The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# First Measurement of Differential Cross Sections for Muon Neutrino Charged
Current Interactions on Argon with a Two-proton Final State in the MicroBooNE
Detector

P. Abratenko  Tufts University, Medford, MA, 02155, USA  D. Andrade Aldana
Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  J. Anthony
University of Cambridge, Cambridge CB3 0HE, United Kingdom  L. Arellano  The
University of Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi
University of Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv
University, Tel Aviv, Israel, 69978  S. Balasubramanian  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  B. Baller  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Barr
University of Oxford, Oxford OX1 3RH, United Kingdom  J. Barrow  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  Tel Aviv University,
Tel Aviv, Israel, 69978  V. Basque  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  L. Bathe-Peters  Harvard University,
Cambridge, MA 02138, USA  O. Benevides Rodrigues  Syracuse University,
Syracuse, NY, 13244, USA  S. Berkman  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  A. Bhanderi  The University of Manchester,
Manchester M13 9PL, United Kingdom  M. Bhattacharya  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Bishai  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster
University, Lancaster LA1 4YW, United Kingdom  B. Bogart  University of
Michigan, Ann Arbor, MI, 48109, USA  T. Bolton  Kansas State University (KSU),
Manhattan, KS, 66506, USA  J. Y. Book  Harvard University, Cambridge, MA
02138, USA  L. Camilleri  Columbia University, New York, NY, 10027, USA  D.
Caratelli  University of California, Santa Barbara, CA, 93106, USA  I. Caro
Terrazas  Colorado State University, Fort Collins, CO, 80523, USA  F. Cavanna
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G.
Cerati  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
Y. Chen  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J.
M. Conrad  Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139,
USA  M. Convery  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025,
USA  L. Cooper-Troendle  Wright Laboratory, Department of Physics, Yale
University, New Haven, CT, 06520, USA  J. I. Crespo-Anadón  Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid
E-28040, Spain  M. Del Tutto  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  S. R. Dennis  University of Cambridge, Cambridge CB3
0HE, United Kingdom  P. Detje  University of Cambridge, Cambridge CB3 0HE,
United Kingdom  A. Devitt  Lancaster University, Lancaster LA1 4YW, United
Kingdom  R. Diurba  Universität Bern, Bern CH-3012, Switzerland  University of
Minnesota, Minneapolis, MN, 55455, USA  R. Dorrill  Illinois Institute of
Technology (IIT), Chicago, IL 60616, USA  K. Duffy  University of Oxford,
Oxford OX1 3RH, United Kingdom  S. Dytman  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  B. Eberly  University of Southern Maine, Portland,
ME, 04104, USA  A. Ereditato  Universität Bern, Bern CH-3012, Switzerland  J.
J. Evans  The University of Manchester, Manchester M13 9PL, United Kingdom  R.
Fine  Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  O. G.
Finnerud  The University of Manchester, Manchester M13 9PL, United Kingdom  W.
Foreman  Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  B. T.
Fleming  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  N. Foppiani  Harvard University, Cambridge, MA 02138, USA  D.
Franco  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  A. P. Furmanski  University of Minnesota, Minneapolis, MN,
55455, USA  D. Garcia-Gamez  Universidad de Granada, Granada E-18071, Spain
S. Gardiner  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  G. Ge  Columbia University, New York, NY, 10027, USA  S. Gollapinni
University of Tennessee, Knoxville, TN, 37996, USA  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  O. Goodwin  The University of
Manchester, Manchester M13 9PL, United Kingdom  E. Gramellini  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  P. Green  The
University of Manchester, Manchester M13 9PL, United Kingdom  H. Greenlee
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W. Gu
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  R. Guenette  The
University of Manchester, Manchester M13 9PL, United Kingdom  P. Guzowski  The
University of Manchester, Manchester M13 9PL, United Kingdom  L. Hagaman
Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  O. Hen  Massachusetts Institute of Technology (MIT), Cambridge,
MA, 02139, USA  R. Hicks  Los Alamos National Laboratory (LANL), Los Alamos,
NM, 87545, USA  C. Hilgenberg  University of Minnesota, Minneapolis, MN,
55455, USA  G. A. Horton-Smith  Kansas State University (KSU), Manhattan, KS,
66506, USA  B. Irwin  University of Minnesota, Minneapolis, MN, 55455, USA  R.
Itay  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  C.
James  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
X. Ji  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  L. Jiang
Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA  J. H.
Jo  Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  R. A. Johnson  University of Cincinnati, Cincinnati, OH, 45221,
USA  Y.-J. Jwa  Columbia University, New York, NY, 10027, USA  D. Kalra
Columbia University, New York, NY, 10027, USA  N. Kamp  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  G. Karagiorgi
Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Kobilarcik
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  I.
Kreslo  Universität Bern, Bern CH-3012, Switzerland  M. B. Leibovitch
University of California, Santa Barbara, CA, 93106, USA  I. Lepetic  Rutgers
University, Piscataway, NJ, 08854, USA  J.-Y. Li  University of Edinburgh,
Edinburgh EH9 3FD, United Kingdom  K. Li  Wright Laboratory, Department of
Physics, Yale University, New Haven, CT, 06520, USA  Y. Li  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  K. Lin  Rutgers University,
Piscataway, NJ, 08854, USA  B. R. Littlejohn  Illinois Institute of Technology
(IIT), Chicago, IL 60616, USA  W. C. Louis  Los Alamos National Laboratory
(LANL), Los Alamos, NM, 87545, USA  X. Luo  University of California, Santa
Barbara, CA, 93106, USA  K. Manivannan  Syracuse University, Syracuse, NY,
13244, USA  C. Mariani  Center for Neutrino Physics, Virginia Tech,
Blacksburg, VA, 24061, USA  D. Marsden  The University of Manchester,
Manchester M13 9PL, United Kingdom  J. Marshall  University of Warwick,
Coventry CV4 7AL, United Kingdom  D. A. Martinez Caicedo  South Dakota School
of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  K. Mason  Tufts
University, Medford, MA, 02155, USA  A. Mastbaum  Rutgers University,
Piscataway, NJ, 08854, USA  N. McConkey  The University of Manchester,
Manchester M13 9PL, United Kingdom  V. Meddage  Kansas State University (KSU),
Manhattan, KS, 66506, USA  K. Miller  University of Chicago, Chicago, IL,
60637, USA  J. Mills  Tufts University, Medford, MA, 02155, USA  A. Mogan
Colorado State University, Fort Collins, CO, 80523, USA  T. Mohayai  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Mooney
Colorado State University, Fort Collins, CO, 80523, USA  A. F. Moor
University of Cambridge, Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin
The University of Manchester, Manchester M13 9PL, United Kingdom  J. Mousseau
University of Michigan, Ann Arbor, MI, 48109, USA  S. Mulleriababu
Universität Bern, Bern CH-3012, Switzerland  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  N. Nayak  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  J. Nowak  Lancaster University,
Lancaster LA1 4YW, United Kingdom  M. Nunes  Syracuse University, Syracuse,
NY, 13244, USA  N. Oza  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  O. Palamara  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  N. Pallat  University of Minnesota, Minneapolis, MN,
55455, USA  V. Paolone  University of Pittsburgh, Pittsburgh, PA, 15260, USA
A. Papadopoulou  Massachusetts Institute of Technology (MIT), Cambridge, MA,
02139, USA  V. Papavassiliou  New Mexico State University (NMSU), Las Cruces,
NM, 88003, USA  H. B. Parkinson  University of Edinburgh, Edinburgh EH9 3FD,
United Kingdom  S. F. Pate  New Mexico State University (NMSU), Las Cruces,
NM, 88003, USA  N. Patel  Lancaster University, Lancaster LA1 4YW, United
Kingdom  Z. Pavlovic  Fermi National Accelerator Laboratory (FNAL), Batavia,
IL 60510, USA  E. Piasetzky  Tel Aviv University, Tel Aviv, Israel, 69978  I.
D. Ponce-Pinto  Wright Laboratory, Department of Physics, Yale University, New
Haven, CT, 06520, USA  I. Pophale  Lancaster University, Lancaster LA1 4YW,
United Kingdom  S. Prince  Harvard University, Cambridge, MA 02138, USA  X.
Qian  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V.
Radeka  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  M.
Reggiani-Guzzo  The University of Manchester, Manchester M13 9PL, United
Kingdom  L. Ren  New Mexico State University (NMSU), Las Cruces, NM, 88003,
USA  L. Rochester  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  J. Rodriguez Rondon  South Dakota School of Mines and Technology
(SDSMT), Rapid City, SD, 57701, USA  M. Rosenberg  Tufts University, Medford,
MA, 02155, USA  M. Ross-Lonergan  Los Alamos National Laboratory (LANL), Los
Alamos, NM, 87545, USA  C. Rudolf von Rohr  Universität Bern, Bern CH-3012,
Switzerland  G. Scanavini  Wright Laboratory, Department of Physics, Yale
University, New Haven, CT, 06520, USA  D. W. Schmitz  University of Chicago,
Chicago, IL, 60637, USA  A. Schukraft  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  W. Seligman  Columbia University, New York,
NY, 10027, USA  M. H. Shaevitz  Columbia University, New York, NY, 10027, USA
R. Sharankova  Fermi National Accelerator Laboratory (FNAL), Batavia, IL
60510, USA  J. Shi  University of Cambridge, Cambridge CB3 0HE, United Kingdom
A. Smith  University of Cambridge, Cambridge CB3 0HE, United Kingdom  E. L.
Snider  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
M. Soderberg  Syracuse University, Syracuse, NY, 13244, USA  S. Söldner-
Rembold  The University of Manchester, Manchester M13 9PL, United Kingdom  J.
Spitz  University of Michigan, Ann Arbor, MI, 48109, USA  M. Stancari  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. St. John
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T.
Strauss  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
S. Sword-Fehlberg  New Mexico State University (NMSU), Las Cruces, NM, 88003,
USA  A. M. Szelc  University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
W. Tang  University of Tennessee, Knoxville, TN, 37996, USA  N. Taniuchi
University of Cambridge, Cambridge CB3 0HE, United Kingdom  K. Terao  SLAC
National Accelerator Laboratory, Menlo Park, CA, 94025, USA  C. Thorpe
Lancaster University, Lancaster LA1 4YW, United Kingdom  D. Torbunov
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  D. Totani
University of California, Santa Barbara, CA, 93106, USA  M. Toups  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y.-T. Tsai
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Tyler
Kansas State University (KSU), Manhattan, KS, 66506, USA  M. A. Uchida
University of Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher  SLAC
National Accelerator Laboratory, Menlo Park, CA, 94025, USA  B. Viren
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  M. Weber
Universität Bern, Bern CH-3012, Switzerland  H. Wei  Louisiana State
University, Baton Rouge, LA, 70803, USA  A. J. White  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  Z. Williams
University of Texas, Arlington, TX, 76019, USA  S. Wolbers  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Wongjirad  Tufts
University, Medford, MA, 02155, USA  M. Wospakrik  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  K. Wresilo  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  N. Wright  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  W. Wu  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E. Yandel
University of California, Santa Barbara, CA, 93106, USA  T. Yang  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  L. E. Yates
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  H. W. Yu
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  G. P. Zeller
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J.
Zennamo  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
C. Zhang  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We present the first measurement of differential cross sections for charged-
current muon neutrino interactions on argon with one muon, two protons, and no
pions in the final state. Such interactions leave the target nucleus in a two-
particle two-hole state; these states are of great interest, but currently
there is limited information about their production in neutrino-nucleus
interactions. Detailed investigations of the production of two-particle two-
hole states are vital to support upcoming experiments exploring the nature of
the neutrino, and the development of the liquid-argon time-projection-chamber
has made possible the isolation of such final states. The opening angle
between the two protons, the angle between the total proton momentum and the
muon, and the total transverse momentum of the final state system are
sensitive to the underlying physics processes as embodied in a variety of
models. Realistic initial-state momentum distributions are shown to be
important in reproducing the data.

††preprint: APS/123-QED

The introduction of the liquid argon time projection chamber (LArTPC) Rubbia
(1977); Chen  _et al._ (1976); Willis and Radeka (1974); Nygren (1974) has
revolutionized the field of accelerator-based neutrino physics, allowing a
more detailed observation of ionizing radiation in the final state than was
previously possible. This development, in turn, has highlighted the need for
advanced modeling of neutrino-nucleus, especially neutrino-argon, interactions
and detailed measurements to benchmark those models. Cross-section
measurements of a variety of different final state topologies for an array of
different nuclei are needed to support the development of neutrino interaction
models Formaggio and Zeller (2012). These models must address both in-medium
nuclear modification of the fundamental neutrino interactions Nieves  _et al._
(2004a) and also final-state interactions (FSI) Dytman  _et al._ (2021)
involving the reaction products as they exit the nucleus.
One process that probes both neutrino interactions and nuclear effects is the
production of two-particle two-hole (2p2h) states in which two nucleons are
removed from the nucleus. These states are primarily produced by neutrino
interactions where the momentum transfer is shared between two nucleons via
the exchange of a virtual meson, known as a meson exchange current (MEC)
Katori (2015). In addition, 2p2h states can also be produced by nuclear
effects, such as short-range nucleon-nucleon correlations (SRC) Hen  _et al._
(2017); Cruz-Torres  _et al._ (2021) and FSI. In the case of SRCs, the
neutrino interacts with a nucleon that is part of a correlated nucleon-nucleon
pair. The momentum transfer is relayed to a single nucleon, but, because this
nucleon is part of a correlated pair, both nucleons are knocked out of the
nucleus. In the case of FSI, it is possible for a single nucleon to knock out
a second nucleon as it exits the nucleus thereby leading to a 2p2h final
state. Despite the known importance of 2p2h interactions for neutrino energy
reconstruction and precision oscillation measurements, their production cross
section has never been measured directly.
A final state topology clearly indicative of the production of a 2p2h state is
a charged-current (CC) muon neutrino ($\nu_{\mu}$) interaction that results in
one muon, two protons, and no charged or neutral pions (CC1$\mu$2p0$\pi$).
While there is an existing measurement of CC1$\mu$2p0$\pi$ events on argon,
the analysis was statistically limited, unlike this measurement, and no cross
sections were extracted Acciarri  _et al._ (2014). In a detector that cannot
count individual protons and must report inclusive CC yields (see for example
Abe  _et al._ (2013)), this final state cannot be distinguished from other CC
interactions, and is therefore particularly important to understand for
neutrino energy reconstruction in oscillation measurements. In this letter, we
present the first differential cross section measurements of CC1$\mu$2p0$\pi$
topologies on argon using data collected from the Micro Booster Neutrino
Experiment (MicroBooNE) Acciarri  _et al._ (2017).
MicroBooNE uses a LArTPC detector located at Fermi National Accelerator
Laboratory. The detector sits on axis to the Booster Neutrino Beam (BNB) which
has an average energy of $\langle E_{\nu}\rangle=0.8$ GeV Aguilar-Arevalo  _et
al._ (2009) and is located approximately 470 m from the neutrino production
target. The detector consists of two components: (1) a TPC, 10.36 m long in
the beam direction, 2.56 m wide in the drift direction, and 2.32 m tall; and
(2) an optical system comprised of 32 eight-inch photomultiplier tubes (PMTs).
The TPC consists of three wire planes, two induction planes and one collection
plane, with the PMTs mounted in an array behind the collection plane. Both
components are enclosed in a cylindrical cryostat containing 170 metric tons
of liquid argon with approximately 85 metric tons within the TPC volume.
Electronic signals from the TPC and PMTs are recorded and subdivided into two
distinct data samples. The first sample, known as on-beam data (BNB data), is
collected coincident to a 1.6 $\mu s$ BNB neutrino spill. The second sample,
known as off-beam data (EXT data), is recorded outside of the BNB spill. The
purpose of this second sample is to account for electronic noise and the large
cosmic muon background caused by the surface location of the MicroBooNE
detector. The BNB data and a portion of the EXT data samples are then filtered
based on the amount of activity measured in the PMTs.
In addition to the two real data samples, MicroBooNE utilizes the GENIE
neutrino event generator Andreopoulos  _et al._ (2010) to create two samples
of simulated neutrino interactions: (1) Overlay Monte Carlo (Overlay MC) which
addresses signal events and beam-related backgrounds; and (2) Dirt Monte Carlo
(Dirt MC) which addresses backgrounds from neutrino interactions in the
material surrounding the cryostat. The simulated events from both samples are
then overlaid onto events from an unbiased EXT sample (which is not filtered
based on PMT activity) in order to simulate MicroBooNE’s cosmic ray background
Adams  _et al._ (2019). Furthermore, MicroBooNE utilizes another sample of
simulated events generated using the NuWro neutrino event generator Golan  _et
al._ (2012) for additional studies. This sample is generated in the same way
as the Overlay MC, but the underlying physics is dictated by NuWro.
The physics in both the Overlay MC and Dirt MC samples is given by the GENIE
MicroBooNE Tune, a version of the GENIE v3.0.6 G18_10a_02_11a model set in
which four parameters are tuned to $\nu_{\mu}$ CC0$\pi$ data from the T2K
experiment Abratenko  _et al._ (2022a). The four parameters are: (1) the CC
quasi-elastic axial mass Alvarez-Ruso  _et al._ (2021); (2) the strength of
the random phase approximation (RPA) in the Nieves CCQE cross section
calculation Nieves  _et al._ (2004b); (3) the absolute normalization of the
CC2p2h cross section Schwehr  _et al._ (2016); (4) the shape of the CC2p2h
cross section. The shape of the CC2p2h cross section is represented by a
parameter that linearly interpolates between two models: the Nieves prediction
Schwehr  _et al._ (2016) and the empirical prediction Katori (2015).
All of the aforementioned samples are processed by the Pandora reconstruction
framework Acciarri  _et al._ (2018) to identify and reconstruct neutrino
interactions. From the reconstructed products, a series of selection
requirements are applied to identify CC1$\mu$2p0$\pi$ events. The initial
selection looks for events that meet three criteria: (1) the reconstructed
neutrino vertex is located within a fiducial volume (FV) defined to be at
least 10 cm away from any TPC face; (2) there are exactly three tracks, and no
showers as determined by Pandora Van De Pontseele (2020); (3) the three tracks
start within 4 cm of the reconstructed neutrino vertex. Particle
identification techniques described in Ref. Abratenko  _et al._ (2021) are
then used to identify events with a single muon candidate and two proton
candidates.
To benchmark the performance of our selection, the number of CC1$\mu$2p0$\pi$
events that pass all selection requirements is compared to the number of
events generated using the GENIE MicroBooNE tune. Using truth-level
information, we select a sample of events that adhere to the following true
signal definition: signal events are required to be $\nu_{\mu}$ CC events that
produce one muon with momentum in the range 0.1 $\leq$ Pμ $\leq$ 1.2 GeV/$c$,
two protons, both with momentum in the range 0.3 $\leq$ Pp $\leq$ 1.0 GeV/$c$,
no charged pions with momentum above 65 MeV/$c$, no neutral pions of any
momentum, and any number of neutrons. For this signal definition, the event
selection described above achieves an efficiency of 13.0%. Details of the
event selection can be found in Ref. Sword-Fehlberg (2022).
In addition to studying theoretical predictions of 2p2h processes using the
GENIE MicroBooNE Tune and NuWro, we consider predictions from three additional
GENIE model sets Alvarez-Ruso  _et al._ (2021): GENIE v3.0.6 G18_02a_00_000
(GENIE Empirical), GENIE v3.0.6 G18_10a_02_11a (GENIE Nieves), and GENIE
v3.2.0 G21_11b_00_000 (GENIE SuSAv2). All three model sets utilize the same
models for CC coherent pion production (CCCOH) Berger and Sehgal (2009), CC
resonant pion production (CCRES), which is based on the Rein-Sehgal
prediction, but uses updated form factors Graczyk and Sobczyk (2008) and
includes lepton mass effects and additional pion production diagrams Berger
and Sehgal (2007); Kuzmin  _et al._ (2004); Nowak (2009), and CC deep
inelastic scattering (CCDIS) Yang and Bodek (2000); Bodek and Yang (2003). The
GENIE Empirical and GENIE Nieves also use the same models to predict the
neutral current (NC) quasi-elastic (QE), MEC, COH, RES, and DIS interactions.
NC interactions are not implemented in GENIE SuSAv2 at this time Alvarez-Ruso
_et al._ (2021). Further, the GENIE Empirical and GENIE Nieves model sets both
utilize the hA2018 FSI model Ashery  _et al._ (1981), while the GENIE SuSAv2
model set utilizes the hN2018 FSI model Harewood and Gran (2019).

In addition to the two different FSI models, each model set uses distinct
models for quasi-elastic (QE) and MEC neutrino interaction modes. These
distinctions in QE, MEC, and FSI models are critical as these components all
have an impact on 2p2h processes. The GENIE Empirical model set uses the
relativistic Fermi gas (RFG) nuclear model Smith and Moniz (1972), the
Llewellyn Smith QE model Llewellyn Smith (1972), and an empirically derived
prediction for MEC interactions Katori (2015). The GENIE Nieves model set uses
the local Fermi gas (LFG) nuclear model Nieves  _et al._ (2004a), which is
similar to the RFG model but includes considerations for the nuclear density.
The Nieves QE Nieves  _et al._ (2012) model is similar to the Llewellyn Smith
QE model, but includes contributions from long range nucleon-nucleon
correlations and Coulombic effects. The Nieves MEC model is derived from first
principles Schwehr  _et al._ (2016). The original SuSAv2 model set utilizes
the relativistic mean field (RMF) approximation nuclear model Serot and
Walecka (1992); Caballero  _et al._ (2005), which considers relativistic
effects in the calculation of the motion of the nucleons in the nucleus.
Relativistic effects are also considered in the calculation of the SuSAv2 QE
Dolan  _et al._ (2020a) and SuSAv2 MEC Donnelly and Sick (1999) predictions by
using scaling functions, derived from relativistic assumptions, to scale the
cross sections at high momentum transfers. Although GENIE lacks the option to
use an RMF nuclear model directly, it achieves approximate consistency with
the RMF-based results by choosing the nucleon initial momentum from an LFG
distribution. On top of that, the default nucleon binding energy used in GENIE
for the LFG model is replaced for SuSAv2 with an effective value tuned to most
closely duplicate the RMF distribution Dolan  _et al._ (2020b); Papadopoulou
_et al._ (2021).

Using samples of simulated $\nu_{\mu}$ CC interactions from each model set, a
study was conducted to determine variables sensitive to physics differences
between the model sets. One such variable is the opening angle between the
protons in the lab frame,
$\gamma_{\vec{\textrm{P}}_{\textrm{L}}\vec{\textrm{,P}}_{\textrm{R}}}$,
defined as:

$\textrm{cos(}\gamma_{\vec{\textrm{P}}_{\textrm{L}}\vec{\textrm{,P}}_{\textrm{R}}}\textrm{)}=\frac{\vec{\textrm{P}}_{\textrm{L}}\cdot\vec{\textrm{P}}_{\textrm{R}}}{|\vec{\textrm{P}}_{\textrm{L}}|\,|\vec{\textrm{P}}_{\textrm{R}}|}$
(1)

where $\vec{\textrm{P}}_{\textrm{L}}$ is the momentum of the leading proton,
the proton with the highest momentum, and $\vec{\textrm{P}}_{\textrm{R}}$ is
the momentum of the recoil proton, the second proton in the event. A second
sensitive variable is the opening angle between the muon and total proton
momentum vector,
$\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$, defined as:

$\textrm{cos(}\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}\textrm{)}=\frac{\vec{\textrm{P}}_{\mu}\cdot\vec{\textrm{P}}_{\textrm{sum}}}{|\vec{\textrm{P}}_{\mu}|\,|\vec{\textrm{P}}_{\textrm{sum}}|}$
(2)

where $\vec{\textrm{P}}_{\mu}$ is the momentum of the muon and
$\vec{\textrm{P}}_{\textrm{sum}}$ is the vector addition of the leading and
recoil proton momenta. A schematic diagram of these angles is found in Fig. 1.
The opening angle between the protons in the lab frame provides information on
the effect of the QE and MEC modeling on the proton momentum, while the
opening angle between the muon and total proton momentum vectors provides
information on the treatment of the outgoing muon momentum in relation to the
2p2h system.

Figure 1: Diagram of the opening angles
$\gamma_{\vec{\textrm{P}}_{\textrm{L}}\vec{\textrm{,P}}_{\textrm{R}}}$
(dashed) and $\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$
(solid).

In addition to these two angles, we also identify the magnitude of the
momentum transverse to the neutrino beam direction of the final state system,
$\delta\textrm{P}_{\textrm{T}}$ Dolan (2016). The transverse momentum vector
of the CC1$\mu$2p0$\pi$ system ($\delta\vec{\textrm{P}}_{\textrm{T}}$) is
defined as:

$\delta\vec{\textrm{P}}_{\textrm{T}}=\vec{\textrm{P}}^{\mu}_{\textrm{T}}+\vec{\textrm{P}}^{\textrm{L}}_{\textrm{T}}+\vec{\textrm{P}}^{\textrm{R}}_{\textrm{T}}$
(3)

where $\vec{\textrm{P}}^{\mu}_{\textrm{T}}$,
$\vec{\textrm{P}}^{L}_{\textrm{T}}$, and $\vec{\textrm{P}}^{R}_{\textrm{T}}$
are the transverse momentum vectors of the muon, leading proton, and recoil
proton respectively. The magnitude $\delta\textrm{P}_{\textrm{T}}$ is
sensitive to nuclear effects, with $\delta\textrm{P}_{\textrm{T}}\neq 0$
indicative of motion between the nucleons in the initial state.

Distributions of selected events as a function of the cosine of
$\gamma_{\vec{\textrm{P}}_{\textrm{L}}\vec{\textrm{,P}}_{\textrm{R}}}$ can be
found in Fig. 2. The colored bands in each histogram represent events selected
from the overlay MC subdivided into different final state topologies [Fig.
2(a)] and different interaction modes based on the GENIE MicroBooNE Tune
prediction [Fig. 2(b)]. The error bars on the data points are the data
statistical uncertainty while the dashed lines represent the uncertainty on
the combination of the Overlay MC, Dirt MC, and EXT data (also known as the
prediction). This uncertainty includes both the statistical uncertainty of the
prediction and systematic uncertainty, which dominates over the data
statistical uncertainty for this measurement. Contributions from flux modeling
and protons-on-target (POT) counting Aguilar-Arevalo  _et al._ (2013), cross
section modeling Abratenko  _et al._ (2022a), re-interaction modeling Calcutt
_et al._ (2021), and detector modeling Abratenko  _et al._ (2022b) are
considered in the calculation of the systematic uncertainties. Uncertainty on
the modeling of dirt events is also considered in the systematic uncertainty
Abratenko  _et al._ (2022c). Of these six sources of systematic uncertainties,
cross section modeling, with the largest contribution from the MEC model, and
detector modeling are the largest. We find that our CC1$\mu$2p0$\pi$ signal,
represented by the pink band in Fig. 2(a), constitutes the majority of events
selected from the Overlay MC sample with a purity of 65.4%. We also find that
the CCMEC process, represented by the magenta band in Fig. 2(b), has a
somewhat different shape compared to our signal. This is expected as MEC is
not the only contributor to CC1$\mu$2p0$\pi$ topologies. The comparison of the
data points to the prediction shows reasonable shape agreement as well as
normalization agreement at the 1$\sigma$ level.

(a)

(b)

Figure 2: Yield for the cosine of the opening angle between the protons in the
lab frame,
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$).
The selected MC events are broken down into (a) final-state topologies and (b)
$\nu$ interaction channels based on the MicroBooNE Tune Abratenko  _et al._
(2022a) truth information. The error on the data represents the data
statistical uncertainty and the dashed lines represent the statistical plus
systematic uncertainty on the prediction. Note that the Out of FV (grey band)
in both plots represents selected events in which the reconstructed neutrino
vertex fell outside of the predefined FV.

At this point, our kinematic variables are those reconstructed from the data;
due to detector resolution and smearing effects, these require unfolding. We
use the D’Agostini iterative Bayesian unfolding procedure D’Agostini (1995),
as implemented in the RooUnfold software framework D’Agostini (2010), for this
purpose. We determine the optimal number of iterations with techniques
described in Ref. Dolan, S.  and therefore use two unfolding iterations for
the distribution of
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\vec{\textrm{,P}}_{\textrm{R}}}$),
four unfolding iterations for the distribution of
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$), and
two unfolding iterations for the distribution of
$\delta\textrm{P}_{\textrm{T}}$. Unfolded distributions are then normalized by
the detector efficiency, number of target nuclei, and the total integrated
neutrino flux to produce a cross section.

(a)

(b)

Figure 3: Single differential cross sections as a function of (a) the cosine
of the opening angle between the protons in the lab frame,
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$),
and of (b) the cosine of the opening angle between the muon and total proton
momentum vector,
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$). The
inner error bands on the data represent the statistical uncertainty while the
outer error bands represent the systematic uncertainty. A
$\chi^{2}/\textrm{DoF}$, considering systematic and statistical uncertainties,
is calculated between the data and each model set curve.

Fig. 3 shows the single differential cross sections as functions of the cosine
of
$\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$
[Fig. 3(a)] and of the cosine of
$\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$ [Fig. 3(b)].
The black points represent the extracted cross section from data with the
inner error bands representing the statistical uncertainty and the outer error
bands representing the systematic uncertainty. In addition to the six sources
of systematic uncertainty mentioned above, an uncertainty on the choice of the
number of unfolding iterations is also included in the systematic error band.
The cross sections for the GENIE Empirical, GENIE Nieves, and GENIE SuSAv2
model sets were derived using methods described in Ref. Gardiner (2021). The
GENIE MicroBooNE Tune and NuWro cross section curves are created by selecting
generated CC1$\mu$2p0$\pi$ signal events from the Overlay MC and NuWro
samples, respectively. Each distribution was then normalized by the number of
target nucleons and the total integrated flux to produce a cross section
curve. The $\chi^{2}$ per degree of freedom ($\chi^{2}/\textrm{DoF}$) is
calculated between the data and each model set curve. The systematic and
statistical uncertainties on the data are considered in this calculation using
the total covariance matrices found in the Supplemental Materials Abratenko
_et al._ .

We find that the GENIE MicroBooNE Tune and the GENIE Nieves models have the
best agreement with our data for
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$),
and the GENIE Nieves and NuWro models have the best agreement with our data
for cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$). We
also find that the GENIE Nieves and NuWro models have the best agreement with
our data across the full ranges of
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$) and
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$). The
GENIE MicroBooNE Tune and GENIE Empirical models tend to predict higher cross
sections than indicated by our data in regions of low
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$)
and high
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$).
Although the GENIE MicroBooNE Tune uses all the same model elements as the
GENIE Nieves sample, we find an overall difference in shape and normalization
between these two curves. This is most likely caused by the increased CC2p2h
cross section normalization and interpolated CC2p2h cross section shape that
are utilized in the GENIE MicroBooNE Tune.

We show the single differential cross sections for the data and the models as
functions of $\delta\textrm{P}_{\textrm{T}}$ in Fig. 4. In this variable the
NuWro prediction is higher than the data and other models in the first bin.
The reason for this is that Nieves and NuWro form their initial hadronic
states in different ways. In the GENIE implementation of the GENIE Nieves
model set, two nucleons are selected from the Fermi sea of the nucleus Katori
(2015). The momentum of each nucleon is then randomly sampled from a
distribution of the initial state nucleon momentum Schwehr  _et al._ (2016)
formed from the LFG nuclear model Nieves  _et al._ (2004a). In NuWro, the
selection of the two nucleons and their momenta is similar to the GENIE
implementation of GENIE Nieves, but the two nucleons are required to have
back-to-back momenta in the initial state Niewczas and Sobczyk (2016). The
over-prediction of NuWro at low $\delta\textrm{P}_{\textrm{T}}$ has also been
observed in Ref. Bathe-Peters (2020), which also finds this over-prediction in
the absence of FSI, indicating that the excess is an initial state effect. Our
data indicates that there is a preference for the GENIE description of the
initial two-nucleon hadronic state. Cross section values for different
kinematic variables can be found in the Supplemental Material Abratenko  _et
al._ .

(a)  Figure 4: Single differential cross section as a function of the
magnitude of the transverse momentum of the final state system,
$\delta\textrm{P}_{\textrm{T}}=\left|\vec{\textrm{P}}^{\mu}_{\textrm{T}}+\vec{\textrm{P}}^{\textrm{L}}_{\textrm{T}}+\vec{\textrm{P}}^{\textrm{R}}_{\textrm{T}}\right|$.
The inner error bands on the data represent the statistical uncertainty while
the outer error bands represent the systematic uncertainty. A
$\chi^{2}/\textrm{DoF}$, considering systematic and statistical uncertainties,
is calculated between the data and each model set curve.

In summary, this letter presents the first measurement of single differential
cross sections of CC1$\mu$2p0$\pi$ events on argon. Events containing exactly
one muon, two protons, and no other mesons are selected from BNB data. We
extract differential cross sections as functions of three kinematic variables,
cos($\gamma_{\vec{\textrm{P}}_{\textrm{L}}\textrm{,}\vec{\textrm{P}}_{\textrm{R}}}$),
cos($\gamma_{\vec{\textrm{P}}_{\mu}\vec{\textrm{,P}}_{\textrm{sum}}}$) and
$\delta\textrm{P}_{\textrm{T}}$, which are defined in such a way as to be
sensitive to the formation of 2p2h pairs through MEC processes. We compare our
extracted cross sections to those predicted by the NuWro and GENIE generators,
including three different GENIE model sets. These cross-section models span a
range of nuclear models, QE and MEC models, and hadron transport models. We
find that the NuWro and GENIE Nieves predictions show the best overall shape
agreement in the kinematic variables describing opening angles. The
$\delta\textrm{P}_{\textrm{T}}$ variable is sensitive to the different way
that NuWro and GENIE Nieves form the initial hadronic state, and we are
therefore able to show that GENIE Nieves gives the best overall description of
the production of CC1$\mu$2p0$\pi$ final states thanks to its more realistic
initial-state momentum distributions. None of the models used for comparisons
in this work include SRCs in the nuclear model; their inclusion could further
improve the agreement with our measured differential cross sections. This is
the first direct measurement of the CC2p2h cross section, and therefore the
first time that these models can be directly compared to data. This will
provide valuable input for future model development toward precision neutrino
physics measurements. In addition, these measured CC1$\mu$2p0$\pi$ cross
sections can be used to reinterpret data from existing experiments that cannot
distinguish 2p2h final states from other CC interaction mechanisms.

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); and the UK
Research and Innovation (UKRI) Future Leaders Fellowship. Additional support
for the laser calibration system and cosmic ray tagger was provided by the
Albert Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses,
without whom this work would not have been possible. For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC BY) public
copyright license to any Author Accepted Manuscript version arising from this
submission.

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# Generalized Donaldson’s functionals and related nonlinear partial
differential equations

Chuanjing Zhang and Xi Zhang  Chuanjing Zhang
School of Mathematical Sciences,
University of Science and Technology of China,
Hefei, 230026,P.R. China
<EMAIL_ADDRESS>Xi Zhang
School of Mathematical Sciences,
University of Science and Technology of China,
Hefei, 230026,P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce a family of generalized Donaldson’s functional on
holomorphic vector bundles, whose Euler-Lagrange equations are a vector bundle
version of the complex $k$-Hessian equations. We also discuss the uniqueness
of solutions to these equations.

###### Key words and phrases:

holomorphic vector bundle, Kähler manifold, nonlinear PDE

######  Mathematics Subject Classification:

53C07, 58E15

The second author is the corresponding author. The authors are partially
supported by NSF in China No.11625106, 11801535 and 11721101. The research was
partially supported by the project ”Analysis and Geometry on Bundle” of
Ministry of Science and Technology of the People’s Republic of China,
No.SQ2020YFA070080.

##  1\. Introduction

Let $(M,\omega)$ be an $n$-dimensional compact Kähler manifold and
$(E,\overline{\partial}_{E})$ be a holomorphic vector bundle over $M$. For any
$B\in\Omega^{2}(\operatorname{End}(E))$, that is any
$\operatorname{End}(E)$-valued 2-form, we denote

(1.1) $B^{k}:=\underbrace{B\wedge B\cdots\wedge B}_{k},$

with $k\in\mathbb{N}$ and $k\leq n$. For any given Hermitian metric $H$ on
$E$, we use $D_{H}$ to denote the Chern connection with respect to the metric
$H$ and the holomorphic structure, and $F_{H}$ is the curvature tensor of the
Chern connection $D_{H}$.

Let $H_{0}$ be a fixed Hermitian metric on $E$ and $H(s)$ a path connecting
metrics $H_{0}$ and another Hermitian metric $H$. For any integer $0<k\leq n$,
we introduce the following generalized Donaldson’s functionals

(1.2) $\mathcal{M}_{E,k}^{0}(H_{0},H)=\int_{0}^{1}\int_{M}\mbox{\rm
tr\,}\left[\left(\sqrt{-1}F_{H(s)}\right)^{k}\,H(s)^{-1}\,\frac{\partial
H(s)}{\partial s}\right]\wedge\frac{\omega^{n-k}}{(n-k)!}\,ds$

and

(1.3)
$\mathcal{M}_{E,k}(H_{0},H)=\mathcal{M}_{E,k}^{0}(H_{0},H)-\int_{M}\lambda_{E,k}\cdot\log\det(H_{0}^{-1}H)\,\frac{\omega^{n}}{n!},$

where the constant
$\lambda_{E,k}=(2\pi)^{k}\,(k!)\,\frac{ch_{k}(E)\cdot\left[\frac{\omega^{n-k}}{(n-k)!}\right]}{{\rm
rank}(E)\cdot{\rm vol}(M,\omega)}$.

As it will be shown in §2, the functional $\mathcal{M}_{E,k}(H_{0},H)$ is well
defined, i.e. it is independent of the choice of path $H(s)$. Indeed, when
$k=1$, the above functional $\mathcal{M}_{E,1}(H_{0},H)$ is just the well-
known Donaldson’s functional, which was introduced by Donaldson ([12], [24])
in his groundbreaking work on the study of the Hermitian-Yang-Mills
connection. In [12], it is an important observation that the Euler-Lagrange
equation of the Donaldson’s functional is just the Hermitian-Yang-Mills
equation

(1.4) $\sqrt{-1}\,\Lambda_{\omega}F_{H}=\lambda_{E,1}{\rm Id}_{E}.$

The solvability of the above equation has many interesting geometric
applications. Motivated by the study of the inequalities between Chern
numbers, we introduce the generalized Donaldson’s functional by increasing the
power for the curvature tensor $F_{H}$, which corresponds to higher order
Chern classes.

In §2, we will show that the Euler-Lagrange equation of the generalized
Donaldson’s functional $\mathcal{M}_{E,k}$ is given by

(1.5)
$\frac{\left(\sqrt{-1}F_{H}\right)^{k}\wedge\frac{\omega^{n-k}}{(n-k)!}}{\frac{\omega^{n}}{n!}}=\lambda_{E,k}\,{\rm
Id}_{E}.$

It is easy to see that, when $k=1$, the above equation is just the standard
Hermitian-Yang-Mills equation (1.4). On the other hand, if
$(E,\overline{\partial}_{E})$ is a line bundle, the equation (1.5) reduces to
the complex $k$-Hessian equation. Given this connection, we will simply call
the above equation as the vector bundle valued $k$-Hessian equation. In fact,
such type of equations were considered in various settings. When $k=n$, the
equation (1.5) is a vector bundle version of the complex Monge-Ampère equation
introduced by Pingali ([40]) from the viewpoint of moment map, which
generalizes the dHYM equations studied in [10, 11] to vector bundles. When
$k=2$, equations as (1.5) also arise naturally in the Hull-Strominger system
from theoretical physics. One crucial equation in the system is the so-called
anomaly cancellation equation, which can be viewed as a constraint on the
second Chern class and it has similar form as (1.5). The system has been
studied extensively in recent years, see for example [14, 15, 21, 41, 42, 43,
44, 47].

For the Hermitian-Yang-Mills equation (1.4), the classical Donaldson-
Uhlenbeck-Yau theorem ([39], [12], [56]) states that a holomorphic bundle
admits the Hermitian-Einstein metric if and only if it is poly-stable in the
sense of Mumford-Takemoto. There are many interesting and important works
related ([1, 4, 6, 13, 20, 22, 25, 29, 30, 31, 26, 32, 33, 34, 45, 46], etc.).
The complex Monge-Ampère equation has been the subject of intensive studies in
the past forty years, since the groundbreaking work by Yau ([55]) on the proof
of the Calabi conjecture ([7]). Many interesting works were carried out by
many people in the past several decades (see e.g. [3, 9, 8, 35, 23, 51, 52,
53, 54, 48, 49, 18, 19]).

Given all the above nice connections mentioned above, we believe that it will
be very interesting to study the vector bundle version of the complex Hessian
equation (1.5). In this paper, we are going to study the uniqueness problem of
the solution. Before doing that, one has to address the issue about the
natural set for the solutions as the Gärding cone for the standard Hessian
equation for functions (see for example [50]). However, the situation here is
much more complicated than the scalar case. In the following, the space of
Hermitian metrics on $E$ will be denoted by $\emph{Herm}(E)$. For any
Hermitian metric $H\in\emph{Herm}(E)$, we denote

(1.6) $S_{H}(E)=\left\\{\eta\in\Omega^{0}(M,End(E))\,|\,\eta^{\ast
H}=\eta\right\\},$ (1.7) $S_{H,\epsilon}(E)=\left\\{\eta\in S_{H}(E)\,|\
\sup_{M}|\eta|_{H}<\epsilon,\
\sup_{M}\left(|\partial_{H}\eta|_{H}+|\overline{\partial}\partial_{H}\eta|_{H}\right)<\epsilon\right\\}$

and

(1.8) $B_{H,\epsilon}(E)=\left\\{K\in\emph{Herm}(E)\,|\ \log H^{-1}K\in
S_{H,\epsilon}(E)\right\\}.$

Now, we introduce a positivity condition, which is a natural generalization
for the $\Gamma_{k}$ cone for scalar Hessian equation.

######  Definition 1.1.

Let $H$ be a Hermitian metric on the holomorphic vector bundle
$(E,\overline{\partial}_{E})$ over a compact Kähler manifold $(M,\omega)$. The
curvature tensor $\sqrt{-1}F_{H}$ is said to be $\sigma_{k}$-$\omega$-positive
on a point $p\in M$ if, for any non-zero $End(E)$-valued $(0,1)$-form $\xi$,
we have

(1.9) $\frac{\omega^{n-k}}{(n-k)!}\wedge\sqrt{-1}\,\mbox{\rm
tr\,}\left[\sum_{i=0}^{k-1}\left(\sqrt{-1}F_{H}\right)^{i}\wedge\xi^{\ast
H}\wedge\left(\sqrt{-1}F_{H}\right)^{k-1-i}\wedge\xi\right]\bigg{|}_{p}>0.$

We denote the set of Hermitian metrics whose curvature are
$\sigma_{k}$-$\omega$-positive on $M$ by $\mathcal{H}_{\omega,k}$.

It is a natural question whether the set $\mathcal{H}_{\omega,k}$ is indeed
convex or not. That is, for any two metrics $K,H\in\mathcal{H}_{\omega,k}$,
let $H(t)$ be the geodesic connecting $K$ and $H$, do we have
$H(t)\in\mathcal{H}_{\omega,k}$? Indeed, if one can give a positive answer to
the above question, by the second variation formula (3) for the generalized
Donaldson’s functional (1.3), one can conclude that a solution
$H\in\mathcal{H}_{\omega,k}$ of the bundle-valued complex $k$-Hessian equation
(1.5) attains an absolute minimum of the generalized Donaldson’s functional
(1.3), and then obtain the uniqueness of the solution. At present we can not
prove the convexity of $\mathcal{H}_{\omega,k}$. Even if we can’t give a
positive answer to the above question, we can still prove local uniqueness of
the solution of bundle version complex $k$-Hessian equation (1.5) and the
local minimum in the following theorem.

######  Theorem 1.2.

Let $(E,\overline{\partial}_{E})$ be a holomorphic vector bundle over an
$n$-dimensional compact Kähler manifold $(M,\omega)$, $H_{0}$ be a Hermitian
metric on $E$. If $H$ is a solution of equation (1.5) and its curvature is
$\sigma_{k}$-$\omega$-positive on $M$, then there exists a small number
$\epsilon$ such that, for any $\tilde{H}\in B_{H,\epsilon}(E)$, we have

(1.10) $\mathcal{M}_{E,k}(H_{0},H)\leq\mathcal{M}_{E,k}(H_{0},\tilde{H}).$

Moreover, if the holomorphic bundle $(E,\overline{\partial}_{E})$ is simple
and $K\in B_{H,\epsilon}(E)$ is another solution of (1.5), then there exists a
constant $a$ such that $K=e^{a}H$.

For the special case $k=2$, under the strongly $\sigma_{2}$-$\omega$-positive
hypothesis (see the definition in §3), we obtain the following uniqueness
theorem.

######  Theorem 1.3.

Let $(E,\overline{\partial}_{E})$ be a simple holomorphic vector bundle over
an $n$-dimensional compact Kähler manifold $(M,\omega)$, $H$ and $K$ be two
Hermitian metrics on $E$ whose curvature are strongly
$\sigma_{2}$-$\omega$-positive on the whole $M$. If both $H$ and $K$ are
solutions of (1.5) for $k=2$, then there must exist a constant $a$ such that
$K=e^{a}H$.

In the following, we want to discuss the $k$-stability of holomorphic vector
bundles. First, let us recall the definition of the Chern classes (Chern
character class) of coherent sheaf $\mathcal{F}$. If the base manifold $M$ is
a projective manifold, there exists a resolution

(1.11) $0\rightarrow E_{1}\rightarrow E_{2}\rightarrow\cdots\rightarrow
E_{N}\rightarrow\mathcal{F}\rightarrow 0$

for each coherent sheaf $\mathcal{F}$ by locally free sheaves. The total Chern
class $c(\mathcal{F})$ is defined by

(1.12) $c(\mathcal{F})=c(E_{N})c(E_{N-1})^{-1}\cdots
c(E_{N-i})^{(-1)^{i}}\cdots c(E_{1})^{(-1)^{N-1}}.$

The definition is indeed independent of the chosen resolution as proved by
Borel and Serre [5]. Over a non-projective compact complex manifold, the Chern
classes of a coherent sheaf can be defined by the classes of Atiyah-
Hirzenbruch ([2], see [17] for details). Note that on a non-projective compact
complex manifold, a coherent sheaf does not necessarily have a global
resolution via locally free coherent sheaves. Atiyah and Hirzebruch [2]
circumvent this problem by using real-analytic functions. Using Grauert’s
results [16], they prove that, by taking tensor with the sheaf
$\mathcal{A}_{M}$ of germs of real analytic functions on $M$, there is a
resolution by real analytic vector bundles

(1.13) $0\rightarrow E_{1}\rightarrow E_{2}\rightarrow\cdots\rightarrow
E_{N}\rightarrow\mathcal{F}\otimes_{\mathcal{O}_{M}}\mathcal{A}_{M}\rightarrow
0,$

where $E_{i}^{\prime}s$ are real analytic complex vector bundles. Then we can
define the total Chern class $c(\mathcal{F})$ as that in (1.12). The
definition is also shown to be independent of the choice of resolution. By the
way, we can obtain the definition of the Chern character classes
$ch_{k}(\mathcal{F})$. We define the $k$-$\omega$-degree of $\mathcal{F}$ by

(1.14)
$deg_{k,\omega}(\mathcal{F})=\int_{M}ch_{k}(\mathcal{F})\wedge\frac{\omega^{n-k}}{(n-k)!}.$

######  Definition 1.4.

Let $(M,\omega)$ be a compact Kähler manifold, and $\mathcal{E}$ be a torsion-
free coherent sheaf over $M$. $\mathcal{E}$ is called $k$-$\omega$-stable
(semi-stable), if for every coherent sub-sheaf
$\mathcal{F}\hookrightarrow\mathcal{E}$ of lower rank, it holds:

(1.15)
$\mu_{k,\omega}(\mathcal{F})=\frac{deg_{k,\omega}(\mathcal{F})}{rank\mathcal{F}}<(\leq)\mu_{k,\omega}(\mathcal{E})=\frac{deg_{k,\omega}(\mathcal{E})}{rank\mathcal{E}}.$

When $k=1$, $k$-$\omega$-stability is just the stability in the sense of
Mumford-Takemoto, and is MA-stability ([40]) when $k=n$. Following the
classical Donaldson-Uhlenbeck-Yau theorem, it is natural to conjecture: Let
$(E,\overline{\partial}_{E})$ be a simple holomorphic vector bundle over a
compact Kähler manifold, and the set $\mathcal{H}_{\omega,k}$ is non empty.
Then $E$ admits a Hermitian metric $H$ satisfying the bundle valued complex
$k$-Hessian equations (1.5) if and only if it is $k$-$\omega$-stable.

This paper is organized as follows. In Section 2, we prove that the
generalized Donaldson’s functional is well defined, and give the first
variation formula of this functional. In section 3, we give the proof of
theorem 1.2 and theorem 1.3.

Acknowledgements: The authors would like to thank Xiangwen Zhang for many
helpful discussion.

##  2\. Generalized Donaldson’s functionals

Let $(M,\omega)$ be an $n$-dimensional compact Kähler manifold and
$(E,\overline{\partial}_{E})$ a holomorphic vector bundle over $M$. Given any
two Hermitian metrics $H_{0}$ and $H$ on bundle $E$, let

(2.1) $\displaystyle h=H_{0}^{-1}H\in\Gamma(End(E)).$

It is easy to check that

(2.2) $h^{\ast H_{0}}=h^{\ast H}=h,$ (2.3)
$D_{H}=D_{H_{0}}+h^{-1}\partial_{H_{0}}h$

and

(2.4) $F_{H}=F_{H_{0}}+\overline{\partial}(h^{-1}\partial_{H_{0}}h).$

Moreover, we can prove the following lemma

######  Lemma 2.1.

The functional $\mathcal{M}_{E,k}(H_{0},H)$ is well defined, i.e. it is
independent of the choice of path $H(s)$.

###### Proof.

Let $H(\tau,s)$ be a two-parameter family of metrics such that

$H(\tau,0)=H_{0}\ \ \textit{\rm and}\ \ H(\tau,1)=H,$

where $\tau\in(-\epsilon,\epsilon)$, $s\in[0,1]$. Then, $H(0,s)$ and $H(1,s)$
are two paths connecting $H_{0}$ and $H$. Let

$h(\tau,s)=H_{0}^{-1}H(\tau,s).$

It is easy to check that

(2.5)
$\frac{\partial}{\partial\tau}F_{H(\tau,s)}=\overline{\partial}\partial_{H(\tau,s)}\left(h^{-1}\frac{\partial
h}{\partial\tau}\right)$

and

(2.6) $\frac{\partial}{\partial
s}F_{H(\tau,s)}=\overline{\partial}\partial_{H(\tau,s)}\left(h^{-1}\frac{\partial
h}{\partial s}\right).$

Claim:

$\displaystyle\frac{\partial}{\partial\tau}\mbox{\rm
tr\,}\left\\{\left(\sqrt{-1}F_{H(\tau,s)}\right)^{k}h^{-1}\frac{\partial
h}{\partial s}\right\\}-\frac{\partial}{\partial s}\mbox{\rm
tr\,}\left\\{\left(\sqrt{-1}F_{H(\tau,s)}\right)^{k}h^{-1}\frac{\partial
h}{\partial\tau}\right\\}$ $\displaystyle=$ $\displaystyle-\partial\mbox{\rm
tr\,}\left\\{\sqrt{-1}\overline{\partial}\eta_{k}h^{-1}\frac{\partial
h}{\partial s}\right\\}-\overline{\partial}\mbox{\rm
tr\,}\left\\{\sqrt{-1}\partial_{H}\phi_{k}h^{-1}\frac{\partial
h}{\partial\tau}\right\\}$

where

$\eta_{k}=\sum_{i=0}^{k-1}\left(\sqrt{-1}F_{H(\tau,s)}\right)^{i}h^{-1}\frac{\partial
h}{\partial\tau}\wedge\left(\sqrt{-1}F_{H(\tau,s)}\right)^{k-1-i}$

and

$\phi_{k}=\sum_{i=0}^{k-1}\left(\sqrt{-1}F_{H(\tau,s)}\right)^{i}h^{-1}\frac{\partial
h}{\partial s}\wedge\left(\sqrt{-1}F_{H(\tau,s)}\right)^{k-1-i}.$

By the equality (2), we have:

$\displaystyle\frac{d}{d\tau}\int_{0}^{1}\int_{M}\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial s}]\wedge\frac{\omega^{n-k}}{(n-k)!}ds$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{M}\frac{\partial}{\partial\tau}\\{\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial s}]\\}\wedge\frac{\omega^{n-k}}{(n-k)!}ds$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\int_{M}\frac{\partial}{\partial
s}\\{\mbox{\rm tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial\tau}]\\}\wedge\frac{\omega^{n-k}}{(n-k)!}ds$
$\displaystyle=$ $\displaystyle\int_{M}\\{\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial\tau}]\\}\wedge\frac{\omega^{n-k}}{(n-k)!}|_{s=0}^{1}$
$\displaystyle=$ $\displaystyle 0.$

This shows that the functional $\mathcal{M}_{E,k}(H_{0},H)$ is independent of
the choice of path $H(s)$. Therefore, we only need to prove the claim (2).

By applying the second Bianchi identity, we have

$\displaystyle\frac{\partial}{\partial\tau}\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial s}]$ $\displaystyle=$ $\displaystyle\sqrt{-1}\mbox{\rm
tr\,}[\sum_{i=0}^{k-1}(\sqrt{-1}F_{H(\tau,s)})^{i}\wedge\overline{\partial}\partial_{H(\tau,s)}(h^{-1}\frac{\partial
h}{\partial\tau})\wedge(\sqrt{-1}F_{H(\tau,s)})^{k-1-i}h^{-1}\frac{\partial
h}{\partial s}]$ $\displaystyle+\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(-h^{-1}\frac{\partial
h}{\partial\tau}h^{-1}\frac{\partial h}{\partial
s}+h^{-1}\frac{\partial^{2}h}{\partial\tau\partial s})],$ $\displaystyle=$
$\displaystyle\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}\partial_{H(\tau,s)}(\eta_{k})h^{-1}\frac{\partial
h}{\partial s}]+\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(-h^{-1}\frac{\partial
h}{\partial\tau}h^{-1}\frac{\partial h}{\partial
s}+h^{-1}\frac{\partial^{2}h}{\partial\tau\partial s})]$

and similarly

$\displaystyle\frac{\partial}{\partial s}\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}H^{-1}(\tau,s)\frac{\partial
H(\tau,s)}{\partial\tau}]$ $\displaystyle=$ $\displaystyle\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}]+\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(-h^{-1}\frac{\partial h}{\partial
s}h^{-1}\frac{\partial h}{\partial\tau}+h^{-1}\frac{\partial^{2}h}{\partial
s\partial\tau})].$

Using (2) and (2), we have

$\displaystyle\frac{\partial}{\partial\tau}\mbox{\rm
tr\,}\\{(\sqrt{-1}F_{H(\tau,s)})^{k}h^{-1}\frac{\partial h}{\partial
s}\\}-\frac{\partial}{\partial s}\mbox{\rm
tr\,}\\{(\sqrt{-1}F_{H(\tau,s)})^{k}h^{-1}\frac{\partial h}{\partial\tau}\\}$
$\displaystyle=$ $\displaystyle-\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}\overline{\partial}(\eta_{k})h^{-1}\frac{\partial
h}{\partial s}]-\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}]$ $\displaystyle+\mbox{\rm
tr\,}\\{(\sqrt{-1}F_{H}\wedge\eta_{k}-\eta_{k}\wedge\sqrt{-1}F_{H})h^{-1}\frac{\partial
h}{\partial s}\\}$ $\displaystyle-\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(h^{-1}\frac{\partial
h}{\partial\tau}h^{-1}\frac{\partial h}{\partial s})]+\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(h^{-1}\frac{\partial h}{\partial
s}h^{-1}\frac{\partial h}{\partial\tau})].$

By direct calculation, one can check that

$\displaystyle\mbox{\rm
tr\,}\\{(\sqrt{-1}F_{H(\tau,s)}\wedge\eta_{k}-\eta_{k}\wedge\sqrt{-1}F_{H(\tau,s)})h^{-1}\frac{\partial
h}{\partial s}\\}$ $\displaystyle-\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(h^{-1}\frac{\partial
h}{\partial\tau}h^{-1}\frac{\partial h}{\partial s})]+\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(\tau,s)})^{k}(h^{-1}\frac{\partial h}{\partial
s}h^{-1}\frac{\partial h}{\partial\tau})]$ $\displaystyle=$ $\displaystyle 0,$

and

$\displaystyle-\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}\overline{\partial}(\eta_{k})h^{-1}\frac{\partial
h}{\partial s}]-\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}]$ $\displaystyle=$ $\displaystyle-\partial\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}(\eta_{k})h^{-1}\frac{\partial h}{\partial
s}]-\overline{\partial}\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}]$ $\displaystyle-\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}(\eta_{k})\wedge\partial_{H(\tau,s)}(h^{-1}\frac{\partial
h}{\partial s})]-\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}(\phi_{k})\wedge\overline{\partial}(h^{-1}\frac{\partial
h}{\partial\tau})]$ $\displaystyle=$ $\displaystyle-\partial\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}(\eta_{k})h^{-1}\frac{\partial h}{\partial
s}]-\overline{\partial}\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}]$ $\displaystyle-\mbox{\rm
tr\,}\\{\sum_{i=0}^{k-1}(\sqrt{-1}F_{H})^{i}\wedge\overline{\partial}(h^{-1}\frac{\partial
h}{\partial\tau})\wedge(\sqrt{-1}F_{H})^{k-1-i}\wedge\partial_{H}(h^{-1}\frac{\partial
h}{\partial s})\\}$ $\displaystyle-\mbox{\rm
tr\,}\\{\sum_{j=0}^{k-1}(\sqrt{-1}F_{H})^{j}\wedge\partial_{H}(h^{-1}\frac{\partial
h}{\partial
s})\wedge(\sqrt{-1}F_{H})^{k-1-j}\wedge\overline{\partial}(h^{-1}\frac{\partial
h}{\partial\tau})\\}$ $\displaystyle=$ $\displaystyle-\partial\mbox{\rm
tr\,}[\sqrt{-1}\overline{\partial}(\eta_{k})h^{-1}\frac{\partial h}{\partial
s}]-\overline{\partial}\mbox{\rm
tr\,}[\sqrt{-1}\partial_{H(\tau,s)}(\phi_{k})h^{-1}\frac{\partial
h}{\partial\tau}].$

So (2), (2) and (2) imply the claim (2).

In the following, we denote

(2.14)
$\Psi_{k}(H)=\frac{(\sqrt{-1}F_{H})^{k}\wedge\frac{\omega^{n-k}}{(n-k)!}}{\frac{\omega^{n}}{n!}}.$

Let $H(t)$ be a family of metrics on $E$. We can consider two parameter family
of metrics $H(t,s)$ such that $H(t,0)=H_{0}$ and $H(t,1)=H(t)$. By (2), we
have the following first variation formula:

$\displaystyle\frac{d}{dt}\mathcal{M}_{E,k}(H_{0},H(t))$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{M}\frac{\partial}{\partial t}\\{\mbox{\rm
tr\,}[(\sqrt{-1}F_{H(t,s)})^{k}H^{-1}(t,s)\frac{\partial H(t,s)}{\partial
s}]\\}\wedge\frac{\omega^{n-k}}{(n-k)!}ds$
$\displaystyle-\frac{d}{dt}\int_{M}\lambda_{E,k}\log\det(h(t))\frac{\omega^{n}}{n!}$
$\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{[\Psi_{k}(H(t))-\lambda_{E,k}Id_{E}]H^{-1}(t)\frac{\partial
H(t)}{\partial t}\\}\frac{\omega^{n}}{n!}.$

Then we have the following lemma.

######  Lemma 2.2.

The Hermitian metric $H$ is a critical of the functional $\mathcal{M}_{E,k}$
if and only if it satisfies the equation

(2.16) $\Psi_{k}(H)-\lambda_{E,k}Id_{E}=0.$

By the definition of $\mathcal{M}_{E,k}$ and choosing a special path, it is
easy to obtain the following lemma.

######  Lemma 2.3.

(1) For any metrics $H_{i}$, $i=0,1,2$, we have:

(2.17)
$\mathcal{M}_{E,k}(H_{0},H_{1})+\mathcal{M}_{E,k}(H_{1},H_{2})=\mathcal{M}_{E,k}(H_{0},H_{2}).$

(2) $\mathcal{M}_{E,k}(H,aH)=0$ for any positive constant $a$.

##  3\. Some unique results

In this section, we consider the uniqueness of the solution to the equation
(1.5). Let $K$ and $H$ be two Hermitian metrics on the bundle $E$. Set

(3.1) $s=\log H^{-1}K.$

The path $H(t)=He^{ts}$ will be called by the geodesic connecting $H$ and $K$.
Along the geodesic, we have

(3.2) $h(t)^{-1}\frac{\partial h(t)}{\partial t}=s,$ (3.3)
$\frac{\partial}{\partial
t}(h(t)^{-1}\partial_{H}h(t))=\partial_{H}s-sh(t)^{-1}\partial_{H}h(t)+h(t)^{-1}\partial_{H}h(t)s$

and

$\displaystyle\frac{\partial}{\partial t}(F_{H(t)}-F_{H})$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
t}(\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t)))=\overline{\partial}_{E}(\partial_{H(t)}s)$
$\displaystyle=$
$\displaystyle\overline{\partial}_{E}(\partial_{H}s)-(\overline{\partial}_{E}s)\wedge(h(t)^{-1}\partial_{H}h(t))-(h(t)^{-1}\partial_{H}h(t))\wedge(\overline{\partial}_{E}s)$
$\displaystyle-s\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))+\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))s,$

where $h(t)=H^{-1}H(t)=e^{ts}$.

From above identities, we obtain

$\frac{\partial}{\partial t}|h(t)^{-1}\partial_{H}h(t)|_{H}^{2}\leq
2|\partial_{H}s|_{H}|h(t)^{-1}\partial_{H}h(t)|_{H}+4|s|_{H}|h(t)^{-1}\partial_{H}h(t)|_{H},$

and

$\displaystyle\frac{\partial}{\partial
t}|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}^{2}$
$\displaystyle\leq$ $\displaystyle
2|\overline{\partial}_{E}(\partial_{H}s)|_{H}|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}+4|\overline{\partial}_{E}s|_{H}|h(t)^{-1}\partial_{H}h(t)|_{H}|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}$
$\displaystyle+4|s|_{H}|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}.$

Similarly, we can also compute

$\displaystyle\frac{\partial}{\partial
t}\left(\sqrt{|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}^{2}+\epsilon}+\sqrt{|h(t)^{-1}\partial_{H}h(t)|_{H}^{2}+\epsilon}\right)$
$\displaystyle\leq$
$\displaystyle|\overline{\partial}_{E}(\partial_{H}s)|_{H}+|\partial_{H}s|_{H}$
$\displaystyle+4(|s|_{H}+|\overline{\partial}_{E}s|_{H})\left(\sqrt{|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}^{2}+\epsilon}+\sqrt{|h(t)^{-1}\partial_{H}h(t)|_{H}^{2}+\epsilon}\right)$

and

$\displaystyle\frac{\partial}{\partial
t}\left[e^{-4t(|s|_{H}+|\overline{\partial}_{E}s|_{H})}\left(\sqrt{|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}^{2}+\epsilon}+\sqrt{|h(t)^{-1}\partial_{H}h(t)|_{H}^{2}+\epsilon}\right)\right]$
$\displaystyle\leq$
$\displaystyle(|\overline{\partial}_{E}(\partial_{H}s)|_{H}+|\partial_{H}s|_{H})e^{-4t(|s|_{H}+|\overline{\partial}_{E}s|_{H})},$

where $\epsilon$ is a positive number. Integrating the equality (3) from $0$
to $t$, and letting $\epsilon\rightarrow 0$, we obtain

(3.7)
$|\overline{\partial}_{E}(h(t)^{-1}\partial_{H}h(t))|_{H}+|h(t)^{-1}\partial_{H}h(t)|_{H}\leq(|\overline{\partial}_{E}(\partial_{H}s)|_{H}+|\partial_{H}s|_{H})\frac{e^{4t(|s|_{H}+|\overline{\partial}_{E}s|_{H})}-1}{4(|s|_{H}+|\overline{\partial}_{E}s|_{H})}.$

###### Proof of Theorem 1.2.

By (2), we have the following second variation formula for the generalized
Donaldson’s functional (1.3):

$\displaystyle\frac{d^{2}}{dt^{2}}\mathcal{M}_{E,k}(H_{0},H(t))$
$\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{[\Psi_{k}(H(t))-\lambda_{E,k}Id_{E}]\frac{\partial}{\partial
t}[H^{-1}(t)\frac{\partial H(t)}{\partial t}]\\}\frac{\omega^{n}}{n!}$
$\displaystyle+\int_{M}\mbox{\rm tr\,}\\{\frac{\partial}{\partial
t}[\Psi_{k}(H(t))]H^{-1}(t)\frac{\partial H(t)}{\partial
t}\\}\frac{\omega^{n}}{n!}$

and

$\displaystyle\frac{\partial}{\partial t}[\Psi_{k}(H(t))]$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{\omega^{n}}{n!}}\\{\frac{\omega^{n-k}}{(n-k)!}\wedge[\sum_{i=0}^{k-1}(\sqrt{-1}F_{H(t)})^{i}\wedge\sqrt{-1}\overline{\partial}\partial_{H(t)}(H^{-1}(t)\frac{\partial
H(t)}{\partial t})\wedge(\sqrt{-1}F_{H(t)})^{k-1-i}]\\}.$

Let $H$ and $K$ be two Hermitian metrics on $E$, and $H(t)=He^{ts}$ the
geodesic connecting $H$ and $K$, where $s=\log H^{-1}K$. By (3), (3) and the
Bianchi identity, we get

$\displaystyle\frac{d^{2}}{dt^{2}}\mathcal{M}_{E,k}(H_{0},H(t))$
$\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{\frac{\partial}{\partial t}[\Psi_{k}(H(t))]s\\}\frac{\omega^{n}}{n!}$
$\displaystyle=$
$\displaystyle\int_{M}\frac{\omega^{n-k}}{(n-k)!}\wedge\mbox{\rm
tr\,}[\sum_{i=0}^{k-1}(\sqrt{-1}F_{H(t)})^{i}\wedge\sqrt{-1}\partial_{H(t)}s\wedge(\sqrt{-1}F_{H(t)})^{k-1-i}\wedge\overline{\partial}_{E}s].$

By the assumption that $H$ is a Hermitian metric whose curvature is
$\sigma_{k}$-$\omega$-positive on $M$, there exists a positive constant
$\delta$ such that

(3.11) $\frac{\omega^{n-k}}{(n-k)!}\wedge\sqrt{-1}\mbox{\rm
tr\,}[\sum_{i=0}^{k-1}(\sqrt{-1}F_{H})^{i}\wedge\xi^{\ast
H}\wedge(\sqrt{-1}F_{H})^{k-1-i}\wedge\xi]\geq\delta|\xi|_{\omega,H}^{2}\frac{\omega^{n}}{n!}$

for any $\operatorname{End}(E)$-valued $(0,1)$-form $\xi$. If $K\in
B_{H,\epsilon}(E)$, by (3.7), it is straightforward to check that

$\displaystyle\frac{\omega^{n-k}}{(n-k)!}\wedge\sqrt{-1}\mbox{\rm
tr\,}[\sum_{i=0}^{k-1}(\sqrt{-1}F_{H(t)})^{i}\wedge\xi^{\ast
H(t)}\wedge(\sqrt{-1}F_{H(t)})^{k-1-i}\wedge\xi]$ $\displaystyle\geq$
$\displaystyle(\delta-C_{\epsilon})|\xi|_{\omega,H}^{2}\frac{\omega^{n}}{n!},$

where $C_{\epsilon}$ is a constant depending only on $\epsilon$ and
$\sup_{M}|F_{H}|_{\omega,H}$, and $C_{\epsilon}\rightarrow 0$ as
$\epsilon\rightarrow 0$. When $\epsilon$ is small enough, (3) implies that
every $H(t)$ is a Hermitian metric whose curvature is also
$\sigma_{k}$-$\omega$-positive on $M$. By (3), we have

(3.13)
$\frac{d^{2}}{dt^{2}}\mathcal{M}_{E,k}(H_{0},H(t))\geq\frac{\delta}{2}\int_{M}|\overline{\partial}_{E}s|_{\omega,H}^{2}\frac{\omega^{n}}{n!}\geq
0$

for every $t\in[0,1]$.

By the assumption that $H$ is a solution of the equation (1.5), (2) implies
that

$\frac{d}{dt}\mathcal{M}_{E,k}(H_{0},H(t))|_{t=0}=0$

and then $\frac{d}{dt}\mathcal{M}_{E,k}(H_{0},H(t))\geq 0$ for every
$t\in[0,1]$. Hence we know that

(3.14) $\mathcal{M}_{E,k}(H_{0},H)\leq\mathcal{M}_{E,k}(H_{0},K)$

for every $K\in B_{H,\epsilon}(E)$ when $\epsilon$ is small enough, i.e. $H$
is a local minimum point of the generalized Donaldson’s functional (1.3).

Furthermore, if $K\in B_{H,\epsilon}(E)$ is another solution of (1.5), (3.13)
implies $\overline{\partial}_{E}s\equiv 0$, i.e. $s$ is a holomorphic section
of $\operatorname{End}(E)$. Under the assumption that the holomorphic bundle
$(E,\overline{\partial}_{E})$ is simple, we know that $s=aId_{E}$, and then
$K=e^{a}H$.

Set $h=K^{-1}H$ and recall that

(3.15) $D_{H}-D_{K}=h^{-1}\partial_{K}h,\quad
F_{H}-F_{K}=\overline{\partial}_{E}(h^{-1}\partial_{K}h).$

By direct calculation, we have

$\displaystyle(\sqrt{-1}F_{H})^{k}-(\sqrt{-1}F_{K})^{k}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\sqrt{-1}F_{H})^{k-i+1}\wedge(\sqrt{-1}F_{K})^{i-1}-\sum_{i=1}^{k}(\sqrt{-1}F_{H})^{k-i}\wedge(\sqrt{-1}F_{K})^{i}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\sqrt{-1}F_{H})^{k-i}\wedge(\sqrt{-1}F_{H}-\sqrt{-1}F_{K})\wedge(\sqrt{-1}F_{K})^{i-1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\sqrt{-1}F_{H})^{k-i}\wedge(\sqrt{-1}\overline{\partial}_{E}(h^{-1}\partial_{K}h))\wedge(\sqrt{-1}F_{K})^{i-1}.$

In the following, we will focus on the case $k=2$.

######  Definition 3.1.

Let $H$ be a Hermitian metric on holomorphic vector bundle
$(E,\overline{\partial}_{E})$ over a compact Kähler manifold $(M,\omega)$. $H$
is said to be strongly $\sigma_{2}$-$\omega$-positive at a point $p\in M$ if
its curvature $\sqrt{-1}F_{H}$ satisfies that, for every non-zero locally
$End(E)$-valued $(0,1)$-form $\xi$, there is

(3.17) $\frac{\omega^{n-2}}{(n-2)!}\wedge\sqrt{-1}\mbox{\rm
tr\,}[(\sqrt{-1}F_{H})\wedge\xi^{\ast H}\wedge\xi]|_{p}>0$

and

(3.18) $\frac{\omega^{n-2}}{(n-2)!}\wedge\sqrt{-1}\mbox{\rm tr\,}[\xi^{\ast
H}\wedge(\sqrt{-1}F_{H})\wedge\xi]|_{p}>0.$

We recall the definition of Nakano positivity and dual Nakano positivity.

######  Definition 3.2.

A Hermitian metric $H$ on the holomorphic vector bundle
$(E,\overline{\partial}_{E})$ is said to be Nakano-positive, if for any
nonzero $n$-tuple $\\{u^{\alpha}\\}_{\alpha=1}^{n}$ of sections of $E$, it
holds that

(3.19) $\langle F_{H}(\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\partial\bar{z}^{\beta}})u^{\alpha},u^{\beta}\rangle_{H}>0.$

A metric $H$ is said to be dual Nakano-positive, if for any nonzero $n$-tuple
$\\{v^{\alpha}\\}_{\alpha=1}^{n}$ of sections of $E$, it holds that

(3.20) $\langle F_{H}(\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\partial\bar{z}^{\beta}})v^{\beta},v^{\alpha}\rangle_{H}>0.$

In [27], Liu, Sun and Yang found some Hermitian metrics that are both Nakano
positive and dual Nakano positive. In the following, we will show that Nakano
positivity implies (3.17) and dual Nakano positivity implies (3.18), i.e., we
obtain the following lemma.

######  Lemma 3.3.

Let $(E,\overline{\partial}_{E})$ be a holomorphic vector bundle over an
$n$-dimensional compact Kähler manifold $(M,\omega)$, $H$ be a Hermitian
metric on $E$. If $H$ is Nakano-positive and dual Nakano-positive, then $H$
must be strongly $\sigma_{2}$-$\omega$-positive.

###### Proof.

Let $\xi$ be an $\operatorname{End}(E)$-valued $(0,1)$-form. Choose suitable
coordinates such that

$\omega=\sqrt{-1}dz^{\alpha}\wedge
d\bar{z}^{\alpha},\quad\xi=\xi_{\bar{\beta}}d\bar{z}^{\beta},\quad
F_{H}=F_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta},$

where
$\xi_{\bar{\beta}},F_{\alpha\bar{\beta}}\in\Gamma(\operatorname{End}(E))$.
After direct calculations, we find

(3.21) $\begin{split}&\sqrt{-1}{\rm tr}((\sqrt{-1}F_{H})\wedge\xi^{\ast
H}\wedge\xi)\wedge\frac{\omega^{n-2}}{(n-2)!}\\\ =&(\sqrt{-1})^{2}{\rm
tr}(F_{\alpha\bar{\beta}}\cdot(\xi_{\bar{\gamma}})^{\ast
H}\cdot\xi_{\bar{\delta}})dz^{\alpha}\wedge d\bar{z}^{\beta}\wedge
dz^{\gamma}\wedge d\bar{z}^{\delta}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
=&(\sqrt{-1})^{2}\sum_{\alpha\neq\beta}{\rm
tr}(F_{\alpha\bar{\beta}}(\xi_{\bar{\beta}})^{\ast
H}\xi_{\bar{\alpha}})dz^{\alpha}\wedge d\bar{z}^{\beta}\wedge dz^{\beta}\wedge
d\bar{z}^{\alpha}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
&+(\sqrt{-1})^{2}\sum_{\alpha\neq\beta}{\rm
tr}(F_{\alpha\bar{\alpha}}(\xi_{\bar{\beta}})^{\ast
H}\xi_{\bar{\beta}})dz^{\alpha}\wedge d\bar{z}^{\alpha}\wedge dz^{\beta}\wedge
d\bar{z}^{\beta}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
=&\sum_{\alpha\neq\beta}{\rm
tr}(F_{\alpha\bar{\alpha}}(\xi_{\bar{\beta}})^{\ast
H}\xi_{\bar{\beta}}-F_{\alpha\bar{\beta}}(\xi_{\bar{\beta}})^{\ast
H}\xi_{\bar{\alpha}})\frac{\omega^{n}}{n!}\\\
=&\sum_{i=1}^{r}\sum_{\alpha\neq\beta}(\langle
F_{\alpha\bar{\alpha}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\beta}})^{\ast H}(e_{i})\rangle-\langle
F_{\alpha\bar{\beta}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\alpha}})^{\ast H}(e_{i})\rangle)\frac{\omega^{n}}{n!}\\\
=&\sum_{i=1}^{r}\sum_{\alpha<\beta}(\langle
F_{\alpha\bar{\alpha}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\beta}})^{\ast H}(e_{i})\rangle-\langle
F_{\alpha\bar{\beta}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\alpha}})^{\ast H}(e_{i})\rangle\\\ &+\langle
F_{\beta\bar{\beta}}(\xi_{\bar{\alpha}})^{\ast
H}(e_{i}),(\xi_{\bar{\alpha}})^{\ast H}(e_{i})\rangle-\langle
F_{\beta\bar{\alpha}}(\xi_{\bar{\alpha}})^{\ast
H}(e_{i}),(\xi_{\bar{\beta}})^{\ast
H}(e_{i})\rangle)\frac{\omega^{n}}{n!},\end{split}$

where $\\{e_{i}\\}_{i=1}^{r}$ is a unitary frame of $E$ with respect to the
metric $H$. For every $\alpha<\beta$, set $u=(u^{1},\cdots,u^{n})$,
$u^{\alpha}=(\xi_{\bar{\beta}})^{\ast H}(e_{i})$,
$u^{\beta}=-(\xi_{\bar{\alpha}})^{\ast H}(e_{i})$, $u^{\gamma}=0$ for
$\gamma\neq\alpha$ and $\gamma\neq\beta$. If the $n$-tuple of section $u$ is
nonzero, by the assumption that $H$ is Nakano positive, we know

(3.22) $\begin{split}0<&\sum_{\gamma,\delta}\langle
F_{\gamma\bar{\delta}}u^{\gamma},u^{\delta}\rangle=\sum_{\delta}(\langle
F_{\alpha\bar{\delta}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),u^{\delta}\rangle-\langle
F_{\beta\bar{\delta}}(\xi_{\bar{\alpha}})^{\ast
H}(e_{i}),u^{\delta}\rangle)\\\ =&\langle
F_{\alpha\bar{\alpha}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\beta}})^{\ast H}(e_{i})\rangle-\langle
F_{\alpha\bar{\beta}}(\xi_{\bar{\beta}})^{\ast
H}(e_{i}),(\xi_{\bar{\alpha}})^{\ast H}(e_{i})\rangle\\\ &-\langle
F_{\beta\bar{\alpha}}(\xi_{\bar{\alpha}})^{\ast
H}(e_{i}),(\xi_{\bar{\beta}})^{\ast H}(e_{i})\rangle+\langle
F_{\beta\bar{\beta}}(\xi_{\bar{\alpha}})^{\ast
H}(e_{i}),(\xi_{\bar{\alpha}})^{\ast H}(e_{i})\rangle.\end{split}$

From (3.21) and (3.22), we see that (3.17).

On the other hand,

(3.23) $\begin{split}&\sqrt{-1}{\rm tr}(\xi^{\ast
H}\wedge(\sqrt{-1}F_{H})\wedge\xi)\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
=&(\sqrt{-1})^{2}{\rm tr}((\xi_{\bar{\alpha}})^{\ast H}\cdot
F_{\gamma\bar{\delta}}\cdot\xi_{\bar{\beta}})dz^{\alpha}\wedge
dz^{\gamma}\wedge d\bar{z}^{\delta}\wedge
d\bar{z}^{\beta}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
=&\sum_{\alpha\neq\beta}(\sqrt{-1})^{2}{\rm tr}((\xi_{\bar{\alpha}})^{\ast
H}F_{\beta\bar{\alpha}}\xi_{\bar{\beta}})dz^{\alpha}\wedge dz^{\beta}\wedge
d\bar{z}^{\alpha}\wedge d\bar{z}^{\beta}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
&+\sum_{\alpha\neq\beta}(\sqrt{-1})^{2}{\rm tr}((\xi_{\bar{\alpha}})^{\ast
H}F_{\beta\bar{\beta}}\xi_{\bar{\alpha}})dz^{\alpha}\wedge dz^{\beta}\wedge
d\bar{z}^{\beta}\wedge d\bar{z}^{\alpha}\wedge\frac{\omega^{n-2}}{(n-2)!}\\\
=&\sum_{\alpha\neq\beta}{\rm tr}((\xi_{\bar{\alpha}})^{\ast
H}F_{\beta\bar{\beta}}\xi_{\bar{\alpha}}-(\xi_{\bar{\alpha}})^{\ast
H}F_{\beta\bar{\alpha}}\xi_{\bar{\beta}})\frac{\omega^{n}}{n!}\\\
=&\sum_{i=1}^{r}\sum_{\alpha\neq\beta}(\langle
F_{\beta\bar{\beta}}\xi_{\bar{\alpha}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle-\langle
F_{\beta\bar{\alpha}}\xi_{\bar{\beta}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle)\frac{\omega^{n}}{n!}\\\
=&\sum_{i=1}^{r}\sum_{\alpha<\beta}(\langle
F_{\beta\bar{\beta}}\xi_{\bar{\alpha}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle-\langle
F_{\beta\bar{\alpha}}\xi_{\bar{\beta}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle\\\
&+\langle
F_{\alpha\bar{\alpha}}\xi_{\bar{\beta}}(e_{i}),\xi_{\bar{\beta}}(e_{i})\rangle-\langle
F_{\alpha\bar{\beta}}\xi_{\bar{\alpha}}(e_{i}),\xi_{\bar{\beta}}(e_{i})\rangle)\frac{\omega^{n}}{n!}.\end{split}$

For any $\alpha<\beta$, set $v=(v^{1},\cdots,v^{n})$,
$v^{\alpha}=\xi_{\bar{\beta}}(e_{i})$, $v^{\beta}=-\xi_{\bar{\alpha}}(e_{i})$,
$u^{\gamma}=0$ for $\gamma\neq\alpha$ and $\gamma\neq\beta$. If $v$ is
nonzero, the dual Nakano positivity implies

(3.24) $\begin{split}0<&\sum_{\gamma,\delta}\langle
F_{\gamma\bar{\delta}}v^{\delta},v^{\gamma}\rangle=\sum_{\delta}(\langle
F_{\alpha\bar{\delta}}v^{\delta},\xi_{\bar{\beta}}(e_{i})\rangle-\langle
F_{\beta\bar{\delta}}v^{\delta},\xi_{\bar{\alpha}}(e_{i})\rangle)\\\ =&\langle
F_{\alpha\bar{\alpha}}\xi_{\bar{\beta}}(e_{i}),\xi_{\bar{\beta}}(e_{i})\rangle-\langle
F_{\alpha\bar{\beta}}\xi_{\bar{\alpha}}(e_{i}),\xi_{\bar{\beta}}(e_{i})\rangle\\\
&-\langle
F_{\beta\bar{\alpha}}\xi_{\bar{\beta}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle+\langle
F_{\beta\bar{\beta}}\xi_{\bar{\alpha}}(e_{i}),\xi_{\bar{\alpha}}(e_{i})\rangle,\end{split}$

then (3.23) and (3.24) mean (3.18).

Now, we consider the uniqueness of the solution of the following bundle valued
complex $2$-Hessian equation

(3.25)
$\frac{(\sqrt{-1}F_{H})^{2}\wedge\frac{\omega^{n-2}}{(n-2)!}}{\frac{\omega^{n}}{n!}}=\lambda_{E,2}Id_{E}.$

###### Proof of Theorem 1.3.

By (3), we have

(3.26) $\begin{array}[]{ll}&(\sqrt{-1}F_{H})^{2}-(\sqrt{-1}F_{K})^{2}\\\
&=\sqrt{-1}F_{H}\wedge(\sqrt{-1}F_{H}-\sqrt{-1}F_{K})+(\sqrt{-1}F_{H}-\sqrt{-1}F_{K})\wedge\sqrt{-1}F_{K}\\\
&=\sqrt{-1}F_{H}\wedge\sqrt{-1}(\overline{\partial}_{E}(\partial_{H}hh^{-1}))+\sqrt{-1}(\overline{\partial}_{E}(h^{-1}\partial_{K}h))\wedge\sqrt{-1}F_{K}.\\\
\end{array}$

Using the strongly $\sigma_{2}$-$\omega$-positivity hypothesis, the Bianchi
identity and Stokes formula, we derive

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{[(\sqrt{-1}F_{H})^{2}-(\sqrt{-1}F_{K})^{2}]h\\}\wedge\omega^{n-2}$
$\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{[\sqrt{-1}F_{H}\wedge\sqrt{-1}(\overline{\partial}_{E}(\partial_{H}hh^{-1}))+\sqrt{-1}(\overline{\partial}_{E}(h^{-1}\partial_{K}h))\wedge\sqrt{-1}F_{K}]h\\}\wedge\omega^{n-2}$
$\displaystyle=$ $\displaystyle\int_{M}\mbox{\rm
tr\,}\\{\sqrt{-1}F_{H}\wedge(\sqrt{-1}\partial_{H}hh^{-\frac{1}{2}})\wedge
h^{-\frac{1}{2}}\overline{\partial}_{E}h\\}\wedge\omega^{n-2}$
$\displaystyle+\int_{M}\mbox{\rm
tr\,}\\{\sqrt{-1}(h^{-\frac{1}{2}}\partial_{K}h)\wedge\sqrt{-1}F_{K}\wedge\overline{\partial}_{E}hh^{-\frac{1}{2}}\\}\wedge\omega^{n-2}$
$\displaystyle\geq$
$\displaystyle\tilde{c}\int_{M}\\{|h^{-\frac{1}{2}}\overline{\partial}_{E}h|_{H}^{2}+|\overline{\partial}_{E}hh^{-\frac{1}{2}}|_{K}^{2}\\}\omega^{n}$

for some positive constant $\tilde{c}$, and then

(3.27) $\overline{\partial}_{E}h=0.$

The simplicity of $(E,\overline{\partial}_{E})$ gives us $h=e^{-a}\,{\rm
Id}_{E}$, and then $K=e^{a}H$.

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