) injected, (b) pinned in the strained
area and (c) continuing propagation for larger magnetic fields. (d) 1D model
calculations of energy profile and the corresponding local magnetoelastic
field for a vortex wall in the strain profile shown in Fig. 2(a) as a function
of the DW position. (e) experimental values of depinning field (black squares)
and maximum strain $\epsilon_{xx}-\epsilon_{yy}$ (blue diamonds) for different
distances between a pair of square openings. The values consider a 500 nm wire
of Co70Fe30 with 30 nm thickness. (f) micromagnetic simulations magnetization
(blue) and experimentally measured Kerr contrast (orange) for a 800 nm wide
nanowire as function of applied field. The averaged wire magnetization along
$x$ direction ($m_{x}$) is proportional to the DW position.

For this study, a 500 nm wide magnetic racetrack of Co70Fe30(30 nm) is
considered together with a pair of square openings in the SiN 10$\times$10 µm2
in size. We use Kerr microscopy in transverse mode to image the magnetic state
of the device, while in-plane magnetic field is applied parallel to the wire
along $x$. Figs. 3 (a)-(c) show the position of a DW along the magnetic
racetrack as a function of the applied magnetic field. When the field is
sufficiently large, the DW is injected from the reservoir (Fig. 3 (a)) into
the magnetic wire. As can be seen in Fig. 3 (b), after injection the wall does
not propagate until the end of the magnetic channel but is pinned in the area
between the SiN openings corresponding to the strained area. The corresponding
surface strain was shown with a simulation in Figs. 2 (a) and (c). Only for
larger magnetic fields the wall can continue to propagate to the other end of
the magnetic channel, as captured by Fig. 3 (c).

For a Ni81Fe19 sample with nearly no magnetostriction no DW pinning was found
within the resolution of our measurement, supporting the idea of a strain-
based pinning. We repeated the same measurement for devices with different
distance $d$ between racetrack and openings. According to FEM simulations, the
value of the (compressive) strain increases as the opening distance $d$ is
reduced (blue diamonds in Fig. 3 (e)). As shown in Fig. 3 (e), the depinning
field (black squares) increases from 11.0(2) mT to 14.5(2) mT for a distance
between the opening and the magnetic racetrack decreasing from 2.5 to 1 µm. A
larger depinning field $B_{dep}$ for smaller opening spacing confirms that the
magnetoelastic energy is indeed the dominant pinning cause in our systemLei
_et al._ (2013); Franken  _et al._ (2013).

To support our experimental findings, we performed micromagnetic simulations
and 1D model calculations where the strain profile from FEM simulations was
used. The results are summarized in Figs. 3 (d) and (f) and consider nanowires
made of 30 nm thick Co40Fe40B20. For more details about the micromagnetic
simulations and the 1D analytical model, see section S2 of the supplementary
information. Fig. 3 (d) shows the DW energy per unit area and the
corresponding magnetoelastic field as a function of the DW position for a
nanowire $w=500$ nm wide, calculated considering the strain profile shown in
Fig. 2 (a) and a rigid profile for the DW. Comparing Fig. 3 (d) with Fig. 3
(b), it is clear that the point where the DW sits is the minimum of DW energy.
At the sides of the pinning site, the effective magnetoelastic field -
proportional to $\frac{d}{dx}\left(\epsilon_{xx}-\epsilon_{yy}\right)$
according to Eq. 2 \- is non-zero, and opposite to the applied external field.
This equivalent force prevents the DW to move forward unless the external
applied field is increased.

For the MumaxVansteenkiste  _et al._ (2014) micromagnetic simulations a wire
of 800 nm width has been considered. The magnetization has been initialized in
the system with a DW on the left side of the strained area and then a magnetic
field has been applied. Multiple dynamic simulations have been performed at
different values of external magnetic field, and the results are summarized in
Fig. 3 (f). As can be seen, the averaged magnetization along the $x$ direction
(proportional to the DW position) coincides with the strained area (state I)
for external fields $B_{ext}<B_{dep}$. When the applied magnetic field is
increased above $B_{dep}$, the domain wall is free to propagate and reaches
the right end of the wire (state II). For comparison, the experimental values
for the DW position as a function of $B_{ext}$ is reported in Fig. 4 (f) for a
800 nm width Co40Fe40B20 wire. The pinning position (where
$\epsilon_{xx}-\epsilon_{yy}\neq 0$) coincides and discrepancies between the
simulations and experiments for the value of $B_{dep}$ can be due to thermally
activated depinning events that are not fully captured by micromagnetic
simulations.

The ability to adjust the maximum value of the strain, and thus the value of
the depinning field, by changing the aperture design - as shown in Fig. 3 (e)
- allows for the realization of a non-volatile magnetic field sensor capable
of detecting discrete values of magnetic fields or current peaks from wires or
coils in the sensor proximity. Previous work Al Bahri  _et al._ (2019),
suggested similar concepts, however, the one proposed here does not require
sub 100 nm lithography resolution for the notches.

The conceptualization of the peak-field sensor is presented in Fig. 4. The
device comprises of a magnetic nanowire for DWs propagation with a number of
pinning sites along it. As shown in Figs. 4 (a) and (b) if the spacing between
the SiN openings - acting as pinning sites - decreases, the strain magnitude
is increased progressively. According to Fig. 3 (e), the depinning field
$B_{dep}$ will increase going from left to right. The device considered here
presents four pair of openings and is therefore able to identify four discrete
levels of external magnetic fields.

Figure 4:  (a) surface strain and strain gradient along the racetrack path,
that is highlighted with a dashed line in (b) calculated with FEM. (c) MOKE
images of the DW position (white arrow) in the device for different magnetic
field steps. (d) magnetic contrast along the racetrack obtained with Kerr
microscopy. The sample is made of Co70Fe30 and the width of the wire is 500
nm. The inset shows a possible contacts configuration for resistivity
measurements of the DW position.

The proof of concept of the realized device is presented in Figs. 4 (c) and
(d). We consider, in this case, a magnetic wire of 500 nm width and realized
with a Co70Fe30 magnetic layer. The shape of the openings is 5$\times$5 µm2
and the distance between the pairs is, in order 5, 3, 2 and 1 µm, as shown in
Fig. 4 (b).

The device is initialized with large, negative magnetic field in the $x$
direction creating a uniform magnetic state. After that, positive magnetic
filed is gradually increased while the magnetic state in the wire is monitored
using Kerr microscopy. The magnetic contrast in Fig. 4 (d) shows how the
magnetization has well defined discrete levels as $B_{ext}$ is increased. This
occurs because a DW propagating into the nanowire occupies only discrete
positions along $x$, as shown in Fig. 4 (c) in the strained area between the
openings. The position of the DW in the magnetic channel (output) will
indicate the maximum field (input) that the device has seen after
initialization. The magnetic state has been measured in Fig. 4 (d) with
optical methods, however, electrical readout of the DW position is possible
using, e.g., Giant Magnetoresistive effect (GMR)Diegel  _et al._ (2009) and
two electrical contacts at the extremity of the magnetic channel. This sensing
solution is particularly suitable for hardly accessible measurement
environments and energy efficient devices as electrical power is required only
for readout and initialization.

In summary, in this work we propose an validate a method for generating a
local strain on a rigid substrate that is compatible with standard CMOS
technologies. The intrinsic stress that occurs at the substrate/layer
interface during SiN deposition can be modified when selected regions of the
passivation layer are removed by etching. The strain is only modified near the
removed material, as shown by FEM simulations. Using in-situ measurements of
the magnetoelastic anisotropy, we experimentally determine the magnitude of
the uniaxial strain up to $0.05(1)\%$. The magnitude and the gradient of the
in-plane strain can be tuned depending on the geometry and position of the
openings in the stress-generating layer. We validate the use of the above-
mentioned strain gradients for the manipulation of magnetic domain walls in
spintronic devices by exploiting magnetoelastic coupling in magnetostrictive
materials. Using Kerr microscopy, we experimentally show how the
magnetoelastic energy landscape enables the creation of engineered pinning
sites which represent local energy minima for in-plane vortex walls. We report
substantial pinning fields of up to 15 mT and support our experimental
findings with micromagnetic simulations and 1D model calculations using a
realistic strain profile. This provides the opportunity to realize an
alternative generation of DW-based devices with technology compatible with
wafer-level production, and an example of a discrete magnetic field or current
sensor using imprinted strain gradients is demonstrated.

## Supplementary Material

See supplementary material for details about the material parameters used, the
finite-element-method and micromagnetic simulations and the anisotropy
measurements.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie Grant
Agreement No 860060 “Magnetism and the effect of Electric Field” (MagnEFi),
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR
173 - 268565370 (project A01 and B02), the DFG funded collaborative research
center (CRC)1261 / project A10 and the Austrian Research Promotion Agency
(FFG). The authors acknowledge support by the chip production facilities of
Sensitec GmbH (Mainz, DE), where part of this work was carried out and the
Max-Planck Graduate Centre with Johannes Gutenberg University.

## Author Declarations

### Conflict of interest

The authors have no conflicts to disclose.

## Data Sharing Policy

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

## References

  * Manipatruni, Nikonov, and Young (2018) S. Manipatruni, D. E. Nikonov, and I. A. Young, “Beyond CMOS computing with spin and polarization,” Nature Physics 14, 338–343 (2018).
  * Ono  _et al._ (1999) T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, “Propagation of a magnetic domain wall in a submicrometer magnetic wire,” Science 284, 468–470 (1999).
  * Atkinson  _et al._ (2003) D. Atkinson, D. A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner, and R. P. Cowburn, “Magnetic domain-wall dynamics in a submicrometre ferromagnetic structure,” Nature materials 2, 85–87 (2003).
  * Hayashi  _et al._ (2008) M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. Parkin, “Current-controlled magnetic domain-wall nanowire shift register,” Science 320, 209–211 (2008).
  * Kumar  _et al._ (2022) D. Kumar, T. Jin, R. Sbiaa, M. Kläui, S. Bedanta, S. Fukami, D. Ravelosona, S.-H. Yang, X. Liu, and S. Piramanayagam, “Domain wall memory: Physics, materials, and devices,” Physics Reports 958, 1–35 (2022).
  * Franken, Swagten, and Koopmans (2012) J. Franken, H. Swagten, and B. Koopmans, “Shift registers based on magnetic domain wall ratchets with perpendicular anisotropy,” Nature nanotechnology 7, 499–503 (2012).
  * Luo  _et al._ (2020) Z. Luo, A. Hrabec, T. P. Dao, G. Sala, S. Finizio, J. Feng, S. Mayr, J. Raabe, P. Gambardella, and L. J. Heyderman, “Current-driven magnetic domain-wall logic,” Nature 579, 214–218 (2020).
  * Zhang  _et al._ (2018) X. Zhang, N. Vernier, Z. Cao, Q. Leng, A. Cao, D. Ravelosona, and W. Zhao, “Magnetoresistive sensors based on the elasticity of domain walls,” Nanotechnology 29, 365502 (2018).
  * Diegel  _et al._ (2009) M. Diegel, S. Glathe, R. Mattheis, M. Scherzinger, and E. Halder, “A new four bit magnetic domain wall based multiturn counter,” IEEE Transactions on Magnetics 45, 3792–3795 (2009).
  * Khan  _et al._ (2021) M. A. Khan, J. Sun, B. Li, A. Przybysz, and J. Kosel, “Magnetic sensors-a review and recent technologies,” Engineering Research Express 3, 022005 (2021).
  * Borie  _et al._ (2017) B. Borie, J. Wahrhusen, H. Grimm, and M. Kläui, “Geometrically enhanced closed-loop multi-turn sensor devices that enable reliable magnetic domain wall motion,” Applied Physics Letters 111, 242402 (2017).
  * Alamdar  _et al._ (2021) M. Alamdar, T. Leonard, C. Cui, B. P. Rimal, L. Xue, O. G. Akinola, T. Patrick Xiao, J. S. Friedman, C. H. Bennett, M. J. Marinella, _et al._ , “Domain wall-magnetic tunnel junction spin–orbit torque devices and circuits for in-memory computing,” Applied Physics Letters 118, 112401 (2021).
  * Yue  _et al._ (2019) K. Yue, Y. Liu, R. K. Lake, and A. C. Parker, “A brain-plausible neuromorphic on-the-fly learning system implemented with magnetic domain wall analog memristors,” Science advances 5, eaau8170 (2019).
  * Boulle, Malinowski, and Kläui (2011) O. Boulle, G. Malinowski, and M. Kläui, “Current-induced domain wall motion in nanoscale ferromagnetic elements,” Materials Science and Engineering: R: Reports 72, 159–187 (2011).
  * Omari  _et al._ (2019) K. A. Omari, T. J. Broomhall, R. W. Dawidek, D. A. Allwood, R. C. Bradley, J. M. Wood, P. W. Fry, M. C. Rosamond, E. H. Linfield, M.-Y. Im, _et al._ , “Toward chirality-encoded domain wall logic,” Advanced Functional Materials 29, 1807282 (2019).
  * Shiu  _et al._ (2021) D.-S. Shiu, K.-F. Lai, Y.-Y. Liu, Y.-T. Li, Z.-E. Gao, Y.-M. Kao, J.-C. Wu, and L. Horng, “Depinning behavior of the vortex domain wall at the asymmetric triangular notch in permalloy wires,” Journal of Physics Communications 5, 075014 (2021).
  * Al Bahri  _et al._ (2019) M. Al Bahri, B. Borie, T. Jin, R. Sbiaa, M. Kläui, and S. Piramanayagam, “Staggered magnetic nanowire devices for effective domain-wall pinning in racetrack memory,” Physical Review Applied 11, 024023 (2019).
  * Fattouhi  _et al._ (2022) M. Fattouhi, F. Garcia-Sanchez, R. Yanes, V. Raposo, E. Martinez, and L. Lopez-Diaz, “Absence of Walker breakdown in the dynamics of chiral Neel domain walls driven by in-plane strain gradients,” Physical Review Applied 18, 044023 (2022).
  * Masciocchi  _et al._ (2021) G. Masciocchi, M. Fattouhi, A. Kehlberger, L. Lopez-Diaz, M.-A. Syskaki, and M. Kläui, “Strain-controlled domain wall injection into nanowires for sensor applications,” Journal of Applied Physics 130, 183903 (2021).
  * Lei  _et al._ (2013) N. Lei, T. Devolder, G. Agnus, P. Aubert, L. Daniel, J.-V. Kim, W. Zhao, T. Trypiniotis, R. P. Cowburn, C. Chappert, _et al._ , “Strain-controlled magnetic domain wall propagation in hybrid piezoelectric/ferromagnetic structures,” Nature communications 4, 1–7 (2013).
  * Zhou  _et al._ (2020) H. Zhou, S. Shi, D. Nian, S. Cui, J. Luo, Y. Qiu, H. Yang, M. Zhu, and G. Yu, “Voltage control of magnetic domain wall injection into strain-mediated multiferroic heterostructures,” Nanoscale 12, 14479–14486 (2020).
  * Hu  _et al._ (2016) J.-M. Hu, T. Yang, K. Momeni, X. Cheng, L. Chen, S. Lei, S. Zhang, S. Trolier-McKinstry, V. Gopalan, G. P. Carman, _et al._ , “Fast magnetic domain-wall motion in a ring-shaped nanowire driven by a voltage,” Nano letters 16, 2341–2348 (2016).
  * Dean  _et al._ (2011) J. Dean, M. Bryan, T. Schrefl, and D. Allwood, “Stress-based control of magnetic nanowire domain walls in artificial multiferroic systems,” Journal of Applied Physics 109, 023915 (2011).
  * Barra  _et al._ (2021) A. Barra, A. Ross, O. Gomonay, L. Baldrati, A. Chavez, R. Lebrun, J. Schneider, P. Shirazi, Q. Wang, J. Sinova, _et al._ , “Effective strain manipulation of the antiferromagnetic state of polycrystalline NiO,” Applied Physics Letters 118, 172408 (2021).
  * Chu  _et al._ (2009) M. Chu, Y. Sun, U. Aghoram, and S. E. Thompson, “Strain: A solution for higher carrier mobility in nanoscale MOSFETs,” Annual Review of Materials Research 39, 203–229 (2009).
  * Doherty  _et al._ (2020) J. L. Doherty, S. G. Noyce, Z. Cheng, H. Abuzaid, and A. D. Franklin, “Capping layers to improve the electrical stress stability of MoS2 transistors,” ACS applied materials & interfaces 12, 35698–35706 (2020).
  * Martin  _et al._ (2009) N. Martin, J. McCord, A. Gerber, T. Strache, T. Gemming, I. Mönch, N. Farag, R. Schäfer, J. Fassbender, E. Quandt, _et al._ , “Local stress engineering of magnetic anisotropy in soft magnetic thin films,” Applied Physics Letters 94, 062506 (2009).
  * McCord (2015) J. McCord, “Progress in magnetic domain observation by advanced magneto-optical microscopy,” Journal of Physics D: Applied Physics 48, 333001 (2015).
  * Soldatov and Schäfer (2017) I. Soldatov and R. Schäfer, “Selective sensitivity in Kerr microscopy,” Review of Scientific Instruments 88, 073701 (2017).
  * Ohring (1992) M. Ohring, “The materials science of thin films,” Appl. Opt 31, 7162 (1992).
  * Marks, Hassan, and Cheong (2014) M. R. Marks, Z. Hassan, and K. Y. Cheong, “Characterization methods for ultrathin wafer and die quality: A review,” IEEE Transactions on components, packaging and manufacturing technology 4, 2042–2057 (2014).
  * COMSOL AB, Stockholm, Sweden (2023) COMSOL AB, Stockholm, Sweden, “Comsol multiphysics® v. 6.1,”  (2023).
  * Finizio  _et al._ (2014) S. Finizio, M. Foerster, M. Buzzi, B. Krüger, M. Jourdan, C. A. Vaz, J. Hockel, T. Miyawaki, A. Tkach, S. Valencia, _et al._ , “Magnetic anisotropy engineering in thin film Ni nanostructures by magnetoelastic coupling,” Physical Review Applied 1, 021001 (2014).
  * Masciocchi  _et al._ (2022) G. Masciocchi, J. W. van der Jagt, M.-A. Syskaki, A. Lamperti, N. Wolff, A. Lotnyk, J. Langer, L. Kienle, G. Jakob, B. Borie, _et al._ , “Control of magnetoelastic coupling in Ni/Fe multilayers using He+ ion irradiation,” Applied Physics Letters 121, 182401 (2022).
  * McCord (2004) J. McCord, “Irregular domain patterns in structured magnetic thick films,” Journal of applied physics 95, 6855–6857 (2004).
  * Urs  _et al._ (2014) N. O. Urs, I. Teliban, A. Piorra, R. Knöchel, E. Quandt, and J. McCord, “Origin of hysteretic magnetoelastic behavior in magnetoelectric 2-2 composites,” Applied Physics Letters 105, 202406 (2014).
  * Thórarinsdóttir  _et al._ (2022) K. Thórarinsdóttir, N. Strandqvist, V. Sigurjónsdóttir, E. Thorsteinsson, B. Hjörvarsson, and F. Magnus, “Finding order in disorder: Magnetic coupling distributions and competing anisotropies in an amorphous metal alloy,” APL Materials 10, 041103 (2022).
  * Wen  _et al._ (2020) D. Wen, Z. Chen, W. Li, M. Qin, D. Chen, Z. Fan, M. Zeng, X. Lu, X. Gao, and J.-M. Liu, “Ultralow-loss domain wall motion driven by a magnetocrystalline anisotropy gradient in an antiferromagnetic nanowire,” Physical Review Research 2, 013166 (2020).
  * Fattouhi  _et al._ (2021) M. Fattouhi, F. García-Sánchez, R. Yanes, V. Raposo, E. Martínez, and L. Lopez-Diaz, “Electric Field Control of the Skyrmion Hall Effect in Piezoelectric-Magnetic Devices,” Physical Review Applied 16, 044035 (2021).
  * Franken  _et al._ (2013) J. Franken, Y. Yin, A. Schellekens, A. van den Brink, H. Swagten, and B. Koopmans, “Voltage-gated pinning in a magnetic domain-wall conduit,” Applied Physics Letters 103, 102411 (2013).
  * Vansteenkiste  _et al._ (2014) A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, “The design and verification of mumax3,” AIP advances 4, 107133 (2014).


# Slope-semistability and moduli of coherent sheaves: a survey

Mihai Pavel, Matei Toma  Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania<EMAIL_ADDRESS>Université de
Lorraine, CNRS, IECL, F-54000 Nancy, France<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

We survey old and new results on the existence of moduli spaces of semistable
coherent sheaves both in algebraic and in complex geometry.

###### Key words and phrases:

semistable coherent sheaves, moduli spaces

###### 2020 Mathematics Subject Classification:

14D20, 32G13

_Dedicated to the memory of Lucian Bădescu_

###### Contents

  1. 1 Introduction
  2. 2 First properties of slope-semistable sheaves
  3. 3 Some key results on slope-semistable sheaves
  4. 4 Families of slope-semistable sheaves
  5. 5 Moduli functors and moduli stacks
  6. 6 Moduli spaces of sheaves
  7. 7 Further topics

##  1\. Introduction

In algebraic geometry and complex geometry the classification of vector
bundles is an important problem that has attracted considerable attention
since the early sixties. One goal of the classification was the construction
of nicely behaved moduli spaces of vector bundles. This had been achieved for
line bundles, but it soon became apparent that for higher rank vector bundles
a restrictive condition was necessary to obtain moduli spaces with good
geometric properties. To this purpose Mumford introduced the concept of slope-
semistability in [Mum63] for vector bundles over algebraic curves. This was
later extended to cover coherent sheaves over higher dimensional bases. The
theory attracted even more interest through Donaldson’s work in four-manifolds
differential topology. In particular the so-called Kobayashi-Hitchin
correspondence relates moduli spaces of stable vector bundles on complex
projective surfaces to moduli spaces of anti-self-dual connections in gauge
theory, thus providing a new perspective in the study of the subject in
complex geometry.

In this paper we survey the existence and construction results of moduli
spaces of semistable coherent sheaves in both algebraic and complex geometry,
with a particular stress on functorial aspects. We do not delve into the vast
domain studying the geometric properties and applications of these moduli
spaces in enumerative geometry, classification of manifolds, hyperkähler
geometry, gauge theory, etc. For this there exists a rich literature, see for
instance [HL10, MS16] and the references therein.

### Acknowlegements:

MP was partly supported by the PNRR grant CF 44/14.11.2022 Cohomological Hall
algebras of smooth surfaces and applications.

##  2\. First properties of slope-semistable sheaves

In this section we recall the notion of slope-semistability for coherent
sheaves, and discuss its first important properties in the context of both
algebraic and complex-analytic geometry. Our main references here are [HL10,
Chapter 1] and [Kob87, Chapter 5].

Setup. Throughout this paper we denote by $(X,\omega)$ a polarized
$n$-dimensional space which will be either

  1. (AG)

a smooth projective variety over an algebraically closed field $k$ with an
integral ample class $\omega\in\operatorname{Amp}^{1}(X)$, or

  2. (CG)

a compact complex manifold endowed with a Hermitian metric whose Kähler form
$\omega$ is such that $\partial\bar{\partial}(\omega^{n-1})=0$; such a metric
is called a Gauduchon metric.

An important special subcase of (CG) that we will frequently refer to in the
sequel is when $(X,\omega)$ is

  1. (KG)

a compact Kähler manifold endowed with a Kähler class $\omega\in
H^{1,1}(X,\mathbb{R})$.

Also we will denote by $\operatorname{Coh}(X)$ the category of coherent
sheaves on $X$. It should be understood that the coherent sheaves we consider
are algebraic or analytic depending on whether $(X,\omega)$ is in the case
(AG) or (CG) respectively.

For a torsion-free sheaf $E\in\operatorname{Coh}(X)$, we define the
$\omega$-slope of $E$ by

$\mu^{\omega}(E)\coloneqq\frac{c_{1}(E)\cdot\omega^{n-1}}{\operatorname{rk}(E)}.$

In the algebraic setting the intersection product $c_{1}(E)\cdot\omega^{n-1}$
is performed in the Chow ring $A^{*}(X)$. Otherwise, in the complex case,
$\omega^{n-1}$ defines a class in Aeppli cohomology which may be integrated
against any class in $(1,1)$-Bott-Chern cohomology, in particular against
$c_{1}(E)$ viewed as a class in $H^{1,1}_{BC}(X,\mathbb{R})$. When
$(X,\omega)$ is Kähler, the intersection product $c_{1}(E)\cdot\omega^{n-1}$
is given by cup product in the cohomology ring $H^{*}(X,\mathbb{R})$.

The following notion of slope-semistability was introduced by Mumford [Mum63]
over curves, and later extended in higher dimensions by Takemoto [Tak72]. We
also recall in Definition 2.3 the Gieseker-Maruyama semistability following
[Mar76, Gie77].

######  Definition 2.1 (Slope-semistability).

A sheaf $E\in\operatorname{Coh}(X)$ is $\omega$-semistable (resp.
$\omega$-stable) if

  1. (1)

$E$ is torsion-free,

  2. (2)

for any subsheaf $F\subset E$ with
$0<\operatorname{rk}(F)<\operatorname{rk}(E)$ we have

$\mu^{\omega}(F)\leq\mu^{\omega}(E)\quad(\text{resp. }<).$

For the terminology, we will also say (semi)stable, for slope-(semi)stable or
for $\omega$-(semi)stable when $\omega$ is clear from the context. A torsion-
free sheaf $E$ is called polystable if it is isomorphic to a direct sum of
stable sheaves of the same slope.

######  Definition 2.2.

A coherent sheaf $E$ on $X$ is called simple if $\operatorname{Hom}(E,E)\cong
k$.

It is immediately shown that stable sheaves are simple.

######  Definition 2.3 (Gieseker-Maruyama-semistability).

In the (AG) and (KG) setups, one defines the Hilbert polynomial of a coherent
sheaf $E$ with respect to $\omega$ by setting

$P_{\omega}(E,m)=\int_{X}\operatorname{ch}(E)e^{m\omega}\operatorname{Todd}_{X}.$

A sheaf $E\in\operatorname{Coh}(X)$ is said to be Gieseker-Maruyama (GM)
semistable (resp. Gieseker-Maruyama-stable) if

  1. (1)

$E$ is torsion-free,

  2. (2)

for any subsheaf $F\subset E$ with
$0<\operatorname{rk}(F)<\operatorname{rk}(E)$ we have

$\frac{P_{\omega}(F,m)}{\operatorname{rk}(F)}\leq\frac{P_{\omega}(E,m)}{\operatorname{rk}(E)}\quad(\text{resp.
}<)\text{ for }m\gg 0.$

######  Remark 2.1.

It is easy to see that we have the following implications for coherent sheaves
in the (AG) or (KG) setups

$\omega\text{-stable}\implies GM\text{-stable}\implies
GM\text{-semistable}\implies\omega\text{-semistable}.$

######  Examples 2.1.

  1. (1)

All torsion-free sheaves of rank one and in particular all line bundles are
stable with respect to any polarization.

  2. (2)

A direct sum of semistable sheaves of the same slope is also semistable.

  3. (3)

The tangent bundle of the complex projective space $\mathbb{P}^{n}$ is stable.

  4. (4)

The tangent bundle of a complex (algebraic or non-algebraic) K3 surface is
stable with respect to any polarization. In positive characteristic the
stability of the tangent bundle is currently unknown [Huy16, Ch. 9, Sect. 4].

  5. (5)

All non-algebraic compact complex surfaces admit irreducible, hence stable,
rank two vector bundles [Brî96]. Recall that by definition a torsion-free
sheaf is irreducible if it admits no coherent subsheaf of intermediate rank.

The notion of slope-semistability fits within the broader context of algebraic
stability conditions introduced by Rudakov [Rud97], and furthemore it
satisfies the Härder-Narasimhan property. That is, for any torsion-free sheaf
$E\in\operatorname{Coh}(X)$ there exists a unique Härder-Narasimhan (HN)
filtration

$0=E_{0}\subset E_{1}\subset\ldots\subset E_{m}=E$

such that the factors $E_{i}/E_{i-1}$ are $\omega$-semistable and

$\mu^{\omega}(E_{1})>\mu^{\omega}(E_{2}/E_{1})>\ldots>\mu^{\omega}(E/E_{m-1}).$

We refer the reader to [HL10, Section 1.3] for a proof in the algebraic case
and to [Kob87, Theorem 5.7.15] in the analytic case.

In general one can “approximate” any semistable sheaf
$E\in\operatorname{Coh}(X)$ by stable sheaves using Seshadri filtrations

$E^{\bullet}:\quad 0=E_{0}\subset E_{1}\subset\ldots\subset E_{m}=E$

with stable factors $E_{i}/E_{i-1}$ of slope
$\mu^{\omega}(E_{i}/E_{i-1})=\mu^{\omega}(E)$. We shall denote by
$\operatorname{gr}^{S}(E^{\bullet})=\bigoplus_{i}E_{i}/E_{i-1}$ the
corresponding graded sheaf of such a filtration.

######  Remark 2.2.

We note that a semistable sheaf $E$ might admit many Seshadri filtrations,
however one can show that the graded module corresponding to any such
filtration is uniquely determined in codimension one. In other words, the
reflexive hull $\operatorname{gr}^{S}(E^{\bullet})^{\vee\vee}$ of the graded
sheaf does not depend on the choice of the Seshadri filtration [HL10, Section
1]. The graded modules of different Seshadri filtrations may however be
distinct, see [BTT17, Example 3.1].

##  3\. Some key results on slope-semistable sheaves

###  3.1. Set-theoretical Kobayashi-Hitchin correspondence

We present here the Kobayashi-Hitchin correspondence which establishes a link
between the algebraic geometric concept of stability and the existence of
Hermite-Einstein metrics in complex differential geometry. It allows the use
of analytic and differential geometric methods in the study of semistable
vector bundles and their moduli spaces in complex geometry. See [LT95] for a
thorough treatment of this subject.

This works in the complex geometrical setup of compact complex manifolds
endowed with a Gauduchon metric. Let us fix a compact Gauduchon manifold
$(X,\omega)$ of dimension $n$. Let $(E,h)$ be a $\mathscr{C}^{\infty}$ complex
vector bundle on $X$, endowed with a Hermitian metric $h$. Then, by
definition, an $h$-unitary connection $A$ on $(E,h)$ is called
$\omega$-Hermite–Einstein if $A$ is integrable and satisfies

$\displaystyle\Lambda_{\omega}F_{A}=-\frac{2\pi
i}{(n-1)!\operatorname{vol}_{\omega}(X)}\mu^{\omega}(E)\text{Id}_{E}.$

Here $\Lambda_{\omega}$ is the adjoint of the Lefschetz operator on forms
given by wedging with $\omega$. Moreover, $A$ is called irreducible if it has
no decomposition $A=A_{1}\oplus A_{2}$ coming from an orthogonal splitting
$E=E_{1}\oplus E_{2}$ of the Hermitian smooth vector bundle $(E,h)$. Note that
the integrability condition on $A$ endows $E$ with a holomorphic structure
$\mathcal{E}_{A}=(E,\overline{\partial}_{A})$ by the Newlander-Nirenberg
theorem.

######  Theorem 3.1.

Let $(E,h_{0})$ be a Hermitian complex vector bundle on the Gauduchon manifold
$(X,\omega)$. If there exists an irreducible $\omega$-Hermite-Einstein
connection on $(E,h_{0})$, then the induced holomorphic structure
$\mathcal{E}_{A}$ on $E$ is $\omega$-stable. Conversely, if $\mathcal{E}$ is
an $\omega$-stable holomorphic structure on $E$, then there exists a Hermitian
metric $h$ on $E$ such that its Chern connection with respect to $\mathcal{E}$
is irreducible $\omega$-Hermite-Einstein on $(E,h)$. This metric $h$ is called
$\omega$-Hermite-Einstein on $\mathcal{E}$ and is unique up to multiplication
by a positive factor.

######  Remark 3.1.

In the setup of Theorem 3.1, if $\mathcal{E}$ is an $\omega$-stable
holomorphic structure on $E$, then there exists an $\omega$-Hermite-Einstein
connection $A$ on $(E,h_{0})$ such that the induced holomorphic structure
$\mathcal{E}_{A}$ on $E$ is isomorphic to $\mathcal{E}$, see [LT95].

###  3.2. Bogomolov inequality

Over a smooth projective variety or compact Kähler manifold, the Bogomolov
inequality expresses a strong topological constraint to which semistable
torsion-free sheaves are subject. It is particularly useful in boundedness
questions, see Theorem 4.4.

We will state the Bogomolov inequality in the framework of algebraic and
Kähler geometry, and then make a remark on its formulation in the Gauduchon
setup.

######  Theorem 3.2 (Bogomolov inequality).

In the zero characteristic (AG) setup and in the (KG) setup, for any
$\omega$-semistable sheaf $E$ on $X$ one has

$\Delta(E)\cdot\omega^{n-2}\geq 0,$

where

$\Delta(E):=2\operatorname{rk}(E)c_{2}(E)-(\operatorname{rk}(E)-1)c_{1}(E)^{2}.$

In the (AG) setup in characteristic $p>0$ and for $E$ $\omega$-semistable, we
have

$\Delta(E)\cdot\omega^{n-2}+\frac{\operatorname{rk}(E)^{2}(\operatorname{rk}(E)-1)^{2}}{(p-1)^{2}}\omega^{n}\geq
0.$

######  Remark 3.2.

In the algebraic case, the theorem was first proved by Bogomolov [Bog79] over
algebraic surfaces in zero characteristic. The general algebraic case in zero
characteristic follows from his result and the Mehta-Ramanathan restriction
theorem 3.3. The positive characteristic case was proved by Langer [Lan04].

######  Remark 3.3.

The inequality $\Delta(E)\cdot\omega^{n-2}\geq 0$ was proved in the Kähler
case by Lübke for holomorphic vector bundles $E$ admitting an
$\omega$-Hermite-Einstein metric, see [Kob87, LT95]. Together with the
Kobayashi-Hitchin correspondence it yields the statement of Theorem 3.2 in the
Kähler case. In fact, Lübke’s proof also applies in the context of a Gauduchon
manifold $(X,\omega)$, leading to a pointwise inequality of $(n,n)$-forms:

$\Delta(E,h)\wedge\omega^{n-2}\geq 0,$

where $h$ is an $\omega$-Hermite-Einstein metric on $E$ and the $(2,2)$-form
$\Delta(E,h)$ is computed using the associated Chern connection.

###  3.3. Restriction theorems

In this subsection we place ourselves in the algebraic setting and present
restriction results for (semi)stable sheaves. These are used in moduli theory
to prove boundedness and general properties of moduli spaces of sheaves.

Let $H$ be an ample divisor representing the polarization $\omega$. We assume
the dimension $n$ of $X$ to be larger than one. Let
$E\in\operatorname{Coh}(X)$ be a torsion-free sheaf. If $D\in|aH|$ is a smooth
divisor for some $a>0$ such that $E|_{D}$ is (semi)stable with respect to
$H|_{D}$, then it is immediate to see that $E$ is also (semi)stable with
respect to $H$. One may wonder if a converse statement holds. The example of
the tangent bundle $\mathcal{T}_{\mathbb{P}_{\mathbb{C}}^{n}}$ shows that some
caution is required. Indeed, its restriction to any hyperplane $D$ is
isomorphic to
$\mathcal{O}_{\mathbb{P}_{\mathbb{C}}^{n-1}}(1)\oplus\mathcal{T}_{\mathbb{P}_{\mathbb{C}}^{n-1}}$,
which is not semistable. However, a positive answer is found if one takes a
general divisor $D\in|aH|$ for $a$ sufficiently large. This is the content of
the Mehta-Ramanathan restriction theorem [MR82, MR84]:

######  Theorem 3.3.

If $E$ is a $H$-(semi)stable sheaf on $X$, then its restriction $E|_{D}$ to a
general divisor $D\in|aH|$ of sufficiently large degree is
$(H|_{D})$-(semi)stable.

More refined restriction theorems which give effective bounds on the degree
$a$ were proved by Flenner [Fle84] in zero characteristic and by Langer
[Lan04, Theorem 5.2 and Corollary 5.4] in mixed characteristic. See also the
recent paper [Fey22] containing effective restriction results. We state here a
variant of Langer’s result.

######  Theorem 3.4.

If $E$ is a $H$-(semi)stable sheaf on $X$, then its restriction $E|_{D}$ to a
general divisor $D\in|aH|$ is $(H|_{D})$-(semi)stable provided that

$a>\frac{\operatorname{rk}(E)-1}{\operatorname{rk}(E)}\Delta(E)H^{n-2}+\frac{1}{\operatorname{rk}(E)(\operatorname{rk}(E)-1)H^{n}}+\frac{(\operatorname{rk}(E)-1)H^{n}}{\operatorname{rk}(E)}\gamma_{\operatorname{rk}(E)},$

where $\gamma_{r}:=0$ if $\operatorname{char}(k)=0$ and
$\gamma_{r}:=\frac{r^{2}(r-1)^{2}}{(p-1)^{2}}$ if
$\operatorname{char}(k)=p>0$.

######  Remark 3.4.

In Theorem 3.4, “general” can be made explicit depending on $E$. More
precisely, if $0=E_{0}\subset\ldots\subset E_{m}=E$ is a Seshadri filtration
of $E$, then the statement holds for any smooth (even normal) divisor
$D\in|aH|$ so that any factor $E_{i}/E_{i-1}$ restricted to $D$ remains
torsion-free.

##  4\. Families of slope-semistable sheaves

In this section we present properties of families of slope-semistable sheaves
which are essential in moduli theory. An $S$-flat family of coherent sheaves
on $X$ is by definition a coherent $\mathcal{O}_{S\times X}$-module
$\mathcal{E}$, flat over $S$. The parameterizing space $S$ is either a
$k$-scheme or a complex analytic space, depending on whether we work in the
algebraic or analytic setup respectively.

###  4.1. Boundedness of sets of coherent sheaves

We recall below what we mean by a bounded set of coherent sheaves on $X$.

######  Definition 4.1.

Let $\mathcal{S}$ be a set of isomorphism classes of coherent sheaves on $X$.
We say that $\mathcal{S}$ is bounded if

  1. (AG)

there exists a scheme $S$ of finite type over $k$ and a coherent sheaf $E$ on
$S\times X$ such that $\mathcal{S}$ is contained in the set of isomorphism
classes of fibers of $E$ over points of $S$ [HL10, Definition 1.7.5].

  2. (CG)

there exists a complex analytic space $S$, a compact subset $K\subset S$ and a
coherent sheaf $E$ on $S\times X$ such that $\mathcal{S}$ is contained in the
set of isomorphism classes of fibers of $E$ over points of $K$ [Tom21,
Definition 5.1].

When $X$ is complex projective, the above two definitions are in fact
equivalent via the GAGA Theorem, cf. [Tom16, Remark 3.3].

Note that if $\mathcal{S}$ is a bounded family of sheaves on $X$, then the
Chern classes (seen in the numerical group of $X$ in the algebraic case, and
in the singular cohomology ring $H^{*}(X,\mathbb{Z})$ in the analytic case
respectively) of the elements in $\mathcal{S}$ range within a finite set.

The following boundedness criterion is due to Grothendieck [Gro61] in the
algebraic case. The analytic version can be found in [Tom21].

######  Proposition 4.1.

Let $\mathcal{S}$ be a set of isomorphism classes of torsion-free sheaves on
$X$. Then $\mathcal{S}$ is bounded if and only if the following two conditions
are fulfilled

  1. (1)

$\mathcal{S}$ is dominated, i.e., there exists a bounded set $\mathcal{T}$ of
classes of coherent sheaves on $X$ such that each element of $\mathcal{S}$ is
a quotient of an element of $\mathcal{T}$,

  2. (2)

the slope function $\mu^{\omega}$ is upper bounded on $\mathcal{S}$.

###  4.2. Openness of semistability

The following result shows that slope-semistability, resp. slope-stability, is
an open property in flat families of sheaves. Its proof is based on the
boundedness criterion stated in Proposition 4.1.

######  Proposition 4.2.

Let $(X,\omega)$ be a polarized space as in our (AG) or (KG) setups. Let
$\mathcal{E}$ be an $S$-flat family of coherent sheaves on $X$. Then the locus
$S^{\circ}$ of closed points $s\in S$ such that $\mathcal{E}|_{\\{s\\}\times
X}$ is $\omega$-semistable (resp. $\omega$-stable) is a Zariski open subset of
$S$.

###### Proof.

See [HL10, Proposition 2.3.1] for a proof in the algebraic case, and [Tom21,
Corollary 6.7] for the analytic case. ∎

######  Remark 4.1.

The above result is in general false for non-Kähler Gauduchon manifolds, see
[Tel08].

###  4.3. Langton’s semistable reduction

We treat the algebraic case. For the analytic case see [Tom20].

######  Proposition 4.3 (Semistable reduction, [Lan75]).

Let $(X,\omega)$ be as in the (AG) setup. Let $R$ be a DVR over $k$ of
quotient field $K$. Let $E$ be an $R$-flat family of coherent sheaves on $X$
such that $E_{K}$ is $\omega$-semistable. Then there exists a subsheaf
$F\subset E$ such that $F_{K}\cong E_{K}$ and such that $F_{k}$ is
$\omega$-semistable.

###  4.4. Boundedness of semistability

The Chern character of a coherent sheaf $E$ on $X$ will determine its
numerical type $\operatorname{ch}(E)$ in the numerical group
$\operatorname{K_{\text{num}}}(X)_{\mathbb{Q}}$ in the (AG) case, respectively
in the singular cohomology group $H^{*}(X,\mathbb{Q})$ in the (CG) case. Given
a class $\gamma$ in $\operatorname{K_{\text{num}}}(X)_{\mathbb{Q}}$,
respectively in $H^{*}(X,\mathbb{Q})$, we consider the following boundedness
statement:

$\textbf{B}_{\gamma}(\omega):$ The set of isomorphism classes of coherent
sheaves $E$ of class $\gamma$ on $X$ that are $\omega$-semistable is bounded.

Boundedness of semistable sheaves was intensively studied and it took the
efforts of many mathematicians to completely solve the algebraic case (e.g.
[Ati57, Kle71, Tak72, Gie77, Mar81, Sim94, Lan04]).

######  Theorem 4.4.

In the (AG) setup, boundedness of semistability $\textbf{B}_{\gamma}(\omega)$
holds for any numerical class $\gamma$.

In the (KG) case, the boundedness statement is not yet known in full
generality. We present here a few results that indicate its validity.

######  Theorem 4.5.

Let $(X,\omega)$ be a compact Kähler manifold and $\gamma$ be a topological
class in $H^{*}(X,\mathbb{Q})$. Then $\textbf{B}_{\gamma}(\omega)$ is known in
the following cases:

  1. (1)

when $\gamma$ is the class of a rank 1 coherent sheaf [Tom16, Corollary 5.5],

  2. (2)

when $X$ is complex projective, but $\omega$ is not necessarily an ample class
[GT17, Proposition 6.3].

  3. (3)

when $X$ is a (not necessarily algebraic) K3 surface or a 2-dimensional torus
and the class $\omega$ is $\gamma$-generic, see [Tom01, Theorem 4.3] and
[PT17, Section 2.2.1] for details.

######  Remark 4.2.

In the non-Kähler case, boundedness of semistability cannot be expected in the
above formulation as can be seen from the following example. Let $X$ be a
class $VII$ surface. Then $b_{1}(X)=1$ and the identity component
$\operatorname{Pic}^{0}(X)$ of the Picard group pf $X$ is isomorphic to
$\mathbb{C}^{*}$. In this case holomorphic line bundles with $c_{1}=0$ form an
unbounded family of semistable sheaves on $X$, with respect to any
polarization.

A possible remedy to this situation would be to fix the first Chern class of
the line bundles in the Bott-Chern cohomology of $X$, and not only in the
singular cohomology. The effect would be to fix the degree of the considered
line bundles. The set parametrizing line bundles of fixed degree in the above
example is a circle in $\mathbb{C}^{*}$ centered at $0$ [LT95, Section 1.3].

##  5\. Moduli functors and moduli stacks

###  5.1. Moduli functors of sheaves

In this subsection we let $\mathcal{C}$ be the category
$(\operatorname{Sch}\\!/k)$ of $k$-schemes in the algebraic geometric setting,
and the category $(\operatorname{An}\\!/\mathbb{C})$ of (not necessarily
separated) complex analytic spaces in the complex geometric setting.

######  Definition 5.1.

Let $F:\mathcal{C}\to(Sets)$ be a contravariant functor and
$\phi:F\to\operatorname{Hom}(-,M)$ a natural transformation of functors where
$M$ is an algebraic space over $k$, respectively an analytic space. We say
that $\phi$ is

  * •

a categorical moduli space for $F$ if any other natural transformation
$\psi:F\to\operatorname{Hom}(-,N)$ with $N$ an algebraic space over $k$,
respectively an analytic space factorizes through $\phi$. One also says that
$M$ corepresents the functor $F$ in this case.

  * •

a coarse moduli space for $F$ if it is a categorical moduli space and moreover
induces a bijection at the level of $k$-points
$F(\operatorname{Spec}k)\to\operatorname{Hom}(\operatorname{Spec}k,M)$,
respectively of $\mathbb{C}$-points.

  * •

a fine moduli space for $F$ if $\phi$ is an isomorphism of functors.

Let $(X,\omega)$ be a polarized space as in the algebraic, respectively Kähler
setup. Given a class $\gamma$ in
$\operatorname{K_{\text{num}}}(X)_{\mathbb{Q}}$, respectively in
$H^{*}(X,\mathbb{Q})$, let
$\operatorname{Coh}_{X,\gamma}:\mathcal{C}\to(Sets)$ denote the functor of
coherent sheaves of class $\gamma$, which sends an object $S\in\mathcal{C}$ to
the set of isomorphism classes of flat families of coherent sheaves of class
$\gamma$ on $X$ parameterized by $S$. In the sequel we will consider the
following subfunctors of the functor $\operatorname{Coh}_{X,\gamma}$:

  * •

$\operatorname{Coh}^{ss}_{X,\omega,\gamma}$ of $\omega$-semistable sheaves,

  * •

$\operatorname{Coh}^{s}_{X,\omega,\gamma}$ of $\omega$-stable sheaves,

  * •

$\operatorname{Coh}^{lf,s}_{X,\omega,\gamma}$ of $\omega$-stable locally free
sheaves,

  * •

$\operatorname{Coh}^{spl}_{X,\gamma}$ of simple torsion-free sheaves [AK80,
KO89],

  * •

$\operatorname{Coh}^{(SLF)}_{X,\omega,\gamma}$ of torsion-free sheaves with
Seshadri locally free graduations [PT24, BS22],

  * •

$\operatorname{Coh}^{(SR)}_{X,\omega,\gamma}$ of torsion-free sheaves with
Seshadri reflexive graduations [PT24].

All these functors are Zariski-open subfunctors of
$\operatorname{Coh}_{X,\gamma}$, see Proposition 4.2 and the above cited
references.

We will discuss the question whether these functors admit a
categorical/coarse/fine moduli space after saying a few words about the
corresponding moduli stacks.

###  5.2. Moduli stacks of sheaves

Since the theory of analytic stacks is less developed, in this subsection we
place ourselves in the algebraic setup and recall some known facts about the
stack of coherent sheaves. For a detailed account on stacks and algebraic
stacks we refer the reader to [LMB00, Ols16, Góm01, Alp24, Sta20].

Consider the category $\mathcal{C}oh_{X}$ whose objects are pairs $(S,E)$,
where $S$ is a scheme over $k$ and $E$ is an $S$-flat family of sheaves on
$X$. A morphism $(S^{\prime},E^{\prime})\to(S,E)$ in $\mathcal{C}oh_{X}$
consists of a map $f:S^{\prime}\to S$ of $k$-schemes together with a morphism
$E\to f_{*}E^{\prime}$ of sheaves whose adjoint is an isomorphism. We
visualize this as a cartesian diagram

$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S}$

This defines a category fibered in groupoids over the category of schemes over
$k$.

######  Proposition 5.1 ([Sta20, Tag 09DS, Tag 0DLY]).

The category $\mathcal{C}oh_{X}$ is an algebraic stack locally of finite type
and with affine diagonal over $k$.

Given a numerical class $\gamma\in\operatorname{K_{\text{num}}}(X)$, we
consider the open substacks

$\mathcal{C}oh_{X,\omega,\gamma}^{ss},\mathcal{C}oh_{X,\omega,\gamma}^{s},\mathcal{C}oh_{X,\omega,\gamma}^{lf,s},\mathcal{C}oh_{X,\gamma}^{spl},\mathcal{C}oh_{X,\omega,\gamma}^{(SLF)},\mathcal{C}oh_{X,\omega,\gamma}^{(SR)}$

of $\mathcal{C}oh_{X}$ corresponding to the functors

$\operatorname{Coh}_{X,\omega,\gamma}^{ss},\operatorname{Coh}_{X,\omega,\gamma}^{s},\operatorname{Coh}_{X,\omega,\gamma}^{lf,s},\operatorname{Coh}_{X,\gamma}^{spl},\operatorname{Coh}_{X,\omega,\gamma}^{(SLF)},\operatorname{Coh}_{X,\omega,\gamma}^{(SR)}.$

Theorem 4.4 implies that $\mathcal{C}oh_{X,\omega,\gamma}^{ss}$ is quasi-
compact, therefore so are $\operatorname{Coh}_{X,\omega,\gamma}^{s},\\\
\operatorname{Coh}_{X,\omega,\gamma}^{lf,s},\operatorname{Coh}_{X,\omega,\gamma}^{(SLF)},\operatorname{Coh}_{X,\omega,\gamma}^{(SR)}$
too. Moreover Proposition 4.3 yields that
$\mathcal{C}oh_{X,\omega,\gamma}^{ss}$ is universally closed over $k$. All
together we obtain:

######  Proposition 5.2.

The substack $\mathcal{C}oh_{X,\omega,\gamma}^{ss}\subset\mathcal{C}oh_{X}$ is
open and a universally closed algebraic stack of finite type and with affine
diagonal over $k$.

As in the case of moduli functors, one can define the notions of
categorical/coarse/fine moduli spaces for algebraic stacks in the following
way.

######  Definition 5.2.

Let $\mathfrak{X}$ be an algebraic stack over $k$ and
$\phi:\mathfrak{X}\to\operatorname{Hom}(-,M)$ a morphism of stacks, where $M$
is an algebraic space over $k$. We say that $\phi$ is

  * •

a categorical moduli space for $\mathfrak{X}$ if any other morphism
$\psi:\mathfrak{X}\to\operatorname{Hom}(-,N)$ with $N$ an algebraic space over
$k$ factorizes through $\phi$.

  * •

a coarse moduli space for $\mathfrak{X}$ if it is a categorical moduli space
and moreover induces a bijection between the set of isomorphism classes of
$k$-points of $\mathfrak{X}$ and $\operatorname{Hom}(\operatorname{Spec}k,M)$.

  * •

a fine moduli space for $\mathfrak{X}$ if $\phi$ is an isomorphism of stacks.

Note that the above moduli stacks never admit fine moduli spaces, since the
automorphism groups of the objects are non-trivial. As to the existence of
categorical or coarse moduli spaces for moduli stacks, this is equivalent to
the existence of categorical or coarse moduli spaces for the corresponding
moduli functors described above.

A more refined version of a categorical moduli space is the following.

######  Definition 5.3 ([Alp13]).

A quasi-compact and quasi-separated morphism $\phi:\mathfrak{X}\to M$ from an
algebraic stack $\mathfrak{X}$ to an algebraic space $M$ is said to be a good
moduli space if

  * •

the pushfoward functor on quasi-coherent sheaves is exact, and

  * •

the induced morphism of sheaves
$\mathcal{O}_{M}\to\phi_{*}\mathcal{O}_{\mathfrak{X}}$ is an isomorphism.

Good moduli spaces are always categorical [Alp13, Theorem 6.6], but not coarse
in general.

A natural question to be discussed next is that of the existence of a
categorical/good/coarse moduli space for the above moduli stacks.

##  6\. Moduli spaces of sheaves

In algebraic and in complex geometry, the first moduli spaces of sheaves of
particular interest were moduli spaces of line bundles and more generally of
vector bundles. In the latter case it was soon observed that in order to
obtain moduli spaces with good geometric properties (such as local-
separatedness) one has to impose some restriction on the class of vector
bundles to be classified. This led Mumford to introduce the slope-stability
condition in [Mum63].

######  Theorem 6.1.

The functor $\operatorname{Coh}_{X,\omega,\gamma}^{lf,s}$ admits a separated
coarse moduli space $M_{X,\omega,\gamma}^{lf,s}$. In the (AG) case,
$M_{X,\omega,\gamma}^{lf,s}$ is a quasi-projective scheme over $k$.

This result was proved in the algebraic geometrical context using Geometric
Invariant Theory methods over curves by Mumford [Mum63], Seshadri [Ses67] (see
also [NS64]), over surfaces by Gieseker [Gie77] and in higher dimensions by
Maruyama [Mar77]. In the analytic setup, the result is a consequence of the
existence of a coarse moduli space of simple vector bundles, proved by Norton
[Nor79] using Banach-analytic techniques, together with the openness of
stability, Proposition 4.2.

######  Remark 6.1.

In general the moduli spaces $M_{X,\omega,\gamma}^{lf,s}$ are rarely fine
[Mar78]. A situation when this is known to happen is when $X$ is a curve,
$\omega$ is the fundamental class of $X$, and the rank and degree of the
concerned vector bundles are coprime.

In complex geometry, moduli spaces of stable vector bundles are related via
the Kobayashi-Hitchin correspondence to moduli spaces of Hermite-Einstein
connections.

######  Theorem 6.2 (Moduli-theoretical Kobayashi-Hitchin correspondence,
[Don87, Kob87, LO87, Miy89, LT95]).

Let $(X,\omega)$ be a Gauduchon compact complex manifold. Let $E$ be a smooth
complex vector bundle on $X$ and $h$ a Hermitian metric on $E$. Then the set-
theoretical Kobayashi-Hitchin correspondence yields a real-analytic
isomorphism

$M^{HE}_{X,\omega,E,h}\to M^{lf,s}_{X,\omega,E}$

between the moduli space of $\omega$-Hermite-Einstein connections on $(E,h)$
and the moduli space of $\omega$-stable holomorphic structures on $E$.

In general the moduli spaces $M_{X,\omega,\gamma}^{lf,s}$ are not compact. If
one aims at constructing natural compactifications, one generally needs to
relax both the locally-freeness and the stability condition of the coherent
sheaves to be parametrized.

We have already mentioned moduli spaces of simple vector bundles. These extend
to (not necessarily separated) moduli spaces of simple torsion-free sheaves.

######  Theorem 6.3 ([AK80, KO89]).

The functor $\operatorname{Coh}_{X,\gamma}^{spl}$ admits a coarse moduli space
$M_{X,\gamma}^{spl}$.

######  Corollary 6.4.

The functor $\operatorname{Coh}_{X,\omega,\gamma}^{s}$ admits a separated
coarse moduli space $M_{X,\omega,\gamma}^{s}$ as an open subset in
$M_{X,\gamma}^{spl}$.

We note that in the algebraic setup one can further show that the moduli space
$M_{X,\omega,\gamma}^{s}$ is quasi-projective over $k$ [Mar77].

When $X$ is a curve, the functor $\operatorname{Coh}_{X,\omega,\gamma}^{ss}$
admits a projective categorical moduli space $M_{X,\omega,\gamma}^{ss}$, which
contains $M_{X,\omega,\gamma}^{s}$ as an open subscheme [Ses67]. The geometric
points of $M_{X,\omega,\gamma}^{ss}$ correspond to isomorphism classes of
Seshadri graduations of semistable sheaves, and therefore
$M_{X,\omega,\gamma}^{ss}$ is not a coarse moduli space in general.

When trying to employ Geometric Invariant Theory to construct
compactifications of $M_{X,\omega,\gamma}^{s}$ in higher dimensions, one is
led to consider the Gieseker-Maruyama-semistability condition, see Definition
2.3.

######  Theorem 6.5 ([Gie77, Mar77, Sim94, Alp13]).

In the algebraic setup and in zero characteristic, the substack
$\mathcal{C}oh_{X,\omega,\gamma}^{GMss}\subset\mathcal{C}oh_{X,\gamma}$ of
Gieseker-Maruyama-semistable sheaves is open and admits a projective good
moduli space $M_{X,\omega,\gamma}^{GMss}$.

######  Remark 6.2.

In positive characteristic, the above statement holds if one replaces “good
moduli” by “adequate moduli”, see Alper [Alp14].

######  Remark 6.3.

In the general (KG) setup, the existence of the Gieseker-Maruyama moduli space
is generally unknown. There exist partial results when $X$ is complex
projective and $\omega$ is a non-ample Kähler class [GT20].

A different way to enlarge the open substack
$\mathcal{C}oh_{X,\omega,\gamma}^{lf,s}$ is to look at
$\mathcal{C}oh_{X,\omega,\gamma}^{(SLF)}$ and
$\mathcal{C}oh_{X,\omega,\gamma}^{(SR)}$. For these one still gets good moduli
spaces in the (AG) setting and characteristic zero [PT24]. See also [BS22] for
the complex analytic case.

##  7\. Further topics

###  7.1. Donaldson-Uhlenbeck compactification

In this subsection we will consider the case when $(X,\omega)$ is a polarized
smooth complex projective surface. Let $E$ be a smooth complex vector bundle
on $X$ and $h$ a Hermitian metric on $E$. Recall that by the moduli-
theoretical Kobayashi-Hitchin correspondence there is a real-analytic
isomorphism

$M^{HE}_{X,\omega,E,h}\to M^{lf,s}_{X,\omega,E}$

between the moduli space of $\omega$-Hermite-Einstein connections on $(E,h)$
and the moduli space of $\omega$-stable holomorphic structures on $E$.

It is important in Donaldson theory to work with suitable compactifications of
moduli spaces of anti-self-dual connections. These were constructed by
Donaldson based on compactness results due to Uhlenbeck [DK90] and lead also
to a compactification $M^{DU}_{{X,\omega,E,h}}$ of $M^{HE}_{X,\omega,E,h}$,
which we call the Donaldson-Uhlenbeck compactification.

On the algebraic geometrical side, we have already seen a compactification of
the moduli space $M^{lf,s}_{X,\omega,E}$ of slope-stable vector bundles by
adding Gieseker-Maruyama-semistable torsion-free sheaves at the boundary,
which is the Gieseker-Maruyama moduli space
$M^{GMss}_{X,\omega,\operatorname{ch}(E)}$. Le Potier [Le 92] and Jun Li
[Li93] constructed a projective morphism

$\varphi:M^{GMss}_{X,\omega,\operatorname{ch}(E)}\to\mathbb{P}_{\mathbb{C}}^{N}$

which is an immersion on $M^{lf,s}_{X,\omega,E}$. Furthermore, Li proved that
the closure of the image $\varphi(M^{lf,s}_{X,\omega,E})$ inside
$\mathbb{P}_{\mathbb{C}}^{N}$ is homeomorphic to the Donaldson-Uhlenbeck
compactification $M^{DU}_{{X,\omega,E,h}}$. This extends the inverse of the
Kobayashi-Hitchin correspondence as a homeomorphism of compact spaces

$\overline{\varphi(M^{lf,s}_{X,\omega,E})}\to M^{DU}_{{X,\omega,E,h}}.$

In particular one can transfer the complex algebraic structure of
$\overline{\varphi(M^{lf,s}_{X,\omega,E})}$ to the Donaldson-Uhlenbeck
compactification.

As a further compactification of $M^{lf,s}_{X,\omega,\gamma}$, Huybrechts and
Lehn constructed in [HL10, Chapter 8] a complex projective moduli space
$M^{\mu ss}_{X,\omega,\gamma}$ of slope-semistable sheaves over a smooth
surface, which comes together with a natural transformation of functors

$\operatorname{Coh}_{X,\omega,\gamma}^{ss}\to\operatorname{Hom}(-,M^{\mu
ss}_{X,\omega,\gamma}).$

However, $M^{\mu ss}_{X,\omega,\gamma}$ does not corepresent the moduli
functor in general.

Similar results were obtained by Greb, Sibley, Toma, Wentworth [GSTW21] in
dimension larger than two using the analogue of the Donaldson-Uhlenbeck
compactification due to Tian [Tia00] and the higher dimensional analogue of
the Huybrechts-Lehn moduli space $M^{\mu ss}_{X,\omega,\gamma}$.

###  7.2. Moduli of pure sheaves

Until now we have only considered classification problems of torsion-free
sheaves. It is however natural to extend this research to lower-dimensional
coherent sheaves. The analogue of the torsion-free condition in this situation
is the purity condition.

######  Definition 7.1.

A coherent sheaf $E$ on $X$ is said to be pure of dimension $d$ if any non-
trivial coherent subsheaf $F\subset E$ has dimension $d$ too.

One defines the notions of $\omega$-(semi)stability and $GM$-(semi)stability
also for pure sheaves in the (AG) and (KG) setups. For this one writes the
Hilbert polynomial of a coherent sheaf $E$ of dimension $d$ on $X$ in the
following form

$P_{\omega}(E,m)=\sum_{i=0}^{d}\alpha_{i}(E)m^{i},\quad\text{where
}\alpha_{i}(E)=\frac{1}{i!}\int_{X}\operatorname{ch}(E)\omega^{i}\operatorname{Todd}_{X}.$

######  Definition 7.2.

A coherent sheaf $E$ of dimension $d$ on $(X,\omega)$ is said to be
$\omega$-(semi)stable if

  1. (1)

$E$ is pure,

  2. (2)

for any coherent subsheaf $F\subset E$ with $0<\alpha_{d}(F)<\alpha_{d}(E)$ we
have

$\frac{\alpha_{d-1}(F)}{\alpha_{d}(F)}\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle($}}\leq\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle)$}}\frac{\alpha_{d-1}(E)}{\alpha_{d}(E)}.$

######  Definition 7.3.

A coherent sheaf $E$ of dimension $d$ on $(X,\omega)$ is said to be
$GM$-(semi)stable if

  1. (1)

$E$ is pure,

  2. (2)

for any coherent subsheaf $F\subset E$ with $0<\alpha_{d}(F)<\alpha_{d}(E)$ we
have

$\frac{P_{\omega}(F,m)}{\alpha_{d}(F)}\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle($}}\leq\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle)$}}\frac{P_{\omega}(E,m)}{\alpha_{d}(E)}\quad\text{for
}m\gg 0.$

One can also consider the following notion of semistability that interpolates
between slope-semistability and GM-semistability.

######  Definition 7.4.

For integers $1\leq\ell\leq d\leq n$, a coherent sheaf $E$ of dimension $d$ on
$(X,\omega)$ is said to be $\ell$-(semi)stable if

  1. (1)

$E$ is pure,

  2. (2)

for any coherent subsheaf $F\subset E$ with $0<\alpha_{d}(F)<\alpha_{d}(E)$ we
have

$\frac{\sum_{i=d-\ell}^{d}\alpha_{i}(F)m^{i}}{\alpha_{d}(F)}\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle($}}\leq\mathrel{\raisebox{-2.15277pt}{$\scriptscriptstyle)$}}\frac{\sum_{i=d-\ell}^{d}\alpha_{i}(E)m^{i}}{\alpha_{d}(E)}\quad\text{for
}m\gg 0.$

For any class $\gamma$ and $\ell$ between $1$ and $\dim(\gamma)$, we obtain a
chain of corresponding open subfunctors of $\ell$-semistable sheaves

$\operatorname{Coh}_{X,\omega,\gamma}^{GMss}\subset\operatorname{Coh}_{X,\omega,\gamma}^{\ell
ss}\subset\operatorname{Coh}_{X,\omega,\gamma}^{ss}\subset\operatorname{Coh}_{X,\gamma}.$

The existence of moduli spaces of pure sheaves has been established in various
situations:

  * •

The case of simple pure sheaves is settled in [AK80] and [KO89].

  * •

In the characteristic zero (AG) context, Simpson [Sim94] proved the existence
of a projective categorical moduli space for
$\operatorname{Coh}_{X,\omega,\gamma}^{GMss}$ using techniques of Geometric
Invariant Theory; the positive characteristic case is to be found in [Mar16].

  * •

Analogues of the Huybrechts-Lehn moduli spaces for $\ell$-semistable sheaves
were constructed in [Pav22].

###  7.3. Change of semistability and wall-crossing

In this subsection we restrict the discussion to the algebraic geometric
setting. When the dimension of $X$ is larger than one, the moduli spaces of
semistable sheaves above depend on the choice of the polarization $\omega$ in
the ample cone $\operatorname{Amp}^{1}(X)$. The situation which seems to
appear in general is that for any numerical class
$\gamma\in\operatorname{K_{\text{num}}}(X)$ there exists a locally finite set
$\mathfrak{W}$ of real algebraic hypersurfaces
$\mathfrak{w}\subset\operatorname{Amp}^{1}(X)$, called walls, leading to a
decomposition into connected components, called chambers, of

$\operatorname{Amp}^{1}(X)\setminus\bigcup_{\mathfrak{w}\in\mathfrak{W}}\mathfrak{w}$

accounting for the change of (semi)stability in the following sense. If
$\omega_{1},\omega_{2}$ are ample classes in the same chamber, then a coherent
sheaf $E$ of class $\gamma$ is $\omega_{1}$-(semi)stable if and only if $E$ is
$\omega_{2}$-(semi)stable. If this is indeed the case, then moduli spaces with
respect to $\omega_{1}$ and $\omega_{2}$ coincide. The next step to understand
the variation of the moduli spaces depending on the polarization would be to
study wall-crossing, i.e. the relation between moduli spaces corresponding to
adjacent chambers.

The existence of a chamber structure as above is guaranteed once the following
stronger boundedness property for semistable sheaves is established.

######  Definition 7.5 (Uniform boundedness of semistable sheaves).

Given a numerical class $\gamma\in\operatorname{K_{\text{num}}}(X)$, we say
that the uniform boundedness of semistability holds for $\gamma$ if for any
compact subset $K\subset\operatorname{Amp}^{1}(X)_{\mathbb{R}}$, the set of
isomorphism classes of coherent sheaves of type $\gamma$ on $X$ that are
$\omega$-semistable with respect to some $\omega\in K$ is bounded.

In the surface case uniform boundedness is established using the Bogomolov
inequality and the Hodge Index Theorem [Qin93, Göt96, FQ95, EG95, MW97]. In
this case the resulting walls are moreover rational linear, a situation which
is propitious to the study of wall-crossing phenomena, see the above
references for such studies.

Another situation where a rational linear chamber structure exists is the case
of two-dimensional pure sheaves [PRT24].

In higher dimensions a wall and chamber structure exists [GT17], however walls
are neither linear nor rational in general [Qin93, Sch00]. This problem was
circumvented in [GRT16b, GRT16a] by introducing a more refined notion of
semistability for which wall-crossing is well-behaved.

## References

  * [AK80] Allen B. Altman and Steven L. Kleiman, _Compactifying the Picard scheme_ , Adv. Math. 35 (1980), 50–112 (English).
  * [Alp13] Jarod Alper, _Good moduli spaces for Artin stacks_ , Ann. Inst. Fourier 63 (2013), no. 6, 2349–2402 (English).
  * [Alp14] by same author, _Adequate moduli spaces and geometrically reductive group schemes_ , Algebr. Geom. 1 (2014), no. 4, 489–531 (English).
  * [Alp24] by same author, _Stacks and moduli_ , 2024, https://sites.math.washington.edu/~jarod/moduli.pdf.
  * [Ati57] Michael F. Atiyah, _Vector bundles over an elliptic curve_ , Proc. Lond. Math. Soc. (3) 7 (1957), 414–452 (English).
  * [Bog79] F. A. Bogomolov, _Holomorphic tensors and vector bundles on projective varieties_ , Math. USSR, Izv. 13 (1979), 499–555 (English).
  * [Brî96] Vasile Brînzănescu, _Holomorphic vector bundles over compact complex surfaces_ , Lect. Notes Math., vol. 1624, Berlin: Springer, 1996 (English).
  * [BS22] Nicholas Buchdahl and Georg Schumacher, _An analytic application of Geometric Invariant Theory. II: Coarse moduli spaces_ , J. Geom. Phys. 175 (2022), 13 (English), Id/No 104467.
  * [BTT17] Nicholas Buchdahl, Andrei Teleman, and Matei Toma, _A continuity theorem for families of sheaves on complex surfaces_ , J. Topol. 10 (2017), no. 4, 995–1028 (English).
  * [DK90] S. K. Donaldson and P. B. Kronheimer, _The geometry of four-manifolds_ , Oxford Math. Monogr., Oxford: Clarendon Press, 1990 (English).
  * [Don87] S. K. Donaldson, _Irrationality and the h-cobordism conjecture_ , J. Differ. Geom. 26 (1987), 141–168 (English).
  * [EG95] Geir Ellingsrud and Lothar Göttsche, _Variation of moduli space and Donaldson invariants under change of polarization_ , J. Reine Angew. Math. 467 (1995), 1–49 (English).
  * [Fey22] Soheyla Feyzbakhsh, _An effective restriction theorem via wall-crossing and Mercat’s conjecture_ , Math. Z. 301 (2022), no. 4, 4175–4199 (English).
  * [Fle84] Hubert Flenner, _Restrictions of semistable bundles on projective varieties_ , Comment. Math. Helv. 59 (1984), 635–650 (English).
  * [FQ95] Robert Friedman and Zhenbo Qin, _Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces_ , Commun. Anal. Geom. 3 (1995), no. 1, 11–83 (English).
  * [Gie77] D. Gieseker, _On the moduli of vector bundles on an algebraic surface_ , Ann. Math. (2) 106 (1977), 45–60 (English).
  * [Góm01] Tomás L. Gómez, _Algebraic stacks_ , Proc. Indian Acad. Sci., Math. Sci. 111 (2001), no. 1, 1–31 (English).
  * [Göt96] Lothar Göttsche, _Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces_ , Math. Z. 223 (1996), no. 2, 247–260 (English).
  * [Gro61] Alexandre Grothendieck, _Techniques de construction et théoremes d’existence en géométrie algébrique. IV: Les schemas de Hilbert_ , Sem. Bourbaki 13(1960/61), No. 221, 28 p. (1961)., 1961.
  * [GRT16a] Daniel Greb, Julius Ross, and Matei Toma, _Moduli of vector bundles on higher-dimensional base manifolds - construction and variation_ , Int. J. Math. 27 (2016), no. 7, 27 (English), Id/No 1650054.
  * [GRT16b] by same author, _Variation of Gieseker moduli spaces via quiver GIT_ , Geom. Topol. 20 (2016), no. 3, 1539–1610 (English).
  * [GSTW21] Daniel Greb, Benjamin Sibley, Matei Toma, and Richard Wentworth, _Complex algebraic compactifications of the moduli space of Hermitian Yang-Mills connections on a projective manifold_ , Geom. Topol. 25 (2021), no. 4, 1719–1818 (English).
  * [GT17] Daniel Greb and Matei Toma, _Compact moduli spaces for slope-semistable sheaves_ , Algebr. Geom. 4 (2017), no. 1, 40–78 (English).
  * [GT20] Daniel Greb and Matei Toma, _Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation_ , J. Éc. Polytech., Math. 7 (2020), 233–261 (English).
  * [HL10] Daniel Huybrechts and Manfred Lehn, _The geometry of moduli spaces of sheaves. 2nd ed_ , Cambridge University Press, 2010 (English).
  * [Huy16] Daniel Huybrechts, _Lectures on $K$3 surfaces_, Camb. Stud. Adv. Math., vol. 158, Cambridge: Cambridge University Press, 2016 (English).
  * [Kle71] S. Kleiman, _The finiteness theorems for the Picard functor._ , Sem. Geom. algebrique Bois Marie 1966/67, SGA 6, Lect. Notes Math. 225, 616-666 (1971)., 1971.
  * [KO89] Siegmund Kosarew and Christian Okonek, _Global moduli spaces and simple holomorphic bundles_ , Publ. Res. Inst. Math. Sci. 25 (1989), no. 1, 1–19.
  * [Kob87] Shoshichi Kobayashi, _Differential geometry of complex vector bundles_ , Publ. Math. Soc. Japan, vol. 15, Princeton, NJ: Princeton University Press; Tokyo: Iwanami Shoten Publishers, 1987 (English).
  * [Lan75] Stacy G. Langton, _Valuative criteria for families of vector bundles on algebraic varieties_ , Ann. Math. (2) 101 (1975), 88–110 (English).
  * [Lan04] Adrian Langer, _Semistable sheaves in positive characteristic_ , Ann. Math. (2) 159 (2004), no. 1, 251–276 (English).
  * [Le 92] Joseph Le Potier, _Fibré déterminant et courbes de saut sur les surfaces algébriques_ , Complex projective geometry, 1989 in Bergen, Cambridge University Press, 1992, pp. 213–240 (French).
  * [Li93] Jun Li, _Algebraic geometric interpretation of Donaldson’s polynomial invariants_ , J. Differ. Geom. 37 (1993), no. 2, 417–466 (English).
  * [LMB00] Gérard Laumon and Laurent Moret-Bailly, _Champs algébriques_ , Ergeb. Math. Grenzgeb., 3. Folge, vol. 39, Berlin: Springer, 2000 (French).
  * [LO87] M. Lübke and C. Okonek, _Moduli spaces of simple bundles and Hermitian-Einstein connections_ , Math. Ann. 276 (1987), 663–674 (English).
  * [LT95] Martin Lübke and Andrei Teleman, _The Kobayashi-Hitchin correspondence_ , Singapore: World Scientific, 1995 (English).
  * [Mar76] Masaki Maruyama, _Openness of a family of torsion free sheaves_ , J. Math. Kyoto Univ. 16 (1976), 627–637 (English).
  * [Mar77] Masaki Maruyama, _Moduli of stable sheaves. I_ , J. Math. Kyoto Univ. 17 (1977), 91–126 (English).
  * [Mar78] Masaki Maruyama, _Moduli of stable sheaves. II_ , J. Math. Kyoto Univ. 18 (1978), 557–614 (English).
  * [Mar81] by same author, _On boundedness of families of torsion free sheaves_ , J. Math. Kyoto Univ. 21 (1981), 673–701 (English).
  * [Mar16] by same author, _Moduli spaces of stable sheaves on schemes: restriction theorems, boundedness and the GIT construction. With collaboration of T. Abe and M. Inaba_ , MSJ Mem., vol. 33, Tokyo: Mathematical Society of Japan, 2016 (English).
  * [Miy89] Kimio Miyajima, _Kuranishi family of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections_ , Publ. Res. Inst. Math. Sci. 25 (1989), no. 2, 301–320 (English).
  * [MR82] V. B. Mehta and A. Ramanathan, _Semistable sheaves on projective varieties and their restriction to curves_ , Math. Ann. 258 (1982), 213–224 (English).
  * [MR84] by same author, _Restriction of stable sheaves and representations of the fundamental group_ , Invent. Math. 77 (1984), 163–172 (English).
  * [MS16] Nicole Mestrano and Carlos Simpson, _Moduli of sheaves_ , Development of moduli theory – Kyoto 2013. Proceedings of the 6th Mathematical Society of Japan-Seasonal Institute, MSJ-SI, Kyoto, Japan, June 11–21, 2013, Tokyo: Mathematical Society of Japan (MSJ), 2016, pp. 77–172 (English).
  * [Mum63] D. Mumford, _Projective invariants of projective structures and applications_ , Proc. Int. Congr. Math. 1962, 526-530 (1963)., 1963.
  * [MW97] Kenji Matsuki and Richard Wentworth, _Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface_ , Int. J. Math. 8 (1997), no. 1, 97–148 (English).
  * [Nor79] V. Alan Norton, _Analytic moduli of complex vector bundles_ , Indiana Univ. Math. J. 28 (1979), 365–387 (English).
  * [NS64] M. S. Narasimhan and C. S. Seshadri, _Holomorphic vector bundles on a compact Riemann surface_ , Math. Ann. 155 (1964), 69–80 (English).
  * [Ols16] Martin Olsson, _Algebraic spaces and stacks_ , Colloq. Publ., Am. Math. Soc., vol. 62, Providence, RI: American Mathematical Society (AMS), 2016 (English).
  * [Pav22] Mihai-Cosmin Pavel, _Moduli spaces of semistable sheaves_ , 2022, PhD thesis.
  * [PRT24] Mihai Pavel, Julius Ross, and Matei Toma, _Uniform boundedness of semistable pure sheaves on projective manifolds_ , 2024.
  * [PT17] Arvid Perego and Matei Toma, _Moduli spaces of bundles over nonprojective $K3$ surfaces_, Kyoto J. Math. 57 (2017), no. 1, 107–146 (English).
  * [PT24] Mihai Pavel and Matei Toma, _Moduli spaces of slope-semistable sheaves with reflexive seshadri graduations_ , 2024.
  * [Qin93] Zhenbo Qin, _Equivalence classes of polarizations and moduli spaces of sheaves_ , J. Differ. Geom. 37 (1993), no. 2, 397–415 (English).
  * [Rud97] Alexei Rudakov, _Stability for an abelian category_ , J. Algebra 197 (1997), no. 1, 231–245 (English).
  * [Sch00] Alexander Schmitt, _Walls of Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases_ , Comment. Math. Helv. 75 (2000), no. 2, 216–231 (English).
  * [Ses67] C. S. Seshadri, _Space of unitary vector bundles on a compact Riemann surface_ , Ann. Math. (2) 85 (1967), 303–336 (English).
  * [Sim94] Carlos T. Simpson, _Moduli of representations of the fundamental group of a smooth projective variety. I_ , Publ. Math., Inst. Hautes Étud. Sci. 79 (1994), 47–129 (English).
  * [Sta20] The Stacks project authors, _The stacks project_ , https://stacks.math.columbia.edu, 2020.
  * [Tak72] Fumio Takemoto, _Stable vector bundles on algebraic surfaces_ , Nagoya Math. J. 47 (1972), 29–48 (English).
  * [Tel08] Andrei Teleman, _Families of holomorphic bundles_ , Commun. Contemp. Math. 10 (2008), no. 4, 523–551 (English).
  * [Tia00] Gang Tian, _Gauge theory and calibrated geometry. I_ , Ann. Math. (2) 151 (2000), no. 1, 193–268 (English).
  * [Tom01] Matei Toma, _Compact moduli spaces of stable sheaves over non-algebraic surfaces_ , Doc. Math. 6 (2001), 9–27 (English).
  * [Tom16] by same author, _Bounded sets of sheaves on Kähler manifolds_ , J. Reine Angew. Math. 710 (2016), 77–93 (English).
  * [Tom20] by same author, _Properness criteria for families of coherent analytic sheaves_ , Algebr. Geom. 7 (2020), no. 4, 486–502 (English).
  * [Tom21] by same author, _Bounded sets of sheaves on relative analytic spaces_ , Ann. Henri Lebesgue 4 (2021), 1531–1563 (English).