# Multiple magneto-ionic regimes in Ta/Co20Fe60B20/HfO2

R. Pachat  Centre de Nanosciences et de Nanotechnologies, CNRS, Université
Paris-Saclay, 91120 Palaiseau, France  D. Ourdani  Laboratoire des Sciences
des Procédés et des Matériaux, CNRS-UPR 3407, Université Paris 13, Sorbonne
Paris Cité, 93430 Villetaneuse, France  J. W. van der Jagt  Spin-Ion
technologies, C2N, 10 Boulevard Thomas Gobert, 91120 Palaiseau, France  M.-A.
Syskaki  Singulus Technologies AG, Hanauer Landstrasse 103, 63796 Kahl am
Main, Germany  A. Di Pietro  Istituto Nazionale di Ricerca Metrologica, Strada
delle Cacce 91, 10135 Torino, Italy  Y. Roussigné  Laboratoire des Sciences
des Procédés et des Matériaux, CNRS-UPR 3407, Université Paris 13, Sorbonne
Paris Cité, 93430 Villetaneuse, France  S. Ono  Central Research Institute of
Electric Power Industry, Yokosuka,Kanagawa 240-0196, Japan  M. S. Gabor
Center for Superconductivity, Spintronics and Surface Science, Physics and
Chemistry Department, Technical University of Cluj-Napoca, Cluj-Napoca
RO-400114, Romania  M. Chérif  Laboratoire des Sciences des Procédés et des
Matériaux, CNRS-UPR 3407, Université Paris 13, Sorbonne Paris Cité, 93430
Villetaneuse, France  G. Durin  Istituto Nazionale di Ricerca Metrologica,
Strada delle Cacce 91, 10135 Torino, Italy  J. Langer  Singulus Technologies
AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany  M. Belmeguenai
Laboratoire des Sciences des Procédés et des Matériaux, CNRS-UPR 3407,
Université Paris 13, Sorbonne Paris Cité, 93430 Villetaneuse, France  D.
Ravelosona  Centre de Nanosciences et de Nanotechnologies, CNRS, Université
Paris-Saclay, 91120 Palaiseau, France  Spin-Ion technologies, C2N, 10
Boulevard Thomas Gobert, 91120 Palaiseau, France  L. Herrera Diez
<EMAIL_ADDRESS>Centre de Nanosciences et de
Nanotechnologies, CNRS, Université Paris-Saclay, 91120 Palaiseau, France

###### Abstract

In Ta/CoFeB/HfO2 stacks a gate voltage drives, in a nonvolatile way, the
system from an underoxidized state exhibiting in-plane anisotropy (IPA) to an
optimum oxidation level resulting in perpendicular anisotropy (PMA) and
further into an overoxidized state with IPA. The IPA$\,\to\,$PMA regime is
found to be significantly faster than the PMA$\,\to\,$IPA regime, while only
the latter shows full reversibility under the same gate voltages. The
effective damping parameter also shows a marked dependence with gate voltage
in the IPA$\,\to\,$PMA regime, going from 0.029 to 0.012, and only a modest
increase to 0.014 in the PMA$\,\to\,$IPA regime. The existence of two magneto-
ionic regimes has been linked to a difference in the chemical environment of
the anchoring points of oxygen species added to underoxidized or overoxidized
layers. Our results show that multiple magneto-ionic regimes can exist in a
single device and that their characterization is of great importance for the
design of high performance spintronics devices.

Controlling magnetic properties with electric (E) fields is of great
importance for spintronics applications due to its potential for lowering
power consumption in novel memory prototypes. It has been shown that E-field
induced charge accumulation can lead to important changes in magnetic
anisotropy Weisheit et al. (2007); Daalderop et al. (1991); Nakamura et al.
(2009); Duan et al. (2008), which can be used to reliably control domain wall
(DW) pinning and velocities Bernand-Mantel et al. (2013); Liu et al. (2017),
and assist the magnetization switching in magnetic tunnel junctions Wang et
al. (2012) in ferromagnetic (FM) metallic films. More complex mechanisms like
switching between ferromagnetic and skyrmionic states Hsu et al. (2017);
Schott et al. (2017) have been also demonstrated, where the E-field control of
the Dzyaloshinskii-Moriya Interaction (DMI) Srivastava et al. (2018); Koyama
et al. (2018) is also at play.
In addition to the effects of charge accumulation, magneto-ionics can also
provide nonvolatility and a more extended effect over the entire magnetic
layer, unlike charge accumulation, limited by electrostatic screening Zhang
(1999). Large magneto-ionic effects in magnetic anisotropy can be achieved,
including a spin reorientation transition, which applied to DW motion can be
used to create nonvolatile and very efficient domain wall traps Bauer et al.
(2013, 2015). In addition, magneto-ionic control of spin accumulation Mishra
et al. (2019) and DMI Herrera Diez et al. (2019) have been demonstrated as
well as DW chirality switching in a controlled oxygen atmosphere Chen et al.
(2020), which is of capital importance for implementing E-field assisted spin-
orbit torques in devices. In these ionic materials, the migration of oxygen
Bauer et al. (2015); Gilbert et al. (2016) or hydrogen species Tan et al.
(2019) is at the heart of the ionic effects observed, which have recently been
shown to offer operation speeds close to 1 ms Lee et al. (2020).
The interest in nonvolatility for spintronics applications goes hand in hand
with magneto-ionic reversibility. Recently, studies have focused on the
magneto-ionic reversibility variations between

Figure 1: Graphic representation of the magneto-ionic stack covered with the
ionic liquid [EMI]+[TFSI]- gate. The gate voltage induces the motion of oxygen
species in HfO2.

oxides Fassatoui (2020), pointing out the importance of the differences in the
mechanisms governing ionic conduction. A deep understanding of the mechanisms
of reversibility in magneto-ionics is therefore needed to design high
performing magneto-electric devices. In this study, we unveil a higher degree
of complexity in HfO2 based devices, where two distinct magneto-ionic regimes
have been identified in a single structure. In addition, we also shine light
on the magneto-ionic effects on the effective damping parameter
$\alpha_{\mathrm{eff}}$, largely overlooked in the literature and of great
importance for fast magnetization dynamics. We show that
$\alpha_{\mathrm{eff}}$ can be significantly reduced by inducing ionic motion
and that its lowest value (0.012) coincides with the appearance of PMA.
The magnetic materials used in this study are amorphous Ta(5 nm)/Co20Fe60B20(1
nm)/HfO2(3 nm) films grown by magnetron sputtering, all samples investigated
here were cut from the same wafer. The ionic liquid (IL) [EMI][TFSI]
(1-Ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide) is added to
the surface of the film to incorporate the IL gate, which can provide high
E-fields that have been shown to induce ionic motion in a variety of materials
Jeong et al. (2013); Shi et al. (2013); Leighton (2019). The thickness of the
IL is macroscopic, in the range of several hundreds of $\mu$m. However, the
effective IL thickness is estimated taking into account only the distance over
which an electric double layer is formed at the side of each one of the
electrodes, which is 1 nm Ono et al. (2008). A counter electrode, a glass
substrate coated with a 100 nm thick indium tin oxide (ITO) layer is
subsequently placed on top of the IL. The size of the E-field biased area is
of about 0.25 cm2. Samples were stored in air and at room temperature before
conducting the gating experiments if not indicated otherwise. The hysteresis
loops have been measured by anomalous Hall effect using a bias current of 400
$\mu$A. All magnetic states presented here are nonvolatile, all measurements
were conducted after switching off the gate voltage. A graphic representation
of the device geometry is presented in Fig. 1.
In the initial state, before exposure to the gate voltage, the

Figure 2: A progressive oxidation is induced upon exposure to a gate voltage
GV=-2 V. Different exposure times t drive the system from IPA (a, initial
state) through regime I (b) into PMA (c) and back to IPA through regime II
(d).

magnetic layers are underoxidized, exhibiting in-plane magnetic anisotropy
(IPA). Under the action of a negative gate voltage applied to the top ITO
electrode a first regime (I) is identified, in which IPA transitions into PMA,
followed by a second regime (II) where further oxidation drives PMA into IPA.
This entire anisotropy evolution is presented in Fig.2 with the corresponding
cumulative biasing times for each case, these values correspond to the
accumulated time of all previous operations. The gate voltage GV=-2 V was
applied for a total, accumulated time of t=1380 s. It is well known that thin
magnetic films can exhibit a window of oxidation levels at the interface with
an oxide which promotes PMA, while under/over oxidation will result in IPA
Manchon et al. (2008). Oxygen species are also known to migrate from the HfO2
toward the CoFeB layer under gate voltages, as reported for other HfO2 based
devices Herrera Diez et al. (2019); Nagata et al. (2010), therefore oxygen
migration is thought to be at the heart of the effects observed here. In the
following, we will describe in detail the two magneto-ionic regimes and the
impact of ionic diffusion on the effective damping parameter
$\alpha_{\mathrm{eff}}$.

##  I Magneto-ionic regimes I and II

Fig.3 (a) shows the time dependence of the out-of-plane remanence percentage,
where 100% corresponds to the Hall voltage at zero applied magnetic field in
the PMA state, for the entire anisotropy rang going from the initial
underoxidized IPA state through regime I (positive slope), PMA (the point of
highest remanence), and regime II (negative slope) to finish in an
overoxidized IPA state. As-grown samples have been investigated under three
different constant gate voltages of GV=-2.3 V (circles), GV=-2 V (squares),
and GV=-1.7 V (triangles). The speed of the entire process critically
increases for higher gate voltages, responding to a stronger driving force for
ionic motion. For all applied gate voltages, the magneto-ionic process in
regime I is significantly faster than in regime II.

Figure 3: Remanence as a function of gating time in (a) as-grown samples under
GV=-2.3 V (circles), GV=-2 V (squares), and GV=-1.7 V (triangles). (b) Samples
annealed at 180 °C (half open squares) and 350 °C (open squares) under GV=-2
V.

This difference in the time evolution as a function of gate voltage between
regime I and II could be linked to a difference in the chemical environment,
and thus of anchoring, of the oxygen species interacting with the CoFeB layer
in regimes I and II. In regime I, the energy barrier associated with adding
oxygen species to the underoxidized CoFeB interface may be lower than in
regime II, where oxygen species have to be added to an already optimally
oxidized CoFeB interface.
Measurements were also conducted using underoxidized samples that were
annealed before applying a gate voltage of -2 V, these measurements are
presented in Fig. 3 (b). An underoxidized sample was annealed at 180 ∘C (open
squares) for one hour to reduce moisture without inducing the crystallization
of the CoFeB. Under GV=-2 V, the speed of the ionic process in both regimes I
and II was critically reduced, while still showing the same difference
previously discussed between regimes I and II. This has been linked to a
contribution to the mobile oxygen species coming from air humidity, a feature
observed in ionic systems based on hydrogen mobility Tan et al. (2019). A
sample annealed at 350 ∘C (half open squares) for one hour was also
investigated under GV=-2 V. These annealing conditions are known to induce the
formation of crystallized grains in CoFeB/MgO stacks, where the MgO layer acts
as a crystallization template Yuasa et al. (2005); Wang et al. (2009), and the
migration of B towards the Ta substrate Sinha et al. (2015). In CoFeB/HfO2, a
similar behaviour could be tentatively proposed, where B diffusion Vermeulen
et al. (2017) and the formation of crystalline grains would have an impact in
ionic mobility. Fig. 3 shows that annealing has an impact in ionic dynamics in
this system, regime I is slower than in the as-grown sample while regime II is
significantly slower with respect to all other samples investigated. As
mentioned earlier, the chemical environment of the mobile oxygen species at
their anchoring points in the CoFeB layer is likely to be non equivalent in
regimes I and II. An already slower regime II further reducing its velocity in
crystallized samples may be an indication that ions in this regime do not only
occupy binding sites at the already oxygen rich interface but that they could
also penetrate into the layer. In this context, the crystalline structure of
the CoFeB grains could hinder the diffusion of ions inside the layer with
respect to the amorphous case, significantly reducing the speed and efficiency
of the incorporation of oxygen species. However, a detailed study of the
impact of annealing on the crystalline structure of Ta/CoFeB/HfO2 is needed to
fully understand the changes observed in the magneto-ionic behaviour.
Regimes I and II do not only show two distinct speeds of the magneto-ionic
process but they also show different reversibility behaviors. The
reversibility of the effects of a gate voltage of -2 V has been tested with
positive gate voltages going up to +4 V in regimes I and II of as-grown
samples. Higher voltages have not been applied due to the electrochemical
limitations of the ionic liquid Seidemann (2017). The effects of the
application of +4 V for 10 minutes and up to one hour to a magnetic state
close to PMA in regime I are shown in Fig.4 (a), where only minor changes are
induced in the magnetic state. The same effect is found for other intermediate
states in regime I and also for the fully perpendicular state, positive gate
voltages up to +4 V and long exposures can not induce a recovery of the
initial under oxidized state. Regime II shows an entirely different behaviour;
a fully reversible transition between IPA and PMA can be observed. Fig. 4(b)
shows the Hall voltage hysteresis loops corresponding to reversibility cycles
number 1 and 10, while Fig. 4(c) shows the remanence variation of the entire
series. The ten cycles show a switching of the remanence between nearly 0 $\%$
(IPA) and 100 $\%$ (PMA) under gate voltages of -2 V and +4V, respectively. It
is interesting to notice that in the first cycle, the PMA$\,\to\,$IPA
transition is induced by the application of -2 V for 1200 s, while for all
subsequent cycles a shorter time of 240 s is required for this transition to
occur. For the IPA$\,\to\,$PMA transition an exposure time of 600 s has been
found to be sufficient for all cycles. This shows that the magneto-ionic
process most likely undergoes

Figure 4: (a) Suppressed reversibility of the IPA$\,\to\,$PMA transition under
GV=+4 V in regime I. (b) Hysteresis loops for the reversible PMA$\,\to\,$IPA
cycles 1 (solid line) and 10 (dotted line) in regime II. (c) Remanence as a
function of the reversibility cycle number in regime II.

a first ’activation’ phase in which the first diffusion of ions involves an
additional energy barrier. This could be related to the energy cost of a first
detachment of the ions from their original anchoring points as well as to the
formation of ionic conduction channels in the CoFeB layer, after which the
ions can be moved reversibly by using significantly lower exposure times.
Regime I is therefore showing a relatively faster dynamics than in regime II
under GV=-2 V and a highly suppressed reversibility at GV=+4 V, while in
regime II, a slower dynamics and full reversibility are observed. This may be
linked to a high degree of stability of the final position of the ions within
regime I, which would create a high energy barrier for the inverse process,
suppressing reversibility under a gate voltage of +4 V. In regime II, although
the energy barrier remains asymmetric between the PMA$\,\to\,$IPA and
PMA$\leftarrow$IPA transitions, a gate voltage of +4 V is sufficient to revert
the effects of a gate voltage of -2 V. This could be linked to a weaker
anchoring of the mobile oxygen species in regime II.

##  II Effective damping parameter

In this section, we will discuss the impact of magneto-ionics on the effective
damping parameter $\alpha_{\mathrm{eff}}$. A series of samples, exposed to
gate voltages -2 V and -1.5 V, has been investigated by Brillouin light
scattering (BLS) in the Damon-Eshbach geometry after removal of the IL gate.
The in-plane magnetic field dependence of the average frequency of the Stokes
and anti-Stokes spin-wave frequencies was measured at a wavenumber k=8.08
$\mu$m-1. Fig. 5(a) shows this dependence for the as-grown sample (black
circles) and for samples exposed to GV=-2 V for 45 s (underoxidized, green
circles), 360 s (PMA, red circles) and 2700 s (overoxidized, cyan circles). A
theoretical modelling of this dependence has been performed according to
Belmeguenai et al. (2015) resulting in the fitting lines shown in Fig. 5(a).
From this fitting, the values of the effective magnetisation
$M_{\mathrm{eff}}=M_{\mathrm{s}}-H_{k}$ have been obtained, where
$M_{\mathrm{s}}$ and $H_{k}$ are the saturation magnetisation and the
anisotropy field, respectively. Fig. 5(b) shows the corresponding plots of the
BLS frequency full linewidth ($\Delta F$) as a function of the magnetic field
applied in the plane of the sample and the corresponding fitting lines to:
$\Delta F=2\alpha_{\mathrm{eff}}(\gamma/2\pi)H+\Delta F_{0}$, where
$\alpha_{\mathrm{eff}}$ can be extracted from the slope. The dependence of
$\alpha_{\mathrm{eff}}$ on exposure time to a gate voltage of -2 V (black
squares) and -1.5 V (yellow squares) is presented in Fig. 5(c), where the
inset shows

Figure 5: (a) BLS average frequency and (b) linewidth as a function of
magnetic field for an as-grown (black circles) sample, and for samples exposed
to GV=-2 V for 45 s (green circles), 350 s (red circles) and 2700 s (cyan
circles), and the corresponding fitting lines. (c) $\alpha_{\mathrm{eff}}$
dependence on exposure time to GV=-2 V (black squares) and GV=-1.5 V (yellow
squares) and on $\mu_{0}M_{\mathrm{eff}}$ (inset). Lines are a guide to the
eye.

also the dependence on $\mu_{0}M_{\mathrm{eff}}$

. In regime I, the value of $\alpha_{\mathrm{eff}}$ rapidly decreases from
0.029 for the as-grown sample to 0.012 for the PMA state. As shown earlier,
the change from GV=-2 V to GV=-1.5 V critically decreases the speed of the
process. In regime II, a relatively small increase in $\alpha_{\mathrm{eff}}$
to 0.014 is observed, however, it does not show to have a marked dependence on
the exposure time. This difference between regimes I and II is also expressed
in the $\alpha_{\mathrm{eff}}$ vs. $\mu_{0}M_{\mathrm{eff}}$ dependence, where
a monotonic decrease is seen in regime I, in contrast with the much less
pronounced increase seen for regime II in a similar $\mu_{0}M_{\mathrm{eff}}$
range.
It is important to mention that the values of $\alpha_{\mathrm{eff}}$ obtained
from the slopes of the data points in Fig. 5(b) contain not only the intrinsic
contribution from the Gilbert damping parameter but also an extrinsic
contribution. This extrinsic contribution can be associated with a variety of
sources including spin pumping Nakayama et al. (2012); Belmeguenai et al.
(2018, 2017), due to the proximity with a heavy metal with high spin-orbit
coupling, two-magnon scattering Kuanr et al. (2005) or the existence of sample
inhomogeneities such as a distribution of anisotropy values across the sample
Capua et al. (2015); Diez et al. (2019). The extrinsic contribution can add a
relatively small factor to the value of the Gilbert damping or in more extreme
cases it can dominate the $\Delta F$ vs $H$ dependence resulting in the loss
of linearity Capua et al. (2015); Diez et al. (2019).
The damping parameter in CoFeB has been shown to depend on the oxidation level
at the interface. In CoFeB/Gd/MgO stacks, the thin Gd layer serving as an
oxygen sink has been shown to modulate the level of oxidation of an
overoxidized CoFeB. In this system, $\alpha_{\mathrm{eff}}$ shows a non-
monotonic dependence on the thickness of the Gd layer where a minimum is
reached for 0.6 nm Yang et al. (2017). However, this has been linked mostly to
changes in the degree of homogeneity and quality of the interface. As
mentioned, the proximity and quality of the interface with a high spin orbit
coupling material has been shown to induce a spin-pumping contribution to the
effective damping which shows as an inverse proportionality between
$\alpha_{\mathrm{eff}}$ and the thickness of the magnetic film Ikeda et al.
(2010); Benguettat-El Mokhtari et al. (2020). This additional damping
contribution has been quantified for Ta by comparing symmetric MgO/CoFeB/MgO
and Ta/CoFeB/Ta stacks to asymmetric Ta/CoFeB/MgO, where the Ta contribution
dominates over the MgO contribution Iihama et al. (2014). This has also been
shown in the case of Hf/CoFeB/MgO stacks, which show for a CoFeB thickness of
1.08 nm, $\alpha_{\mathrm{eff}}$ values that are higher than in MgO/CoFeB/MgO
stacks by about a factor of five Lourembam et al. (2018).

Figure 6: Proposed magneto-ionic mechanisms for regimes I and II. (a) For
G${}_{V}<$ 0 oxygen species migrate toward the CoFeB layer achieving an
optimum surface coverage and PMA at the end of regime I (b). If G${}_{V}<$ 0
continues to be applied, oxygen species will continue to be incorporated into
the CoFeB layer in regime II (c) resulting in IPA. Under G${}_{V}>$ 0 mobile
oxygen species can migrate back to the top electrode (d). For the same applied
GV, there is no reversibility in regime I ((b) and (e)), while full
reversibility is achieved in regime II ((c) and (f)).

Taking into account these considerations, it can be concluded that in the
present case the underoxidized samples could have a significantly larger
contact surface with Hf in HfO2 than in the optimally oxidised case. A
progression in regime I toward PMA could therefore decrease the contact
surface with Hf and significantly reduce the associated spin pumping
contribution to $\alpha_{\mathrm{eff}}$. Interestingly, the changes in
$\alpha_{\mathrm{eff}}$ are minor in regime II, which supports the idea
mentioned earlier that in this regime oxygen species may migrate mostly into
the layer rather than binding at the already optimally oxidized interface
reached after regime I. This is considered as the main contribution to the
observed variations in $\alpha_{\mathrm{eff}}$, however, a reordering of the
interface structure throughout regime I is also a likely scenario. A sharper
interface may be reached at the end of regime I, also leading to the highest
PMA state, which could also play a role in the observed reduction of
$\alpha_{\mathrm{eff}}$.

##  III Discussion

The proposed magneto-ionic mechanisms involved in regimes I and II are
depicted in Fig. 6. For G${}_{V}<$ 0, mobile oxygen species inside HfO2,
represented as green dots, migrate toward the CoFeB layer and after GV is
switched off (b), they remain bound to the CoFeB, achieving an optimum surface
coverage and completing the IPA$\,\to\,$PMA transition at the end of regime I.
If G${}_{V}<$ 0 continues to be applied, the magneto-ionic mechanism will move
into regime II. After the voltage is switched off (c), the incorporated oxygen
species will contribute to the PMA$\,\to\,$IPA transition. As the starting
point of regime II is an optimally oxidized CoFeB surface, oxygen species
incorporated in regime II will be less likely to bind at the same binding
sites available in regime I and can potentially penetrate further into the
layer (light green dots).
Under high enough positive gate voltages, mobile oxygen species are expected
to migrate back to the top electrode (d). However, for the same applied
G${}_{V}>$ 0, reversibility is highly suppressed for oxygen species
incorporated in regime I ((b) and (e)), while full reversibility is achieved
in regime II ((c) and (f)). This discrepancy in the degree of reversibility of
regimes I and II is attributed to the differences in the binding sites, and
the associated anchoring strength, of the oxygen species incorporated in each
regime. The difference in chemical environment for these two types of binding
sites is proposed as the origin of magneto-ionic regimes I and II.
A preferential occupation of a limited number of binding sites at the
CoFeB/HfO2 interface in regime I could also explain the observed larger impact
of ionic migration on the value of $\alpha_{\mathrm{eff}}$ with respect to
regime II. As mentioned, the effects observed may be due to a decoupling of
the CoFeB surface from the Hf atoms in HfO2, mostly linked to ion
incorporation at the surface of an underoxidized CoFeB rather than to ionic
migration into the layer in regime II. This is valid in a scenario where ionic
migration into the layer is limited to the vicinity of the CoFeB/HfO2
interface and does not extend up to the Ta/CoFeB interface as seen in other
systems where higher gate voltages are applied Herrera Diez et al. (2019). It
could be anticipated that ionic migration up to the Ta/CoFeB interface could
induce a significant decoupling from the Ta layer, an effect observed in Pt/Co
reflected as a reduction of the DMI Herrera Diez et al. (2019), resulting in a
further reduction of $\alpha_{\mathrm{eff}}$. Another important indication
supporting the proposed mechanism is found in the measurements done on
crystallized samples. In this case, the time evolution of magnetic anisotropy
under a negative gate voltage is strikingly reduced in regime II compared to
regime I. This could be due to an increased difficulty of ionic diffusion into
a crystallized CoFeB layer in regime II, compared to ionic incorporation at
the surface in regime I.
Surface and bulk components to the magneto-ionic effect have already been
introduced in the literature, in particular, in Pd/Co/GdOx and Pt/Co/HfO2
films, where ion migration deep down into the magnetic layer and beyond can
cause partial or total irreversibility Gilbert et al. (2016); Herrera Diez et
al. (2019). As mentioned, in the present case the bulk component is thought to
be limited only to the vicinity of the CoFeB/HfO2 interface. The observed
suppressed reversibility seen in regime I happens at low oxygen content,
ruling out the anchoring of the oxygen species deep into the layer as a
potential source of irreversibility. This is also well supported by the
subsequent observation of regime II, exhibiting full reversibility, in the
overoxidized state.
In this context, studies in the literature addressing the incorporation of
oxygen species from the gas phase into crystalline Co and Fe systems provide
an interesting perspective. In Co, the profile of the oxygen intake as a
function of oxygen pressure has a distinct change in slope that is related to
a two stage oxidation process. The first step involves the formation of CoO up
to a defined surface coverage threshold, after which a higher oxygen and high
temperature exposure leads to the formation of Co3O4 Bridge and Lambert
(1979). In Fe crystals, a similar behaviour has been observed at room
temperature. The oxygen sticking coefficient as a function of surface coverage
is seen to decrease drastically up to a coverage of 0.5, where an inflection
point occurs followed by a less pronounced decrease. The initial phase of
rapid decrease has been linked to the filling of preferential empty binding
sites, while additional oxygen incorporation leads to the inflection point
identified as the onset of oxide growth. The oxygen incorporation is proposed
to entail a progression from initially adsorbed oxygen species to the
nucleation and expansion of a two dimensional layer of FeO at the sticking
coefficient inflection point, where it remains fairly constant until the
completion of the FeO layer formation. Further oxygen incorporation happens in
the presence of FeO, which is responsible for the observation of a reduction
in the sticking coefficient after the inflection point. This last process
ultimately leads to the formation of Fe2O3/ Fe304 together with the appearance
of a three dimensional oxide structure Brundle (1977); Simmons and Dwyer
(1975).
In amorphous systems, the distinction between the well defined oxide phases
present in crystalline systems could be less evident, however, a distinction
between purely surface incorporation and a subsequent oxygen intake more
extended into the magnetic layer could be evidenced. The films used in this
study are Fe rich, with a composition of Co20Fe60B20, where variations of the
Fe content at the interface with an oxide have been shown to drive large PMA
variations Herrera Diez et al. (2015). It is therefore worth considering a
scenario where the formation of different Fe oxide types is at the origin of
regimes I and II. Regime I is likely to reflect the initial oxygen adsorption
and formation of FeO, which involves a change in valence from Fe0 to Fe2+. A
significantly high energy barrier is therefore introduced for the inverse
process which would largely suppress reversibility under a low energy
stimulus. On the other hand, the full completion of a two dimensional FeO
layer could well lead to an optimum and homogeneous oxidation at the
interface, allowing for the highest PMA values observable in this system. As
mentioned, further oxygen incorporation in the presence of FeO becomes less
favourable, which is in line with the observed slower dynamics seen in regime
II. The Fe2O3/ Fe3O4 oxide phase involves a mixture of Fe2+ and Fe3+ states,
therefore it can be proposed that the path toward a Fe2+ $\,\to\,$ Fe3+
transition involves the progressive addition of oxygen species to the chemical
environment of the Fe2+ centres. This phase is thought to describe regime II,
where a relatively low energy barrier is involved in the inverse process due
to the conservation of the Fe2+ valence state. An ultimate change to Fe3+
would also be expected to introduce a large energy barrier for the inverse
process and hinder reversibility, as seen in regime I. This is well in line
with the observed behaviour under GV= -3 V (120 s exposure time), where the
system enters an irreversible IPA overoxidized state.

##  IV Conclusion

In conclusion, we have shown the existence of two distinct nonvolatile
magneto-ionic regimes in Ta/CoFeB/HfO2 stacks where oxygen species migrate
under negative/positive gate voltages toward/away from the CoFeB layer. This
voltage driven ionic motion induces first an IPA$\,\to\,$PMA transition in
regime I, corresponding to a transition from an underoxidized to an optimally
oxidized state. In regime II, a PMA$\,\to\,$IPA transition occurs and it is
correlated to a transition from an optimally oxidized to an overoxidized
state. Regime I shows a much faster dynamics and highly suppressed
reversibility for positive gate voltages with respect to regime II. In
addition, it also shows a marked decrease of the effective damping parameter
from 0.029 to 0.012 compared to the relatively small increase to 0.014
observed in regime II. The existence of regimes I and II is proposed to be the
result of a difference in the binding strength of the migrated oxygen species,
that can be correlated with different binding sites on the surface and inside
the CoFeB layer, respectively.
The results presented here reveal the complexity of magneto-ionics and the
importance of a deep understanding of the ionic mechanisms involved in order
to design robust and reliable devices for spintronics applications. This
system could be easily transferred to a solid state device where the two
regimes could be probed at higher gate voltages in order to re-evaluate
reversibility and explore fast operation times. This will allow to choose an
IPA to PMA transition, either in regime I or II, that best fits the
requirements for practical applications and to design strategies to favour
operation in one of the two magneto-ionic regimes.

###### Acknowledgements.

We gratefully acknowledge financial support from the European Union H2020
Program (MSCA ITN grant No. 860060), from the French National Research Agency
(project ELECSPIN), and the program PHC Sakura. This work was in part
supported by JSPS KAKENHI Grant Number 20H05304.

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# On the moduli spaces of framed logarithmic connections on a Riemann surface

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>,
Michi-aki Inaba  Department of Mathematics, Kyoto University, Kyoto 606-8502,
Japan<EMAIL_ADDRESS>,  Arata Komyo  Center for Mathematical and
Data Sciences, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501,
Japan<EMAIL_ADDRESS>and  Masa-Hiko Saito  Department of
Mathematics, Graduate School of Science, Kobe University, Kobe, Rokko,
657-8501, Japan<EMAIL_ADDRESS>
###### Abstract.

We describe some results on moduli space of logarithmic connections equipped
with framings on a $n$-pointed compact Riemann surface.

###### Key words and phrases:

Logarithmic connection, framing, residue, parabolic structure

###### 2010 Mathematics Subject Classification:

53D30, 14D20, 53B15

##  1\. Introduction

Our aim here is to initiate the study of logarithmic connections, on a Riemann
surface, equipped with a framing. To describe these objects, let $X$ be a
compact connected Riemann surface and $D\,\subset\,X$ a finite subset. Given a
holomorphic vector bundle $E$ on $X$ of rank $r$, a framing of it is an
isomorphism $\phi_{x}\,:\,{\mathbb{C}}^{r}\,\longrightarrow\,E_{x}$ for every
$x\,\in\,D$ (see [7]). A framed logarithmic connection of rank $r$ on
$(X,\,D)$ is a triple of the form $(E,\,\nabla,\,\phi)$, where $E$ is a
holomorphic vector bundle on $X$ of rank $r$, $\nabla$ is a logarithmic
connection on $E$ whose polar part is contained in $D$ and $\phi$ is a framing
of $E$.

It can be shown that a moduli space of framed logarithmic connections exists
as a Deligne-Mumford stack. Furthermore, this moduli stack is equipped with a
natural algebraic symplectic structure. We note that the moduli space of
logarithmic connections has a natural Poisson structure. The forgetful map,
from the moduli space of framed logarithmic connections to the moduli space
logarithmic connection that simply forgets the framing, is in fact Poisson.

Actually, the above results holds in a more general setting; see Section 2.
Also, the above results can be extended to the set-up of holomorphic principal
bundles with a logarithmic connection.

It is known that the moduli space of connections is not an affine variety.
This should be contrasted with the fact that moduli space of connections is
canonically biholomorphic to an affine scheme. Indeed, the character variety
is affine, and the Riemann-Hilbert correspondence produces a biholomorphism of
it with the moduli space of connections. This result is extended to the
context of vector bundles equipped with parabolic structure (see Theorem 3.3).

The full details of the proofs will appear in [4].

##  2\. Symplectic form on the moduli space of framed connections

Let $X$ be a compact connected Riemann surface of genus $g$, and let $D$ be a
reduced effective divisor on $X$. The holomorphic line bundle
$K_{X}\otimes{\mathcal{O}}_{X}(D)$, where $K_{X}$ is the holomorphic cotangent
bundle of $X$, will be denoted by $K_{X}(D)$. Fix a connected complex
algebraic proper subgroup $H_{x}\,\subsetneq\,\mathrm{GL}(r,\mathbb{C})$ for
each $x\in D$, and set $H\,=\,\\{H_{x}\\}_{x\in D}$.

Let $E$ be a holomorphic vector bundle on $X$ of rank $r$. A framing of $E$ is
an ${\mathcal{O}}_{D}$–linear isomorphism
$\phi\,\colon\,E|_{D}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\mathcal{O}}_{D}^{\oplus
r}$. Two framings $\phi$ and $\phi^{\prime}$ are called equivalent if
$\phi^{\prime}\circ\phi^{-1}|_{x}\,\in\,H_{x}$ for all $x\,\in\,D$. A framing
of $E$ with respect to $H$ is an equivalence class of framings of $E$.

Let $\mathfrak{h}_{x}$ be the Lie algebra of $H_{x}$. The orthogonal
complement of $\mathfrak{h}_{x}$, with respect to the trace pairing
$(A,\,B)\,\longmapsto\,\text{trace}(AB)$, will be denoted by
$\mathfrak{h}_{x}^{\bot}$.

Let $F(E)\,\longrightarrow\,X$ be the frame bundle for $E$; it is also the
principal $\text{GL}(r,{\mathbb{C}})$–bundle associated to $E$. Giving a
framing of $E$ with respect to $H$ is equivalent to giving a reduction of
structure group of $F(X)_{x}\,\longrightarrow\,x$ to $H_{x}$ for every
$x\,\in\,D$. Given a framing of $E$ with respect to $H$, choose a framing
$\phi$ in the equivalence class, and consider the corresponding isomorphism
$\text{Lie}(\text{GL}(r,{\mathbb{C}}))\,=\,{\rm
M}(r,{\mathbb{C}})\,\longrightarrow\,\text{End}(E)_{x}$ for every $x\,\in\,D$.
The restriction of this homomorphism to
$\mathfrak{h}^{\bot}_{x}\,\subset\,{\rm M}(r,{\mathbb{C}})$ depends on the
choice of $\phi$ in the equivalence class, but the subspace
$\text{image}(\mathfrak{h}^{\bot}_{x})\,\subset\,\text{End}(E)_{x}$ is
independent of the choice of $\phi$. This subspace of $\text{End}(E)_{x}$ will
be denoted by $[\phi](\mathfrak{h}^{\bot}_{x})$. Similarly, the subspace
$\text{image}(\mathfrak{h}_{x})\,\subset\,\text{End}(E)_{x}$ is independent of
the choice of $\phi$. This subspace of $\text{End}(E)_{x}$ will be denoted by
$[\phi](\mathfrak{h}_{x})$.

######  Definition 2.1.

A framed connection with respect to $H$ is a triple $(E,\,[\phi],\,\nabla)$,
where

  * •

$E$ is a holomorphic vector bundle on $X$ of rank $r$ equipped with a framing
$[\phi]$ with respect to $H$, and

  * •

$\nabla\,\colon\,E\,\longrightarrow\,E\otimes K_{X}(D)$ is a logarithmic
connection on $E$ such that $\mathop{\sf
res}\nolimits_{x}(\nabla)\,\in\,[\phi](\mathfrak{h}^{\bot}_{x})$ for every
$x\,\in\,D$.

The group of all automorphisms $T$ of $E$ preserving $[\phi]$ such that
$(T\otimes{\rm Id}_{K_{X}(D)})\circ\nabla\,=\,\nabla\circ T$ will be denoted
by $\mathop{\rm Aut}\nolimits(E,[\phi],\nabla)$.

######  Definition 2.2.

A framed connection $(E,\,[\phi],\,\nabla)$ with respect to $H$ is simple if
the quotient group $\mathop{\rm
Aut}\nolimits(E,[\phi],\nabla)\;/\;(\mathbb{C}^{*}\mathrm{id}_{E}\cap\mathop{\rm
Aut}\nolimits(E,[\phi],\nabla))$ is a finite group.

Let $\mathcal{M}^{H}_{\rm{FC}}(d)$ denote the moduli space of all framed
connections $(E,\,[\phi],\,\nabla)$ with respect to $H$ such that $\deg
E\,=\,d$.

######  Theorem 2.3.

The moduli space $\mathcal{M}^{H}_{\rm{FC}}(d)$ exists as a Deligne-Mumford
stack.

Given a framed connection $(E,\,[\phi],\,\nabla)$ with respect to $H$, we
construct a complex of sheaves ${\mathcal{D}}^{\bullet}$ as follows:

$\displaystyle{\mathcal{D}}^{0}\,$
$\displaystyle=\,\left\\{u\in{\mathcal{E}nd}(E)\,\,\mid\,\,u(x)\,\in\,[\phi](\mathfrak{h}_{x})\,\
\forall\ x\,\in\,D\right\\}$ $\displaystyle{\mathcal{D}}^{1}\,$
$\displaystyle=\,\left\\{v\in{\mathcal{E}nd}(E)\otimes
K_{X}(D)\,\,\mid\,\,\mathop{\sf
res}\nolimits_{x}(v)\,\in\,[\phi](\mathfrak{h}_{x}^{\bot})\,\ \forall\
x\,\in\,D\right\\}$ $\displaystyle{\mathcal{D}}^{0}\ $ $\displaystyle\ni\ u\
\longmapsto\ \nabla\circ u-(u\otimes\mathrm{id})\circ\nabla\ \in\
{\mathcal{D}}^{1}.$

######  Theorem 2.4.

The tangent space of $\mathcal{M}^{H}_{\rm{FC}}(d)$ at any point
$(E,\,[\phi],\,\nabla)\,\in\,\mathcal{M}^{H}_{\rm{FC}}(d)$ is identified with
the hypercohomology $\mathbb{H}^{1}({\mathcal{D}}^{\bullet})$ of the above
complex ${\mathcal{D}}^{\bullet}$.

Consider the following homomorphism of complexes
${\mathcal{D}}^{\bullet}\otimes{\mathcal{D}}^{\bullet}\,\longrightarrow\,\Omega^{\bullet}_{X}$:

$\displaystyle({\mathcal{D}}^{\bullet}\otimes{\mathcal{D}}^{\bullet})^{0}={\mathcal{D}}^{0}\otimes{\mathcal{D}}^{0}\
\ni\ f\otimes g\ \longmapsto\ {\rm trace}(fg)\ \in{\mathcal{O}}_{X}$
$\displaystyle({\mathcal{D}}^{\bullet}\otimes{\mathcal{D}}^{\bullet})^{1}={\mathcal{D}}^{0}\otimes{\mathcal{D}}^{1}\oplus{\mathcal{D}}^{1}\otimes{\mathcal{D}}^{0}\
\ni\ (f\otimes\omega,\eta\otimes g)\ \longmapsto\ {\rm trace}(f\omega-g\eta)\
\in\ \Omega^{1}_{X}$

It produces a pairing

$\Theta^{H}\,\colon\,\mathbb{H}^{1}({\mathcal{D}}^{\bullet})\otimes\mathbb{H}^{1}({\mathcal{D}}^{\bullet})\,\longrightarrow\,\mathbb{H}^{2}({\mathcal{D}}^{\bullet}\otimes{\mathcal{D}}^{\bullet})\,\longrightarrow\,\mathbf{H}^{2}(\Omega^{\bullet}_{X})\,=\,\mathbb{C}\,.$
(2.1)

We have the following theorem.

######  Theorem 2.5.

The pairing $\Theta^{H}$ in (2.1) defines a symplectic form on
$\mathcal{M}^{H}_{\rm{FC}}(d)$.

We give an outline of the proof of Theorem 2.5. We can check that $\Theta^{H}$
is the skew-symmetric by a calculation in C̆ech cohomology. The homomorphism
$\mathbb{H}^{1}({\mathcal{D}}^{\bullet})\,\longrightarrow\,\mathbb{H}^{1}({\mathcal{D}}^{\bullet})^{*}$
induced by $\Theta^{H}$ is isomorphic, because the vertical arrows except the
middle one in the exact commutative diagram

$\setcounter{MaxMatrixCols}{11}\begin{CD}H^{0}({\mathcal{D}}^{0})@>{}>{}>H^{0}({\mathcal{D}}^{1})@>{}>{}>\mathbb{H}^{1}({\mathcal{D}}^{\bullet})@>{}>{}>H^{1}({\mathcal{D}}^{0})@>{}>{}>H^{1}({\mathcal{D}}^{1})\\\
@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V\\\
H^{1}({\mathcal{D}}^{1})^{*}@>{}>{}>H^{1}({\mathcal{D}}^{0})^{*}@>{}>{}>\mathbb{H}^{1}({\mathcal{D}}^{\bullet})^{*}@>{}>{}>H^{0}({\mathcal{D}}^{1})^{*}@>{}>{}>H^{0}({\mathcal{D}}^{0})^{*}\end{CD}$

are Serre duality isomorphisms. In other words, the $2$-form $\Theta^{H}$ is
nondegenerate.

It remains to prove that the $2$-form $\Theta^{H}$ is $d$-closed. Let us
consider the moduli space ${\mathcal{M}}^{e}_{\rm FC}(d)$ of framed
connections with respect to the trivial group. When $H_{x}\,=\,\\{e\\}$, then
${\mathfrak{h}}^{\bot}_{x}\,=\,{\rm M}(r,{\mathbb{C}})$. Let
${\mathcal{M}}^{e}_{\rm
FC}(d)^{\mathfrak{h}^{\bot}}\,\subset\,{\mathcal{M}}^{e}_{\rm FC}(d)$ be the
locus of all $(E,\,\phi,\,\nabla)$ such that $\mathrm{res}_{x}(\nabla)$ lies
in the image of $\mathfrak{h}_{x}^{\bot}$ for all $x\,\in\,D$. Then we have
the following diagram

$\textstyle{{\mathcal{M}}^{e}_{\rm FC}(d)^{\mathfrak{h}^{\bot}}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$$\textstyle{\mathcal{M}^{e}_{\rm{FC}}(d)}$$\textstyle{\mathcal{M}^{H}_{\rm{FC}}(d)\hbox
to0.0pt{.\hss}}$

The above morphism $\pi$ sends any $(E,\,\phi,\,\nabla)$ to
$(E,\,[\phi],\,\nabla)$, and it is a smooth morphism. So $\Theta^{H}$ is
$d$-closed if $\pi^{*}\Theta^{H}$ is $d$-closed. On the other hand, we can
check the equality $\pi^{*}\Theta^{H}=\iota^{*}\Theta^{e}$. So it suffices to
prove the theorem for the case $H\,=\,\\{e\\}$. Since the moduli space
${\mathcal{M}}^{e}_{\rm FC}(d)$ is irreducible, it is enough to prove that the
restriction of $\Theta^{e}$ to some non-empty open subset of
${\mathcal{M}}^{e}_{\rm FC}(d)$ is $d$-closed.

For simplicity, we consider the case where $g\geq 2$. Let us consider the
following moduli spaces

$\displaystyle{\mathcal{N}}(d)$ $\displaystyle=\left\\{E\ \,\mid\,\ \text{$E$
is a stable bundle of rank $r$ and degree $d$}\right\\}$
$\displaystyle{\mathcal{N}}^{e}(d)$ $\displaystyle=\left\\{(E,\,\phi)\
\middle|\ \begin{array}[]{l}\text{$E$ is a stable bundle of rank $r$ and
degree $d$}\\\ \text{and $\phi\colon
E|_{D}\xrightarrow{\sim}{\mathcal{O}}_{D}^{\oplus r}$ is an
isomorphism}\end{array}\right\\}$ $\displaystyle{\mathcal{M}}(d)_{0}$
$\displaystyle=\left\\{(E,\,\nabla)\,\middle|\,\begin{array}[]{l}\text{$E$ is
a stable bundle of rank $r$ and degree $d$,}\\\ \text{$\nabla\colon
E\longrightarrow E\otimes K_{X}(D)$ is a logarithmic connection such that}\\\
\text{$\mathrm{res}_{x_{i}}(\nabla)\,=\,0$ for $1\leq i\leq n-1$ and
$\mathrm{res}_{x_{n}}(\nabla)\,=\,-\frac{d}{r}\mathrm{id}$}\end{array}\right\\}$
$\displaystyle{\mathcal{M}}^{e}_{\rm FC}(d)_{0}$
$\displaystyle=\left\\{(E,\,\phi,\,\nabla)\,\in\,{\mathcal{M}}^{e}_{FC}(d)\,\middle|\,\text{$(E,\,\phi)\,\in\,{\mathcal{N}}^{e}(d)$\,
and\, $(E,\nabla)\,\in\,{\mathcal{M}}(d)_{0}$}\right\\}.$

Then we have the following diagram

$\textstyle{{\mathcal{M}}(d)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\textstyle{\mathcal{M}^{e}_{\rm{FC}}(d)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{q}}$$\scriptstyle{\iota}$$\scriptstyle{p^{e}_{0}}$$\textstyle{\mathcal{M}^{e}_{\rm{FC}}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{e}}$$\textstyle{{\mathcal{N}}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{{\mathcal{N}}^{e}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox
to0.0pt{.\hss}}$$\scriptstyle{q}$

whose left square is a Cartesian diagram. We can see that $p_{0},\,p^{e}_{0}$
are smooth morphisms and $\iota$ is a locally closed immersion. So we can take
a non-empty analytic open subset (or an étale neighborhood)
$U\,\subset\,{\mathcal{N}}(d)$ with a section
$s\,\colon\,U\,\longrightarrow\,p_{0}^{-1}(U)$ of $p_{0}$. Let $U^{e}$ be the
pullback of $U$ to ${\mathcal{N}}^{e}(d)$. Then $s$ induces a section

$s^{e}\,\colon\,U^{e}\longrightarrow(p^{e}_{0})^{-1}(U^{e})\,\hookrightarrow\,\mathcal{M}^{e}_{\rm{FC}}(d)$
(2.2)

of $p^{e}_{0}$. Using the section $s^{e}$, we obtain the following isomorphism

$P_{1}\colon T^{*}U^{e}\,\xrightarrow{\ \cong\ }\,(p^{e})^{-1}(U^{e}),\ \
(y,\,v)\,\longmapsto\,s^{e}(y)+v.$

######  Lemma 2.6.

Let $\Phi_{U}$ be the Liouville $2$-form on the cotangent bundle $T^{*}U^{e}$.
Then,

$\Theta^{e}-(P_{1}^{-1})^{*}\Phi_{U}\,=\,(p^{e})^{*}((s^{e})^{*}\Theta^{e}).$

In view of Lemma 2.6 it is enough to prove that $(s^{e})^{*}\Theta^{e}$ is
$d$-closed, because the Liouville form is $d$-closed. We can construct a two
form $\Theta_{0}$ on the moduli space ${\mathcal{M}}(d)_{0}$ in the same way
as done in (2.1). By construction, it is just the pullback of the Goldman
symplectic form ([9], [3]) via the Riemann-Hilbert morphism. So we have
$d\Theta_{0}\,=\,0$.

######  Lemma 2.7.

The equality $\iota^{*}\Theta^{e}\,=\,\tilde{q}^{*}\Theta_{0}$ holds.

From the above lemma we can see that $\iota^{*}\Theta^{e}$ is $d$-closed.
Therefore, so is $(s^{e})^{*}\iota^{*}\Theta^{e}\,=\,(s^{e})^{*}\Theta^{e}$,
and we are done.

###  2.1. A Poisson map

In this subsection we assume that $H_{x}\,=\,\\{e\\}$ for all $x\,\in\,D$. Let
$\mathcal{M}^{H}_{\rm{C}}(d)$ be the moduli space of pairs $(E,\,\nabla)$,
where $E$ is a holomorphic vector bundle on $X$ of rank $r$ and degree $d$,
and $\nabla$ is a logarithmic connection on $E$ whose singular part is
contained in $D$. Given any $(E,\,\nabla)\,in\,\mathcal{M}^{H}_{\rm{C}}(d)$,
construct a complex $\widetilde{\mathcal{D}}^{\bullet}$ as follows:

$\widetilde{\mathcal{D}}^{0}\,=\,{\mathcal{E}nd}(E),\ \
\widetilde{\mathcal{D}}^{1}\,=\,{\mathcal{E}nd}(E)\otimes K_{X}(D)\,,$

and $\widetilde{\mathcal{D}}^{0}\,\ni\,u\,\longmapsto\,\nabla\circ
u-(u\otimes\mathrm{id})\circ\nabla\,\in\,\widetilde{\mathcal{D}}^{1}$. Then we
have

$T_{(E,\nabla)}\mathcal{M}^{H}_{\rm{C}}(d)\,=\,{\mathbb{H}}^{1}(\widetilde{\mathcal{D}}^{\bullet})\,.$

The homomorphism of hypercohomologies corresponding to the natural inclusion
of the Serre dual complex $\widetilde{\mathcal{D}}^{\bullet}$ in
$\widetilde{\mathcal{D}}^{\bullet}$ produces a Poisson structure on the moduli
space $\mathcal{M}^{H}_{\rm{C}}(d)$.

For $(E,\,\nabla,\,\phi)\,\in\,\mathcal{M}^{H}_{\rm{FC}}(d)$, the differential
$T_{(E,\nabla,\phi)}\mathcal{M}^{H}_{\rm{FC}}(d)\,\longrightarrow\,T_{(E,\nabla)}\mathcal{M}^{H}_{\rm{C}}(d)$
of the natural forgetful map

$\mathcal{M}^{H}_{\rm{FC}}(d)\,\longrightarrow\,\mathcal{M}^{H}_{\rm{C}}(d)$
(2.3)

that forgets the framing coincides with the homomorphism of hypercohomologies
given by the inclusion map of the complex ${\mathcal{D}}^{\bullet}$ in
$\widetilde{\mathcal{D}}^{\bullet}$. Using this it follows that the forgetful
map in (2.3) is Poisson.

##  3\. Moduli space of parabolic connections

It is known that there is no non-constant global algebraic function on the
moduli space of logarithmic connections with central residues on a curve of
genus at least $3$ [8]. On the other hand, the character variety, which is a
moduli space of representations of a fundamental group, is affine. So we can
see that the Riemann-Hilbert morphism, from the moduli space of connections to
the character variety, is not algebraic. This non-algebraic map preserves the
algebraic symplectic forms on these two moduli spaces [5].

We replace the moduli space ${\mathcal{M}}^{B}_{\rm{FC}}$ with the moduli
space of the more general objects of parabolic connections. Let $X$ be a
compact Riemann surface and $D$ be a reduced effective divisor on $X$. We fix
data $\boldsymbol{\nu}\,=\,\\{\nu^{x}_{j}\,\in\,\mathbb{C}\\}_{1\leq j\leq
r;\,x\in D}$ satisfying

$\sum_{x\in D}\sum_{j=1}^{r}\nu^{x}_{j}\,=\,0.$

######  Definition 3.1.

We say that a triple $(E,l,\nabla)$ is a $\boldsymbol{\nu}$-parabolic
connection if

  * (i)

$E$ is a vector bundle of rank $r$ and degree $0$,

  * (ii)

$\nabla\colon E\longrightarrow E\otimes K_{X}(D)$ is a connection admitting
poles along $D$ and

  * (iii)

$l$ is a filtration $E|_{D}=l_{1}\supset l_{2}\supset\cdots\supset
l_{r}\supset l_{r+1}=0$ satisfying the condition
$\left(\mathrm{res}_{D}(\nabla)-\nu_{j}\mathrm{id}\right)(l_{j})\,\subset\,l_{j+1}$
for all $j\,=\,1,\,\cdots,\,r$.

If we take $\boldsymbol{\nu}\,=\,\boldsymbol{0}$ so that all $\nu_{j}=0$, then
a $\boldsymbol{0}$-parabolic connection is equivalent to a framed connection
with respect to the Borel subgroup $B$.

For simplicity we adopt the following genericity assumption on
$\boldsymbol{\nu}$.

######  Assumption 3.2.

For any integer $1\,\leq\,s\,<\,r$ and for any choice of $s$ elements
$\\{j^{x}_{1},\,\cdots,\,j^{x}_{s}\\}$ in $\\{1,\,\cdots,\,r\\}$ for each
$x\,\in\,D$, the following holds:

$\sum_{x\in D}\sum_{k=1}^{s}\nu_{j^{x}_{k}}\,\notin\,\mathbb{Z}.$

Let ${\mathcal{M}}(\boldsymbol{\nu})$ be the moduli space of
$\boldsymbol{\nu}$-parabolic connections. Assumption 3.2 ensures that any
$\boldsymbol{\nu}$-parabolic connection is irreducible. So it is stable with
respect to any parabolic weight. Let
${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ be the moduli space of
$\boldsymbol{\nu}$-parabolic connections $(E,\,\nabla,\,l)$ with
$\det(E,\,\nabla)\,\cong\,({\mathcal{O}}_{X},\,d)$. There is a closed
immersion
$\iota\,\colon\,{\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})\,\hookrightarrow\,{\mathcal{M}}(\boldsymbol{\nu})$,
and also ${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ is smooth.

######  Theorem 3.3.

Assume that $r\geq 2$, $n\geq 1$ and $g\geq 2$. Then the moduli space
${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ of $\boldsymbol{\nu}$-connections
with the trivial determinant is not affine.

######  Remark 3.4.

  * (1)

There is a Riemann-Hilbert morphism from the moduli space of
$\boldsymbol{\nu}$-parabolic connections to the character variety
parameterizing the representations of the fundamental group
$\pi_{1}(X\setminus D)$ with a fixed local monodromy data. If we assume, in
addition to Assumption 3.2, that $\nu_{j}^{x}-\nu_{k}^{x}\,\notin\,\mathbb{Z}$
for $j\neq k$, then the Riemann-Hilbert morphism is an analytic isomorphism.
Since the character variety is affine, the property that
${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ is not affine implies that the
Riemann-Hilbert morphism is not algebraic.

  * (2)

For a moduli space of framed connections with respect to the Borel subgroup
$B$ it is better to consider a stability with respect to a parabolic weight.
In that case, Assumption 3.2 does not hold nor the assumption in (1) above.
The moduli space of $\boldsymbol{\nu}$-parabolic connections in the case of
non-generic exponent $\boldsymbol{\nu}$, involving the case of $g=0$ is also
interesting. Indeed it contains loci of some special solutions of the
isomonodromy equations, which is like Riccatti loci in Painlevé VI equations
(see [12], [13] and [16]). The property of global algebraic functions on the
moduli space of connections in a special case of $g=0$ is also used in [1].

  * (3)

The formulation of the isomonodromy equation is given in [12], [13] and [11],
but it is also formulated in [6] and [10]. Classically the isomonodromy
equation is known to be characterized by the isomonodromy $2$-form ([15],
[14]). Since the isomonodromy equation gives an algebraic splitting of the
tangent bundle of the moduli space of connections, we can construct the
isomonodromy $2$-form from the symplectic form constructed in Section 2.

We will give an outline of the proof of Theorem 3.3, using the following
proposition.

######  Proposition 3.5.

Under the assumption in Theorem 3.3, consider the locus $Z$ in
${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ consisting of
$\boldsymbol{\nu}$-parabolic connections $(E,\,\nabla,\,l)$ whose underlying
quasi-parabolic vector bundle $(E,\,l)$ is not simple. Then the codimension of
$Z$ in ${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ is at least $2$.

We denote by ${\mathcal{N}}^{\rm{spl}}_{\rm{par}}$ the moduli space of simple
quasi-parabolic bundles $(E,\,l)$ such that $\det E\,=\,{\mathcal{O}}_{X}$.
Let ${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\rm{SL}}(\boldsymbol{\nu})$ be
the moduli space of $\boldsymbol{\nu}$-parabolic connections with the trivial
determinant such that the underlying quasi-parabolic bundle $(E,\,l)$ is
simple. Then there is a morphism
${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\rm{SL}}(\boldsymbol{\nu})\longrightarrow{\mathcal{N}}^{\rm{spl}}_{\rm
par}$ which is an affine space bundle. There is a universal family
$(\widetilde{E},\,\widetilde{l})$ on $X\times\mathcal{N}_{\rm
par}^{\mathrm{spl}}$. Let $\mathrm{At}(\widetilde{E})$ be the Atiyah bundle of
$\widetilde{E}$ introduced in [2], which fits in the short exact sequence

$0\,\longrightarrow\,{\mathcal{E}nd}(\widetilde{E})\,\longrightarrow\,\mathrm{At}(\widetilde{E})\,\longrightarrow\,T_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}/{\mathcal{N}}^{\mathrm{spl}}_{\rm par}}\,\longrightarrow\,0.$

We set

$\displaystyle{\mathcal{E}nd}_{\mathfrak{sl}}(\widetilde{E},\,\widetilde{l})$
$\displaystyle:=\left\\{a\in{\mathcal{E}nd}(\widetilde{E})\,\middle|\,\text{$\mathrm{Tr}(a)=0$
and $a|_{D\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}(\widetilde{l}_{j})\subset\widetilde{l}_{j}$ for any $j$}\right\\}$
$\displaystyle{\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\,\widetilde{l})$
$\displaystyle=\left\\{a\in{\mathcal{E}nd}(\widetilde{E})\,\middle|\,\text{$\mathrm{Tr}(a)=0$
and $a|_{D\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}(\widetilde{l}_{j})\,\subset\,\widetilde{l}_{j+1}$ for any
$j$}\right\\}.$

We can define a subsheaf
$\mathrm{At}(\widetilde{E},\widetilde{l})\,\subset\,\mathrm{At}(\widetilde{E})$
fitting in the exact commutative diagram

$\begin{CD}0\,\longrightarrow\
&{\mathcal{E}nd}_{\mathfrak{sl}}(\widetilde{E},\,\widetilde{l})@>{}>{}>\mathrm{At}(\widetilde{E},\,\widetilde{l})@>{}>{}>T_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}/{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}(-D\times{\mathcal{N}}^{\mathrm{spl}}_{\rm par})&\ \longrightarrow\,0\\\
&@V{}V{}V@V{}V{}V@V{}V{}V&\\\ 0\,\longrightarrow\
&{\mathcal{E}nd}(\widetilde{E})@>{}>{}>\mathrm{At}(\widetilde{E})@>{}>{}>T_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}/{\mathcal{N}}^{\mathrm{spl}}_{\rm par}}&\ \longrightarrow\,0.\end{CD}$

Tensoring $K_{X}(D)$, we get an exact sequence

$0\longrightarrow{\mathcal{E}nd}_{\mathfrak{sl}}(\widetilde{E},\,\widetilde{l})\otimes
K_{X}(D)\longrightarrow\mathrm{At}(\widetilde{E},\,\widetilde{l})\otimes
K_{X}(D)\longrightarrow{\mathcal{O}}_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}\longrightarrow 0,$

from which we get a short exact sequence

$0\longrightarrow\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}(\widetilde{E},\,\widetilde{l})\otimes
K_{X}(D)\right)\longrightarrow\pi_{*}\left(\mathrm{At}(\widetilde{E},\,\widetilde{l})\otimes
K_{X}(D)\right)\longrightarrow\pi_{*}\left({\mathcal{O}}_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}\right)\longrightarrow 0.$

We put
${\mathcal{Q}}\,:=\,\pi_{*}\left(\mathrm{At}(\widetilde{E},\,\widetilde{l})\otimes
K_{X}(D)\right)$, and consider the projective bundle

$\mathbb{P}_{*}({\mathcal{Q}})\,=\,\mathrm{Proj}\left(\mathrm{Sym}^{*}\left({\mathcal{Q}}^{\vee}\right)\right)$

where $\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})$ is the symmetric algebra of
${\mathcal{Q}}^{\vee}$ over ${\mathcal{O}}_{{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}$. There is a tautological sub line bundle

${\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}(-1)\,\hookrightarrow\,Q\otimes{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}.$

There are induced sections

$\displaystyle\epsilon$
$\displaystyle:{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}(-1)\hookrightarrow
Q\otimes{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}\longrightarrow\pi_{*}\left({\mathcal{O}}_{X\times{\mathcal{N}}^{\mathrm{spl}}_{\rm
par}}\right)\otimes{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}={\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}$
$\displaystyle\widetilde{\nu}_{j}$
$\displaystyle\colon{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}(-1)\hookrightarrow
Q\otimes{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}\xrightarrow{\mathrm{res}_{D}}{\mathcal{E}nd}_{\mathfrak{sl}}(\widetilde{E},\widetilde{l})\big{|}_{D\times\mathbb{P}_{*}({\mathcal{Q}})}\longrightarrow{\mathcal{E}nd}(\widetilde{l}_{j}/\widetilde{l}_{j+1})\otimes{\mathcal{O}}_{\mathbb{P}_{*}({\mathcal{Q}})}={\mathcal{O}}_{D\times\mathbb{P}_{*}({\mathcal{Q}})}$

Let ${\mathcal{I}}$ be the ideal sheaf of the graded
${\mathcal{O}}_{{\mathcal{N}}^{\mathrm{spl}}_{\rm par}}$-algebra
$\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})$ generated by

$\left\\{\widetilde{\nu}_{j}|_{x}-\nu_{j}|_{x}\,\epsilon\>\middle|\ x\in
D,1\leq j\leq r\right\\}\,.$

Then there is a short exact sequence

$0\,\longrightarrow\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\,\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\,\longrightarrow\,\mathrm{Sym}^{*}\left(\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\widetilde{l})\otimes
K_{X}(D)\right)\right)\,\longrightarrow\,0.$ (3.1)

We put

$\overline{{\mathcal{M}^{\prime}}}:=\mathrm{Proj}(\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}})\subset\mathbb{P}_{*}({\mathcal{Q}}).$

Let $Y\,\subset\,\overline{\mathcal{M}^{\prime}}$ be the effective divisor
defined by $\epsilon\,=\,0$. There is a canonical isomorphism
${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\rm{SL}}(\boldsymbol{\nu})\,\cong\,\overline{\mathcal{M}^{\prime}}\setminus
Y\,\cong\,\mathrm{Spec}\left(\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)_{(\epsilon)}$,
where
$\left(\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)_{(\epsilon)}$
is the degree zero component of the localized graded ring
$\left(\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)_{\epsilon}$.

Suppose that ${\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})$ is affine. Then the
ring of global algebraic functions

$A\,=\,\Gamma\left({\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu}),\
{\mathcal{O}}_{{\mathcal{M}}_{\rm{SL}}(\boldsymbol{\nu})}\right)$

is a finitely generated $\mathbb{C}$-algebra of Krull dimension
$2(r^{2}-1)(g-1)+r(r-1)n$. By Proposition 3.5,

$A\,=\,\Gamma\left({\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\rm{SL}}(\boldsymbol{\nu}),\,{\mathcal{O}}_{{\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\rm{SL}}(\boldsymbol{\nu})}\right)\,=\,\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\left(\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)_{(\epsilon)}\right)\,=\,\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)_{(\epsilon)}.$

Since $A$ is a finitely generated $\mathbb{C}$-algebra, it can be proved that
$\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)$ is a
finitely generated graded $\mathbb{C}$-algebra. So the Krull dimension of
$\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)$ should be
$\text{Krull-dim}(A)+1$. From the exact sequence (3.1), we get the exact
sequence

$\displaystyle 0\,\longrightarrow\,\Gamma\left({\mathcal{N}}^{\rm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)$
$\displaystyle\,\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\,\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)$
$\displaystyle\longrightarrow\,\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}\left(\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\,\widetilde{l})\otimes\Omega^{1}_{X}(D)\right)\right)\right).$

So we have

$\displaystyle\begin{split}\text{Krull-dim}(A)&\,=\,\text{Krull-
dim}\left(\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)\big{/}\epsilon\;\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)\right)\\\
&=\,\mathrm{tr.deg}_{\mathbb{C}}\left(\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)\big{/}\epsilon\;\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}({\mathcal{Q}}^{\vee})/{\mathcal{I}}\right)\right)\\\
&\leq\,\mathrm{tr.deg}_{\mathbb{C}}\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}\left(\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\,\widetilde{l})\otimes\Omega^{1}_{X}(D)\right)\right)\right)\end{split}$
(3.2)

On the other hand, let ${\mathcal{M}}^{\rm{spl}}_{\mathrm{SL},\mathrm{Higgs}}$
be the moduli space of simple parabolic Higgs bundles $(E,\,\Phi,\,l)$ such
that $\Phi|_{D}(l_{j})\subset l_{j+1}$ for $1\leq j\leq r$,
$\det(E)\,=\,{\mathcal{O}}_{X}$ and ${\rm trace}(\Phi)\,=\,0$. It contains the
moduli space
${\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}}$ of
$\boldsymbol{\alpha}$-stable parabolic Higgs bundles for any choice of
parabolic weight $\boldsymbol{\alpha}$. Let
${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\mathrm{SL},\mathrm{Higgs}}$ be the
open subspace of ${\mathcal{M}}^{\rm{spl}}_{\mathrm{SL},\mathrm{Higgs}}$
consisting of those whose underlying quasiparabolic bundle $(E,\,l)$ is
simple. Then
${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\mathrm{SL},\mathrm{Higgs}}$ is
isomorphic to the cotangent bundle of ${\mathcal{N}}^{\mathrm{spl}}_{\rm
par}$, which means that

${\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\mathrm{SL},\mathrm{Higgs}}\,\cong\,\mathrm{Spec}\left(\mathrm{Sym}^{*}\left(\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\,\widetilde{l})\otimes\Omega^{1}_{X}(D)\right)\right)\right).$

We have the codimension estimation for the moduli space of parabolic Higgs
bundles which is similar to Proposition 3.5. So we have

$\displaystyle\begin{split}\Gamma\left({\mathcal{N}}^{\mathrm{spl}}_{\rm
par},\,\mathrm{Sym}^{*}\left(\pi_{*}\left({\mathcal{E}nd}_{\mathfrak{sl}}^{\mathrm{nil}}(\widetilde{E},\,\widetilde{l})\otimes\Omega^{1}_{X}(D)\right)\right)\right)&\,\cong\,\Gamma\left({\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\mathrm{SL},\mathrm{Higgs}},\,{\mathcal{O}}_{{\mathcal{M}}^{\text{$\mathcal{N}$-spl}}_{\mathrm{SL},\mathrm{Higgs}}}\right)\\\
&=\,\Gamma\left({\mathcal{M}}^{\rm{spl}}_{\mathrm{SL},\mathrm{Higgs}},\,{\mathcal{O}}_{{\mathcal{M}}^{\rm{spl}}_{\mathrm{SL},\mathrm{Higgs}}}\right)\\\
&\subseteq\,\Gamma\left({\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}},\,{\mathcal{O}}_{{\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}}}\right).\end{split}$
(3.3)

Choosing $\boldsymbol{\alpha}$ generically, we may assume that the
$\boldsymbol{\alpha}$-semistability implies $\boldsymbol{\alpha}$-stability.
So the Hitchin map

${\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}}\,\longrightarrow\,\bigoplus_{k=2}^{r}H^{0}\left(X,\,K_{X}^{\otimes
k}((k-1)D)\right)$

is a proper morphism. Hence the $\mathbb{C}$-algebra
$\Gamma\left({\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}},\,{\mathcal{O}}_{{\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}}}\right)$
is a finite algebra over

$\Gamma\left(\bigoplus_{k=2}^{r}H^{0}\left(X,\,K_{X}^{\otimes
k}((k-1)D)\right),{\mathcal{O}}_{\bigoplus_{k=2}^{r}H^{0}\left(X,K_{X}^{\otimes
k}((k-1)D)\right)}\right),$

whose transcendence degree is the same as its Krull dimension
$r^{2}(g-1)-g+1+nr(r-1)/2$. Combining (3.2) and (3.3) we get the inequality

$\displaystyle 2(r^{2}-1)(g-1)+r(r-1)n\,=\,\text{Krull-dim}(A)$
$\displaystyle\,\leq\,\mathrm{tr.deg}_{\mathbb{C}}\Gamma\left({\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}},\,{\mathcal{O}}_{{\mathcal{M}}^{\boldsymbol{\alpha}}_{\mathrm{SL},\mathrm{Higgs}}}\right)$
$\displaystyle\leq\,(r^{2}-1)(g-1)+\frac{nr(r-1)}{2},$

which is a contradiction.

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