# On a fully nonlinear elliptic equation with differential forms

Hao Fang<EMAIL_ADDRESS>and  Biao Ma<EMAIL_ADDRESS>
###### Abstract.

We introduce a fully nonlinear PDE with a differential form, which unifies
several important equations in $\rm{K\ddot{a}hler}$ geometry including Monge-
Ampère equations, $J$-equations, inverse $\sigma_{k}$ equations, and the
deformed Hermitian Yang-Mills (dHYM) equation. We pose some natural positivity
conditions on $\Lambda$, and prove analytical and algebraic criterions for the
solvability of the equation. Our results generalize previous works of G.Chen,
J.Song, Datar-Pingali and others. As an application, we prove a conjecture of
Collins-Jacob-Yau for the dHYM equation with small global phase.

###### Contents

  1. 1 Introduction
  2. I Analytical Criterion
    1. 2 Assumptions and main results
    2. 3 Preliminary set-up
    3. 4 Ellipticity and convexity
    4. 5 Cone condition
    5. 6 Equations with almost positive volume forms
    6. 7 Continuity method
  3. II Numerical Criterion
    1. 8 Notations and technical preparation
    2. 9 Initiation of induction argument
    3. 10 Mass concentration
    4. 11 Completing the induction
    5. 12 An application to supercritical dHYM equations
    6. A Functional and uniqueness
    7. B Proof of Proposition 7.11
    8. C List of notations

###  1\. Introduction

In this paper, we study a general form of fully non-linear partial
differential equations of Monge-Ampère type on $\rm{K\ddot{a}hler}$ manifolds.

Let $(M,\omega_{0})$ be a $\rm{K\ddot{a}hler}$ manifold of dimension $n$, and
let $[\omega_{0}]$ be the corresponding $\rm{K\ddot{a}hler}$ class. We fix a
closed differential form of the following format:

$\Lambda=\sum_{k=1}^{n}\Lambda^{[k]},$

where for each $1\leq k\leq n$, $\Lambda^{[k]}$ is a real $(k,k)$-form. Hereby
we use $\alpha^{[k]}$ to denote the $(k,k)$ component of a differential form
$\alpha$. Let $\omega=\omega_{0}+i\partial\bar{\partial}u\in[\omega_{0}]$ be a
$\rm{K\ddot{a}hler}$ metric. We denote

$\Omega=\Omega(\omega)=\exp\omega=\sum_{k=0}^{n}\frac{\omega^{k}}{k!}.$

In this paper, we study the following partial differential equation of
function $u$:

(1.1) $\kappa\Omega^{[n]}=(\Lambda\wedge\Omega)^{[n]},$

where $\kappa$ is a positive constant such that
$\kappa\int_{M}\exp\omega_{0}=\int_{M}\Lambda\wedge\exp\omega_{0}.$

When $\Lambda$ has only the positive $(n,n)$ component, it is thus a volume
form of $M$. (1.1) is the well-known complex Monge-Ampère equation, first
completely solved by Yau [38].

For future convenience, we write $\Lambda^{[n]}$ as $\frac{f\rho^{n}}{n!}$,
where $f$ is a smooth function on $M$and $\rho$ is a fixed background
$\rm{K\ddot{a}hler}$ metric. Then (1.1) can be expanded as:

(1.2)
$\kappa\frac{\omega^{n}}{n!}=\sum_{k=1}^{n-1}\Lambda^{[k]}\wedge\frac{\omega^{n-k}}{(n-k)!}+f\frac{\rho^{n}}{n!}.$

In addition to the original Monge-Ampère equation, other special cases of
equation (1.2) include:

  1. (1)

When $\Lambda=\rho$ is a $\rm{K\ddot{a}hler}$ form, (1.1) is the $J$-equation,
which was first introduced by Donaldson [15] and has been extensively studied
by many authors. See [15, 6, 37, 33, 10, 4, 32] and references therein.

  2. (2)

When $\Lambda$ is $\rho^{k}$ or $\sum_{i=k}^{n-1}c_{k}\rho^{k}$, where $\rho$
is a $\rm{K\ddot{a}hler}$ form and $c_{k}\geq 0$, (1.1) is the inverse
$\sigma_{k}$-equation, which was first raised in Fang-Lai-Ma [19]. An
incomplete list of works on these equations includes [19, 17, 18, 10, 35, 12].

  3. (3)

Given a specific global phase $\theta>0$ and
$\Lambda=\sin\theta\cos\rho-\cos\theta\sin\rho$, where $\rho$ is a Kähler
form, (1.1) is the deformed Hermitian Yang-Mills (dHYM) equation. dHYM
equation was introduced in Marino-Minasian-Moore-Strominger [26] and Leung-
Yau-Zaslow [23] and has been extensively studied; See [21, 8, 29, 4, 7, 11,
16, 20, 24] and references therein. See also Section 12 for details.

PDEs in the general form of (1.1) are inspired by above-mentioned works, which
have roots in rich geometric and mathematical physics background, and are
important in the field of fully non-linear PDEs with significant geometric
implications. Known results indicate that the existence of unique smooth
solutions can be interpreted as analytical and algebraic properties of the
underlying manifold. Therefore, it is our intention to systematically explore
more general forms of such PDEs. In particular, we would like to study
necessary and sufficient conditions that ensure the existence of solutions.

We realize from past works that, aside from the unknown Kähler metric to be
solved, given geometric data often involve another Kähler form or its powers.
The exact format of (1.1) suggests some point-wise positivity requirement for
the differential form $\Lambda.$ In addition, the important fact of
ellipticity of Monge-Ampère type equations poses similar requirements. One of
our main goals is to find natural positivity conditions under which necessary
a priori estimates can be established and deep connections between PDE and
geometry can be reconstructed. Inspired by Gao Chen’s work [4], we also
consider (1.1) in a bit more general form, in which we allow $\Lambda^{[n]}$
to be slightly negative at some points.

For convenience of further discussion, we decompose $\Lambda$ as follows

$\Lambda=\mathring{\Lambda}+\Lambda^{[n]},$

and discuss constraints on $\mathring{\Lambda}$ and $\Lambda^{[n]}$
separately.

######  Definition 1.1.

We call a real even closed differential form $\mathring{\Lambda}$_on M
$k$-uniformly positive _($k$-UP) if there exists an integer $1\leq k\leq n$, a
reference $\rm{K\ddot{a}hler}$ metric $\rho$ of $M$, and a uniform constant
$m>0$ such that $\Lambda^{[l]}=0$ for $l<k$ ; and

(1.3) $\mathring{\Lambda}-m\frac{\rho^{k}}{k!}\geq 0,$

which (1.3) means that each $(l,l)$-component of the left hand side is a
positive form for $1\leq l\leq n$. See Definition 3.1.

Notice that Definition 1.1 is independent of the choice of reference metric
$\rho$. An $n$-UP form $\Lambda$ is a positive volume form, which is required
for solving the classic Monge-Ampère equation. $J$-equation is a special case
where $\Lambda$ is $1$-UP. Furthermore, positive linear combination of powers
of a given Kähler form is uniformly positive. Therefore, inverse $\sigma_{k}$
equations also fall into this category. However, as uniform positivity is an
open condition, it is far more general than these special cases. In future
sections, we will also give more general and technical conditions under which
various conclusions hold.

Since the classic Monge-Ampère equation is well known in the field, we will
discuss only cases when $\mathring{\Lambda}$ is $k$-UP for some $1\leq k\leq
n-1$. When we consider $\Lambda^{[n]},$ in spirit of the work of G. Chen [4],
we pose a slightly weaker condition.

######  Definition 1.2.

Given $1\leq k_{0}\leq n-1$, $m>0$, and a reference $\rm{K\ddot{a}hler}$
metric $\rho$, we call $(n,n)$-form $\alpha$ _an almost positive volume form_
with respect to $(k_{0},m,\rho)$, if $\int_{M}\alpha\geq 0$ and there exists
$\epsilon=\epsilon(n,m,\kappa,k_{0},\omega_{0},\rho)>0$ such that

(1.4) $\frac{\alpha}{\rho^{n}/n!}>-\epsilon.$

######  Remark 1.3.

We will choose a specific constant
$\epsilon<\min\left\\{\frac{m}{4n+2}\gamma_{\min}(\frac{2\kappa}{m},1,n,k_{0}),\frac{\kappa\int_{M}\omega_{0}^{n}}{2\int_{M}\rho^{n}}\right\\},$where
$\gamma_{\min}$ is a positive number, defined later in (6.2) in the rest of
the paper.

We state the following hypothesis on $\Lambda$, which will be a key assumption
for our paper.

######  Definition 1.4.

$\Lambda=\mathring{\Lambda}+\Lambda^{[n]}$ is called to satisfy the condition
H1, if and only if $\mathring{\Lambda}$ is $k_{0}$-uniformly positive for some
$1\leq k_{0}\leq n-1$ with a reference metric $\rho$ and a uniform constant
$m>0$; and $\Lambda^{[n]}$ is almost positive with respect to a specific
$\epsilon$(as in Remark 1.3) and $(k_{0},m,\rho)$.

In order to state our main analytical result, we define the following concept
based on previous works ([33, 19, 17, 35] ).

######  Definition 1.5.

A $\rm{K\ddot{a}hler}$ form $\omega\in[\omega_{0}]$ is called to satisfy the
_cone condition_ for equation (1.2) if and only if pointwisely it satisfies
the following

(1.5)
$\kappa(\exp\omega)^{[n-1]}-\left(\Lambda\wedge\exp\omega\right)^{[n-1]}>0;$

Or equivalently,

(1.6)
$\frac{\kappa\omega^{n-1}}{(n-1)!}-\sum_{k=1}^{n-1}\frac{\Lambda^{[k]}\wedge\omega^{n-k-1}}{(n-k-1)!}>0,$

which means that the left hand side of (1.5) or 1.6) is a strictly positive
$(n-1,n-1)$ form. We also call $\omega$ a _subsolution_ of the equation (1.1)
or (1.2) if (1.5) holds everywhere.

In order to state our main geometrical result, inspired by the works of [22,
35], we propose the following numerical positivity conditions:

######  Definition 1.6.

Let $[\Lambda]$ be the cohomology class of $\Lambda$. Let $\kappa>0$ be a
constant. Let $[\alpha]$ be a $\rm{K\ddot{a}hler}$ class. We call $[\alpha]$ a
_$([\Lambda],\kappa)$ -positive_ class if

(1.7) $[\exp\alpha]\cdot[\kappa-\Lambda]\cdot[M]\geq 0,$

and for any subvariety $Y$ of $M$ with $\dim Y<n$ it holds that

(1.8) $[\exp\alpha]\cdot[\kappa-\Lambda]\cdot[Y]>0.$

With key definitions ready, we state our main theorems, which consist 2 parts.
The first is an analytic criterion for the existence of solution to (1.2).

######  Theorem 1.7.

Let $M$ be a connected compact Kähler manifold of dimension $n$ with a fixed
Kähler class $[\omega_{0}]$. Suppose that $\Lambda$ is a closed real
differential form satisfying H1. Then, there exists a smooth solution of (1.2)
if and only if there exists a smooth subsolution of (1.2).

Special cases of Theorem 1.7 have been established by Song-Weinkove [33],
Fang-Lai-Ma [19], Guan [17], Guan-Sun [18], Szèkelyhidi [35], Chen [4], Datar-
Pingali [12].

######  Remark 1.8.

We also establish the uniqueness result regarding (1.2). See Appendix A. In
fact, solutions will be shown to be the unique minimizer of a global
functional $\mathcal{F}$ defined in (A.7). We will further explore the
variational structure of equation (1.2) in the Appendix A and some future
works.

######  Remark 1.9.

Following works of [17, 18, 35], it is likely that Theorem 1.7 may be extended
to Hermitian manifolds, when similar subsolution definition may be
established.

Our second main theorem relates the solvability of equation (1.2) to
Definition 1.6.

######  Theorem 1.10.

Let $M$ be a connected compact Kähler manifold of dimension $n$ with a fixed
Kähler class $[\omega_{0}]$. Suppose that $\Lambda$ is a closed real
differential form satisfying H1. Then $[\omega_{0}]$ is
$([\Lambda],\kappa)$-positive if and only if there exists a smooth Kähler
metric $\omega$ solves (1.2).

Theorem 1.10 generalizes several existing works. In the context of
$J$-equation, G. Chen [4] proved the equivalence of $J$-uniformly positivity
and the solvability of the $J$-equation; Later, J. Song [32] reduced the
$J$-uniformly positivity to only $J$-positivity. In the context of inverse
$\sigma_{k}$ type equations, using different methods, similar numerical
results have been proved by Collins-Szèkelyhidi [10] for toric manifolds and
Datar-Pingali [12] for projective manifolds.

While many of past works are raised from strong geometric background, our work
aims at a general form of non-linear PDEs of Monge-Ampère type on
$\rm{K\ddot{a}hler}$ manifolds. In particular, our equation is one of the few
fully non-linear PDEs that involve general differential forms of different
degrees. It is hopeful that future works will reveal more geometry of positive
differential forms on complex manifolds, which has not been studied much from
the non-linear PDE point of view.

As an application of our main results, we study supercritical deformed Yang-
Mills equations to get the following

######  Theorem 1.11.

Let $M$ be a connected compact Kähler manifold of dimension $n$. Let
$[\omega_{0}]$ be a real $(1,1)$-cohomology class and let $\rho$ be a Kähler
form. Let

$\cot\theta=\frac{\int_{M}\text{Re}(\omega+\sqrt{-1}\rho)^{n}}{\int_{M}\text{Im}(\omega+\sqrt{-1}\rho)^{n}}.$

If $\theta\in(0,\frac{\pi}{n-1}]$, then there exists a smooth solution to the
supercritical dHYM equation

(1.9)
$\text{Re}(\omega+\sqrt{-1}\rho)^{n}=\cot\theta\text{Im}(\omega+\sqrt{-1}\rho)^{n},$

if and only if $[\omega_{0}-\cot\theta\rho]$ is Kähler and for any analytic
subvariety $V$ of dimension $d\leq n-1$, we have

(1.10)
$\int_{V}\left(\text{Re}(\omega_{0}+\sqrt{-1}\rho)^{d}-\cot\theta\text{Im}(\omega_{0}+\sqrt{-1}\rho)^{d}\right)>0.$

For supercritical dHYM equations, the equivalence of the existence of a smooth
solution to (1.9) and the numerical condition (1.10) was conjectured by
Collins-Jacob-Yau [8] and confirmed by the work of G. Chen [4], Chu-Lee-
Takahashi [7], and A. Ballal [1] under various stronger assumptions. See more
details in Section 12. The original conjecture in [8] in its full generality
does not hold due to counterexamples constructed by Zhang [39]. However,
Theorem 1.11 confirms the original conjecture when
$\theta\in(0,\frac{\pi}{n-1}]$. Notice that for Zhang’s examples[39] ,
$\theta>\frac{\pi}{n-1}$. It is therefore interesting to find the sharp range
of $\theta$ in which the conjecture of Collins-Jacob-Yau holds.

We make some comments on our proofs.

We have utilized numerous ideas and techniques of existing results, many have
been cited above. On the analytic side: We employ Yau’s continuity method
which has been successfully used by many to attack Monge-Ampère type and other
types of equations. We follow works of many [33, 19, 17, 10, 35, 4, 12] to
derive a priori estimates. We use Szèkelyhidi’s argument to derive $C^{0}$
estimate [35]. We use arguments in G. Chen [4] and Datar-Pingali [12] to treat
almost positive volume forms. The definition of subsolution is inspired by
similar concepts in Song-Weinkove [33], Fang-Ma-Lai [19], Guan [17], and
Szèkelyhidi [35]. On the algebraic side: The numerical condition in Definition
1.6 is inspired by [22, 35]. We follow G. Chen’s induction method and overall
approach in [4] to tackle the numerical criterion. We make use of Demailly-
Păun’s mass concentration technique [14]. We follow J. Song’s treatment of
singular subvarieties [32].

We have made a conscious effort to examine existing methods and explore their
maximal applicability, which has lead us to the current formulation of general
equations and proper positivity conditions. As mentioned earlier, our equation
includes many existing important special cases. We apply a new set of local
notations and carry out detailed multi-linear algebraic computation involving
inverse $\sigma_{k}$ type quantities.

One key component of our proof is the definition of the uniform positivity and
the subsequent H1 condition, which ensure that the important concept of
subsolutions can be defined and key a priori estimates can be carried out. We
would like to remark that the uniform positivity condition can not be used
directly to prove Theorem 1.10. From the algebraic point of view, proper
positivity conditions are required to be preserved under two important
geometric procedures which are used to prove corresponding numerical results:
the first is a product manifold construction; the second is passing from a
$\rm{K\ddot{a}hler}$ manifold to its sub-varieties. The uniform positivity
condition can be preserved under the second procedure, but unfortunately it is
not preserved under the product manifold procedure. In order to overcome this
difficulty, we introduce a more technical positivity condition, called H2,
which is a similar but more general version of the H1 condition and will be
stated in later sections. H2 works under the product manifold procedure, but
does not descend to sub-varieties. We carefully establish all necessary
algebraic and analytic results that are needed in our proofs under these
technical assumptions by fully exploring the algebraic structure of
differential forms. The final write-up is technical but self-contained for the
convenience of readers. We believe that our main theorems, Theorems 1.7 and
1.10, are cleanest in format, still incorporate all known special cases and
will be of most use in future applications.

Another key component of our proof is a new geometric PDE (10.8) on the
product manifold, which is simpler and more flexible than the one used by Chen
[4], even in the J-equation case. Under our general setting, the new equation
is compatible with proper cone conditions, and still carries essential
algebraic information. See Example 10.5 for further elaboration in a simple
yet illuminating case.

There are several future research directions of interest.

First of all, our results may be extended further. The general form of dHYM
equations do not satisfy any of our positivity assumptions, while its special
cases have been studied successfully. In [25], Lin also studied the convexity
of inverse $\sigma_{k}$ type equations with some negative coefficients. This
may indicate that some weaker positivity conditions may lead to existence
results.

Similar to the J-equation and dHYM equations ( [15, 9, 11]), there is also a
proper moment map interpretation of our equation, which involves an infinite
dimensional symplectic space. Such point of view is helpful to consider
further generalizations and applications. We would like to address this topic
in a future work.

From the PDE point of view, our current approach, as well as several important
previous works, heavily depends on the continuity method. It is interesting to
explore corresponding geometric flows, which has been successfully used in the
setting of J-equation and dHYM equations but missing in the more general
setting. See[6, 37, 33, 19] and [8, 16, 4, 7, 20, 24, 29, 11, 36] for an
incomplete list. Overall, the parabolic approach will expose finer geometric
information.

From the geometry point of view, as mentioned earlier, one may explore the
geometry of properly defined positive differential forms using non-linear
PDEs, which will be a new research direction. Also mentioned earlier, some of
our results is likely to be extended to the Hermitian setting. This may
further open applications in the field of special geometry and mathematical
physics. From another prospective, indicated by works of Datar-Pingali [12],
Chu-Lee-Takahashi [7] and a recent lecture note of G. Chen [5], one may
explore the more restrictive setting where $M$ is projective. Studies of
generalized Hodge conjecture explore fine distinction between
$\rm{K\ddot{a}hler}$ and projective manifolds. Our general form of PDE may be
of help.

The rest of the paper is organized as follows: In Part 1, we establish main
analytical results of this paper; In Part 2, we prove our main algebraic
result. We give more details: In Section 2, we state our main technical
analytical results; In Section 3, we describe our local setup and establish
proper notations; In Section 4, when $\Lambda^{[n]}$ is non-negative, we
compute variations of our non-linear differential operators and derive its
ellipticity and concavity; In Section 5, we discuss the cone condition and
related properties; In Section 6, we extend results of previous sections to
include the almost positive $\Lambda^{[n]}$ ; In Section 7, we establish
proper a priori estimate and use the continuity method to prove our main
analytical results; In Section 8, we set up notations and make technical
preparations for Part 2; In Section 9, we initiate the induction argument to
prove Theorem 1.10; In Section 10, we prove the key mass concentration
theorem; In Section 11, we complete the proof of Theorem 1.10; In Section 12,
we study dHYM equations and prove Theorem 1.11.

#### Acknowledgements

We thank Jian Song and Jianchun Chu for helpful discussions. We thank Gang
Tian and Xiaohua Zhu for comments and suggestions. We thank Vamsi Pritham
Pingali and Ved Datar for their interest and comments that are helpful.

##  Part I Analytical Criterion

###  2\. Assumptions and main results

In this part we prove Theorem 1.7, the first of our main theorems, by
establishing a stronger result.

We begin introduce a positivity assumption, called H2, which is weaker than H1
and allows us to work with differential forms which do not satisfy $k$-uniform
positivity. In particular, this new positivity condition can be applied to
product manifolds, which will be crucial in later proofs.

We start by defining a pointwise structure on $M$. Let $\mathcal{T}_{p}M$ be
the holomorphic tangent space of $M$ at $p.$ We fix a reference
$\rm{K\ddot{a}hler}$ metric as $\rho$ and for future convenience, define

(2.1) $P:=\exp\rho=\sum_{k=0}^{n}\frac{\rho^{k}}{k!}.$

As before, we define $\mathring{\Lambda}=\Lambda-\Lambda^{[n]}$.

######  Definition 2.1.

_A labeled orthogonal splitting_ at $p\in M$ with respect to $\rho$ is a tuple

$\mathcal{O}_{p}=(n_{p},\mathbf{d}_{p},\mathbf{V}_{p},\mathbf{k}_{p}),$

where $n_{p}\in\mathbb{Z}_{+}$,
$\mathbf{d}_{p}=(d_{1},\cdots,d_{n_{p}})\in\mathbb{Z}_{\geq 0}^{n_{p}}$ ,
$\mathbf{k}_{p}=(k_{1},\cdots,k_{n_{p}})\in\mathbb{Z}_{\geq 0}^{n_{p}}$ and
$\mathbf{V}_{p}=\\{\mathcal{V}_{i}:\text{linear subspaces in
}\mathcal{T}_{p}M\\}$ such that

  1. (1)

$d_{i}=\dim_{\mathbb{C}}\mathcal{V}_{i}$ and $1\leq k_{i}\leq d_{i}-1$;

  2. (2)

$\mathcal{V}_{i}$ are mutually orthogonal with respect to $\rho$ and
$\mathcal{T}_{p}M=\oplus_{i=1}^{n_{p}}\mathcal{V}_{i}.$

Given $\mathcal{O}_{p}$ at $p$, we denote $\iota_{i},\pi_{i}$ be the standard
embedding and orthogonal projection for each $\mathcal{V}_{i}$, respectively.
Denote

(2.2) $\rho_{i}=\pi_{i}^{*}\iota_{i}^{*}\rho.$

If at each $p\in M$, there is a labeled orthogonal splitting
$\mathcal{O}_{p}$, we call $\mathcal{O}:=\\{\mathcal{O}_{p}:p\in M\\}$ a
labeled orthogonal splitting on $M$.

We propose the following

######  Definition 2.2.

Suppose that there exists a labeled orthogonal splitting
$\mathcal{O}=\\{(n_{p},\mathbf{d}_{p},\mathbf{V}_{p},\mathbf{k}_{p}):p\in
M\\}$ with respect to some reference $\rm{K\ddot{a}hler}$ metric $\rho$.
$\mathring{\Lambda}$ is called $\mathcal{O}$-_uniformly positive_
(_$\mathcal{O}$_ -UP) if there exists a uniform constant $m>0$ such that at
each point $p$, we have

(2.3)
$\mathring{\Lambda}-m\sum_{i=1}^{n_{p}}\frac{\rho_{i}^{k_{i}}}{k_{i}!}\geq 0,$

which means the $(l,l)$-component of the left hand side is positive (See
Definition 3.1) for $l=1,\cdots,n-1$.

$\mathcal{O}$-uniform positivity is a weaker condition compared to $k$-UP, and
the later corresponds to $\\{(1,n,\mathcal{T}_{p}M,k)\\}$-uniform positivity.
However, as $\mathcal{O}$ depends on the choice of $\rho$,_ $\mathcal{O}$_-UP
property also depends on the choice of $\rho$. Moreover, the $\mathcal{O}$-UP
property is not preserved when descending to a smooth subvariety, which is why
we assume $k$-UP in Theorem 1.10.

For $(n,n)$-form $\Lambda^{[n]}$, as discussed in the introduction, we allow
it to be slightly negative:

######  Definition 2.3.

Given a labeled orthogonal splitting
$\mathcal{O}=\\{(n_{p},\mathbf{d}_{p},\mathbf{V}_{p},\mathbf{k}_{p}):p\in
M\\}$ with respect to $\rho$, we call $(n,n)$-form $\alpha$ _an almost
positive volume form or almost positive_ with respect to
$(\mathcal{O},m,\rho)$, if $\int_{M}\alpha\geq 0$ and there exists
$\epsilon=\epsilon(n,m,\kappa,n_{p},\mathbf{d}_{p},\mathbf{k}_{p},\omega_{0},\rho)>0$
such that

(2.4) $\frac{\alpha}{\rho^{n}/n!}|_{p}>-\epsilon.$

######  Remark 2.4.

We will chose the precise number
$\epsilon=\min\left\\{\frac{m}{4n+2}\gamma_{\min}(\frac{2\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}),\frac{\kappa\int_{M}\omega_{0}^{n}}{2\int_{M}\rho^{n}}\right\\}$,
where $\gamma_{\min}$ is a positive number defined in (6.2). Notice that
$\epsilon$ can be chosen independent of $p$, since variables
$\mathbf{d}_{p},\mathbf{k}_{p}$ in $\gamma_{\min}$ only takes finitely many
values for all possible local labeled orthogonal splitting.

We impose the following hypothesis on $\Lambda$, which can be decomposed into
$\mathring{\Lambda}+\Lambda^{[n]}.$

######  Definition 2.5.

We call $\Lambda=\mathring{\Lambda}+\Lambda^{[n]}$ satisfies the condition H2,
if $\mathring{\Lambda}$ is $\mathcal{O}$-uniformly positive for some uniform
constant $m$, and $\Lambda^{[n]}$ is almost positive with respect to a
specific $\epsilon$ and $(\mathcal{O},m,\rho)$; Furthermore, there exists a
constant $C_{H2}=C_{H2}(\Lambda,M)$ such that for any $p\in M$,
$\xi\in\mathcal{T}_{p}M$ with $\|\xi\|_{\rho}\leq 1$, and each $1\leq k\leq
n$, we have

(2.5) $\displaystyle-
C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right)$
$\displaystyle\leq\text{Re}\left(\nabla_{\xi}\Lambda^{[k]}\right)\leq
C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right),$
(2.6) $\displaystyle-
C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right)$
$\displaystyle\leq\nabla_{\xi\bar{\xi}}^{2}\Lambda^{[k]}\leq
C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right),$

where $\boldsymbol{l}_{k}=\\{(l_{1},\cdots,l_{n_{p}}):\sum_{i}l_{i}=k,\
l_{i}=0\ \text{or}\ l_{i}\geq k_{i}\\}$ and $\nabla$ is the Levi-Civita
connection of $\rho$.

######  Remark 2.6.

If $k\geq\max_{1\leq i\leq n_{p}}\\{n-d_{i}+k_{i}\\}$, then (2.5) and (2.6)
hold automatically, since $\rho^{k}\leq
c_{0}(n_{p},k)\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}$
(see proof in Lemma 6.3) and we may choose
$C_{H2}=(\|\Lambda\|_{C^{2}}+1)\cdot\sup_{p}c_{0}(n_{p},k)$.

######  Remark 2.7.

H1 is a stronger condition comparing to H2. First, the $k_{0}$-UP condition is
equivalent to the $\mathcal{O}$-UP condition for
$\mathcal{O}=\\{(1,n,\mathcal{T}_{p}M,k_{0}):p\in M\\}$. Second, since
$\Lambda^{[k]}=0$ for $k<k_{0}$, (2.5) and (2.6) hold for $k<k_{0}$. By Remark
2.6, (2.5) and (2.6) hold for $k\geq k_{0}$.

Comparing to H1, condition H2 allows more general choices of geometric data.
However, condition H2 may be more difficult to verify. A crucial fact is
following:

######  Example 2.8.

Let $\\{(M_{i},\rho_{i},\mathcal{O}_{i})\\}_{i=1}^{l}$ be several
$\rm{K\ddot{a}hler}$ manifolds, each with labeled orthogonal splitting
_$\mathcal{O}_{i}$_. Let $\Lambda_{i}$ be positive differential forms on
$M_{i}$ satisfying H2 for some uniform constant $m$, respectively. Let
$\mathcal{M}:=\prod_{i=1}^{l}M_{i}$ be the product manifold and
$\text{pr}_{i}$ be the canonical projection from $\mathcal{M}$ to $M_{i}$. Let
$\rho=\sum_{i=1}^{l}\text{pr}_{i}^{*}\rho_{i}$. Then $\mathcal{M}$ has a
natural labeled orthogonal splitting at each
$\mathbf{p}=(p_{1},\cdots,p_{l})\in\mathcal{M}$ as

$\mathcal{O}_{(p_{1},\cdots,p_{l})}=\left(\sum_{i=1}^{l}n_{p_{i}},(\mathbf{d}_{p_{1}},\cdots,\mathbf{d}_{p_{l}}),\cup_{i=1}^{l}\mathbf{V}_{p_{i}},(\mathbf{k}_{p_{1}},\cdots,\mathbf{k}_{p_{l}})\right).$

Let

$\boldsymbol{\Lambda}:=\sum_{i=1}^{l}\text{pr}_{i}^{*}\Lambda_{i}.$

Then $\mathbf{\Lambda}$ satisfies $\mathcal{O}$-uniformly positivity. Note the
$\nabla_{\xi}\text{pr}_{i}^{*}\Lambda_{i}=\nabla_{\xi^{\top}}\text{pr}_{i}^{*}\Lambda_{i}$
where $\xi^{\top}$ is the tangential part of $\xi$ to $M_{i}$. Thus,
$\text{pr}_{i}\Lambda_{i}$ satisfies (2.5) and (2.6), and so does
$\boldsymbol{\Lambda}$ (with $C_{H2}=\max\\{C_{H2}(\Lambda_{i},M_{i})\\}$). On
the other hand, if each $\Lambda_{i}$ is $k_{0i}$-uniformly positive for some
$k_{0i}\geq 2$, i.e. $\Lambda_{i}^{[k_{0i}]}\geq m\rho_{i}^{k_{0i}}/k_{0i}!$
on $M_{i}$, one does not expect $\boldsymbol{\Lambda}\geq m\rho^{k}/k!$ for
some $k\geq 2$. Therefore, H1 does not necessarily hold on $\mathcal{M}$.

We are now ready to state the main theorem of this part.

######  Theorem 2.9.

Let $M$ be a connected compact $\rm{K\ddot{a}hler}$ manifold of dimension $n$
with a fixed $\rm{K\ddot{a}hler}$ class $[\omega_{0}]$. Suppose $\Lambda$ is a
closed real differential form satisfying H2. Then, there is a smooth solution
of (1.2) if and only if there is a smooth subsolution of (1.2).

From discussions above, it is clear that Theorem 1.7 is a direct consequence
of Theorem 2.9. The rest of this part is devoted to prove Theorem 2.9. It is
organized as follows. In Section 3, we introduce necessary notations and
definitions. In Section 4, we compute variations of the local functional and
verify its ellipticity and convexity. In Section 5, we study the cone
condition and state several equivalent expressions. In Section 6, we extend
results of previous Sections to PDEs with almost positive volume forms. In
Section 7, we use the continuity method to prove Theorem 2.9 by establishing
key a priori estimates.

###  3\. Preliminary set-up

In this section, we introduce some notations and discuss some basic properties
of our equation (1.2).

####  3.1. Positivity of differential forms

We recall several definitions of positivity for $(k,k)$-forms from Demailly
[13], III,1,A. Our presentation is slightly different due to our choice of
notations.

######  Definition 3.1.

$u\in\bigwedge^{k,k}T_{p}^{*}M$ (resp. $\bigwedge^{k,k}T_{p}M$) is said to be
a _strongly positive_ _form (resp. vector)_ if it is of the form

$u=\sum_{s\in
I}\lambda_{s}\sqrt{-1}^{k}\alpha_{s,1}\wedge\overline{\alpha}_{s,2}\wedge\cdots\wedge\alpha_{s,k}\wedge\overline{\alpha}_{s,k},$

where $\alpha_{s,i}\in\bigwedge^{1,0}T_{p}^{*}M$ (resp.
$\bigwedge^{0,1}T_{p}M$) and $\lambda_{s}\geq 0$.

An element $u\in\bigwedge^{k,k}T_{p}^{*}M$ (resp. $\bigwedge^{k,k}T_{p}M$) is
a _positive form (resp. vector)_ if

$\langle u,v\rangle\geq 0$

for any strongly positive $v\in\bigwedge^{k,k}T_{p}M$ (resp.
$\bigwedge^{k,k}T_{p}^{*}M$). Here $\langle\cdot,\cdot\rangle$ is the
canonical pairing (complex linear) between $\bigwedge^{*}T_{p}M$ and
$\bigwedge^{*}T_{p}^{*}M$. We denote $\alpha\geq\beta$ (resp.
$\alpha\leq\beta$) if for each $k=0,\cdots,n$, $\alpha^{[k]}-\beta^{[k]}$
(resp. $\beta^{[k]}-\alpha^{[k]}$) is a positive $(k,k)$-form/vector.

We denote $\alpha\geq_{s}\beta$ (resp. $\alpha\leq_{s}\beta$) if
$\alpha-\beta$ (resp. $\beta-\alpha$) is a strongly positive form/vector.

We list several direct conclusions from Definition 3.1 and compare these to
our uniform positivity concept. A strongly positive form (vector) is also
positive. The converse is true for $k=0,1,n-1,n$; However it is false for
$k=2,\cdots,n-2$ if $n\geq 4.$ Also, the cone of positive $(k,k)$-forms is the
dual cone of strongly positive $(k,k)$-vectors. It is obvious that both
positivity and strongly positivity are invariant under coordinate change. If
$\rho$ is a $\rm{K\ddot{a}hler}$ form, then $\rho^{k}$ is a strongly positive
$(k,k)$-form. Moreover, $\rho^{k}$ is $k$-uniformly positive by 1.3. A
uniformly positive form is positive, but not necessarily strongly positive.
For instance, pick a weakly positive but not strongly positive $(k,k)$-form
$\alpha$. Then $\rho^{k}+\alpha$ is uniformly positive but not strongly
positive.

The following lemma shows that $\exp\omega$ is strongly positive if $\omega$
is a non-negative $(1,1)$-form. The same argument applies to non-negative
$(1,1)$-tangent vectors.

######  Lemma 3.2.

Let $A$ be a non-negative Hermitian matrix. Let
$\omega=A_{i\bar{j}}\frac{\sqrt{-1}}{2}dz^{i}\wedge d\bar{z}^{j}$. Then
$\exp\omega$ is a strongly positive form. Moreover, if both $A$,$B$ are non-
negative Hermitian matrix,
$\omega^{\prime}=B_{i\bar{j}}\frac{\sqrt{-1}}{2}dz^{i}\wedge d\bar{z}^{j}$ and
$A\geq B$, then $\exp\omega\geq_{s}\exp\omega^{\prime}$ as a strongly positive
form.

###### Proof.

We may assume that $A$ is non-singular; otherwise, we just restrict to the
column space of $A$. Since strongly positivity is invariant under linear
transform, we may assume that $A$ is the identity matrix. Furthermore, we may
assume that $B=\sum_{i=1}^{n}\lambda_{i}\frac{\sqrt{-1}}{2}dz^{i}\wedge
d\bar{z}^{i}$. Then, clearly, $A\geq B$ which implies $0\leq\lambda_{i}\leq
1$. We then compute

$\exp\omega=1+\sum_{k=1}^{n}\sum_{|I|=k}\prod_{i\in
I}\frac{\sqrt{-1}}{2}dz^{i}\wedge d\bar{z}^{i},$
$\exp\omega-\exp\omega^{\prime}=\sum_{I}(1-\prod_{i\in
I}\lambda_{i})\prod_{i\in I}\frac{\sqrt{-1}}{2}dz^{i}\wedge d\bar{z}^{i},$

where $I$ runs over all ordered subsets of $\\{1,\cdots,n\\}$. It is clear
that the coefficient $1-\prod_{i\in I}\lambda_{i}$ is always non-negative,
which proves that $\exp\omega\geq_{s}\exp\omega^{\prime}$ is strongly
positive. ∎

####  3.2. Point-wise setup

We use $\Gamma_{n\times n},\Gamma_{n\times n}^{+},\overline{\Gamma_{n\times
n}^{+}}$ to denote the set of all Hermitian matrices, the set of positive
definite Hermitian matrices, and the set of non-negative Hermitian matrices,
respectively.

For $p\in M,$ we pick a local normal coordinate near $p$ with respect to
$\rho.$ Therefore,
$\rho=\frac{\sqrt{-1}}{2}\sum_{i,j}\delta_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}$
at $p$. We may write

$\omega=\frac{\sqrt{-1}}{2}\sum_{i,j}A_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j},$

where $A=(A_{i\bar{j}})$ is a positive definite Hermitian matrix. Let
$(A^{\bar{j}i}$) be the inverse matrix of $A$; i.e.
$A_{i\bar{j}}A^{\bar{j}k}=\delta_{i}^{k}.$ We define the following local
functional

(3.1) $F(A)=F(A,\Lambda):=\frac{(\Lambda\wedge\Omega)^{[n]}}{\Omega^{[n]}}.$

A coordinate change preserves
$\frac{(\Lambda\wedge\Omega)^{[n]}}{\Omega^{[n]}}$, and hence $F$ is invariant
under a coordinate change. Equation (1.1) can be re-written as

(3.2) $F(A)=\kappa.$

######  Definition 3.3.

Notations as above. We define a canonical $(1,1)$-vector
$\chi\in\bigwedge^{1,1}T_{p}M$ as

(3.3)
$\chi=2\sqrt{-1}\sum_{i,j}A^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}.$

$\chi$ is a strongly positive $(1,1)$-vector which induces a Hermitian metric
on $T^{*}M$ that is dual to $\omega$.

Let $\langle\cdot,\cdot\rangle$ be the complex bi-linear pairing
$\bigwedge^{*}(T_{p}^{*}M)\times\bigwedge^{*}(T_{p}M)\to\mathbb{C}.$ A direct
computation shows that

(3.4)
$\displaystyle\langle\frac{\omega^{k}}{k!},\frac{\chi^{k}}{k!}\rangle=\frac{n!}{k!(n-k)!},\
\ \langle\frac{\rho^{k}}{k!},\frac{\chi^{k}}{k!}\rangle=\sigma_{k}(A^{-1}),$

where $\sigma_{k}(\cdot)$ is the $k$-th elementary symmetric function of
eigenvalues of a Hermitian matrix.

With a labeled orthogonal splitting
$\mathcal{O}_{p}=\\{n_{p},\mathbf{d}_{p},\\{\mathcal{V}_{i}\\},\mathbf{k}_{p}\\}$
at $p$, we first fix a normal coordinate $\\{z^{i}\\}$ at $p$ of $\rho$ such
that $\\{\sqrt{2}\frac{\partial}{\partial z^{i}}\\}$ restricts to a unitary
basis on each $\mathcal{V}_{i}$. Suppose that
$\rho_{i}=\pi_{i}^{*}\iota_{i}^{*}\rho$ with respect to $\mathcal{O}_{p}$. We
may write under this frame

$A^{-1}=\left(\begin{array}[]{cccc}A^{-1}|_{\mathcal{V}_{1}}&*&*&*\\\
*&A^{-1}|_{\mathcal{V}_{2}}&*&*\\\ *&*&\ddots&*\\\
*&*&*&A^{-1}|_{\mathcal{V}_{n_{p}}}\end{array}\right),$

where $A_{i}\in\Gamma_{d_{i}\times d_{i}}$ . Then

(3.5)
$\langle\frac{\rho_{i}^{k}}{k!},\frac{\chi^{k}}{k!}\rangle=\langle\iota_{i}^{*}\frac{\rho^{k}}{k!},(\pi_{i})_{*}\frac{\chi^{k}}{k!}\rangle=\sigma_{k}(A^{-1}|_{\mathcal{V}_{i}}).$

To future convenience, we define, for $k\leq n,$

(3.6) $F_{k}(A)=\frac{n!\Lambda^{[k]}\wedge\omega^{n-k}}{(n-k)!\omega^{n}}.$

Denote $\exp\chi=\sum_{k=0}^{n}\frac{\chi^{k}}{k!}.$ The following lemma shows
that $F_{k}$ and $F$ can be represented by $\chi$.

######  Lemma 3.4.

Notations as above. We have

(3.7) $F_{k}(A)=\langle\Lambda^{[k]},\frac{\chi^{k}}{k!}\rangle.$ (3.8)
$F(A)=\langle\Lambda,\exp\chi\rangle.$

###### Proof.

It is enough to prove (3.7)) for $k\leq n-1$. For two ordered index sets
$I=\\{i_{1}<i_{2}<\cdots<i_{k}\\},\ J=\\{j_{1}<\cdots<j_{k}\\}$, we denote

$\displaystyle\sqrt{-1}^{k^{2}}dz^{I}\wedge d\bar{z}^{J}$
$\displaystyle=\sqrt{-1}^{k}dz^{i_{1}}\wedge
d\bar{z}^{j_{1}}\wedge\cdots\wedge dz^{i_{k}}\wedge d\bar{z}^{j_{k}},$
$\displaystyle\sqrt{-1}^{k^{2}}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{I}}$
$\displaystyle=\sqrt{-1}^{k}\frac{\partial}{\partial\bar{z}^{j_{1}}}\wedge\frac{\partial}{\partial
z^{i_{1}}}\wedge\cdots\wedge\frac{\partial}{\partial\bar{z}^{j_{k}}}\wedge\frac{\partial}{\partial
z^{i_{k}}}.$

Thus, we have the following

(3.9)
$\Lambda^{[k]}=\frac{\sqrt{-1}^{k^{2}}}{2^{k}}\sum_{|I|=|J|=k}\Lambda_{I,\bar{J}}dz^{I}\wedge
d\bar{z}^{J},$

and

(3.10)
$\frac{\omega^{k}}{k!}=\frac{\sqrt{-1}^{k^{2}}}{2^{k}}\sum_{|I|=|J|=k}A_{I,\bar{J}}dz^{I}\wedge
d\bar{z}^{J},\
\frac{\chi^{k}}{k!}=2^{k}\sqrt{-1}^{k^{2}}\sum_{|I|=|J|=k}A^{\bar{J},I}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{I}},$

where

$A_{I,\bar{J}}:=\det\left(\begin{array}[]{cccc}A_{i_{1}\overline{j_{1}}}&A_{i_{1}\overline{j_{2}}}&\cdots&A_{i_{1}\overline{j_{k}}}\\\
A_{i_{2}\overline{j_{1}}}&A_{i_{2}\overline{j_{2}}}&\cdots&A_{i_{2}\overline{j_{k}}}\\\
\vdots&\vdots&\ddots&\vdots\\\
A_{i_{k}\overline{j_{1}}}&A_{i_{k}\overline{j_{2}}}&\ldots&A_{i_{k}\overline{j_{k}}}\end{array}\right),\
A^{\bar{J},I}:=\det\left(\begin{array}[]{cccc}A^{\overline{j_{1}}i_{1}}&A^{\overline{j_{1}}i_{2}}&\cdots&A^{\overline{j_{1}}i_{k}}\\\
A^{\overline{j_{2}}i_{1}}&A^{\overline{j_{2}}i_{2}}&\cdots&A^{\overline{j_{2}}i_{k}}\\\
\vdots&\vdots&\ddots&\vdots\\\
A^{\overline{j_{k}}i_{1}}&A^{\overline{j_{k}}i_{2}}&\ldots&A^{\overline{j_{k}}i_{k}}\end{array}\right).$

As a consequence of (3.9) and (3.10), we have

$\Lambda^{[k]}\wedge\frac{\omega^{n-k}}{(n-k)!}=\sum_{|I|=|J|=k}\epsilon_{J,J^{c}}^{I,I^{c}}\Lambda_{I,\bar{J}}A_{I^{c},\bar{J^{c}}}\frac{\rho^{n}}{n!}$

where index sets $I^{c}$, $J^{c}$ are ordered complement of $I$ and $J$; and
$\epsilon_{J,J^{c}}^{I,I^{c}}=\frac{\sqrt{-1}^{n^{2}}dz^{I}\wedge
dz^{I^{c}}\wedge d\bar{z}^{J}\wedge d\bar{z}^{J^{c}}}{2^{n}\rho^{n}/n!}$.

We make the following claim:

(3.11) $\frac{A_{I^{c},\bar{J^{c}}}}{\det
A}\epsilon_{J,J^{c}}^{I,I^{c}}=A^{\bar{J},I}.$

After proper row and column permutations, we may assume without loss of
generality that $I=\\{1,2,\cdots,k\\}$ and $J=\\{1,\cdots,k\\}$. We decompose
as follows

$A=\left(\begin{array}[]{cc}A_{1}&C^{\prime}\\\ C&A_{2}\end{array}\right),\
A^{-1}=\left(\begin{array}[]{cc}B_{1}&D^{\prime}\\\
D&B_{2}\end{array}\right),$

where $A_{1}$ is an $(n-k)\times(n-k)$ matrix and $B_{1}$ is a $k\times k$
matrix. Since the sign change of the permutation is
$\epsilon_{J,J^{c}}^{I,I^{c}}$, in order to prove (3.11), it suffices to show
$\det(A_{1})=\det A\cdot\det B_{2}$. Suppose first that $A_{1}$ is
nonsingular. Let $E=-CA_{1}^{-1}$, $E^{\prime}=-A_{1}^{-1}C^{\prime}$. Since

$\left(\begin{array}[]{cc}I&0\\\
E&I\end{array}\right)A\left(\begin{array}[]{cc}I&E^{\prime}\\\
0&I\end{array}\right)=\left(\begin{array}[]{cc}A_{1}&0\\\
0&A_{2}^{\prime}\end{array}\right).$

We may write

$\left(\begin{array}[]{cc}I&-E^{\prime}\\\
0&I\end{array}\right)A^{-1}\left(\begin{array}[]{cc}I&0\\\
-E&I\end{array}\right)=\left(\begin{array}[]{cc}\tilde{B_{1}}&\tilde{D}^{\prime}\\\
\tilde{D}&B_{2}\end{array}\right).$

Thus

(3.12) $\left(\begin{array}[]{cc}\tilde{B_{1}}&\tilde{D}^{\prime}\\\
\tilde{D}&B_{2}\end{array}\right)\left(\begin{array}[]{cc}A_{1}&0\\\
0&A_{2}^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}I&0\\\
0&I\end{array}\right).$

(3.12) implies

$B_{2}A_{2}^{\prime}=I.$

Hence $\det B_{2}\det A_{2}^{\prime}=1$. Notice that $\det A=\det A_{1}\det
A_{2}^{\prime}$. Therefore, $\det(A_{1})=\det(A)\det(B_{2})$. We have proved
the claim (3.11) for non-singular $A_{1}$. If $A_{1}$ is singular, we consider
the perturbation $A_{\epsilon}=A+\epsilon I$ for suitable $\epsilon$ such that
both $A_{1}$ and $A$ are non-singular. Then the claim follows from the
continuity of the inversion and determinant functions. We have now established
(3.11).

To finish the proof of our lemma, we observe that from (3.9)

(3.13)
$\left\langle\Lambda^{[k]},\left(2^{k}\sqrt{-1}^{k^{2}}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{I}}\right)\right\rangle=\Lambda_{I,\bar{J}}.$

Thus, by (3.11),

(3.14) $\displaystyle\langle\Lambda^{[k]},\chi^{k}/k!\rangle$
$\displaystyle=\langle\Lambda^{[k]},A^{\bar{J},I}2^{k}\sqrt{-1}^{k^{2}}(\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{I}})\rangle$
$\displaystyle=\left(\sum_{|I|=|J|=k}\Lambda_{I,\bar{J}}A^{\bar{J},I}\right)$
$\displaystyle=\sum_{|I|=|J|=k}\Lambda_{I,\bar{J}}\frac{A_{I^{c},\bar{J^{c}}}}{\det
A}\epsilon_{J,J^{c}}^{I,I^{c}}$ $\displaystyle=F_{k}(A).$

We have finished the proof. ∎

For future use, we record the following linear algebraic fact.

######  Lemma 3.5.

Given Hermitian matrices $V,H$, and $A=\left(\begin{array}[]{cc}H&D\\\
D^{\dagger}&V\end{array}\right)$ which are invertible, then

(3.15) $A^{-1}=\left(\begin{array}[]{cc}\hat{H}^{-1}&-\hat{H}{}^{-1}DV^{-1}\\\
-V^{-1}D^{\dagger}\hat{H}^{-1}&V^{-1}+V^{-1}D^{\dagger}\hat{H}^{-1}DV^{-1}\end{array}\right),$

where $\hat{H}=\left(H-DV^{-1}D^{\dagger}\right)$,
$\hat{V}=V-D^{\dagger}H^{-1}D$, and
$V^{-1}+V^{-1}D^{\dagger}\hat{H}^{-1}DV^{-1}=\hat{V}^{-1}$. Furthermore, if
$\hat{H}>0$ then

(3.16) $A^{-1}\geq\left(\begin{array}[]{cc}0&0\\\ 0&V^{-1}\end{array}\right).$

###### Proof.

(3.15) can be proved by column and row operation as in the proof of Lemma 3.4.
By symmetry, we have also the following

$A^{-1}=\left(\begin{array}[]{cc}H^{-1}+H^{-1}D\hat{V}^{-1}D^{\dagger}H^{-1}&-H^{-1}D\hat{V}^{-1}\\\
-\hat{V}^{-1}D^{\dagger}H^{-1}&\hat{V}^{-1}\end{array}\right).$

For (3.16), let $C=-\hat{H}{}^{-1}DV^{-1}=-H^{-1}D\hat{V}^{-1}$. We have

$\displaystyle A^{-1}-\left(\begin{array}[]{cc}0&0\\\
0&V^{-1}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\hat{H}^{-1}&C\\\
C^{\dagger}&C^{\dagger}\hat{H}C\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}I&0\\\
C^{\dagger}&I\end{array}\right)\left(\begin{array}[]{cc}\hat{H}^{-1}&0\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}I&C\\\ 0&I\end{array}\right).$

Since $\left(\begin{array}[]{cc}\hat{H}^{-1}&0\\\ 0&0\end{array}\right)\geq
0$, $A^{-1}\geq\left(\begin{array}[]{cc}0&0\\\ 0&V^{-1}\end{array}\right)$. ∎

###  4\. Ellipticity and convexity

In this section, we discuss the monotonicity and convexity of $F$, when
certain positivity condition on $\Lambda$ holds. First, we compute the first
and second variations of $F(A)$ using notations set earlier. We use
abbreviations $F^{i\bar{j}}$, $F^{i\bar{j},r\bar{s}}$, $F_{k}^{i\bar{j}}$, and
$F_{k}^{i\bar{j},r\bar{s}}$ to represent $\frac{\partial F}{\partial
A^{i\bar{j}}}$, $\frac{\partial^{2}F}{\partial A^{i\bar{j}}\partial
A^{r\bar{s}}}$, $\frac{\partial F_{k}}{\partial A^{i\bar{j}}}$, and
$\frac{\partial^{2}F_{k}}{\partial A^{i\bar{j}}\partial A^{r\bar{s}}}$,
respectively.

######  Lemma 4.1.

Notations as above.

(4.1)
$F_{k}^{i\bar{j}}=-\sum_{ab}A^{\bar{a}i}A^{\bar{j}b}\langle\Lambda^{[k]},\frac{\chi^{k-1}}{(k-1)!}\wedge
2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\rangle.$ (4.2) $\displaystyle F_{k}^{i\bar{j},r\bar{s}}$
$\displaystyle=\sum_{a,b,c,d}A^{\bar{a}i}A^{\bar{j}b}A^{\bar{c}r}A^{\bar{s}d}\langle\Lambda^{[k]},\left(2\sqrt{-1}\right)^{2}\frac{\chi^{k-2}}{(k-2)!}\wedge\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\wedge\frac{\partial}{\partial\bar{z}^{c}}\wedge\frac{\partial}{\partial
z^{d}}\rangle$
$\displaystyle+\sum_{a,b,c,d}\left(A^{\bar{a}r}A^{\bar{s}i}A^{\bar{j}b}+A^{\bar{a}i}A^{\bar{j}r}A^{\bar{s}b}\right)\langle\Lambda^{[k]},\left(2\sqrt{-1}\frac{\chi^{k-1}}{(k-1)!}\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\right)\rangle.$

###### Proof.

Recall
$\chi=2\sqrt{-1}A^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}$. We use Lemma 3.4 to find

$\frac{\partial F_{k}}{\partial
A^{\bar{j}i}}=\langle\Lambda^{[k]},\frac{\chi^{k-1}}{(k-1)!}\wedge
2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}\rangle.$

Therefore

(4.3) $\displaystyle F_{k}^{i\bar{j}}$ $\displaystyle=\frac{\partial
A^{\bar{a}b}}{\partial A_{i\bar{j}}}\cdot\frac{\partial F_{k}}{\partial
A^{\bar{a}b}}$
$\displaystyle=-\sum_{a,b}A^{\bar{a}i}A^{\bar{j}b}\langle\Lambda^{[k]},\frac{\chi^{k-1}}{(k-1)!}\wedge
2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\rangle.$

Furthermore,

(4.4) $\displaystyle F_{k}^{i\bar{j},r\bar{s}}$
$\displaystyle=\sum_{a,b,c,d}A^{\bar{a}i}A^{\bar{j}b}A^{\bar{c}r}A^{\bar{s}d}\langle\Lambda^{[k]},\left(\frac{1}{k!}\frac{\partial^{2}\chi^{k}}{\partial
A^{\bar{c}d}\partial A^{\bar{a}b}}\right)\rangle$ $\displaystyle\
+\sum_{a,b,c,d}\left(A^{\bar{a}r}A^{\bar{s}i}A^{\bar{j}b}+A^{\bar{a}i}A^{\bar{j}r}A^{\bar{s}b}\right)\langle\Lambda^{[k]},\left(\frac{\partial}{\partial
A^{\bar{a}b}}\left(\frac{\chi^{k}}{k!}\right)\right)\rangle$
$\displaystyle=\sum_{a,b,c,d}A^{\bar{a}i}A^{\bar{j}b}A^{\bar{c}r}A^{\bar{s}d}\langle\Lambda^{[k]},\left(\left(2\sqrt{-1}\right)^{2}\frac{\chi^{k-2}}{(k-2)!}\right)\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\wedge\frac{\partial}{\partial\bar{z}^{c}}\wedge\frac{\partial}{\partial
z^{d}}\rangle$
$\displaystyle+\sum_{a,b,c,d}\left(A^{\bar{a}r}A^{\bar{s}i}A^{\bar{j}b}+A^{\bar{a}i}A^{\bar{j}r}A^{\bar{s}b}\right)\langle\Lambda^{[k]},2\sqrt{-1}\frac{\chi^{k-1}}{(k-1)!}\wedge\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}\rangle.$

∎

######  Lemma 4.2.

Notations as above. We assume that $\Lambda$ satisfies condition H2. For any
covector $b=b_{i}dz^{i}\not=0$, we have

(4.5) $-F_{k}^{i\bar{j}}b_{i}\overline{b_{j}}\geq 0.$

Moreover,

(4.6) $-\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\overline{b}_{j}>0.$

###### Proof.

Let $b^{\sharp}=\bar{b_{i}}A^{\bar{i}a}\frac{\partial}{\partial z^{a}}$ be the
dual of $b$ raised by $\omega$. Then by (4.1)

(4.7) $\displaystyle-F_{k}^{i\bar{j}}b_{i}\overline{b_{j}}$
$\displaystyle=\langle\Lambda^{[k]},2\sqrt{-1}\overline{b^{\sharp}}\wedge
b^{\sharp}\wedge\exp\chi\rangle.$ $\displaystyle\geq 0.$

We have used the fact that $2\sqrt{-1}\overline{b^{\sharp}}\wedge
b^{\sharp}\wedge\exp\chi$ is strongly positive.

By condition H2, we may choose a subspace
$\mathcal{V}_{i}\subset\mathcal{T}_{p}M$ such that
$(\pi_{i})_{*}(b^{\sharp})\not=0$ and $\Lambda^{[k_{i}]}\geq
m\frac{\left(\rho_{i}\right)^{k_{i}}}{k_{i}!}.$Then

(4.8) $\Lambda^{[k_{i}]}\geq m\frac{\rho_{i}^{k_{i}}}{k_{i}!}\geq
m^{\prime}(p)\frac{(\pi_{i}^{*}\iota_{i}^{*}\omega)^{k_{i}}}{k_{i}!},$

where $m^{\prime}(p)$ is a positive constant at $p$. Then

(4.9)
$\displaystyle\langle\Lambda^{[k_{i}]},2\sqrt{-1}\frac{\overline{b^{\sharp}}\wedge
b^{\sharp}\wedge\chi^{k_{i}-1}}{(k_{i}-1)!}$ $\displaystyle\rangle\geq
m^{\prime}(p)\langle\frac{(\pi_{i}^{*}\iota_{i}^{*}\omega)^{k_{i}}}{k_{i}!},2\sqrt{-1}\frac{\overline{b^{\sharp}}\wedge
b^{\sharp}\wedge\chi^{k_{i}-1}}{(k_{i}-1)!}\rangle>0.$

∎

######  Proposition 4.3.

Notations as above. If $\Lambda$ satisfies condition H2, for any local complex
$n\times n$ matrix $B_{i\bar{j}}$,

(4.10) $\sum_{i,j,r,s}\left(\frac{\partial^{2}F_{k}(A)}{\partial
A_{i\bar{j}}\partial A_{r\bar{s}}}+\frac{\partial F_{k}(A)}{\partial
A_{i\bar{s}}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}\geq 0.$

###### Proof.

Define the matrix $C$ and the corresponding element

(4.11) $\displaystyle C^{\bar{a}b}$
$\displaystyle:=\sum_{i,j}A^{\bar{a}i}B_{i\bar{j}}A^{\bar{j}b},$ (4.12)
$\displaystyle\zeta$
$\displaystyle:=2\sqrt{-1}\sum_{a,b}C^{\bar{a}b}\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}.$

In addition, we use $C^{\dagger}$ to denote the adjoint matrix
$\left(C^{\dagger}\right)^{\bar{a}b}:=\overline{C^{\bar{b}a}}$. Define

(4.13)
$\displaystyle\left(D\right)^{\bar{a}b}:=\sum_{i,j,r,s}A^{\bar{a}i}A^{\bar{s}b}A^{\bar{j}r}B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=\sum_{r,s}C^{\bar{a}r}A_{r\bar{s}}\overline{C^{\bar{b}s}}.$

Then,

(4.14)
$\displaystyle\sum_{i,j,r,s}A^{\bar{a}r}A^{\bar{s}i}A^{\bar{j}b}B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=\overline{C^{\bar{r}a}}A_{r\bar{s}}C^{\bar{s}b}=(D^{\dagger})^{\bar{a}b},$
(4.15)
$\displaystyle\sum_{i,j,r,s}A^{\bar{a}i}A^{\bar{j}b}A^{\bar{c}r}A^{\bar{s}d}B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=C^{\bar{a}b}\overline{C^{\bar{d}c}}.$

We define

(4.16)
$\xi:=2\sqrt{-1}\sum_{a,b}D^{\bar{a}b}\frac{\partial}{\partial\bar{z}^{a}}\wedge\frac{\partial}{\partial
z^{b}}.$

Therefore

(4.17)
$\displaystyle\sum_{i,j,r,s}\left(F_{k}^{i\bar{j},r\bar{s}}+\frac{\partial
F_{k}}{\partial
A_{i\bar{s}}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=\langle\Lambda^{[k]},\frac{\chi^{k-2}}{(k-2)!}\wedge\bar{\zeta}\wedge\zeta\rangle$
$\displaystyle+\langle\Lambda^{[k]},\frac{\chi^{k-1}}{(k-1)!}\wedge\xi\rangle.$

Define

(4.18)
$\Theta_{k}(B,B)=\frac{\chi^{k-2}}{(k-2)!}\wedge(2\sqrt{-1}\bar{\zeta}\wedge\zeta)+\frac{\chi^{k-1}}{(k-1)!}\wedge\xi.$

From (4.17), we see that

(4.19) $\sum_{i,j,r,s}\left(F_{k}^{i\bar{j},r\bar{s}}+\frac{\partial
F_{k}}{\partial
A_{i\bar{s}}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}=\langle\Lambda^{[k]},\Theta_{k}(B,B)\rangle.$

Now, it suffices to check the (strongly) positivity of $\Theta_{k}(B,B)$.
Since the positivity is invariant under coordinate change, we may change the
local coordinate such that at a point $p$, it holds

(4.20)
$\chi=2\sqrt{-1}\sum_{i}^{n}\frac{\partial}{\partial\bar{z}^{i}}\wedge\frac{\partial}{\partial
z^{i}},\ \
\zeta=2\sqrt{-1}\sum_{i=1}^{n}a_{i}\frac{\partial}{\partial\bar{z}^{i}}\wedge\frac{\partial}{\partial
z^{i}}.$

Therefore, by (4.20),

(4.21)
$\frac{\chi^{k-2}}{(k-2)!}\wedge\bar{\zeta}\wedge\zeta=\sum_{|J|=k-2}\sum_{i,j\not\in
J}a_{i}a_{j}2^{k}\sqrt{-1}^{k^{2}}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial\bar{z}^{i}}\wedge\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{J}}\wedge\frac{\partial}{\partial z^{i}}\wedge\frac{\partial}{\partial
z^{j}},$ (4.22)
$\frac{\chi^{k-1}}{(k-1)!}\wedge\left(\xi\right)=\sum_{|J|=k-1}\sum_{i\not\in
J}a_{i}^{2}2^{k}\sqrt{-1}^{k^{2}}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial\bar{z}^{i}}\wedge\frac{\partial}{\partial
z^{J}}\wedge\frac{\partial}{\partial z^{i}}.$

Let $J$ be any ordered subset of $\\{1,\cdots,n\\}$ such that $|J|=k$. The
coefficient of
$\sqrt{-1}^{k^{2}}2^{k}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{J}}$ in $\Theta_{k}(B,B)$ is given by

$\sum_{i\in J}a_{i}^{2}+\sum_{i\in J}\sum_{j\not=i,j\in
J}a_{i}a_{j}=\left(\sum_{i\in J}a_{i}\right)^{2}.$

Since each
$\sqrt{-1}^{k^{2}}2^{k}\frac{\partial}{\partial\bar{z}^{J}}\wedge\frac{\partial}{\partial
z^{J}}$ is strongly positive, we conclude that $\Theta_{k}(B,B)$ is strongly
positive.

For any general complex matrix $B$, we consider the matrix decomposition

$B=B^{R}+\sqrt{-1}B^{I},$

where

$B^{R}=\frac{B+B^{\dagger}}{2},\ B^{I}=\frac{B-B^{\dagger}}{2\sqrt{-1}}.$

Thus, $B^{R}$ and $B^{I}$ are both Hermitian. The Hermitian bilinear form
$h(B,B)=\langle\Lambda^{[k]},\Theta_{k}(B,B)\rangle$ is given by

$\displaystyle h(B,B)$
$\displaystyle=h(B^{R},B^{R})+h(B^{I},B^{I})+\sqrt{-1}\left(h(B^{I},B^{R})-h(B^{R},B^{I})\right)$
$\displaystyle=h(B^{R},B^{R})+h(B^{I},B^{I}).$

Hence $h$ is positive definite. Therefore, we have finished the proof. ∎

Now, we establish the ellipticity and convexity of $F$. In this section, we
consider the simpler case when $\Lambda^{[n]}=f\frac{\rho^{n}}{n!}\geq 0.$

######  Corollary 4.4.

Notations as above. Suppose that $\Lambda^{[n]}\geq 0$ at some point $p$. Then
at $p$, $F(A)$ is a strictly decreasing function in $A$, i.e.

(4.23) $F(A+B)<F(A)$

for any non-zero semi-positive Hermitian matrix $B$. Furthermore, for any
complex matrix $B_{i\bar{j}}$, we have

$\sum_{i,j,r,s}\left(F^{i\bar{j},r\bar{s}}+F^{i\bar{s}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}\geq
0.$

In particular, $F$ is a strictly convex function in $\Gamma_{n\times n}^{+}$.

###### Proof.

Notice that

$F^{i\bar{j}}=\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}-\frac{f}{\det A}A^{\bar{j}i}$

Thus, (4.23) follows from Lemma 4.2 and assumption H2.

For convexity, by Proposition 4.3

$\displaystyle\sum\left(F^{i\bar{j},r\bar{s}}+F^{i\bar{s}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=\sum_{k=1}^{n-1}\langle\Lambda^{[k]},\Theta_{k}(B,B)\rangle+\frac{f}{\det
A}A^{\bar{s}r}A^{\bar{j}i}B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle\geq 0.$

Here, we have used the lower bound for $f.$ By Lemma 4.2 and assumption H2,
$-F^{i\bar{s}}A^{\bar{j}r}B_{i\bar{j}}\overline{B_{s\bar{r}}}>0$ for non-zero
$B$. Thus $F$ is strictly convex. ∎

###  5\. Cone condition

We have introduced the concept of cone condition/subsolution in (1.5). In this
section, we state more criteria for subsolutions and prove some properties
that will be used later. We will focus on the local cone condition near a
point $p\in M$.

######  Definition 5.1.

Let

$\mathcal{C}_{\Lambda}^{\kappa}:=\\{\omega:\omega\ \text{is\
}\rm{K\ddot{a}hler};\ (\ref{eq:cone condition})\text{ holds}\\}.$

At a point $p\in M$, let

$\mathcal{C}_{\Lambda}^{\kappa}(p):=\\{\omega:\omega\text{ is a positive
}(1,1)\text{-form};\ (\ref{eq:cone condition})\text{ holds at }p\\}.$

In a local coordinates at $p$, where $\omega=\frac{\sqrt{-1}}{2}\sum
A_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}$, we say the matrix
$A\in\mathcal{C}_{\Lambda}^{\kappa}(p)$ if $\omega$ satisfies (1.5) at $p$. By
(1.5), $\mathcal{C}_{\Lambda}^{\kappa}(p)$ can be viewed as an open set in
$\Gamma_{n\times n}^{+}$.

The cone condition (1.5) in the study of $J$-equation was first explored in
Song-Weinkove [33]. Later Fang-Lai-Ma [19] extended the discussion to inverse
$\sigma_{k}$ type equations. The notion of subsolution for a class of fully
nonlinear equations was introduced by Guan [17]. See also Székelyhidi [35].

We have the following criterions for subsolutions. Here we do not assume the
sign of $\Lambda^{[n]}$.

######  Proposition 5.2.

Notations as above. Suppose that $\underline{A}$ is a positive Hermitian
matrix. The followings are equivalent

  1. (1)

$\underline{A}\in\mathcal{C}_{\Lambda}^{\kappa}(p)$.

  2. (2)

There is a constant $R=R(\underline{A},\Lambda,n,|f(p)|)$ s.t. if $B$ is a
non-negative Hermitian matrix satisfying

$F(\underline{A}+B)=\kappa,$

then

(5.1) $|B|\leq R.$

Here $|B|=\left(\sum_{i,j}|B_{i\bar{j}}|^{2}\right)^{\frac{1}{2}}.$

  3. (3)

For any non-zero semi-positive Hermitian matrix $B$ the following holds

(5.2) $\lim_{t\to\infty}F(\underline{A}+tB)<\kappa.$

###### Proof.

Pick a local unit covector $b=\sum_{i=1}^{n}b_{i}dz^{i}$ such that
$\|b\|_{\rho}=1.$ We may identify $b$ with an $1\times n$ matrix
$(b_{1},\cdots,b_{n})$. Define
$\beta=\frac{\sqrt{-1}}{2}\sum_{i,j}b_{i}\bar{b}_{j}dz^{i}\wedge
d\bar{z}^{j}$. We have the following identity

(5.3)
$\displaystyle\frac{\left(\omega+t\beta\right)^{n}}{n!}\left(\kappa-F(\underline{A}+tb^{\dagger}b)\right)$
$\displaystyle=t\left(\kappa\Omega-\Lambda\wedge\Omega\right)^{[n-1]}\wedge\beta+\left(\kappa-F(\underline{A})\right)\frac{\omega^{n}}{n!}.$

$(\ref{enu:.TFAE})\Rightarrow(\ref{enu:TFAE 2})$. Suppose that
$\underline{A}\in\mathcal{C}_{\Lambda}^{\kappa}$. Then there is a positive
$\delta=\delta(\underline{A},\Lambda)$ s.t.

$\left(\kappa\Omega-\Lambda\wedge\Omega\right)^{[n-1]}\wedge\beta\geq\delta
P^{[n-1]}.$

Then from (5.3),

(5.4)
$\frac{\left(\omega+t\beta\right)^{n}}{n!}\left(\kappa-F(\underline{A}+tb^{\dagger}b)\right)\geq
t\delta\|b\|_{\rho}^{2}P^{[n]}+O(1).$

Here $O(1)$ represents a $(n,n)$-form with bounded norm with respect to
$\rho$, depending on $\underline{A}$ and $\Lambda$ but not $b$. As

(5.5)
$F(\underline{A}+tb^{\dagger}b)=\sum_{k=1}^{n-1}F_{k}(\underline{A}+tb^{\dagger}b)+\frac{f(p)}{\det(\underline{A}+tb^{\dagger}b)},$

By Lemma 4.2, the first term in the right hand side of (5.5) is decreasing in
$t$ and is non-negative. The second term is bounded and approaches 0 when
$t\to\infty$. Thus $\lim_{t\to\infty}F(A+tb^{\dagger}b)$ exists. Hence from
(5.4),

$\displaystyle t\delta P^{[n]}+O(1)$
$\displaystyle\leq\left(t\frac{\omega^{n-1}}{(n-1)!}\wedge\beta\right)\left(\kappa-F(\underline{A}+tb^{\dagger}b)\right)+O(1)$
$\displaystyle=t\|b\|_{\omega}\det\underline{A}\left(\kappa-F(\underline{A}+tb^{\dagger}b)\right)P^{[n]}+O(1)$
$\displaystyle\leq
t\lambda_{max}\det\underline{A}\left(\kappa-F(\underline{A}+tb^{\dagger}b)\right)P^{[n]}+O(1).$

Here $\lambda_{max}$ is the maximal eigenvalue of $\underline{A}$. Therefore,
there is a $R=R(\underline{A},\Lambda)$ such that if $t>R$, then

(5.6)
$\kappa-F(\underline{A}+tb^{\dagger}b)\geq\frac{\delta}{2\lambda_{max}\det\underline{A}}.$

For any non-zero semi-positive Hermitian matrix $B$ with
$\|B\|=\sqrt{\sum_{i,j}|B_{i\bar{j}}|^{2}}=N$, let $\lambda$ be the biggest
non-zero eigenvalue of $B$ and
$b^{\prime}=(b_{1}^{\prime},\cdots,b_{n}^{\prime})$ be the unit eigenvector
corresponding to $\lambda$. Notice $\lambda\geq\frac{N}{n}$ and

$B-\frac{N}{2n}(b^{\prime})^{\dagger}b^{\prime}\geq 0.$

Suppose that $N/(2n)\geq R$.

(5.7)
$\kappa-F(\underline{A}+\frac{N}{2n}(b^{\prime})^{\dagger}b^{\prime})\geq\frac{\delta}{2\lambda_{max}\det\underline{A}}.$

Now

(5.8) $\displaystyle F(\underline{A}+B)$
$\displaystyle=\sum_{k=1}^{n-1}F_{k}(\underline{A}+B)+\frac{f(p)}{\det(\underline{A}+B)}$
$\displaystyle\leq
F(\underline{A}+\frac{N}{2n}(b^{\prime})^{\dagger}b^{\prime})+|f(p)|\cdot\left|\frac{1}{\det(\underline{A}+B)}-\frac{1}{\det(\underline{A}+\frac{N}{2n}(b^{\prime})^{\dagger}b^{\prime})}\right|.$

Take $N$ large enough so that
$\det(\underline{A}+\frac{N}{2n}(b^{\prime})^{\dagger}b^{\prime})^{-1}<\frac{\delta}{4\lambda_{max}(\det\underline{A})(|f(p)|+1)}$.
Then by (5.7) and (5.8),

$\kappa-F(\underline{A}+B)>\frac{\delta}{4\lambda_{max}\det\underline{A}}.$

We then have a contradiction. We have proved (5.1).

$(\ref{enu:TFAE 2})\Rightarrow(\ref{enu:TFAE 3})$ is obvious since
$F_{k}(\underline{A}+tB)$ is strictly monotonic decreasing and
$\frac{f}{\det(\underline{A}+tB)}$ tends to $0$ as $t\to\infty$.

$(\ref{enu:TFAE 3})\Rightarrow(\ref{enu:.TFAE})$. If
$\lim_{t\to\infty}F(\underline{A}+tB)<\kappa$, we can test each
$B=(B_{i\bar{j}})=(\delta_{i\bar{j}})$ in (5.3) to see that each
$\left(\kappa\Omega-\Lambda\wedge\Omega\right)^{[n-1]}\wedge(\frac{\sqrt{-1}}{2}dz^{i}\wedge
d\bar{z}^{i})$ is positive. Thus,
$\left(\kappa\Omega-\Lambda\wedge\Omega\right)^{[n-1]}$ is positive, which
implies $\underline{A}\in\mathcal{C}_{\Lambda}^{\kappa}(p)$. ∎

######  Remark 5.3.

The criterion (2) in Proposition 5.2 is the definition of subsolution given by
Székelyhidi [35]. The equivalence of (2) and (3) is suggested in [35] Remark
8.

######  Definition 5.4.

Let $B$ be a non-zero semi-positive Hermitian matrix. We define

(5.9) $F_{\Lambda}(A:B):=\lim_{t\to\infty}F(A+tB,\Lambda),$ (5.10)
$\mathcal{P}_{\Lambda}(A):=\max_{B\in\overline{\Gamma_{n\times
n}^{+}},\|B\|=1}\lim_{t\to+\infty}F(A+tB,\Lambda).$

Clearly, $A\in\mathcal{C}_{\Lambda}^{\kappa}(p)$ if and only if for any non-
zero $B\in\overline{\Gamma_{n\times n}^{+}}$, $F_{\Lambda}(A:B)<\kappa$.
Equivalently, $A\in\mathcal{C}_{\Lambda}^{\kappa}(p)$ if and only if

$\mathcal{P}_{\Lambda}(A)<\kappa.$

We use the notation $\mathcal{P}_{\Lambda}(\omega)=\mathcal{P}_{\Lambda}(A)$
if $\omega$ is represented by matrix $A$ in a local coordinate. There is
another perspective where $F_{\Lambda}(A:B)$ and $\mathcal{P}_{\Lambda}(A)$
are given by the restrictions of $F$ on subspaces, which relies on the the
general inverse matrix in the sense of Moore-Penrose [27, 28].

######  Definition 5.5.

Suppose that $A$ is a $n\times n$ matrix. A Moore-Penrose inverse matrix
$A^{-1}$ is the unique matrix that satisfies the following conditions:

  1. (1)

$AA^{-1}A=A$, $A^{-1}AA^{-1}=A^{-1}.$

  2. (2)

Both $AA^{-1}$ and $A^{-1}A$ are Hermitian matrices.

For any matrix, the Moore-Penrose inverse of $A$ exists and can be constructed
via singular value decomposition. The following lemma 5.6 will be used to
relate $F_{\Lambda}(A:B)$ with a Moore-Penrose inverse.

######  Lemma 5.6.

Let $V\geq 0$ be a Hermitian matrix of rank $r$. Let $\mathcal{V}$ be the
linear spaces spaced by the non-zero eigenvectors of $V$ and let $\mathcal{H}$
be the orthogonal complement of $\mathcal{V}$. Let $\Pi_{\mathcal{H}}$ be the
orthogonal projection matrix to the subspace
$\mathcal{H}\subset\mathcal{T}_{p}M$. Denote

(5.11) $(A|_{\mathcal{H}})^{-1}:=\Pi_{\mathcal{H}}A\Pi_{\mathcal{H}}.$

Then

(5.12)
$\lim_{t\to+\infty}\left(A+tV\right)^{-1}=\left(A|_{\mathcal{H}}\right)^{-1},$

######  Remark 5.7.

The inverse on the right hand side of (5.12) is in the sense of Moore-Penrose
and we denote it as $(A|_{\mathcal{H}})^{-1}$.

###### Proof.

Since Moore-Penrose inverse is invariant under unitary transform, we may
choose a unitary basis $\\{e_{i}\\}$ so that $e_{1},\cdots,e_{r}$ spans
$\mathcal{V}$. Then

$V=\sum_{i=1}^{r}V_{i\bar{i}}e_{i}e_{i}^{\dagger},\
A=\sum_{i,j}A_{i\bar{j}}e_{i}e_{j}^{\dagger}.$

Then, under this basis, we have

$V=\left(\begin{array}[]{cc}v&0\\\ 0&0\end{array}\right),\
\Pi_{\mathcal{H}}=\left(\begin{array}[]{cc}0&0\\\ 0&I_{n-r}\end{array}\right)$

where $v$ is a $r\times r$ positive definite matrix and $I_{n-r}$ is an
identity matrix. We may write

$A+tV=\left(\begin{array}[]{cc}A_{1}+tv&C\\\
C^{\dagger}&A_{2}\end{array}\right).$

The inverse is given by

$(A+tV)^{-1}=\left(\begin{array}[]{cc}(\hat{A}_{1}(t))^{-1}&-(\hat{A_{1}}(t))^{-1}CA_{2}^{-1}\\\
-A_{2}^{-1}C^{\dagger}(\hat{A_{1}}(t))^{-1}&A_{2}^{-1}+A_{2}^{-1}C^{\dagger}(\hat{A_{1}}(t))^{-1}CA_{2}^{-1}\end{array}\right)$

where $\hat{A}_{1}(t)=A_{1}+tv-CA_{2}^{-1}C^{\dagger}$. As $t\to+\infty$,
$\left(\hat{A}_{1}(t)\right)^{-1}\to 0$ uniformly. Thus,

$\lim_{t\to+\infty}(A+tV)^{-1}=\left(\begin{array}[]{cc}0&0\\\
0&A_{2}^{-1}\end{array}\right).$

Notice that
$\Pi_{\mathcal{H}}A\Pi_{\mathcal{H}}=\left(\begin{array}[]{cc}0&0\\\
0&A_{2}\end{array}\right)$. Hence, we easily see that
$(\Pi_{\mathcal{H}}A\Pi_{\mathcal{H}})^{-1}=\lim_{t\to+\infty}(A+tV)^{-1}$. ∎

Given an orthogonal splitting
$\mathcal{T}_{p}M=\mathcal{H}\oplus\mathcal{H}^{\perp}$ with respect to
$\rho$. Following the notation in Lemma 5.6, we denote

(5.13) $\chi_{\mathcal{H}}:=\sum_{i,j\geq
d+1}(A|_{\mathcal{H}})^{\bar{j}i}2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}.$

Then $\chi_{\mathcal{H}}\in\bigwedge^{1,1}\mathcal{H}$ is dual to
$\omega|_{\mathcal{H}}$. Now we characterize $\mathcal{P}_{\Lambda}$ using
subspaces.

######  Lemma 5.8.

Notations as above, the following statements are true:

  1. (1)

If $B\geq 0$ and its all the non-zero eigenvectors span $\mathcal{H}^{\perp}$,
then

$F_{\Lambda}(A:B)=\langle\Lambda,\exp\chi_{\mathcal{H}}\rangle.$

  2. (2)

$\mathcal{P}_{\Lambda}(A)=\max_{\mathcal{H}\subset\mathcal{T}_{p}M:\dim\mathcal{H}=d\leq
n-1}\langle\Lambda,\exp\chi_{\mathcal{H}}\rangle.$

  3. (3)

$\mathcal{P}_{\Lambda}(A)=\max_{\mathcal{H}\subset\mathcal{T}_{p}M:\dim\mathcal{H}=n-1}\langle\Lambda,\exp\chi_{\mathcal{H}}\rangle.$

  4. (4)

If $\omega\in\mathcal{C}_{\Lambda}^{\kappa}(p)$, then for any $d$-dimensional
subspace $\mathcal{H}$ of $\mathcal{T}_{p}M$ with $d\leq n-1$, we have

$\left((\kappa-\Lambda)\wedge\exp\omega\right)^{[d]}|_{\mathcal{H}}>0,$

as a positive $(d,d)$-form.

###### Proof.

(1) follows from the definition of $F(A:B)$ and Lemma 5.6.

(2) follows immediately from (1).

(3) follows from the monotonicity of $F$. In fact, $F(A)$ is monotonic in $A$,
which implies
$\lim_{t\to+\infty}F(A+tB,\Lambda)\leq\lim_{t\to\infty}F(A+tb^{\dagger}b)$
where $b$ is a unit eigenvector of $B$ with positive eigenvalue. Thus, we may
restrict to rank one Hermitian matrix when computing
$\mathcal{P}_{\Lambda}(A)$ in (5.10).

(4) follows from (2) immediately. ∎

The following corollary gives the easy part of Theorem 1.10.

######  Corollary 5.9.

Notations as above. If $\omega\in\mathcal{C}_{\Lambda}^{\kappa}$, for any
$d$-dimensional subvariety $Z\subset M$ and $d\leq n-1$, we have

(5.14) $\int_{Z}\left(\kappa-\Lambda\right)\wedge\exp\omega>0.$

Furthermore, if $\omega\in[\omega_{0}]$ solves (1.2), then $[\omega_{0}]$ is
$([\Lambda],\kappa)$-positive.

###### Proof.

For any $d$-dimensional subvariety $Z\subset M$ with its regular component as
$Z_{reg},$ $Z\backslash Z_{\text{reg}}$ is of zero measure. Then by Lemma 5.8
(4)
$\left(\left(\kappa-\Lambda|_{Z}\right)\wedge\exp\left(\omega|_{Z}\right)\right)^{[d]}>0$
for any point $p\in Z_{\text{reg}}$. Hence (5.14) is established.

We will prove that any solution $\omega$ to (1.2) is a subsolution in Lemma
6.1. Thus, by (5.14), we conclude that $[\omega]=[\omega_{0}]$ is
$([\Lambda],\kappa)$-positive. ∎

For future use, we collect some properties of _$\mathcal{P}_{\Lambda}(A)$ in
the following lemma._

######  Lemma 5.10.

Notations as above. The following properties of $\mathcal{P}_{\Lambda}(A)$
hold:

  1. (1)

$\mathcal{P}_{\Lambda}(A)$ is a continuous convex function in $\Gamma_{n\times
n}^{+}$.

  2. (2)

$\mathcal{P}_{\Lambda}(A)$ is decreasing in $A$, i.e.
$\mathcal{P}_{\Lambda}(A^{\prime})\leq\mathcal{P}_{\Lambda}(A)$ if
$A^{\prime}-A\geq 0$.

  3. (3)

$\mathcal{P}_{\Lambda}(A)$ is continuous in $\Lambda$.

  4. (4)

$\mathcal{P}_{\Lambda}(A)$ is increasing in $\Lambda$, i.e.
$\mathcal{P}_{\Lambda}(A)\leq\mathcal{P}_{\Lambda^{\prime}}(A)$ if
$\Lambda^{\prime}-\Lambda\geq 0.$

  5. (5)

_$\mathcal{P}_{\Lambda}(\omega)$ as a function of $p$ on $M$ is continuous._

  6. (6)

$\mathcal{P}_{\Lambda}(A)$ is sub-linear in $\Lambda$, i.e.
$\mathcal{P}_{\Lambda+\Lambda^{\prime}}(A)\leq\mathcal{P}_{\Lambda}(A)+\mathcal{P}_{\Lambda^{\prime}}(A)$.

  7. (7)

$\mathcal{P}_{\Lambda}(A)=\mathcal{P}_{\mathring{\Lambda}}(A)$, for
$\mathring{\Lambda}=\Lambda-\Lambda^{[n]}$,

  8. (8)

_Suppose that $\Lambda$ satisfies _H2 _. If $\\{A_{l}\\}\subset\Gamma_{n\times
n}^{+}$ and $A_{l}\to A_{\infty}\in\partial\Gamma_{n\times n}^{+}$ as
$l\to\infty$, then $\lim_{l\to\infty}\mathcal{P}_{\Lambda}(A_{l})=\infty.$_

###### Proof.

(1) follows from the convexity of $F$ on $\Gamma_{n\times n}^{+}$. (2) follows
from the monotonicity of $F$. See Corollary 4.4.

For (3), let $\Lambda$ and $\Lambda^{\prime}$ be two differential forms. We
may choose $\epsilon^{n}=n!\|\Lambda-\Lambda^{\prime}\|_{\rho}$ so that

$-\exp\epsilon\rho+1\leq\Lambda-\Lambda^{\prime}\leq\exp\epsilon\rho-1.$

By Lemma 5.8 (3), we have

(5.15)
$\displaystyle|\mathcal{P}_{\Lambda}(A)-\mathcal{P}_{\Lambda^{\prime}}(A)|$
$\displaystyle\leq\max_{\mathcal{H}\subset
T^{1,0}M:\dim\mathcal{H}=n-1}\left|\langle\Lambda-\Lambda^{\prime},\exp\chi_{\mathcal{H}}\rangle\right|$
$\displaystyle\leq\max_{\mathcal{H}\subset
T^{1,0}M:\dim\mathcal{H}=n-1}\langle\exp\left(\epsilon\rho\right)-1,\exp\left(\chi_{\mathcal{H}}\right)\rangle$
$\displaystyle\leq\langle\exp\left(\epsilon\rho\right)-1,\exp\chi\rangle.$

Then, as $\|\Lambda-\Lambda^{\prime}\|_{\rho}\to 0$, $\epsilon\to 0$, it holds
that $|\mathcal{P}_{\Lambda}(A)-\mathcal{P}_{\Lambda^{\prime}}(A)|\to 0$ as
well. Hence, $\mathcal{P}_{\Lambda}(A)$ is continuous in $\Lambda$.

(4) follows from Lemma 5.8 (1). (5) follows from (1) and (3).

For (6), we have the following

$\displaystyle\mathcal{P}_{\Lambda+\Lambda^{\prime}}(A)$
$\displaystyle=\max_{\mathcal{H}\subset
T^{1,0}M:\dim\mathcal{H}=n-1}\langle\Lambda+\Lambda^{\prime},\exp\chi_{\mathcal{H}}\rangle$
$\displaystyle\leq\mathcal{P}_{\Lambda}(A)+\mathcal{P}_{\Lambda^{\prime}}(A).$

For (7), we assume that $\Lambda^{[n]}(p)=f(p)P^{[n]}$. Then (7) follows from
the fact that $\lim_{t\to\infty}\frac{f(p)}{\det(A+tB)}=0$ for any $B\geq 0$
and $B\not=0$.

For (8), note we may assume that $\Lambda\geq 0$ since
$\mathcal{P}_{\mathring{\Lambda}}(A)=\mathcal{P}_{\Lambda}$ if
$\mathring{\Lambda}=\Lambda-\Lambda^{[n]}$ by (7). Let
$\chi_{A_{l}}=(A_{l})^{\bar{ji}}2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}$. We first prove that

(5.16) $\lim_{l\to\infty}\langle\Lambda,\exp\chi_{A_{l}}\rangle=\infty$

as $A_{l}\to A_{\infty}\in\partial\Gamma_{n\times n}^{+}$. By H2,

(5.17) $\displaystyle\langle\Lambda,\exp\chi_{A_{i}}\rangle$
$\displaystyle\geq
m\langle\sum_{i=1}^{n_{p}}\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi_{A_{l}}\rangle$
$\displaystyle\geq
m\sum_{i=1}^{n_{p}}\sigma_{k_{i}}(A_{l}^{-1}|_{\mathcal{V}_{i}})$
$\displaystyle\geq
mC(n,\mathbf{d}_{p},\mathbf{k}_{p})\det(A_{l}^{-1})^{\frac{1}{\sum_{i=1}^{n_{p}}\frac{d_{i}}{k_{i}}}}.$

We have used Newton-Maclaurin inequality and the mean value inequality in the
last inequality. (5.16) follows from the fact that
$\lim_{l\to\infty}\det(A_{l}^{-1})=\infty.$ To prove (8), we choose a
hyperplane $\mathcal{H}$ which contains the kernel of $A_{\infty}$. Then,
Lemma 5.8 (3) and (5.16) implies (8). ∎

###  6\. Equations with almost positive volume forms

With the cone condition in place, we extend results of Section 4 to include
cases where $\Lambda$ contains an almost positive volume form. Many ideas are
from works of Chen [4] and Datar-Pingali [12]. For simplicity, we write

(6.1) $\Lambda^{[n]}=fP^{[n]}.$

In this section, we allow $f$ to be almost positive.

The following lemma implies that a solution to (1.2) is also a subsolution if
H2 is satisfied.

######  Lemma 6.1.

Assume that $\mathring{\Lambda}$ is $\mathcal{O}$-uniformly positive with
constant $m$. Suppose that for some $p_{0}\in M$, $f(p_{0})\geq 0$. For any
$p\in M$, let

(6.2)
$\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}):=\frac{\min_{i}\left\\{\binom{d_{i}}{k_{i}-1}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}-1}\right\\}\prod_{i=1}^{n_{p}}\binom{d_{i}}{k_{i}}^{\frac{d_{i}}{k_{i}}}}{\max_{i}\\{d_{i}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}}\\}(\frac{\kappa}{m})^{\sum_{i=1}^{n_{p}}\frac{d_{i}}{k_{i}}}\cdot\left(\sum_{i=1}^{n_{p}}\left(\frac{\kappa}{m}\right)^{\frac{1}{k_{i}}}\right)}.$

If $\omega$ is a solution to (1.2), and for all $p\in M$,
$f(p)>-m\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})$,
then $\omega\in\mathcal{C}_{\Lambda}^{\kappa}$.

We first state some lemmas before proving Lemma 6.1.

######  Lemma 6.2.

Notations as above. Let $A$ be a positive Hermitian matrix. Then the following
statements hold.

  1. (1)

If $k\geq l\geq 0$, $r\geq s\geq 0$, the function

(6.3) $A\mapsto\frac{\sigma_{l}(A)}{\sigma_{k}(A)}$

is decreasing in $A$.

  2. (2)

Let $\mathcal{H}$ be a linear subspace with codimension $r$. Then

(6.4)
$\frac{\sigma_{l-r}(A|_{\mathcal{H}})}{\sigma_{k-r}(A|_{\mathcal{H}})}\leq\frac{\sigma_{l}(A)}{\sigma_{k}(A)}.$

  3. (3)

Let $T_{k-1}(A)$ be the linearized operator of $\sigma_{k}(A)$, i.e. $\langle
T_{k-1}(A),B\rangle=\frac{d}{dt}\sigma_{k}(A+tB)|_{t=0}$. Then

(6.5) $\sigma_{r-1}(A)T_{k-1}(A)\geq\sigma_{k-1}(A)T_{r-1}(A),$

if $r\geq k$.

###### Proof.

(1) is well known in the literature of quotient Hessian equations. See for
instance, [34].

To prove (2), we may assume that
$\mathcal{H}=\text{span}\\{\frac{\partial}{\partial
z^{1}},\cdots,\frac{\partial}{\partial z^{n-r}}\\}$. We define

(6.6) $A_{t}=A+t\left(\begin{array}[]{cc}0&0\\\
0&I_{r}\end{array}\right)=\left(\begin{array}[]{cc}A|_{\mathcal{H}}&C\\\
C^{\dagger}&A|_{\mathcal{H}^{\perp}}+tI_{r}\end{array}\right).$

Then $A_{t}\geq A$ and

(6.7) $\sigma_{l}(A_{t})=t^{r}\sigma_{l-r}(A|_{\mathcal{H}})+o(t^{r}),\
\sigma_{k}(A_{t})=t^{r}\sigma_{k-r}(A|_{\mathcal{H}})+o(t^{r}).$

Hence,

(6.8)
$\frac{\sigma_{l}(A)}{\sigma_{k}(A)}\geq\lim_{t\to\infty}\frac{\sigma_{l}(A_{t})}{\sigma_{k}(A_{t})}=\frac{\sigma_{l-r}(A|_{\mathcal{H}})}{\sigma_{k-r}(A|_{\mathcal{H}})}.$

To prove (3), we may assume that $A$ is diagonal after a unitary transform.
Write $A=\text{diag}\\{\lambda_{1},\cdots,\lambda_{n}\\}.$Then

(6.9) $\left(T_{k-1}(A)\right)^{i\bar{j}}=\sigma_{k-1}(A|i)\delta^{i\bar{j}}.$

Here, $(A|i)$ denote a matrix obtained by deleting $i$-th row and $i$-th
column of $A$. Thus, (6.5) is equivalent to

(6.10) $\sigma_{r-1}(A)\sigma_{k-1}(A|i)\geq\sigma_{k-1}(A)\sigma_{r-1}(A|i),$

for each $i$. If $\sigma_{r-1}(A|i)=0$, then (6.10) holds trivially. Otherwise
(6.10) is equivalent to

(6.11)
$\frac{\sigma_{k-1}(A|i)}{\sigma_{r-1}(A|i)}\geq\frac{\sigma_{k-1}(A)}{\sigma_{r-1}(A)}.$

By 1), $\frac{\sigma_{k-1}(A)}{\sigma_{r-1}(A)}$ is decreasing in $A$. Thus
(6.11) holds since $(A|i)\leq A$. ∎

With a labeled orthogonal splitting
$\mathcal{O}_{p}=\\{n_{p},\mathbf{d}_{p},\\{\mathcal{V}_{i}\\},\mathbf{k}_{p}\\}$
at $p$, we fix a normal coordinate $\\{z^{i}\\}$ at $p$ of $\rho$ such that
$\\{\sqrt{2}\frac{\partial}{\partial z^{i}}\\}$ restricts to a unitary frame
on each $\mathcal{V}_{i}$. For any non-negative Hermitian matrix $A$, we may
write under this frame

(6.12) $A=\left(\begin{array}[]{cccc}A_{1}&*&*&*\\\ *&A_{2}&*&*\\\
*&*&\ddots&*\\\ *&*&*&A_{n_{p}}\end{array}\right),$

where $A_{i}\in\Gamma_{d_{i}\times d_{i}}$ .

######  Lemma 6.3.

Notations as above. For any non-negative $A$, Let

(6.13) $A^{\prime}=\left(\begin{array}[]{cccc}A_{1}&0&0&0\\\ 0&A_{2}&0&0\\\
0&0&\ddots&0\\\ 0&0&0&A_{n_{p}}\end{array}\right).$

Let $\omega^{\prime}=(A^{\prime})_{i\bar{j}}\frac{\sqrt{-1}}{2}dz^{i}\wedge
d\bar{z}^{j}.$ Then

  1. (1)

$\det(A)\leq\det(A^{\prime})=\prod_{i=1}^{n_{p}}\det A_{i};$

  2. (2)

(6.14) $\displaystyle\sigma_{k}(A)$
$\displaystyle\leq\sigma_{k}(A^{\prime})=\sum_{l\in\boldsymbol{l}_{k}}\prod_{j=1}^{n_{p}}\sigma_{l_{j}}(A_{j}),$

where
$\boldsymbol{l}_{k}=\\{(l_{1},\cdots,l_{n_{p}})\in\mathbb{N}:\sum_{j=1}^{n_{p}}l_{j}=k\\}$;

  3. (3)

(6.15) $\displaystyle T_{k-1}(A)$ $\displaystyle\leq
2^{(k-1)(n_{p}-1)}T_{k-1}(A^{\prime})=2^{(k-1)(n_{p}-1)}\text{diag}(T_{1},\cdots,T_{n_{p}}),$

where

(6.16)
$T_{i}=\sum_{l\in\boldsymbol{l}_{k,i}}\prod_{j\not=i}\sigma_{l_{j}}(A_{j})T_{l_{i}-1}(A_{i}),$

and
$\boldsymbol{l}_{k,i}=\\{(l_{1},\cdots,l_{n_{p}})\in\mathbb{N}^{n_{p}}:\sum_{j=1}^{n_{p}}l_{j}=k,\
l_{i}\geq 1\\}$.

###### Proof.

For claim (1), we first prove the case when $n_{p}=2$. We may write

$A=\left(\begin{array}[]{cc}A_{1}&C\\\ C^{\dagger}&A_{2}\end{array}\right).$

Therefore $\det(A)=\det\left(A_{1}-C^{\dagger}A_{2}^{-1}C\right)\det A_{2}$ if
$A_{2}$ is invertible. If $A_{2}$ is not invertible, we replace $A$with
$A_{\epsilon}=A+\epsilon I$ and let $\epsilon\to 0$. Then, (1) follows from
the continuity of the determinant function. The general case follows from
induction.

For claim (2), for any $1\leq i_{1}<\cdots<i_{k}\leq n$, by (1), we have

(6.19) $\displaystyle\sigma_{k}(A)$ $\displaystyle=\sum_{1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq
n}A\left(\begin{array}[]{cccc}i_{1}&i_{2}&\cdots&i_{k}\\\
i_{1}&i_{2}&\cdots&i_{k}\end{array}\right)$ (6.22)
$\displaystyle\leq\sum_{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq
n}A^{\prime}\left(\begin{array}[]{cccc}i_{1}&i_{2}&\cdots&i_{k}\\\
i_{1}&i_{2}&\cdots&i_{k}\end{array}\right)$
$\displaystyle=\sigma_{k}(A^{\prime}).$

Thus the inequality in (6.14) holds. The equality in (6.14) can be calculated
directly.

For claim (3), if $n_{p}=2$, we have

(6.23) $2A^{\prime}-A=\left(\begin{array}[]{cc}2A_{1}&0\\\
0&2A_{2}\end{array}\right)-\left(\begin{array}[]{cc}A_{1}&C\\\
C^{\dagger}&A_{2}\end{array}\right)=\left(\begin{array}[]{cc}A_{1}&-C\\\
-C^{\dagger}&A_{2}\end{array}\right)\geq 0.$

If $n_{p}\geq 2$, by iteration, we have

(6.24) $A\leq 2^{n_{p}-1}A^{\prime},\
\omega\leq_{s}2^{n_{p}-1}\omega^{\prime},\
\frac{\omega^{k}}{k!}\leq_{s}2^{k(n_{p}-1)}\frac{\left(\omega^{\prime}\right)^{k}}{k!}.$

Now, for any $b=b_{i}dz^{i}$, by (6.24), we have

(6.25)
$\displaystyle\left(T_{k-1}(A)\right)^{i\bar{j}}b_{i}\bar{b}_{j}\frac{\rho^{n}}{n!}$
$\displaystyle=\frac{\rho^{n-k}}{(n-k)!}\wedge\frac{\omega^{k-1}}{(k-1)!}\wedge\frac{\sqrt{-1}}{2}b\wedge\bar{b}$
$\displaystyle\leq
2^{(k-1)(n_{p}-1)}\frac{\rho^{n-k}}{(n-k)!}\wedge\frac{(\omega^{\prime})^{k-1}}{(k-1)!}\wedge\frac{\sqrt{-1}}{2}b\wedge\bar{b}$
$\displaystyle=2^{(k-1)(n_{p}-1)}T_{k-1}(A^{\prime})^{i\bar{j}}b_{i}\bar{b}_{j}\frac{\rho^{n}}{n!}.$

Thus, by (6.25),

$T_{k-1}(A)\leq 2^{(k-1)(n_{p}-1)}T_{k-1}(A^{\prime}).$

(6.16) can be verified by direct calculation. ∎

######  Lemma 6.4.

Suppose $\mathring{\Lambda}$ satisfies the $\mathcal{O}$-UP condition. Let $b$
be a covector in $\mathcal{T}_{p}^{*}M$. Let
$\mathcal{B}=\\{\zeta\in\mathcal{T}_{p}M:\langle b,\zeta\rangle=0\\}$ be a
complex hyperplane. Denote

(6.26)
$\chi_{\mathcal{B}}:=(A|_{\mathcal{B}})^{\bar{j}i}2\sqrt{-1}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}.$

If at $p$, $\mathcal{P}_{\Lambda}(A)\leq\kappa$, then we have

(6.27)
$\sum_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi_{\mathcal{B}}\rangle\leq\frac{\kappa}{m}.$

Furthermore,

(6.28)
$\sum_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi\rangle\leq\frac{n\kappa}{m}.$

###### Proof.

Let $\Lambda^{\prime}=\sum_{i=1}^{n_{p}}(\exp\rho_{i})^{[k_{i}]}$. By the
$\mathcal{O}$-UP condition, at $p$,

$\mathring{\Lambda}\geq m\Lambda^{\prime}.$

As a result, we have

(6.29) $\mathcal{P}_{\Lambda^{\prime}}(A)\leq\frac{\kappa}{m}.$

Let $B_{i\bar{j}}=b_{i}\bar{b}_{j}$. Then,

(6.30)
$F_{\Lambda^{\prime}}(A:B)\leq\mathcal{P}_{\Lambda^{\prime}}(A)\leq\frac{\kappa}{m}.$

By Lemma 5.8, (6.30) implies (6.27).

Choose a normal coordinate of $\omega$ at $p$. Let
$\mathcal{B}_{i}=\\{\xi:dz^{i}(\xi)=0\\}$. Then

$\chi=\sum_{i=1}^{n}\frac{\sqrt{-1}}{2}dz^{i}\wedge d\bar{z}^{i},\
\chi_{\mathcal{B}_{j}}=\sum_{i\not=j}\frac{\sqrt{-1}}{2}dz^{i}\wedge
d\bar{z}^{i}.$

Then, for $k\leq n-1$,

$\frac{\chi^{k}}{k!}\leq_{s}\sum_{j=1}^{n}\frac{\chi_{\mathcal{B}_{j}}^{k}}{k!}.$

Thus,

(6.31)
$\sum_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi\rangle\leq\sum_{j=1}^{n}\sum_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi_{\mathcal{B}_{j}}\rangle\leq\frac{n\kappa}{m}.$

We have proved 6.28. ∎

The following lemma gives an explicit estimate of $-F^{i\bar{j}}$ when the
cone condition holds.

######  Lemma 6.5.

Suppose $\mathring{\Lambda}$ satisfies $\mathcal{O}$-UP and for all $p\in M$,
$f(p)>-m\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})$.
Let $b$ be a covector in $\mathcal{T}_{p}^{*}M$. Let $\mathcal{B}$ and
$\chi_{\mathcal{B}}$ be given as in Lemma 6.4. Let $\xi\in\mathcal{T}_{p}M$.
If at $p$, $\mathcal{P}_{\Lambda}(A)\leq\kappa$ , then we have

(6.32)
$\langle\mathring{\Lambda},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle\geq
m\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})\frac{|\langle\xi,b\rangle|^{2}}{\left(\det
A\right)\|b\|_{\omega}^{2}}.$

###### Proof.

If $\langle b,\xi\rangle=0$ then (6.32) holds trivially. Otherwise, we may
assume by rescaling that $\|b\|_{\rho}^{2}=2$ and $|\langle b,\xi\rangle|=1$.
We may choose a normal coordinate $\\{z^{i}\\}$ of $\rho$ at $p$ such that

  1. (1)

$dz^{1}|_{p}=b$;

  2. (2)

$\\{\sqrt{2}e_{j}^{i}\\}_{j=1}^{d_{i}}$ is an orthornormal frame of
$\mathcal{V}_{i}$ with respect to $\rho_{i}$ ;

  3. (3)

$\frac{\partial}{\partial z^{\sum_{l=1}^{i-1}(d_{l}-1)+n_{p}+j}}=e_{j+1}^{i}$
for $j=1,\cdots,d_{i}-1$;

  4. (4)

$e_{1}^{i}=\sum_{j=1}^{n_{p}}\alpha^{ij}\frac{\partial}{\partial z^{j}}$ and
$\alpha^{-1}=(\alpha^{ij})$ is a unitary matrix of dimension $n_{p}$.

The construction of the coordinate can be done as follows: First, we construct
the unitary frame. Take $dz^{1}|_{p}=b$. In each $\mathcal{V}_{i}$, let
$\tilde{e}_{1}^{i}=(\pi_{i})_{*}\frac{\partial}{\partial z^{1}}|_{p}$. If
$\tilde{e}_{1}^{i}=0$, we pick any norm $\sqrt{1/2}$ vector in
$\mathcal{V}_{i}$ to be $e_{1}^{i}$; if $\tilde{e}_{1}^{i}\not=0$, let
$e_{1}^{i}=\frac{\tilde{e}_{1}^{i}}{\sqrt{2}\|\tilde{e}_{1}^{i}\|_{\rho}}$. We
then pick the rest vectors so that $\\{\sqrt{2}e_{j}^{i}\\}_{j=1}^{d_{i}}$ is
a unitary frame of $\mathcal{V}_{i}$ with respect to $\rho_{i}$. At $p$, we
choose $\frac{\partial}{\partial z^{2}}|_{p},\cdots,\frac{\partial}{\partial
z^{n_{p}}}|_{p}$ together with $\frac{\partial}{\partial z^{1}}|_{p}$ to span
the space $\text{span}\\{e_{1}^{i}\\}_{i=1}^{n_{p}}$. Let
$\frac{\partial}{\partial
z^{\sum_{l=1}^{i-1}(d_{l}-1)+n_{p}+j}}|_{p}=e_{j+1}^{i}$ for
$j=1,\cdots,d_{i}-1$. Then we extend $\\{z^{i}\\}$ to be a normal coordinate.

By our choice of coordinate,
$\mathcal{B}=\text{span}\\{\frac{\partial}{\partial
z^{2}},\cdots,\frac{\partial}{\partial z^{n}}\\}$. Use $(A|1)$ to denote the
matrix obtained by deleting 1st row and 1st column of $A$. We may write

(6.33)
$\chi=2\sqrt{-1}\sum_{i,j}A^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}},\
\chi_{\mathcal{B}}=2\sqrt{-1}\sum_{i,j>1}(A|1)^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}},$

where $((A|1)^{\bar{j}i})$ is the inverse matrix of $(A|1)$. Let
$\\{\check{e}_{k}^{j}\\}\subset\bigwedge^{1,0}T_{p}^{*}M$ be the dual frame of
$\\{e_{j}^{k}\\}$, $k=1,2,\cdots,n_{p}$. Therefore, we have the following
decompositions:

$A=\left(\begin{array}[]{cc}a&q\\\
q^{\dagger}&(A|1)\end{array}\right),A^{-1}=\left(\begin{array}[]{cc}\hat{a}^{-1}&\hat{q}\\\
\hat{q}^{\dagger}&(A|1)^{-1}+\hat{q}^{\dagger}\hat{a}\hat{q}\end{array}\right),$

where $a=A_{1\bar{1}}$,

(6.34)
$\hat{a}=a-q(A|1)^{-1}q^{\dagger}=\frac{1}{A^{\bar{1}1}}=\frac{1}{\|b\|_{\omega}^{2}},$

and $\hat{q}=-\hat{a}^{-1}q(A|1)^{-1}$. We may write

(6.35)
$(A|1)^{-1}=\left(\begin{array}[]{ccccc}V&Q_{01}&Q_{02}&\ldots&Q_{1n_{p}}\\\
Q_{01}^{\dagger}&\hat{A}_{1}^{-1}&Q_{12}&\ldots&Q_{1n_{p}}\\\
Q_{02}^{\dagger}&Q_{12}^{\dagger}&\hat{A}_{2}^{-1}&\cdots&Q_{2n_{p}}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
Q_{0n_{p}}^{\dagger}&Q_{1n_{p}}^{\dagger}&Q_{2n_{p}}^{\dagger}&\ldots&\hat{A}_{n_{p}}^{-1}\end{array}\right),$

where $V$ is a matrix of dimension $n_{p}-1$. For $l=1,2,\cdots,d_{1}$, we
have

(6.36) $\displaystyle\frac{\rho_{1}^{l}}{l!}$
$\displaystyle=\sum_{|I|=l,I\subset\\{n_{p}+1,\cdots,n_{p}+d_{1}-1\\}}\frac{\sqrt{-1}^{l^{2}}}{2^{l}}dz^{I}\wedge
d\bar{z}^{I}$
$\displaystyle+\frac{\sqrt{-1}}{2}\check{e}_{1}^{1}\wedge\overline{\check{e}}_{1}^{1}\wedge\sum_{|I|=l-1,I\subset\\{n_{p}+1,\cdots,n_{p}+d_{1}-1\\}}\frac{\sqrt{-1}^{(l-1)^{2}}}{2^{l-1}}dz^{I}\wedge
d\bar{z}^{I}.$

Let $\alpha_{ij}=\overline{\alpha^{ji}}$. Define
$\alpha_{i}^{\prime}:=(\alpha_{2i},\cdots,\alpha_{n_{p}i})$, and

(6.37) $\tilde{A}_{i}^{-1}:=\left(\begin{array}[]{ccc}0&0&0\\\
0&\left(\alpha_{i}^{\prime}\right)^{\dagger}V\alpha_{i}^{\prime}&(\alpha_{i}^{\prime})^{\dagger}Q_{0i}\\\
0&Q_{0i}^{\dagger}\alpha_{i}^{\prime}&\hat{A}_{i}^{-1}\end{array}\right).$

Notice

(6.38)
$\check{e}_{i}^{1}\wedge\overline{\check{e}}_{i}^{1}=\sum_{l,j=2}^{n_{p}}\alpha_{li}\overline{\alpha_{ji}}dz^{l}\wedge
d\bar{z}^{j}+\text{terms with }dz^{1}\text{ or }d\bar{z}^{1}.$

Thus by (6.36), (6.38), (6.37), and (3.4),
$\langle\frac{\rho_{1}^{l}}{l!},\frac{\chi_{\mathcal{B}}^{l}}{l!}\rangle=\sigma_{l}(\tilde{A}_{1}^{-1})$
and similarly

(6.39)
$\langle\frac{\rho_{i}^{l}}{l!},\frac{\chi_{\mathcal{B}}^{l}}{l!}\rangle=\sigma_{l}(\tilde{A}_{i}^{-1}).$

Let

(6.40) $\tilde{\xi}=\langle\check{e}_{1}^{1},\xi\rangle
e_{1}^{1}+\sum_{i=n_{p}+1}^{n_{p}+d_{1}-1}\xi^{i}\frac{\partial}{\partial
z^{i}}.$

Then, by (6.5), we have

(6.41)
$\displaystyle\langle\frac{\rho_{1}^{k_{1}}}{k_{1}!},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle$
$\displaystyle=\langle
T_{k_{1}-1}(\tilde{A}_{1}^{-1}),\tilde{\xi}^{\dagger}\tilde{\xi}\rangle$
$\displaystyle\geq\frac{\sigma_{k_{1}-1}(\tilde{A}_{1}^{-1})}{\sigma_{d_{1}-1}(\tilde{A}_{1}^{-1})}\langle
T_{d_{1}-1}(\tilde{A}_{1}^{-1}),\tilde{\xi}^{\dagger}\tilde{\xi}\rangle.$

Notice that

(6.42)
$T_{d_{1}-1}(\tilde{A}_{1}^{-1})=\left(\begin{array}[]{cc}\sigma_{d_{1}-1}(\tilde{A}_{1}^{-1})&0\\\
0&T^{\prime}\end{array}\right),$

where $T^{\prime}\geq 0$ is a non-negative Hermitian matrix. Thus, from (6.41)
and (6.42), we have

(6.43)
$\displaystyle\langle\frac{\rho_{1}^{k_{1}}}{k_{1}!},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle$
$\displaystyle=\geq\frac{\sigma_{k_{1}-1}(\tilde{A}_{1}^{-1})}{\sigma_{d_{1}-1}(\tilde{A}_{1}^{-1})}\sigma_{d_{1}-1}(\tilde{A}_{1}^{-1})|\langle\check{e}_{1}^{1},\xi\rangle|^{2}$
$\displaystyle=\sigma_{k_{1}-1}(\tilde{A}_{1}^{-1})|\langle\check{e}_{1}^{1},\xi\rangle|^{2}.$

Apply the same argument on each $\mathcal{V}_{i}$ to obtain

(6.44)
$\langle\Lambda^{\prime},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle\geq\sum_{i=1}^{n_{p}}\sigma_{k_{i}-1}(\tilde{A}_{i}^{-1})|\langle\check{e}_{1}^{i},\xi\rangle|^{2},$

where $\Lambda^{\prime}=\sum_{i=1}^{n_{p}}(\exp\rho_{i})^{[k_{i}]}.$ On the
other hand, by applying Lemma 6.3 (2), we have

(6.45)
$\sigma_{n-1}\left((A|1)^{-1}\right)\leq\sum_{i=1}^{n_{p}}\sigma_{d_{i}-1}(\tilde{A}_{1}^{-1})\prod_{j\not=i}\sigma_{d_{j}}(\tilde{A}_{j}^{-1}).$

Let $s_{i}=\sigma_{k_{i}}(\tilde{A}_{i}^{-1})^{\frac{1}{k_{i}}}$. Then by
Newton-Maclaurin inequality,

(6.46)
$\sigma_{k_{i}-1}(\tilde{A}_{i}^{-1})\geq\binom{d_{i}}{k_{i}-1}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}-1}s_{i}^{k_{i}-1},$
(6.47)
$\sum_{i=1}^{n_{p}}\sigma_{d_{i}-1}(\tilde{A}_{i}^{-1})\prod_{j\not=i}\sigma_{d_{j}}(\tilde{A}_{j}^{-1})\leq\frac{\max_{i}\\{d_{i}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}}\\}}{\prod_{i=1}^{n_{p}}\binom{d_{i}}{k_{i}}^{\frac{d_{i}}{k_{i}}}}\prod_{i=1}^{n_{p}}s_{i}^{d_{i}}\sum_{j=1}^{n_{p}}s_{j}^{-1}.$

Thus, by (6.44), (6.45), (6.46), and (6.47), we get

(6.48)
$\displaystyle\langle\Lambda^{\prime},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle$
$\displaystyle\geq\sigma_{n-1}((A|1))^{-1}\cdot\frac{\sum_{i=1}^{n_{p}}\sigma_{k_{i}-1}(\tilde{A}_{i}^{-1})|\langle\check{e}_{1}^{i},\xi\rangle|^{2}}{\sum_{i=1}^{n_{p}}\sigma_{d_{i}-1}(\tilde{A}_{1}^{-1})\prod_{j\not=i}\sigma_{d_{j}}(\tilde{A}_{j}^{-1})}$
$\displaystyle=\det(A)^{-1}\hat{a}\cdot\frac{\sum_{i=1}^{n_{p}}\sigma_{k_{i}-1}(\tilde{A}_{i}^{-1})|\langle\check{e}_{1}^{i},\xi\rangle|^{2}}{\sum_{i=1}^{n_{p}}\sigma_{d_{i}-1}(\tilde{A}_{1}^{-1})\prod_{j\not=i}\sigma_{d_{j}}(\tilde{A}_{j}^{-1})}$
$\displaystyle\geq
c_{0}(\mathcal{O}_{p})\cdot\hat{a}\cdot\det(A)^{-1}\cdot\frac{\sum_{i=1}^{n_{p}}s_{i}^{k_{i}-1}|\langle\check{e}_{1}^{i},\xi\rangle|^{2}}{\prod_{i=1}^{n_{p}}s_{i}^{d_{i}}\sum_{i=1}^{n_{p}}s_{i}^{-1}},$

where

(6.49)
$c_{0}(\mathcal{O}_{p})=\frac{\min_{i}\left\\{\binom{d_{i}}{k_{i}-1}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}-1}\right\\}\prod_{i=1}^{n_{p}}\binom{d_{i}}{k_{i}}^{\frac{d_{i}}{k_{i}}}}{\max_{i}\\{d_{i}\binom{d_{i}}{k_{i}}^{\frac{1}{k_{i}}}\\}}.$

On the other hand, by the cone condition and Lemma 6.4, we have
$s_{i}\leq(\frac{\kappa}{m})^{\frac{1}{k_{i}}}$. Since $\langle\xi,b\rangle=1$
and $b\in\text{span}\\{\check{e}_{1}^{i}\\}_{i=1}^{n_{p}}$, we have
$\sum_{i=1}^{n_{p}}|\langle\check{e}_{1}^{i},\xi\rangle|^{2}\geq 1$. Define a
function

(6.50)
$\gamma(s_{1},\cdots,s_{n_{p}}):=c_{0}(\mathcal{O}_{p})\frac{\sum_{i=1}^{n_{p}}s_{i}^{k_{i}-1}|\langle\check{e}_{1}^{i},\xi\rangle|^{2}}{\prod_{i=1}^{n_{p}}s_{i}^{d_{i}}\sum_{i=1}^{n_{p}}s_{i}^{-1}}.$

Then $\gamma$ is decreasing in each $s_{i}$. Since
$s_{i}\leq(\frac{\kappa}{m})^{\frac{1}{k_{i}}}$, we have

(6.51) $\displaystyle\gamma$ $\displaystyle\geq
c_{0}\frac{\sum_{i=1}^{n_{p}}\left(\frac{\kappa}{m}\right)^{1-\frac{1}{k_{i}}}|\langle\check{e}_{1}^{i},\xi\rangle|^{2}}{(\frac{\kappa}{m})^{\sum_{i=1}^{n_{p}}\frac{d_{i}}{k_{i}}}\sum_{i=1}^{n_{p}}\left(\frac{\kappa}{m}\right)^{-\frac{1}{k_{i}}}}$
$\displaystyle\geq
c_{0}\frac{\min\\{\left(\frac{\kappa}{m}\right)^{1-\frac{1}{k_{i}}}\\}}{(\frac{\kappa}{m})^{\sum_{i=1}^{n_{p}}\frac{d_{i}}{k_{i}}}\sum_{i=1}^{n_{p}}\left(\frac{\kappa}{m}\right)^{-\frac{1}{k_{i}}}}$
$\displaystyle=\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}).$

Thus, by (6.48) and (6.51), we have

(6.52)
$\langle\Lambda^{\prime},2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\mathcal{B}}\rangle\geq\hat{a}\cdot\det
A^{-1}\cdot\gamma_{\min}.$

Notice $m\Lambda^{\prime}\leq\mathring{\Lambda}$ and
$\hat{a}=\frac{1}{\|b\|_{\omega}^{2}}.$ Hence, (6.32) follows from (6.52)
immediately. ∎

An immediate consequence of Lemma 6.5 is the monotonicity of $F(A)$.

######  Lemma 6.6.

Suppose $\mathring{\Lambda}$ satisfies $\mathcal{O}$-UP and for all $p\in M$,
$f(p)>-m\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})$.
If $\omega$ satisfies the cone condition (1.5) at a point $p$, then

(6.53)
$-F^{i\bar{j}}(A)b_{i}\bar{b_{j}}\geq-\frac{1}{2}\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}>0,$

for any non-zero covector $b=b_{i}dz^{i}$. As a result, $F(A)$ is strictly
decreasing in $\mathcal{C}_{\Lambda}^{\kappa}$.

###### Proof.

From (4.1), we have

(6.54)
$-F_{k}^{i\bar{j}}(A)b_{i}\bar{b}_{j}=\langle\Lambda^{[k]},2\sqrt{-1}b^{\sharp}\wedge\bar{b^{\sharp}}\wedge\exp\chi\rangle.$

Let $\mathcal{B}=\\{\xi\in\mathcal{T}_{p}M:b(\xi)=0\\}$. Since
$A^{-1}|_{\mathcal{B}}\geq\left(A|_{\mathcal{B}}\right)^{-1}$ by Lemmas 3.5
and 3.2, $\exp\chi\geq_{s}\exp\chi_{\mathcal{B}}$ . Thus, if we take
$\xi=b^{\sharp}$ in (6.32), then we have

(6.55)
$-\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}\geq\langle\Lambda^{[k]},2\sqrt{-1}b^{\sharp}\wedge\bar{b^{\sharp}}\wedge\exp\chi_{\mathcal{B}}\rangle\geq
m\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})\frac{\|b\|_{\omega}^{2}}{\det
A}.$

Direct computation shows that

$\displaystyle-F^{i\bar{j}}b_{i}\bar{b}_{j}$
$\displaystyle=-\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}+\frac{f(p)}{\det
A}A^{\bar{j}i}b_{i}\overline{b_{j}}$

Then, from Lemma 6.5, we have

(6.56) $\displaystyle-F^{i\bar{j}}b_{i}\bar{b}_{j}$
$\displaystyle\geq-\frac{1}{2}\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}+\left(\frac{m}{2}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})+f(p)\right)\frac{\|b\|_{\omega}^{2}}{\det
A}$
$\displaystyle\geq-\frac{1}{2}\sum_{k=1}^{n-1}F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}>0.$

∎

Now we prove Lemma 6.1.

###### Proof of Lemma 6.1.

We consider a generic point $p_{1}\in\mathcal{M}$. If $f(p_{1})\geq 0$, the
cone condition holds automatically. Thus, we may assume $f(p_{1})<0.$ Let
$\gamma(s),$$s\in[0,1]$ be a curve connecting $p_{0}$ and $p_{1}$. Let
$s_{0}=\min\\{s\in(0,1],\
\omega(\gamma(s))\notin\mathcal{C}_{\Lambda}^{\kappa}\\}$ and let
$p=\gamma(s_{0})$. By Lemma 5.8, the degeneracy of cone condition implies that
we can find a rank 1 Hermitian matrix $B=b^{\dagger}b$ such that

(6.57) $F_{\Lambda}(A:B)=\mathcal{P}_{\Lambda}(A)=\kappa.$

At $p$, we pick a normal coordinate of $\rho$ as in Lemma 6.5. Let

(6.58)
$\chi=2\sqrt{-1}\sum_{i,j}A^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}},\
\chi_{\infty}=2\sqrt{-1}\sum_{i,j>1}(A|1)^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}},$

where $((A|1)^{\bar{j}i})$ is the inverse matrix of $(A|1)$. From (6.57),
$\langle\Lambda,\exp(\chi_{\infty})\rangle=\kappa.$ From equation (1.2),
$\langle\Lambda,\exp(\chi)\rangle=\kappa.$ Thus, we have

$\sum_{k=1}^{n-1}\frac{\langle\Lambda^{[k]},\left(\chi^{k}-\chi_{\infty}^{k}\right)\rangle}{k!}+f(p)\langle\frac{\rho^{n}}{n!},\frac{\chi^{n}}{n!}\rangle=0.$

Hence

(6.59)
$\displaystyle-f(p)\langle\frac{\rho^{n}}{n!},\frac{\chi^{n}}{n!}\rangle$
$\displaystyle=\sum_{k=1}^{n-1}\frac{\langle\Lambda^{[k]},\left(\chi^{k}-\chi_{\infty}^{k}\right)\rangle}{k!}.$

We may write

(6.60) $A=\left(\begin{array}[]{cc}a&Q\\\
Q^{\dagger}&(A|1)\end{array}\right),A^{-1}=\left(\begin{array}[]{cc}\hat{a}^{-1}&\hat{Q}\\\
\hat{Q}^{\dagger}&(A|1)^{-1}+\hat{Q}^{\dagger}\hat{a}\hat{Q}\end{array}\right),$

where $\hat{a}=a-Q(A|1)^{-1}Q^{\dagger}=1/\|dz^{1}\|_{\omega}^{2}$ and
$\hat{Q}=-\hat{a}^{-1}Q(A|1)^{-1}$. Let

(6.61) $\xi=\frac{1}{\sqrt{\hat{a}}}\frac{\partial}{\partial
z^{1}}+\sqrt{\hat{a}}\sum_{i=2}^{n}\hat{Q}_{i}\frac{\partial}{\partial
z^{i}}.$

Then

(6.62) $\displaystyle\chi$
$\displaystyle=2\sqrt{-1}\left(\xi\wedge\bar{\xi}+(A|1)^{\bar{j}i}\frac{\partial}{\partial\bar{z}^{j}}\wedge\frac{\partial}{\partial
z^{i}}\right),$

Thus, we have

(6.63)
$\exp\chi-\exp\chi_{\infty}=2\sqrt{-1}\xi\wedge\bar{\xi}\wedge\exp\chi_{\infty}.$

By (6.59), (6.63), and Lemma 6.5, we have

(6.64) $\displaystyle-\frac{f(p)}{\det A}$ $\displaystyle\geq\frac{1}{\det
A}\frac{|\langle\xi,dz^{1}\rangle|^{2}}{\|dz^{1}\|_{\omega}^{2}}m\gamma_{\min}$
$\displaystyle=\frac{1}{\det A}\cdot m\gamma_{\min}.$

However, it is impossible as $|f(p)|<m\gamma_{\min}$. We have finished the
proof.

Finally, we show that $F$ is strict convex in
$\mathcal{C}_{\Lambda}^{\kappa}(p)$. ∎

######  Lemma 6.7.

Notations as above. If $\mathring{\Lambda}$ satisfies $\mathcal{O}$-UP then

(6.65)
$\sum_{r,s,i,j}\left(F^{i\bar{j},r\bar{s}}(A)+F^{i\bar{s}}(A)A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}\geq\frac{f(p)}{\det
A}A^{\bar{s}r}A^{\bar{j}i}B_{i\bar{j}}\overline{B_{s\bar{r}}}.$

Suppose further that for all $p\in
M$,$f(p)>-\frac{m}{2n+1}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}).$
Then, $F$ is a strictly convex function in $\mathcal{C}_{\Lambda}^{\kappa}$.

###### Proof.

In a local normal coordinate of $\rho$ at $p$, we may assume that
$A=\text{diag}\\{\lambda_{1},\cdots,\lambda_{n}\\}$. By Proposition 4.3,

(6.66)
$\displaystyle\sum_{i,j,r,s}\left(F^{i\bar{j},r\bar{s}}+F^{i\bar{s}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle\geq\frac{f(p)}{\det
A}\left(A^{\bar{s}r}A^{\bar{j}i}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle=\frac{f(p)}{\det
A}\left|\sum_{j}\frac{B_{j\bar{j}}}{\lambda_{j}}\right|^{2}.$

By (6.56), we have

(6.67)
$\displaystyle-\sum_{i,j,r,s}F^{i\bar{s}}(A)A^{\bar{j}r}B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle\geq\frac{1}{\det
A}\left(\frac{m}{2}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})+f(p)\right)\sum_{i,j}\frac{|B_{i\bar{j}}|^{2}}{\lambda_{i}\lambda_{j}}.$

Then by (6.67), Cauchy inequality, and the assumption on $f$, we have

(6.68)
$\displaystyle\sum_{i,j,r,s}\left(F^{i\bar{j},r\bar{s}}+\frac{1}{2}F^{i\bar{s}}A^{\bar{j}r}\right)B_{i\bar{j}}\overline{B_{s\bar{r}}}$
$\displaystyle\geq\frac{1}{\det
A}\sum_{j}\frac{|B_{j\bar{j}}|^{2}}{\lambda_{j}^{2}}\left(\frac{m}{2}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})-(n+\frac{1}{2})|f(p)|\right)$
$\displaystyle\geq 0.$

Since $F^{i\bar{j}}$ is strictly monotone in $\mathcal{C}_{\Lambda}^{\kappa}$
by Lemma 6.6, by (6.68) $F^{i\bar{j},r\bar{s}}$ is positive definite in
$\mathcal{C}_{\Lambda}^{\kappa}$. Thus, $F$ is strictly convex in
$\mathcal{C}_{\Lambda}^{\kappa}$. ∎

###  7\. Continuity method

In this section, we prove the existence of a solution to (1.2) assuming the
existence of a subsolution $\omega_{\text{sub}}\in[\omega_{0}].$ The main
result of this section is the following

######  Theorem 7.1.

Let $M$ be a connected compact Kähler manifold. Suppose $\Lambda$ satisfies
H2. If there is a Kähler metric $\omega_{\text{sub}}\in[\omega_{0}]$ satisfies
the cone condition (1.5), then there exists a unique smooth solution to
equation (1.2).

######  Remark 7.2.

Since a solution itself is a subsolution by Lemma 6.1, Theorem 1.7 and Theorem
2.9 are immediate corollaries of Theorem 7.1. The uniqueness will be addressed
in Appendix A.

To prove Theorem 7.1, we use the continuity method. Consider the following
continuity path depending on the parameter $t\in[0,1]$. Consider
$\Omega_{t}=\exp\omega_{t}$ which solves the PDE

(7.1)
$\kappa(\Omega_{t})^{[n]}=\left(t\mathring{\Lambda}\wedge\Omega_{t}\right)^{[n]}+(tf+\left(1-t\right)\kappa_{0})P^{[n]},$

where the constant $\kappa_{0}$ is chosen such that

$\kappa[\omega_{0}]^{n}=\kappa_{0}[\rho]^{n}.$

Denote $f_{t}(p):=tf(p)+(1-t)\kappa_{0}$, and
$\Lambda_{t}=t\mathring{\Lambda}+f_{t}P^{[n]}$, then we may rewrite (7.1) as

(7.2)
$\kappa(\Omega_{t})^{[n]}=\left(\Lambda_{t}\wedge\Omega_{t}\right)^{[n]}.$

Let

$\mathbf{I}=\\{t\in[0,1]:(\ref{eq:contin path})\ \text{has a smooth
solution}\\}.$

Then, from Yau’s theorem [38], there is a smooth $\rm{K\ddot{a}hler}$ metric
$\hat{\omega}_{0}$ which solves (7.2) at $t=0$. Without confusion, we replace
$\omega_{0}$ by $\hat{\omega}_{0}$ and consider
$\omega_{t}=\omega_{0}+i\partial\bar{\partial}\varphi_{t}$ which solves (7.1)
for $t\in\mathbf{I}\subset[0,1]$. Notice the linearization of (7.1) is

(7.3) $\left(\kappa\Omega_{t}-t\Lambda\wedge\Omega_{t}\right)^{[n-1]}\wedge
i\partial\bar{\partial}u=(t\mathring{\Lambda}\wedge\Omega_{t})^{[n]}+\left(f(p)-\kappa_{0}\right)P^{[n]}.$

By Lemma 6.1,
$\left(\kappa\Omega_{t}-t\Lambda\wedge\Omega_{t}\right)^{[n-1]}>0$; Therefore,
(7.3) is strictly elliptic, which implies the openness of $\mathbf{I}$. If
$\Lambda$ satisfies H2, then for $t>0$, $\Lambda_{t}$ satisfies H2 with
respect to proper positive constants. In fact, $f_{t}(p)\geq 0$ if
$t\leq\frac{1}{2}$, and since $m_{t}\geq\frac{m}{2}$ for $t\geq\frac{1}{2}$,
we have

$f_{t}(p)\geq-\min\left\\{\frac{m_{t}}{2n+1}\gamma_{\min}(\frac{\kappa}{m_{t}},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}),\frac{\kappa_{0}}{2}\right\\}.$

Thus, if $\Lambda$ satisfies H2, $\Lambda_{t}$ satisfies the following H2’
condition.

######  Definition 7.3.

We say $\Lambda$ satisfies H2’ if $\mathring{\Lambda}$ satisfies H2 with some
uniform constant $m>0$, and $\Lambda^{[n]}$ is almost positive with respect to
$(\mathcal{O},2m,\rho)$, i.e. for any $p\in M$,

$\frac{\Lambda^{[n]}}{P^{[n]}}(p)\geq-\min\left\\{\frac{m}{2n+1}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p}),\frac{\kappa_{0}}{2}\right\\}.$

######  Remark 7.4.

Readers may check that all arguments in section 6 are valid if condition H2’
is assumed.

In the following, by replacing $\Lambda$ with $\Lambda_{t}$, we will suppress
the subscript $t$ and derive a priori estimates for equation (1.2) assuming
H2’.

We proceed to prove a priori $C^{0}$-estimate of equation (1.2). The idea of
using Alexandroff-Bakelman-Pucci type estimates based on [2] follows [35]. Let

(7.4) $\omega=\omega_{0}+i\partial\bar{\partial}\varphi$

be a solution to (1.2). Suppose that in the $\rm{K\ddot{a}hler}$ class
$[\omega_{0}]$ there is a
$\omega_{\text{sub}}\in\mathcal{C}_{\Lambda}^{\kappa}$. We denote

(7.5)
$\omega_{\text{sub}}=\omega_{0}+i\partial\bar{\partial}\varphi_{\text{sub}}$

Let $u=\varphi-\varphi_{\text{sub}}$.

######  Proposition 7.5.

Suppose that $\Lambda$ satisfies H2’. If
$\omega=\omega_{\text{sub}}+i\partial\bar{\partial}u$ is a solution to (1.2)
and $\sup u=0$, then there is a constant $C$ depends on $n$, $M$, $\Lambda$,
$\omega_{\text{sub}}$, $\rho$ s.t.

$\sup_{M}|u|<C.$

Suppose that $\underline{A}\in\mathcal{C}_{\Lambda}^{\kappa}(p)$. Since the
$\mathcal{C}_{\Lambda}^{\kappa}(p)$ is open,
$\text{dist}(\underline{A},\partial\mathcal{C}_{\Lambda}^{\kappa}(p))>0$. Thus
there is a $0<r<\text{dist}(\underline{A},\mathcal{C}_{\Lambda}^{\kappa}(p))$
s.t. the radius $r$ ball $\mathcal{B}_{r}(\underline{A})$ in Hermitian matrix
space is contained in $\mathcal{C}_{\Lambda}^{\kappa}(p)$ . As a result, we
have

######  Lemma 7.6.

If $\underline{A}\in\mathcal{C}_{\Lambda}^{\kappa}(p)$, then there is a
constant $r=r(\underline{A},\Lambda)$ s.t.
$\underline{A}-r\text{Id}\in\mathcal{C}_{\Lambda}^{\kappa}(p)$.

###### Proof of Proposition 7.5.

Since $\omega$ is positive, we have

(7.6)
$\Delta_{\rho}u>-\text{tr}_{\rho}\omega_{\text{sub}}>-C(\omega_{\text{sub}},\rho).$

Thus, we can use Green function representation to obtain that
$\|u\|_{L^{1}(\rho^{n})}<C(\omega_{\text{sub}},\rho)$.

Let $L=-\inf_{M}u$ and assume that $L$ is achieved at $x_{0}$. Pick a normal
coordinate of $\rho$ at $x_{0}$. After a proper rescaling of $\rho,$ we may
assume that the chosen coordinate exists in the unit ball
$B_{1}(0)\subset\mathbb{C}^{n}$. For $x\in B_{1}(0)$, we pick a uniform
$r=r(\omega_{\text{sub}},\Lambda)$ s.t.
$\omega_{\text{sub}}-r\sqrt{-1}\partial\bar{\partial}|x|^{2}$ belongs to
$\mathcal{C}_{\Lambda}^{\kappa}(x)$ for all $x\in B_{1}(0)$. Let $a>0$ and
$a<r/2$. Let $w=u+a|x|^{2}$. Note $w>-L+a$ on $\partial B_{1}(0)$. Define the
following set:

$W=\\{x\in B_{1}(0):|Dw(x)|<a,\ w(y)\geq w(x)+Dw(x)\cdot(y-x)\\}.$

We use Alexandroff-Bakelman-Pucci maximum principle (Gilbarg-Trudinger Lemma
9.2) to claim that

$B_{a}(0)\subset Dw(W).$

In $W$, we have $D^{2}w\geq 0$ and hence

(7.7) $c(n)a^{2n}\leq\int_{W}\det(D^{2}w)\leq 2^{2n}\int_{W}\left(\det
w_{i\bar{j}}\right)^{2}.$

As $D^{2}w\geq 0$ in $W$, $D^{2}u\geq-2a\text{Id}_{2n}$ which implies that
$u_{i\bar{j}}+a\delta_{i\bar{j}}\geq 0$ as a Hermitian matrix. Since $a<r/2$,
$\omega_{\text{sub}}-a\sqrt{-1}\partial\bar{\partial}|x|^{2}\in\mathcal{C}_{\Lambda}^{\kappa}(x)$,
we apply Proposition 5.2 to
$\omega_{\text{sub}}-a\sqrt{-1}\partial\bar{\partial}|x|^{2}$ to conclude that
$|w_{i\bar{j}}|<R(\Lambda,\omega_{\text{sub}})$. By (7.7), we have

(7.8) $c(n)a^{2n}\leq C(\Lambda,\omega_{\text{sub}},n)\text{vol}(W).$

On the other hand, it is obvious that

(7.9)
$\text{vol}(W)\leq\frac{\|w\|_{L^{1}}}{|L-a|}\leq\frac{C(\omega_{\text{sub}},\rho)}{|L-a|}.$

Thus by (7.8) and (7.9),

$L<C(\Lambda,\omega_{\text{sub}},\rho,n)\left(\frac{1}{a^{2n}}+1\right)<C(\Lambda,\omega_{\text{sub}},\rho,n).$

∎

Next, we state a $C^{2}$-estimate for solutions to (7.1).

######  Proposition 7.7.

Let $\omega_{\text{sub}}\in\mathcal{C}_{\Lambda}^{\kappa}$. Let
$u=\varphi-\varphi_{\text{sub}}$ where $\varphi,\varphi_{\text{sub}}$ are
given in (7.4),(7.5) and
$\omega=\omega_{\text{sub}}+\sqrt{-1}\partial\bar{\partial}u$ solves (1.2). If
$\Lambda$ satisfies H2’, and $F^{i\bar{j},r\bar{s}},F^{i\bar{j}}$ satisfies
inequality (6.65) then it holds that

$|\partial\bar{\partial}u|_{\rho}<C.$

where $C$ depends on
$\|u\|_{C^{0}},M,\Lambda,\omega_{\text{sub}}$,$n,m,\kappa,C_{H2}$, $\rho$.

We first prove some technical lemmas.

######  Lemma 7.8.

Notations as above. We have

$-F^{i\bar{j}}A_{i\bar{j}}\leq nF(A).$

###### Proof.

Since each $F_{k}$ is homogeneous of degree $-k$, we have

$\displaystyle-\frac{\partial F(A)}{\partial A_{i\bar{j}}}A_{i\bar{j}}$
$\displaystyle=\sum_{k=1}^{n-1}kF_{k}(A)+\frac{nf}{\det A}$
$\displaystyle\leq\sum_{k=1}^{n-1}nF_{k}(A)+\frac{nf}{\det A}$
$\displaystyle=nF(A).$

∎

Let $\\{z^{i}\\}$ be a normal coordinate at $p$. Pick the direction
$\frac{\partial}{\partial z^{1}}$ and denote
$\Lambda_{,1},\Lambda_{,1\bar{1}}$ to be the corresponding covariant
derivatives of $\Lambda$ with respect to the Chern connection of $\rho$. We
denote

(7.10) $\displaystyle F_{,1\bar{1}}$ $\displaystyle=F(A,\Lambda_{,1\bar{1}}),\
F_{,\bar{1}}^{i\bar{j}}=\frac{\partial
F\left(A,\Lambda_{,\bar{1}}\right)}{\partial A_{i\bar{j}}},$ (7.11)
$\displaystyle F_{k,1\bar{1}}$ $\displaystyle=F_{k}(A,\Lambda_{,1\bar{1}}),\
F_{k,\bar{1}}^{i\bar{j}}=\frac{\partial
F_{k}\left(A,\Lambda_{,\bar{1}}\right)}{\partial A_{i\bar{j}}}.$

######  Lemma 7.9.

Notations as above. If $\Lambda$ satisfies H2’, and $F(A)\geq\kappa$, we have

(7.12) $\left|F_{,1\bar{1}}\right|<C_{\ref{lem:tech2}}F(A),$ (7.13)
$\left|2\text{Re}\left(F_{k,\bar{1}}^{i\bar{j}}B_{i\bar{j}}\right)\right|\leq\frac{C_{\ref{lem:tech2}}}{\epsilon}F(A)+C_{\ref{lem:tech2}}\epsilon
F^{i\bar{s}}A^{r\bar{j}}B_{i\bar{j}}\overline{B_{s\bar{r}}},$

for some $C_{\ref{lem:tech2}}=C_{\ref{lem:tech2}}(\kappa,n,m,C_{H2})$.

###### Proof.

From the cone condition, Lemma 6.4, and Newton-Maclaurin inequality, we have

(7.14)
$\langle\frac{\rho_{i}^{l_{i}}}{l_{i}!},\exp\chi\rangle\leq\binom{d_{i}}{l_{i}}\binom{d_{i}}{k_{i}}^{-\frac{l_{i}}{k_{i}}}(\frac{n\kappa}{m})^{\frac{l_{i}}{k_{i}}}.$

By H2’, there is a constant $C_{H2}$ s.t. for $k=1,\cdots,n-1$, it holds

$-C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right)\leq\left(\Lambda^{[k]}\right)_{,1\bar{1}}\leq
C_{H2}\left(\Lambda^{[k]}+\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}}\right),$

where $\boldsymbol{l}_{k}=\\{(l_{1},\cdots,l_{n_{p}}):\sum_{i}l_{i}=k,\
l_{i}=0\ \text{or}\ l_{i}\geq k_{i}\\}$. Hence

(7.15) $\displaystyle|F_{k,1\bar{1}}|$ $\displaystyle\leq
C_{H2}\left(F_{k}+\sum_{l\in\boldsymbol{l}_{k}}\prod_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{l_{i}}}{l_{i}!},\exp\chi\rangle\right)$
$\displaystyle=C_{H2}\left(F_{k}+\sum_{l\in\boldsymbol{l}_{k}}C_{1}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})\right)$
$\displaystyle\leq
C_{H2}\left(F_{k}+C_{2}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})\right)$

By the cone condition and Lemma 6.4,

(7.16) $\det A^{-1}\leq C_{3}(m,\kappa,n_{p},\mathbf{d}_{p},\mathbf{k}_{p}).$

By (7.15) and (7.16), we have

(7.17) $\displaystyle\left|F_{,1\bar{1}}\right|$
$\displaystyle<\sum_{k}|F_{k,1\bar{1}}|+|f_{,1\bar{1}}|\frac{1}{\det A}$
$\displaystyle<C_{4}F(A),$

for some $C_{4}=C_{4}(\kappa,m,n_{p},\mathbf{d}_{p},\mathbf{k}_{p},C_{H2})$.

We use H2’ to deduce that

(7.18)
$\left|\text{Re}\left(F_{k,\bar{1}}^{i\bar{j}}b_{i}\bar{b}_{j}\right)\right|\leq
C_{H2}\left(-F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}+\langle\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}},\exp\chi\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge b^{\sharp}\rangle\right).$

From (6.16) in Lemma 6.3, we have

(7.19)
$\langle\sum_{l\in\boldsymbol{l}_{k}}\rho_{1}^{l_{1}}\cdots\rho_{n_{p}}^{l_{n_{p}}},\exp\chi\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge
b^{\sharp}\rangle=\sum_{i=1}^{n_{p}}\sum_{l\in\boldsymbol{l}_{k,i}}\left(\prod_{j\not=i}\langle\frac{\rho_{j}^{l_{j}}}{l_{j}!},e^{\chi}\rangle\right)\langle\frac{\rho_{i}^{l_{i}}}{l_{i}!},e^{\chi}\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge b^{\sharp}\rangle,$

where
$\boldsymbol{l}_{k,i}=\\{(l_{1},\cdots,l_{n_{p}})\in\boldsymbol{l}_{k}:l_{i}\geq
1\\}$. By (6.5) in Lemma 6.2 and Newton-Maclaurin inequality, for $l\geq
k_{i}$ and any positive $d_{i}\times d_{i}$ Hermitian matrix $D$, we have

(7.20) $\displaystyle T_{l-1}(D)$
$\displaystyle\leq\frac{\sigma_{l-1}(D)}{\sigma_{k_{i}-1}(D)}T_{k_{i}-1}(D)$
$\displaystyle\leq
C(d_{i},l,k_{i})\frac{\sigma_{k_{i}-1}(D)(\sigma_{k_{i}}(D))^{\frac{l-k_{i}}{k_{i}}}}{\sigma_{k_{i}-1}(D)}T_{k_{i}-1}(D)$
$\displaystyle=C(d_{i},l,k_{i})(\sigma_{k_{i}}(D))^{\frac{l-k_{i}}{k_{i}}}T_{k_{i}-1}(D).$

We apply (7.20) to $A^{-1}|_{\mathcal{V}_{i}}$ and use (7.14) to obtain

(7.21) $\langle\frac{\rho_{i}^{l-1}}{(l-1)!},\exp\chi\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge b^{\sharp}\rangle\leq
C_{5}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p},l)\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge b^{\sharp}\rangle.$

Thus, by (7.18), (7.19), and (7.21), we have

(7.22)
$\displaystyle\left|\text{Re}\left(F_{k,\bar{1}}^{i\bar{j}}b_{i}\bar{b}_{j}\right)\right|$
$\displaystyle\leq
C_{H2}\left(-F_{k}^{i\bar{j}}b_{i}\bar{b}_{j}+C_{6}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})\sum_{i=1}^{n_{p}}\langle\frac{\rho_{i}^{k_{i}}}{k_{i}!},\exp\chi\wedge
2\sqrt{-1}\overline{b^{\sharp}}\wedge b^{\sharp}\rangle\right)$
$\displaystyle\leq-C_{7}F^{i\bar{j}}b_{i}\bar{b}_{j},$

where $C_{7}=C_{7}(\kappa,m,n_{p},\mathbf{d}_{p},\mathbf{k}_{p},C_{H2})$.

Now for matrix $B$, by (7.22) and mean value inequality,

(7.23)
$\displaystyle\left|\text{Re}\left(F_{,1}^{i\bar{s}}A^{r\bar{j}}(A_{i\bar{j}}+\epsilon
B_{i\bar{j}})\left(\overline{A_{s\bar{r}}+\epsilon
B_{s\bar{r}}}\right)\right)\right|$ $\displaystyle\leq-
C_{9}F^{i\bar{s}}A^{r\bar{j}}(A_{i\bar{j}}+\epsilon
B_{i\bar{j}})\left(\overline{A_{s\bar{r}}+\epsilon B_{s\bar{r}}}\right)$
$\displaystyle\leq
2C_{9}\left(F(A)-\epsilon^{2}F^{i\bar{s}}A^{r\bar{j}}B_{i\bar{j}}\overline{B_{s\bar{r}}}\right),$

where $C_{9}=C_{9}(n,\kappa,m,n_{p},\mathbf{d}_{p},\mathbf{k}_{p},C_{H2})$. As
a result of (7.23), we have

$\left|2\text{Re}\left(F_{,\bar{1}}^{i\bar{j}}B_{i\bar{j}}\right)\right|\leq\frac{C_{10}}{\epsilon}F(A)-C_{10}\epsilon
F^{i\bar{s}}A^{r\bar{j}}B_{i\bar{j}}\overline{B_{s\bar{r}}},$

where $C_{10}=C_{10}(n,\kappa,m,n_{p},\mathbf{d}_{p},\mathbf{k}_{p},C_{H2})$.

Finally, we choose

$C_{\ref{lem:tech2}}=\max\\{C_{2},C_{4},C_{10}:p\in M\\}.$

Notice that for a fixed labeled __ orthogonal splitting $\mathcal{O}$, the set
$\\{(n_{p},\mathbf{d}_{p},\mathbf{k}_{p}):p\in M\\}$ is finite. Thus
$C_{\ref{lem:tech2}}$ has a uniform upper bound which only depends on
$\kappa,m,n$ and $C_{H2}$. ∎

######  Lemma 7.10.

Notations as above. If $\Lambda$ satisfies H2’, and $F(A)\geq\kappa$, there is
a constant $C_{\ref{lem:ddbar F}}$ depends on $C_{\ref{lem:tech2}}$, and the
bisectional curvature $\text{Rm}_{\rho}$ of $\rho$ s.t. for any $\epsilon>0$

$\displaystyle\partial_{1\bar{1}}F(A)$ $\displaystyle\geq
A_{1\bar{1}}F^{i\bar{j}}\left(\log
A_{1\bar{1}}\right)_{,i\bar{j}}+A^{\bar{1}1}F^{i\bar{j}}A_{i\bar{1},1}\overline{A_{j\bar{1},1}}+F^{i\bar{j},r\bar{s}}A_{i\bar{j},1}A_{r\bar{s},\bar{1}}$
$\displaystyle+\epsilon
F^{i\bar{s}}A^{\bar{j}r}A_{i\bar{j},1}\overline{A_{s\bar{r},1}}-\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)+C_{\ref{lem:ddbar F}}A_{1\bar{1}}\sum_{i}F^{i\bar{i}}.$

###### Proof.

We have

(7.24)
$\partial_{1\bar{1}}F(A)=F^{i\bar{j}}A_{i\bar{j},1\bar{1}}+F^{i\bar{j},r\bar{s}}A_{i\bar{j},1}A_{r\bar{s},\bar{1}}+2\text{Re}\left(F_{,\bar{1}}^{i\bar{j}}A_{i\bar{j},1}\right)+F_{1\bar{1}}.$

Note

(7.25)
$F^{i\bar{j}}A_{i\bar{j},1\bar{1}}=F^{i\bar{j}}A_{1\bar{1},i\bar{j}}+F^{i\bar{j}}(\rho^{a\bar{b}}A_{a\bar{j}}R_{1\bar{1},i\bar{b}}-\rho^{a\bar{b}}A_{a\bar{1}}R_{i\bar{j},1\bar{b}}),$

where $R_{i\bar{j},k\bar{l}}$ is the bisectional curvature of $\rho$. By
(7.24) and (7.25),

(7.26) $\displaystyle\partial_{1\bar{1}}F(A)$
$\displaystyle=F^{i\bar{j}}A_{1\bar{1},i\bar{j}}+F^{i\bar{j}}\left(\rho^{a\bar{b}}A_{a\bar{j}}R_{1\bar{1},i\bar{b}}-\rho^{a\bar{b}}A_{a\bar{1}}R_{i\bar{j},1\bar{b}}\right)$
$\displaystyle+F^{i\bar{j},r\bar{s}}A_{i\bar{j},1}A_{r\bar{s},\bar{1}}+2\text{Re}\left(F_{,\bar{1}}^{i\bar{j}}A_{i\bar{j},1}\right)+F_{1\bar{1}}.$

The first term in (7.26) is

(7.27) $F^{i\bar{j}}A_{1\bar{1},i\bar{j}}=A_{1\bar{1}}F^{i\bar{j}}\left(\log
A_{1\bar{1}}\right)_{,i\bar{j}}+A^{\bar{1}1}F^{i\bar{j}}A_{i\bar{1},1}\overline{A_{j\bar{1},1}}.$

The second term in (7.26) is controlled by

(7.28) $\displaystyle
F^{i\bar{j}}\left(\rho^{a\bar{b}}A_{a\bar{j}}R_{1\bar{1},i\bar{b}}-\rho^{a\bar{b}}A_{a\bar{1}}R_{i\bar{j},1\bar{b}}\right)$
$\displaystyle\geq
C_{1}F^{i\bar{j}}A_{i\bar{j}}+C_{1}A_{1\bar{1}}\sum_{i}F^{i\bar{i}}$
$\displaystyle\geq-C_{2}F(A)+C_{2}A_{1\bar{1}}\sum_{i}F^{i\bar{i}},$

where the constants $C_{1},C_{2}$ depend a bound of the bisectional curvature
$|\text{Rm}_{\rho}|$. Apply Lemma 7.9 to (7.26) to obtain

$\displaystyle\partial_{1\bar{1}}F(A)$ $\displaystyle\geq
A_{1\bar{1}}F^{i\bar{j}}\left(\log
A_{1\bar{1}}\right)_{,i\bar{j}}+A^{\bar{1}1}F^{i\bar{j}}A_{i\bar{1},1}\overline{A_{j\bar{1},1}}+F^{i\bar{j},r\bar{s}}A_{i\bar{j},1}A_{r\bar{s},\bar{1}}$
$\displaystyle+\epsilon
F^{i\bar{s}}A^{\bar{j}r}A_{i\bar{j},1}\overline{A_{s\bar{r},1}}-C_{3}\left(1+\frac{1}{\epsilon}\right)F(A)+C_{3}A_{1\bar{1}}\sum_{i}F^{i\bar{i}}.$

We have proved the claim. ∎

The following proposition is a key ingredient in the proof of $C^{2}$
estimate. It has been proved in several context. See Song-Weinkove [33], Fang-
Lai-Ma [19], Guan [17], Guan-Sun [18], Collins-Szèkelyhidi [10], Szèkelyhidi
[35], Datar-Pingali [12]. The current form of Proposition 7.11 is adapted from
Fang-Lai-Ma [19] and Datar-Pingali [12]. We will put the proof in the
appendix.

######  Proposition 7.11.

Suppose that $\Lambda$ satisfies H2’. Let
$\omega_{\text{sub}}\in\mathcal{C}_{\Lambda}^{\kappa}$ and
$\omega=\omega_{\text{sub}}+i\partial\bar{\partial}u$ be a solution to (1.1).
Then there is a $N=N(\omega_{\text{sub}},\Lambda,M)$ and
$\mu=\mu(\omega_{\text{sub}},\Lambda,M)>0$ s.t. if
$|\partial\bar{\partial}u|_{\rho}>N$,

$F^{i\bar{j}}(A)\left(u_{i\bar{j}}\right)\geq\mu\left(1-\sum_{i}F^{i\bar{i}}(A)\right).$

Now with all the preparations, we prove the $C^{2}$-estimate.

###### Proof of Proposition 7.7.

We use maximum principle to deduce Proposition 7.7. Let
$u=\varphi-\varphi_{\text{sub}}$ and $g$ be the metric tensor of $\omega$. We
may assume that $\inf_{M}u=0$, $|\partial\bar{\partial}u|>N$ so that we can
apply Proposition 7.11. Pick the test function

(7.29) $G(x,\xi)=\log(g_{i\bar{j}}\xi^{i}\xi^{\bar{j}})-\phi(u),$

where $\xi\in T_{x}^{1,0}M,\rho_{i\bar{j}}\xi^{i}\xi^{\bar{j}}=1$.
$\phi:\mathbb{R}_{\geq 0}\to\mathbb{R}$ is a smooth function:

(7.30) $\phi(x)=2Lx-\frac{L\tau}{2}x^{2}.$

The choice of $\tau$ relies on $\sup u$ so that

(7.31) $L\leq\phi^{\prime}\leq 2L,\ \phi^{\prime\prime}=-L\tau.$

For instance, we may start by assuming that $\tau=\frac{1}{\sup u+1}$. Suppose
that $G(x,\xi)$ achieves maximum at $(p,\xi_{0})$. We choose a normal
coordinate $\\{z^{i}\\}$ of $\rho$ at $p$ so that $\xi_{0}$ is along the
direction of $\frac{\partial}{\partial z^{1}}$ and $\omega$ is diagonal at
$p$. Locally

(7.32) $H=\log g_{1\bar{1}}-\phi(u)$

also achieves maximum at $p$. At $p$, we have

(7.33) $0=H_{,i}=g_{1\bar{1}}^{-1}g_{1\bar{1},i}-\phi^{\prime}u_{,i},$ (7.34)
$0\leq F^{i\bar{j}}H_{,i\bar{j}}.$

At $p$, we have

(7.35) $0=H_{,i}=g_{1\bar{1}}^{-1}g_{1\bar{1},i}-\phi^{\prime}u_{,i}.$

We have $g_{1\bar{1},i}=g_{i\bar{1},1}$ and (7.35) implies that

(7.36) $u_{,i}=\frac{g^{\bar{1}1}}{\phi^{\prime}}g_{i\bar{1},1}.$

Now

(7.37) $H_{,i\bar{j}}=\left(\log
g_{1\bar{1}}\right)_{,i\bar{j}}-\phi^{\prime\prime}u_{,i}u_{,\bar{j}}-\phi^{\prime}u_{,i\bar{j}}.$

We have $g_{1\bar{1},i}=g_{i\bar{1},1}$. Thus, by Lemma 7.10,

(7.38) $\displaystyle 0$ $\displaystyle\leq
F^{i\bar{j}}H_{,i\bar{j}}=F^{i\bar{j}}\left(\left(\log
g_{1\bar{1}}\right)_{,i\bar{j}}-\phi^{\prime\prime}u_{,i}u_{,\bar{j}}-\phi^{\prime}u_{,i\bar{j}}\right)$
$\displaystyle\leq-g^{\bar{1}1}\left(F^{i\bar{j},r\bar{s}}g_{i\bar{j},1}g_{r\bar{s},\bar{1}}+F^{i\bar{j}}g^{\bar{1}1}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}+\epsilon
F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}\right)-\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)$ $\displaystyle\ -C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-F^{i\bar{j}}\left(\phi^{\prime\prime}u_{,i}u_{,\bar{j}}+\phi^{\prime}u_{,i\bar{j}}\right).$

We apply (6.65) to (7.38) to get

(7.39) $\displaystyle 0$ $\displaystyle\leq
g^{\bar{1}1}\left((1-\epsilon)F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}-\frac{f(p)}{\det
g}\left|\sum_{j}g^{\bar{j}j}g_{j\bar{j},1}\right|^{2}-F^{i\bar{j}}g^{\bar{1}1}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}\right)$
$\displaystyle+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-F^{i\bar{j}}\left(\phi^{\prime\prime}u_{,i}u_{,\bar{j}}+\phi^{\prime}u_{,i\bar{j}}\right).$

Substitute $u_{,i}=\frac{g^{\bar{1}1}}{\phi^{\prime}}g_{i\bar{1},1}$ into
(7.39) to get

(7.40) $\displaystyle 0$
$\displaystyle\leq(1-\epsilon)g^{\bar{1}1}F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}-\frac{f(p)}{\det
g}\left|\sum_{j}g^{\bar{j}j}g_{j\bar{j},1}\right|^{2}-F^{i\bar{j}}(g^{\bar{1}1})^{2}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}$
$\displaystyle+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)-\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}}F^{i\bar{j}}\left(g^{\bar{1}1}\right)^{2}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}F^{i\bar{j}}u_{,i\bar{j}}$
$\displaystyle=g^{\bar{1}1}\left[(1-\epsilon)F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}-\frac{f(p)}{\det
g}\left|\sum_{j}g^{\bar{j}j}g_{j\bar{j},1}\right|^{2}-\left(1+\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}}\right)F^{i\bar{j}}g^{\bar{1}1}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}\right]$
$\displaystyle+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}F^{i\bar{j}}u_{,i\bar{j}}.$

Take
$\epsilon\leq\min\\{-\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}},\frac{1}{2}\\}$,
then we have

(7.41)
$\displaystyle(1-\epsilon)F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}-\left(1+\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}}\right)F^{i\bar{j}}g^{\bar{1}1}g_{i\bar{1},1}\overline{g_{j\bar{1},1}}$
$\displaystyle\leq\frac{1}{2}\sum_{l\geq
2}F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}.$

If $f(p)\geq 0$, then by (7.41), (7.41), and the fact that $F^{i\bar{j}}\leq
0$ in Lemma 6.6,

(7.42) $0\leq g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}F^{i\bar{j}}u_{,i\bar{j}}.$

If $f(p)<0$, then by (6.56), we have

(7.43) $-\sum_{l\geq
2}F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}\geq\left(\frac{m}{2}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})-|f(p)|\right)\frac{1}{\det
g}\sum_{l\geq
2}\sum_{i=1}^{n}\frac{|g_{i\bar{l},1}|^{2}}{g_{l\bar{l}}g_{i\bar{i}}}.$

We argue similarly as in Lemma 6.7:

(7.44) $\displaystyle\frac{1}{2}\sum_{l\geq
2}F^{i\bar{j}}g^{\bar{l}l}g_{i\bar{l},1}\overline{g_{j\bar{l},1}}-\frac{f(p)}{\det
g}\left|\sum_{j}g^{\bar{j}j}g_{j\bar{j},1}\right|^{2}$
$\displaystyle\leq-\frac{1}{\det
g}\left(\left(\frac{m}{4}\gamma_{\min}(\frac{\kappa}{m},n_{p},\mathbf{d}_{p},\mathbf{k}_{p})-\frac{1}{2}|f(p)|\right)\sum_{l\geq
2}\sum_{i=1}^{n}\frac{|g_{i\bar{l},1}|^{2}}{g_{l\bar{l}}g_{i\bar{i}}}-|f(p)|n\sum_{j=1}^{n}\frac{|g_{j\bar{j},1}|^{2}}{g_{j\bar{j}}^{2}}\right)$
$\displaystyle\leq\frac{n|f(p)|}{\det
g}\frac{|g_{1\bar{1},1}|^{2}}{g_{1\bar{1}}^{2}}.$

The last line is due to the almost positive volume condition.

By (7.40), (7.41), (7.44), and (7.42), we have

(7.45) $\displaystyle 0$
$\displaystyle\leq-\frac{\min\\{0,f(p)\\}n(g^{\bar{1}1})^{3}}{\det
g}|g_{1\bar{1},1}|^{2}+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}F^{i\bar{j}}u_{,i\bar{j}}$
$\displaystyle\leq
C_{0}(m,\kappa,n)(g^{\bar{1}1})^{3}|g_{1\bar{1},1}|^{2}+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}F^{i\bar{j}}u_{,i\bar{j}},$

where the last line is due to $\det g^{-1}\leq
C_{1}(m,\kappa,n_{p},\mathbf{d}_{p},\mathbf{k}_{p})$ by Lemma 6.4. Since
$u_{,i\bar{j}}=g_{i\bar{j}}-g_{i\bar{j}}^{\text{sub}},$ if
$|\partial\bar{\partial}u|>N$, from Proposition 7.11, we have

(7.46) $\displaystyle F^{i\bar{j}}\left(u_{,i\bar{j}}\right)$
$\displaystyle=F^{i\bar{j}}\left(g_{i\bar{j}}-g_{i\bar{j}}^{\text{sub}}\right)$
$\displaystyle\geq\mu(1-\sum_{i}F^{i\bar{i}}).$

Thus, use 7.45) and $g_{1\bar{1},1}=\phi^{\prime}u_{,1}g_{1\bar{1}}$ to get

(7.47) $\displaystyle 0$ $\displaystyle\leq
C_{1}g^{\bar{1}1}(\phi^{\prime})^{2}\left(1+|\nabla
u|^{2}\right)+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)-C_{\ref{lem:ddbar
F}}\sum_{i}F^{i\bar{i}}-\phi^{\prime}\mu(1-\sum_{i}F^{i\bar{i}})$
$\displaystyle\leq C_{1}g^{\bar{1}1}(\phi^{\prime})^{2}\left(1+|\nabla
u|^{2}\right)+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)-\phi^{\prime}\mu+\left(\phi^{\prime}\mu-C_{\ref{lem:ddbar
F}}\right)\sum_{i}F^{i\bar{i}}.$

We choose $\phi^{\prime}\mu>C_{\ref{lem:ddbar F}}$ and then by (7.47) ,

(7.48) $0\leq C_{1}g^{\bar{1}1}(\phi^{\prime})^{2}\left(1+|\nabla
u|^{2}\right)+g^{\bar{1}1}\frac{C_{\ref{lem:ddbar
F}}}{\epsilon}F(A)-\phi^{\prime}\mu.$

Therefore, by (7.48), we have $g_{1\bar{1}}\leq C_{2}(1+|\nabla u|^{2}),$ and
hence at $p$

(7.49) $|\partial\bar{\partial}u|<C_{3}(1+|\nabla u|^{2}),$

for some $C_{3}=C_{3}(C_{\ref{lem:ddbar F}},L,\epsilon,\mu,m,\kappa,n)$.

Now, we fix choices of $L,\tau,\epsilon$. First, we need
$\phi^{\prime}\mu>C_{\ref{lem:ddbar F}}$ which requires
that$L>\frac{C_{\ref{lem:ddbar F}}}{\mu}+1$. Next, we need
$\epsilon\leq-\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}}.$ Note

(7.50)
$-\frac{\phi^{\prime\prime}}{(\phi^{\prime})^{2}}\geq\frac{L\tau}{4L^{2}}=\frac{\tau}{4L}.$

Therefore, we may pick
$\epsilon=\min\\{\frac{\tau}{4L},\frac{1}{2}\\}=\frac{1}{4L(\sup u+1)}$. Note
$L,\epsilon,\tau$ only depend on
$M,\Lambda,\omega_{\text{sub}},n,m,\kappa,$$C_{H2},\sup|u|,|\text{Rm}_{\rho}|$.

Once we have arranged $L,\epsilon,\tau$ properly, from (7.49),

(7.51) $|\partial\bar{\partial}u|<C_{4}\sup(|\nabla u|^{2}+1),$

for some
$C_{4}=C_{4}(M,\Lambda,\omega_{\text{sub}},n,m,\kappa,$$C_{H2},\sup|u|,|\text{Rm}_{\rho}|)$.
Finally, we apply the blow-up technique in [8] (Proposition 5.1) to (7.51) to
deduce that $|\partial\bar{\partial}u|\leq
C_{5}(M,\Lambda,\omega_{\text{sub}},n,$$m,\kappa,C_{H2},\sup|u|,|\text{Rm}_{\rho}|)$
as desired. ∎

We now give the proof of Theorem 7.1.

###### Proof of Theorem 7.1.

Suppose that $\Lambda$ satisfies H2 with uniform constant $m>0$. From
discussions above, we only need to show that $\mathbf{I}$ is closed. Since
$\mathbf{I}$ is open $(0,t_{0})\subset\mathbf{I}$ for some $t_{0}>0$. Suppose
that $t_{1}=\sup\\{t:[0,t)\subset\mathbf{I}\\}>t_{0}>0$. Let
$t\in[t_{0},t_{1}]$, and denote

$\Lambda_{t}=t\mathring{\Lambda}+f_{t}P^{[n]}.$

Then as explained before, $\Lambda_{t}$ satisfies H2’. It is straightforward
to see that if $\omega_{\text{sub}}$ is a subsolution for (1.2), then it is a
subsolution for (7.1) at all $t\in[0,1]$. So we can apply the a priori
$C^{0}$, $C^{2}$ estimates derived in Proposition 7.5 and 7.7. $C^{1}$
estimate is obtained by the blow-up technique as in [8] (Proposition 5.1).
Together with the complex version of Evans-Krylov theory we obtain
$C^{2,\alpha}$ bounds for solutions of (7.1) for all $t\in\mathbf{I}$. The
standard Schauder theory implies $C^{k,\alpha}$ bounds for every $k$. The
closedness of $\mathbf{I}$ can then be obtained from Arzela-Ascoli theorem.
Thus, $\mathbf{I}$ is closed which implies the existence of smooth solution of
(1.2). ∎

##  Part II Numerical Criterion

The second part of this paper aims to give a proof of Theorem 1.10. The main
approach to prove Theorem 1.10 is an induction argument on dimension
introduced first by G. Chen [4]. A key component is the so called mass
concentration technique, which was originally due to Demailly-Păun [14] and
has been successfully employed by Chen [4] to the $J$-equation and the
supercritical dHYM equation. Our work is also based on Song [32], in which
Song improved Chen’s argument to treat singular sub-varieties.

The rest of this part is organized as follows. In Section 9, we set up the
above-mentioned induction process and establish the base case. In Section 10,
we prove a mass concentration result, Theorem 10.1. In Section 11, we complete
the induction argument, and prove Theorem 1.10. In Section 12, we apply
Theorem 1.10 to study special cases of dHYM equations.

###  8\. Notations and technical preparation

In this section, we set up some further notations and definitions. Assume that
$M$, $[\omega_{0}]$, $\Lambda$, $\kappa$ are given as before.

We first define regularized maximum functions. Readers may consult [13] I.5.E.
for general discussion.

######  Definition 8.1.

(Regularized Maximum). For any $\eta\in(0,\infty)^{l}$, the regularized
maximum function is defined as

$\widetilde{\max}_{\eta}(t_{1},\cdots,t_{l}):=\int_{\mathbb{R}^{l}}\max(t_{1}+h_{1},\cdots,t_{l}+h_{l})\prod_{1\leq
j\leq n}\theta\left(\frac{h_{j}}{\eta_{j}}\right)\frac{dh_{1}\cdots
dh_{l}}{\eta_{1}\cdots\eta_{l}},$

for $\eta=(\eta_{1},\cdots,\eta_{l})\in\left(0,\infty\right)^{l}$. Here
$\theta$ is a smooth non-negative function supported on $(-1,1)$ s.t.
$\int_{\mathbb{R}}\theta(t)dt=1$ and $\int_{\mathbb{R}}t\theta(t)dt=0$.

We will use the regularized maximum function and Richberg’s technique [30] to
glue local PSH functions. Some related known facts are collected in the
following lemma. See [13] I.5.18 for proofs.

######  Lemma 8.2.

For any $\eta\in(0,\infty)^{l}$, $\widetilde{\max}_{\eta}$ possesses the
following properties:

  1. (1)

$\widetilde{\max}_{\eta}(t_{1},\cdots,t_{l})$ is non-decreasing in all
variables, smooth and convex on $\mathbb{R}^{l}$;

  2. (2)

$\max\\{t_{1},\cdots,t_{l}\\}\leq\widetilde{\max}_{\eta}(t_{1},\cdots,t_{l})\leq\max\\{t_{1}+\eta_{1},\cdots,t_{l}+\eta_{l}\\}$;

  3. (3)

if $\ t_{j}+\eta_{j}\leq\max_{k\not=j}\\{t_{k}-\eta_{k}\\}$, then

$\widetilde{\max}_{\eta}(t_{1},\cdots,t_{l})=\widetilde{\max}_{(\eta_{1},\cdots,\hat{\eta}_{j},\cdots,t_{l})}(t_{1},\cdots,\hat{t}_{j},\cdots,t_{l});$

  4. (4)

$\widetilde{\max}_{\eta}(t_{1}+a,\cdots,t_{l}+a)=\widetilde{\max}_{\eta}(t_{1},\cdots,t_{l})+a$.

We prove the following technical lemma.

######  Lemma 8.3.

If $g=g(A_{1},\cdots,A_{l})$ is a convex function on the spaces of Hermitian
matrices and is monotone decreasing in each $A_{i}$, then

(8.1)
$g(\sqrt{-1}\partial\bar{\partial}\widetilde{\max}_{\eta}(u_{1},\cdots,u_{l}))\leq\sum_{i}\frac{\partial\widetilde{\max}_{\eta}}{\partial
u_{i}}g(\sqrt{-1}\partial\bar{\partial}u_{i}).$

In particular, if $u_{1},\cdots,u_{l}$ are in $\text{PSH}(M,\omega_{0})$ and
satisfies
$\mathcal{P}_{\Lambda}(\omega_{0}+\sqrt{-1}\partial\bar{\partial}u_{i})<\kappa,$
then

(8.2)
$\mathcal{P}_{\Lambda}(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\widetilde{\max}_{\eta}(u_{1},\cdots,u_{l}))\leq\mathcal{P}_{\Lambda}(\omega_{0}+\sqrt{-1}\partial\bar{\partial}u_{i})<\kappa.$

###### Proof.

Suppose that $u_{1},\cdots,u_{l}$ are $C^{2}$ functions on $\mathbb{C}^{n}$.
Then

(8.3)
$\displaystyle\sqrt{-1}\partial\bar{\partial}\widetilde{\max}_{\eta}(u_{1},\cdots,u_{l})$
$\displaystyle=\sum_{i}\frac{\partial\widetilde{\max}_{\eta}}{\partial
u_{i}}\sqrt{-1}\partial\bar{\partial}u_{i}+\sum_{i,j}\frac{\partial^{2}\widetilde{\max}_{\eta}}{\partial
u_{i}\partial u_{j}}\sqrt{-1}\partial u_{j}\wedge\bar{\partial}u_{i}.$

From Lemma 8.2 (4), we see that
$\sum_{i}\frac{\partial\widetilde{\max}_{\eta}}{\partial u_{i}}=1$ and each
$\frac{\partial\widetilde{\max}}{\partial u_{i}}$ is non-negative. Thus, from
(8.3),

$\sqrt{-1}\partial\bar{\partial}\widetilde{\max}_{\eta}(u_{1},\cdots,u_{l})\geq\sum_{i}\frac{\partial\widetilde{\max}_{\eta}}{\partial
u_{i}}\sqrt{-1}\partial\bar{\partial}u_{i}.$

The right hand side is a convex combination of
$\sqrt{-1}\partial\bar{\partial}u_{i}$. Thus, (8.1) holds by the monotonicity
and the convexity of $g$. (8.2) holds by the corresponding properties of
$\mathcal{P}_{\Lambda}$ in Lemma 5.10. ∎

We use Richberg technique ([13], I. Corollary 5.19) to patch up PSH functions
on manifolds.

######  Corollary 8.4.

Let $u_{\alpha}\in C^{\infty}(\overline{U}_{\alpha})\cap
PSH(U_{\alpha},\omega)$ where $U_{\alpha}\subset\subset M$ is a finite open
covering of $M$. Assume that $u_{\beta}<\max\\{u_{\alpha}(z)\\}$ at every
point $z\in\partial U_{\beta}$ when $\alpha$ runs over the indices s.t. $z\in
U_{\alpha}$. Choose a family $\\{\eta_{\alpha}\\}$ of positive numbers so
small that $u_{\beta}(z)+\eta_{\beta}\leq\max_{U_{\alpha}\ni
z}\\{u_{\alpha}-\eta_{\alpha}\\}$ for all $\beta$ s.t. $z\in\partial
U_{\beta}$. Then the function

$\tilde{u}(z)=\widetilde{\max}_{(\eta_{\alpha})}(u_{\alpha}(z))$

is in $C^{\infty}(M)\cap\text{PSH}(M,\omega)$.

Next, we recall some well known definitions in complex geometry.

######  Definition 8.5.

Let $T$ be a closed positive $(1,1)$-current on $M$. We call $T$ a
$\rm{K\ddot{a}hler}$ current if

$T-\epsilon\gamma\geq 0,$

for some $\epsilon>0$ and $\gamma$ is a Hermitian metric on $M$.

We recall the definition of Lelong number:

######  Definition 8.6.

For $p\in B_{R}\subset V\subset\mathbb{C}^{d}$, and $\varphi\in\text{PSH}(V)$,
define

$\nu_{\varphi}(p,r)=\frac{\bar{\varphi}(p,R)-\bar{\varphi}(p,r)}{\log R-\log
r}$

where $0<r<R$, and $\bar{\varphi}(p,r)=\sup_{B_{r}(p)}\varphi(z)$. The Lelong
number of $\varphi$ at $p$ is given by

$\nu_{\varphi}(p)=\lim_{r\to 0^{+}}\nu_{\varphi}(p,r).$

The Lelong number of a closed positive $(1,1)$-current $T$, denoted as
$\nu_{T}(p)$, is defined to be the Lelong number of the local potential
function.

The classic result of Y.-T. Siu shows that the Lelong number is upper semi-
continuous with respect to analytic Zariski topology.

######  Theorem 8.7 (Siu’s semi-continuity theorem [31]).

If $T$ is a closed positive $(1,1)$-current on $M$, then the upper level sets

$E_{c}(T)=\\{p:\nu_{T}(p)\geq c\\}$

are analytic subsets of $M$ of dimension $\leq n-1$.

Finally we define the local regularization of a current.

######  Definition 8.8.

Let $T$ be a closed positive $(1,1)$-current on a smooth variety $Y$ of