# Geometry of solutions to the c-projective metrizability equation

Keegan J. Flood and A. Rod Gover  K.J. F.:Department of Mathematics and
Statistics
Masaryk University
Kotlaářská 2
Brno 611-37
Czech Republic<EMAIL_ADDRESS>A.R.G.:Department of Mathematics
The University of Auckland
Private Bag 92019
Auckland 1142
New Zealand<EMAIL_ADDRESS>
###### Abstract.

On an almost complex manifold, a quasi-Kähler metric, with canonical
connection in the c-projective class of a given minimal complex connection, is
equivalent to a non-degenerate solution of the c-projectively invariant
metrizability equation. For this overdetermined equation, replacing this
maximal rank condition on solutions with a nondegeneracy condition on the
prolonged system yields a strictly wider class of solutions with non-vanishing
(generalized) scalar curvature. We study the geometries induced by this class
of solutions. For each solution, the strict point-wise signature partitions
the underlying manifold into strata, in a manner that generalizes the model, a
certain Lie group orbit decomposition of $\mathbb{CP}^{m}$. We describe the
smooth nature and geometric structure of each strata component, generalizing
the geometries of the embedded orbits in the model. This includes a quasi-
Kähler metric on the open strata components that becomes singular at the
strata boundary. The closed strata inherit almost CR-structures and can be
viewed as a c-projective infinity for the given quasi-Kähler metric.

###### 2010 Mathematics Subject Classification:

Primary 32J05, 32Q60, 53B15, 53B35, 53C21, 53C55; Secondary 32J27, 53C15,
53A20, 53C25, 53B10, 35N10, 58J60

K. J. Flood gratefully acknowledges support from the Czech Science Foundation
(GAČR) Grant 20-11473S and A. R. Gover gratefully acknowledges support from
the Royal Society of New Zealand via Marsden Grants 16-UOA-051 and 19-UOA-008.

##  1\. Introduction

Given a smooth manifold $M$, a projective structure is an equivalence class of
torsion-free affine connections $\boldsymbol{p}$ that have the same geodesics
as unparametrized curves. A projective manifold is a smooth manifold equipped
with a projective class $(M,\boldsymbol{p})$. A natural question is whether
such a structure is metrizable, i.e., is there is a metric $g$ on $M$ whose
Levi-Civita connection $\nabla^{g}$ lies in the projective class
$\boldsymbol{p}$. By [64, 69] this non-linear problem can instead be rephrased
in terms of solutions to the projectively invariant linear PDE

(1) $\operatorname{trace-free}(\nabla_{a}\zeta^{bc})=0,$

where we employ the Penrose abstract index notation and
$\nabla\in\boldsymbol{p}$. The projective manifold is metrizable if and only
if there is a non-degenerate symmetric contravariant 2-tensor $\zeta^{bc}$
satisfying (1), with inverse of the metric given by
$g^{bc}=\operatorname{sgn}(\tau)\tau\zeta^{bc}$, where $\tau$ is a suitable
determinant of $\zeta$. The study of this equation and related topics has led
to considerable recent progress [6, 10, 27, 28, 29, 37, 51, 57, 59, 62]. There
is growing interest in an analogue in the setting of what is called
c-projective geometry, see e.g. [12, 15, 61], and this is what we take up
here.

On an almost complex $2m$-manifold $(M,J)$, an _almost c-projective structure_
is an equivalence class of affine connections $\boldsymbol{D}$ which preserve
$J$, which have minimal torsion in the sense that the only non-vanishing
component of their torsion is the Nijenhuis tensor $N^{J}$, and which have (up
to reparametrization) the same $J$-planar curves (a complex analogue of
geodesics). Here and throughout $m\geq 2$. The analogue of metrizability, in
the almost c-projective setting, then, is whether there is exists a Hermitian
metric on $M$ which is preserved by a connection in the c-projective class
$\boldsymbol{D}$. Equivalently, an almost c-projective manifold is metrizable
if there exists a non-degenerate solution to the c-projectively invariant
linear PDE

(2) $\operatorname{trace-free}(\nabla_{a}\zeta^{bc})=0,$

where this is a complex trace and $\nabla\in\boldsymbol{D}$. Explicitly, in
real terms, this is given by

(3)
$\displaystyle\nabla_{c}\zeta^{ab}-\frac{1}{m}\delta_{c}^{(a}\nabla_{d}\zeta^{b)d}-\frac{1}{m}J_{c}^{(b}J^{a)}_{e}\nabla_{d}\zeta^{ed}=0.$

where $\zeta$ is a density weighted Hermitian form on $T^{*}M$. Equation (3)
is termed the c-projective metrizability equation. The inverse metric on $M$
is then given by

$g^{bc}=\operatorname{sgn}(\operatorname{det}(\zeta))\operatorname{det}(\zeta)\zeta^{bc},$

where, again, a suitable notion of determinant is involved.

Here we are interested in more general solutions to this c-projective
metrizability equation (3). In particular, we obtain a result that extends, to
generic solutions, a result from [19] concerning the restricted class of so-
called normal solutions. Specifically our aim is to identify and understand
the smooth structure and geometry induced on the different sets where a
solution to (3) is non-degenerate and, respectively, degenerate - we shall
extend the terminology from [19] and call this a curved orbit decomposition.
At points where the solution is non-degenerate it induces a metric as
previously noted. But at points where a solution is degenerate there is not,
in general, a metric, since the metric becomes singular on this set. But,
under suitable assumptions, the degeneracy locus of the solution does inherit
a rich geometric structure. In particular, it has a hypersurface type CR
structure, for which the Levi-form (arising as usual for an embedded
hypersurface) can be seen to be compatible with the metric defined away from
the degeneracy locus of the solution. This work gives an alternative approach
to the c-projective compactification of complete non-compact pseudo-Hermitian
metrics, as developed and studied in [15]. The problem we address is a special
case of a more general phenomenon which we describe below.

Natural overdetermined partial differential equations govern a huge variety of
geometric structures [7, 17, 24, 28, 37, 43, 66] on smooth manifolds. It has
long been known that features of solutions to such equations can partition the
manifold [4, 26, 52, 53], but only recently tools have been developed for
fully understanding the geometries on the more singular components in a way
that relates them to the ambient structures. In fact the components of the
partition can appear radically different to each other, but the link between
them becomes clear when viewing them via prolongations of the solution to the
relevant geometric PDE, see e.g [38, 39]. A reason that this is important is
that one can exploit these relationships to smoothly relate the distinct
components of the partition and thus study the geometry on one component by
means of an adjacent component, as seen in the geometric holography program
e.g. [1, 30, 34, 41, 44, 60].

Hence, given a solution to an overdetermined partial differential equation on
a smooth manifold, the key problem is to determine the basic data of the
components of the partition (e.g. are they smoothly embedded submanifolds of
some dimension or rather more complicated variety type structures?), then to
determine the geometric structures thereon. Finally, one wants to usefully
understand the relationship between the geometric structures on neighboring
components of the partition.

It turns out that for a broad class of natural overdetermined linear partial
differential equations, and then a class of solutions to these equations , one
can obtain remarkably general results. These are for what are called normal
solutions to first BGG equations . In [18, 19] it is shown that the
stratifications arising from solutions to these must be locally diffeomorphic
to stratifications arising from group orbit decompositions of homogeneous
model geometries. Moreover, the components of the partition carry Cartan
geometries that are curved analogues of the homogeneous geometries on the
corresponding partition of components of the model. Unfortunately, the methods
utilized in these sources applies only to solutions which correspond to Cartan
holonomy reductions. Thus it is important to establish to what extent similar
results might be deduced, by different methods, for more general solutions.
This question is treated for the equation (65) (which is an example of a first
BGG equation) in the present article, following to an extent the ideas and the
progress in [31, 39].

A standard approach to studying and treating overdetermined equations is via
differential prolongation, see e.g. [8]. The c-projective tractor calculus
(cf. [12, 15]) is a natural tool for developing and organizing the prolonged
system of the c-projectively invariant equation (65). One reason for this is
that, since it is a first BGG equation [18, 19, 22, 23], the (first) BGG
splitting operator (a canonical invariant differential operator) maps, loosely
speaking, a potential solution of (65) to its prolonged variable system. We
denote this c-projectively invariant second order operator $\zeta\mapsto
L(\zeta)$, where $L(\zeta)$ takes values in the bundle $\mathcal{H}^{*}$ of
Hermitian forms on the standard c-projective cotractor bundle. If $L(\zeta)$
is parallel for the tractor connection then the solution $\zeta$ is said to be
normal, but we consider a more general class of solutions here. This BGG
machinery is introduced in Section 2.10 below.

There is a canonical (c-projectively invariant) real-valued determinant on
sections of $\mathcal{H}^{*}$ so we consider the composite map

(4) $\zeta\mapsto L(\zeta)\mapsto\det(L(\zeta)),$

which takes a solution $\zeta$ of (65) to the determinant of its prolonged
system. If $\zeta$ is a non-degenerate solution of (65) then, up a non-zero
constant, $\det(L(\zeta))$ is the scalar curvature of the metric $g$ with
inverse $g^{-1}=\operatorname{sgn}(\tau)\tau\zeta$ [15], where $\tau$ a
suitable determinant of $\zeta$. But (4) is well-defined even where $\zeta$ is
degenerate. Thus it is natural to consider solutions $\zeta$ of equation (1)
satisfying the condition that $\det(L(\zeta))$ is nowhere zero, i.e. with
$L(\zeta)$ non-degenerate, but with no a priori restriction on the rank of
$\zeta$. This generic condition is a generalization of constant scalar
curvature, where $\zeta$ can have a non-empty degeneracy locus. Such
considerations lead to the following result.

######  Theorem 1.1.

Let $(M,J,\boldsymbol{D})$ be a almost c-projective manifold with real
dimension $2m$ equipped with a solution
$\zeta\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}})$
of the metrizability equation such that $L(\zeta)\in\Gamma(\mathcal{H}^{*})$
is non-degenerate as a pseudo-Hermitian form on the cotractor bundle. If
$L(\zeta)$ is definite then the degeneracy locus $\mathcal{D}(\zeta)$ is empty
and $(M,J,\boldsymbol{D},\zeta)$ is a quasi-Kähler manifold with inverse
Hermitian metric $g^{-1}=\operatorname{sgn}(\tau)\tau\zeta$ where
$\tau={\textbf{det}}(\zeta)$. If $L(\zeta)$ has signature $(p+1,q+1)$, with
$p,q\geq 0$, then $\mathcal{D}(\zeta)$ is either empty or it is a smoothly
embedded separating real hypersurface such that the following hold:

  1. (i)

$M$ is stratified by the strict signature of $\zeta$ as a (density weighted)
Hermitian form on $T^{*}M$ with curved orbit decomposition given by

$M=\coprod\limits_{i\in\\{+,0,-\\}}M_{i}$

where $\zeta$ has signature $(p+1,q)$, $(p,q+1)$,and $(p,q,1)$ on $TM$
restricted to $M_{+}$, $M_{-}$, and $M_{0}$, respectively.

  2. (ii)

On $M_{\pm}$, $\zeta$ induces a quasi-Kähler metric $g_{\pm}$ with
nonvanishing scalar curvature $R^{g_{\pm}}$, with the same signature as
$\zeta$, with inverse
$g^{-1}_{\pm}=\operatorname{sgn}(\tau)\tau\zeta|_{M_{\pm}}$ where
$\tau={\textbf{det}}(\zeta)$.

  3. (iii)

If $M$ is closed, then the components $(M\backslash M_{\mp},J,\boldsymbol{D})$
are c-projective compactifications of $(M_{\pm},J,\nabla^{\zeta})$, with
boundary $M_{0}$.

  4. (iv)

$M_{0}$ inherits a signature $(p,q)$ almost CR structure of hypersurface type.

A smoothly embedded submanifold of real codimension 1 will be referred to as a
hypersurface. Note that each of the components $M_{+}$, $M_{0}$, and $M_{-}$
in the above theorem need not be connected. We denote the signature of a real
symmetric bilinear form by $(p,q,r)$, where $p,q$ and $r$ are the number,
counting multiplicity, of positive, negative, and zero eigenvalues,
respectively, of any matrix representing the form once a basis has been
chosen. When $r=0$ we omit it.

The Fubini-Study metric is a compact homogeneous model for Hermitian geometry.
There are corresponding compact models for the geometries discussed in Theorem
1.1 demonstrating that c-projective manifolds equipped with solutions of (65)
satisfying the given constant rank conditions on their prolonged systems
$L(\zeta)$ exist and are of interest. The models for the structures in Theorem
1.1 are treated in Section 3.2 and from them we glean deeper insight into the
result.

Further motivation comes from [15], wherein it is shown that, given a manifold
with boundary whose interior is equipped with a pseudo-Hermitian metric
satisfying a non-vanishing scalar curvature condition and whose its
c-projective structure extends to the boundary but whose canonical connection
does not extend to any neighborhood of the boundary, then the metric is
c-projectively compact of order 2. Examples of c-projectively compactified
metrics discussed in [15] demonstrates the existence of curved examples of the
structures considered in Theorem 1.1.

The non-degeneracy assumption on $L(\zeta)$, in Theorem 1.1, is a constant
$G$-type assumption, where we have used the terminology of [18, 19]. Constancy
of $G$-type holds for normal solutions on connected manifolds, but it is not
known to hold for general solutions. As discussed in [31] (using results from
[46, 47]), the fixed $G$-type assumption is necessary to get a coherent
theory, as the zero locus of the scalar curvature can be very poorly behaved.
In particular, it need not be a submanifold.

The structure of the article is as follows. In the Section 2 we briefly review
c-projective tractor calculus, c-projective compactification, and BGG
machinery. These provide the framework and computational tools we utilize. In
Section 3 we state and prove the main results.

##  2\. C-projective geometry

In this section, we describe the necessary background from c-projective
geometry. We draw from the main monograph on the subject [12] as well as from
[15] since we will need both the (predominantly) complex viewpoint of the
former as well as the real viewpoint of the latter. Let $(M,J)$ be an almost
complex manifold of dimension $n=2m\geq 4$.

The complexified tangent bundle $\mathbb{C}TM$ and complexified cotangent
bundle $\wedge^{1}M$ decompose into the following direct sums

(5) $\displaystyle\mathbb{C}\mathcal{E}^{a}\colonequals$
$\displaystyle\mathbb{C}TM=T^{1,0}M\oplus T^{0,1}M$ (6)
$\displaystyle\mathbb{C}\mathcal{E}_{a}\colonequals$
$\displaystyle\wedge^{1}M=\wedge^{0,1}M\oplus\wedge^{1,0}M$

where

$\displaystyle\mathcal{E}^{\alpha}\colonequals$ $\displaystyle
T^{1,0}M=\\{X\in\Gamma(TM):JX=iX\\}$
$\displaystyle\mathcal{E}^{\overline{\alpha}}\colonequals$ $\displaystyle
T^{0,1}M=\\{X\in\Gamma(TM):JX=-iX\\}$
$\displaystyle\mathcal{E}_{\overline{\alpha}}\colonequals$
$\displaystyle\wedge^{0,1}M=\\{\alpha\in\Gamma(T^{*}M):J\alpha=-i\alpha\\}$
$\displaystyle\mathcal{E}_{\alpha}\colonequals$
$\displaystyle\wedge^{1,0}M=\\{\alpha\in\Gamma(T^{*}M):J\alpha=i\alpha\\}$

are the vector fields of type $(1,0)$ and $(0,1)$ and $1$-forms of type
$(0,1)$ and $(1,0)$, respectively. There are conjugate linear isomorphisms
$T^{1,0}M=\overline{T^{0,1}M}$ and $\wedge^{0,1}M=\overline{\wedge^{1,0}}M$.
Observe that there are canonical pairings of $\mathcal{E}^{\alpha}$ and
$\mathcal{E}^{\overline{\alpha}}$ with their respective duals
$\mathcal{E}_{\alpha}$ and $\mathcal{E}_{\overline{\alpha}}$, which is
compatible with the canonical complex pairing of $\mathbb{C}\mathcal{E}^{a}$
with $\mathbb{C}\mathcal{E}_{a}$. Note that we will be using lower case latin
indices for real and complex vector fields and $1$-forms.

We have the following complex linear projection maps:

$\displaystyle\mathbb{C}TM$ $\displaystyle\twoheadrightarrow T^{1,0}M\ \ \ \ \
\ \ \ \ \ \ $ $\displaystyle
X^{a}\mapsto\Pi_{a}^{\alpha}X^{a}\colonequals\frac{1}{2}(X-iJX)$
$\displaystyle\mathbb{C}TM$ $\displaystyle\twoheadrightarrow T^{0,1}M\ \ \ \ \
\ \ \ \ \ \ $ $\displaystyle
X^{a}\mapsto\overline{\Pi}_{a}^{\overline{\alpha}}X^{a}\colonequals\frac{1}{2}(X+iJX),$

and their duals

$\displaystyle\wedge^{1,0}M$ $\displaystyle\hookrightarrow\wedge^{1}M\ \ \ \ \
\ \ \ \ \ \ $
$\displaystyle\omega_{\alpha}\mapsto\Pi_{a}^{\alpha}\omega_{\alpha}$
$\displaystyle\wedge^{0,1}M$ $\displaystyle\hookrightarrow\wedge^{1}M\ \ \ \ \
\ \ \ \ \ \ $
$\displaystyle\omega_{\overline{\alpha}}\mapsto\overline{\Pi}_{a}^{\overline{\alpha}}\omega_{\overline{\alpha}}.$

Similarly, we have the inclusions:

$\displaystyle T^{1,0}M$ $\displaystyle\hookrightarrow\mathbb{C}TM\ \ \ \ \ \
\ \ \ \ \ $ $\displaystyle X^{\alpha}\mapsto\Pi^{a}_{\alpha}X^{\alpha}$
$\displaystyle T^{0,1}M$ $\displaystyle\hookrightarrow\mathbb{C}TM\ \ \ \ \ \
\ \ \ \ \ $ $\displaystyle
X^{\overline{\alpha}}\mapsto\overline{\Pi}^{a}_{\overline{\alpha}}X^{\overline{\alpha}},$

and their duals

$\displaystyle\wedge^{1}M$ $\displaystyle\twoheadrightarrow\wedge^{1,0}M\ \ \
\ \ \ \ \ \ \ \ $
$\displaystyle\omega_{a}\mapsto\Pi^{a}_{\alpha}\omega_{\alpha}$
$\displaystyle\wedge^{1}M$ $\displaystyle\twoheadrightarrow\wedge^{0,1}M\ \ \
\ \ \ \ \ \ \ \ $
$\displaystyle\omega_{a}\mapsto\overline{\Pi}^{a}_{\overline{\alpha}}\omega_{\overline{\alpha}}.$

These lead to the following identities

$\displaystyle\Pi_{\alpha}^{a}\Pi^{\beta}_{a}=\delta^{\beta}_{\alpha}\ \ \ \ \
\ \ \ \ \ $
$\displaystyle\Pi^{\alpha}_{a}\Pi^{b}_{\alpha}=\frac{1}{2}(\delta^{b}_{a}-iJ^{b}_{a})\
\ \ \ $
$\displaystyle\overline{\Pi}^{\overline{\alpha}}_{a}\overline{\Pi}^{b}_{\overline{\alpha}}=\frac{1}{2}(\delta^{b}_{a}+iJ^{b}_{a})$
$\displaystyle\Pi_{\alpha}^{a}J_{a}^{b}=i\Pi^{b}_{\alpha}\ \ \ \ \ $
$\displaystyle\overline{\Pi}^{a}_{\overline{\alpha}}J^{b}_{a}=-i\overline{\Pi}^{b}_{\overline{\alpha}}$
$\displaystyle J^{b}_{a}\Pi^{\alpha}_{b}=i\Pi_{a}^{\alpha}$ $\displaystyle
J^{b}_{a}\overline{\Pi}^{\overline{\alpha}}_{b}=-i\overline{\Pi}_{a}^{\overline{\alpha}}.$

Complex valued differential forms can be naturally decomposed according to
type e.g.

$\wedge^{2}M=\wedge^{0,2}M\oplus\wedge^{1,1}M\oplus\wedge^{0,2}M.$

Although such characterizations quickly grow cumbersome for higher forms,
$2$-forms are characterized as follows:

$\displaystyle
J_{a}^{c}\omega_{bc}=-i\omega_{ab}\Longleftrightarrow\omega_{ab}\
\operatorname{is\ type\ }(2,0)$ $\displaystyle
J_{[a}^{c}\omega_{b]c}=0\Longleftrightarrow\omega_{ab}\ \operatorname{is\
type\ }(1,1)$ $\displaystyle
J_{a}^{c}\omega_{bc}=i\omega_{ab}\Longleftrightarrow\omega_{ab}\
\operatorname{is\ type\ }(0,2)$

This characterization extends to appropriate almost complex vector bundle
valued real forms. E.g., the Nijenhuis tensor $N^{J}\in\Omega^{2}(M,TM)$,
which satisfies $N^{J}(X,JY)=JN^{J}(Y,X)$, is type $(0,2)$.

###  2.1. Complex Connections

Affine connections preserving $J$, i.e. satisfying
$\nabla_{\mu}J\eta=J\nabla_{\mu}\eta$ for all $\mu,\eta\in\mathfrak{X}(M)$,
are termed complex connections. It follows that an affine connection is
complex if and only if its extension to a linear connection on
$\mathbb{C}\mathcal{E}^{a}$ preserves types. The torsion $T_{ab}^{c}$ of a
complex connection $\nabla$ naturally splits into types, with the $(0,2)$
component being precisely $-\frac{1}{4}N^{J}$. So a complex connection cannot
be torsion-free unless its Nijenhuis tensor vanishes identically i.e. the
almost complex structure is integrable. Given an almost complex manifold,
there always exists a complex connection on it with torsion of type $(0,2)$ by
[48]. The $(2,0)$ and $(1,1)$ components of the torsion can be removed via a
suitable modification to the complex connection, but as an almost complex
invariant the $(0,2)$ component may not be eliminated.

The pseudo-Riemannian metrics of interest on almost complex manifolds are
those which are Hermitian for J, i.e. satisfying
$g_{ab}J^{a}_{c}J^{b}_{d}=g_{cd}$. Pseudo-Kähler metrics are precisely the
Hermitain metrics whose Levi-Civita connections are complex. Projective
equivalence of two such pseudo-Kähler metrics on $(M,J)$ implies that they are
in fact affinely equivalent [5]. Thus we must introduce a broader class of
curves, the so-called $J$-planar curves. A $J$-planar curve is a curve
$c:I\rightarrow M$ satisfying

$\nabla_{\dot{c}}\dot{c}=\alpha\dot{c}+\beta J\dot{c}$

for some $\alpha,\beta:I\rightarrow\mathbb{R}$. These are also commonly termed
holomorphically flat curves [65] or $h$-planar curves [58]. Clearly all curves
are J-planar on almost complex manifolds of real dimension $2$.

Consider $(\mathbb{CP}^{n},J_{Can},g^{FS})$, where $J_{Can}$ denotes the
canonical complex structure and $g^{FS}$ denotes the Fubini-Study metric.
Observing that the embedding of any complex line
$\mathbb{CP}^{1}\hookrightarrow\mathbb{CP}^{n}$ is totally geodesic with
respect to $\nabla^{FS}$, it follows (for details see Example 1 of [58]) that
the $J$-planar curves on $(\mathbb{CP}^{n},J_{Can},[\nabla^{FS}])$ are
precisely the curves in these linearly embedded copies of $\mathbb{CP}^{1}$.

We say that two complex connections $\nabla$ and $\tilde{\nabla}$ on an almost
complex manifold $(M,J)$ are c-projectively equivalent if they have the same
$J$-planar curves and the same torsion. Two such complex connections are
explicitly related by

$\displaystyle\tilde{\nabla}_{a}\eta^{b}=\nabla_{a}\eta^{b}+\Upsilon_{c}\eta^{b}-\Upsilon_{c}J^{c}_{a}J^{b}_{d}\eta^{d}+\Upsilon_{c}\eta^{c}\delta_{a}^{b}-\Upsilon_{c}J^{c}_{d}\eta^{d}J^{b}_{a}$
$\displaystyle\tilde{\nabla}_{\alpha}\eta^{\gamma}=\nabla_{\alpha}\eta^{\gamma}+2\Upsilon_{\alpha}\eta^{\gamma}+2\delta^{\gamma}_{\alpha}\Upsilon_{\beta}\eta^{\beta}$
$\displaystyle\tilde{\nabla}_{\overline{\alpha}}\eta^{\gamma}=\nabla_{\overline{\alpha}}\eta^{\gamma}$
$\displaystyle\tilde{\nabla}_{a}\nu_{c}=\nabla_{a}\nu_{c}-\Upsilon_{a}\nu_{c}+\Upsilon_{b}J^{b}_{a}J^{d}_{c}\nu_{d}-\Upsilon_{c}\nu_{a}+\Upsilon_{d}J^{b}_{a}\nu_{b}J^{d}_{c}$
$\displaystyle\tilde{\nabla}_{\alpha}\nu_{\gamma}=\nabla_{\alpha}\nu_{\gamma}-2\Upsilon_{\alpha}\nu_{\gamma}-2\nu_{\alpha}\Upsilon_{\gamma}$
$\displaystyle\tilde{\nabla}_{\overline{\alpha}}\nu_{\gamma}=\nabla_{\overline{\alpha}}\nu_{\gamma}$

for some one form $\Upsilon\in\Omega^{1}(M)$, where
$\Upsilon_{\alpha}\colonequals\Pi_{\alpha}^{a}\Upsilon_{a}$ and
$\Upsilon_{\overline{\alpha}}\colonequals\Pi_{\overline{\alpha}}^{a}\Upsilon_{a}$.
We write $\tilde{\nabla}=\nabla+\Upsilon$ as a brief notation to indicate
connections related as in the above formulae. Note that we follow the
convention of [15] in (2.1) rather than that of [12].

In fact, we will only consider complex connections $\nabla$ with minimal
torsion $T^{\nabla}=-\frac{1}{4}N^{J}$, we term these minimal complex
connections. We write $\boldsymbol{D}$ for an equivalence class of
c-projectively related minimal complex affine connections and we call it an
almost c-projective structure. We call a triple $(M,J,\boldsymbol{D})$ an
almost c-projective manifold. If $J$ is integrable we call
$(M,J,\boldsymbol{D})$ a c-projective manifold.

###  2.2. C-projective densities

We write $\mathcal{E}(m+1,0)\colonequals\Lambda^{m}_{\mathbb{C}}TM$ for the
top complex exterior power of the tangent bundle. We will assume the existence
of $(m+1)^{\operatorname{th}}$ roots of this bundle. In particular, this holds
on the model $(\mathbb{CP}^{n},J_{Can},[\nabla^{FS}])$ and hence locally for
all almost c-projective manifolds. Assuming a choice $\mathcal{E}(1,0)$ of
$(m+1)^{\operatorname{th}}$ root of $\mathcal{E}(m+1,0)$, denote the dual,
conjugate, and dual conjugate to $\mathcal{E}(1,0)$ by $\mathcal{E}(-1,0)$,
$\mathcal{E}(0,1)$, and $\mathcal{E}(0,-1)$, respectively. Forming tensor
powers of these bundles gives complex density bundles $\mathcal{E}(k,l)$ of
weight $(k,l)$ where $k,l\in\mathbb{Z}$.

There is a natural inclusion
$\mathcal{E}(-2m-2)\hookrightarrow\mathcal{E}(-m-1,-m-1)$ of the real
densities of weight $(-2m-2)$ into the complex densities of weight
$(-m-1,-m-1)$ as the real subbundle fixed by conjugation. The orientation on
$(M,J)$, induced by the almost complex structure $J$, induces an orientation
on the trivial bundle $\mathcal{E}(-2m-2)$ and so allows us to take arbitrary
real roots of $\mathcal{E}(-2m-2)$ which give the usual real densities
$\mathcal{E}(w)$ of weight $w\in\mathbb{R}$. Thus, for
$w,w^{\prime}\in\mathbb{R}$ such that $w-w^{\prime}\in\mathbb{Z}$, we can
define complex density bundle $\mathcal{E}(w,w^{\prime})$. We denote the image
of $\mathcal{E}(2w)$ under this inclusion map by
$\mathcal{E}(w,w)_{\mathbb{R}}$.

It is shown in [15] that for $\upsilon\in\Gamma(\mathcal{E}(w,w^{\prime}))$
with $w,w^{\prime}\in\mathbb{R}$ and $w-w^{\prime}\in\mathbb{Z}$

(7)
$\displaystyle\tilde{\nabla}_{a}\upsilon=\nabla_{a}\upsilon+(w+w^{\prime})\Upsilon_{a}\upsilon-(w-w^{\prime})i\Upsilon_{b}J^{b}_{a}\upsilon.$

In particular, for $\tau\in\Gamma(\mathcal{(}w,w)_{\mathbb{R}})$ with
$w\in\mathbb{R}$ this reduces to

(8) $\displaystyle\tilde{\nabla}_{a}\tau=\nabla_{a}\tau+2w\Upsilon_{a}\tau.$

###  2.3. C-projective compactness

A local defining function for a hypersurface $\Sigma$ is a smooth function
$r:U\rightarrow\mathbb{R}$, defined on an open subset $U$ of $M$, satisfying
$\mathcal{Z}(r)=\Sigma\cap U$ and $\mathcal{Z}(dr)\cap\Sigma=\varnothing$ on
$\Sigma\cap U$, where $\mathcal{Z}(-)$ denotes the zero locus. Then, extending
this concept, a local defining density of weight w is a local section $\sigma$
of $\mathcal{E}(w)$ such that $\sigma=r\hat{\sigma}$, where $r$ is a defining
function for $\Sigma$ and $\hat{\sigma}$ is a section of $\mathcal{E}(w)$ that
is nonvanishing on $U$.

Consider a smooth manifold with boundary, $\overline{M}=M\cup\partial M$, such
that the interior $M$ is equipped with an almost complex structure $J$ and a
minimal complex affine connection $\nabla$ on $TM$. The (minimal) complex
connection $\nabla$ on $TM$ is said to be c-projectively compact of order
$\alpha\in\mathbb{R}_{+}$ if and only if for any $x\in\partial M$ there is a
neighborhood $U$ of $x$ in $\overline{M}$ and a defining function
$\rho:U\rightarrow\mathbb{R}_{\geq 0}$ for $U\cap\partial M$ such that the
c-projectively equivalent connection
$\overline{\nabla}=\nabla+\frac{d\rho}{\alpha\rho}$ on $U\cap M$ smoothly
extends to all of $U$, i.e $\overline{\nabla}_{\mu}\eta$ is smooth up to the
boundary for all $\mu,\eta\in\mathfrak{X}(\overline{U})$. In what follows we
will only be concerned with the case where $\alpha=2$, so we will often omit
the order of the c-projective compactification. As in the case of projective
compactification (cf. [14]) this notion is independent of choice of defining
function. A connection $\nabla$ is said to be special if and only if there is
a section $\tau\in\Gamma(w,w)_{\mathbb{R}}$ with $w\neq 0$ such that $\tau$ is
parallel for $\nabla$. This leads to the following, which is Proposition 6 of
[15].

######  Proposition 2.1.

Let $\overline{M}=M\cap\partial M$ be a smooth manifold of dimension $n=2m$
equipped with a special affine connection $\nabla$ on $TM$. Then the following
hold:

  1. (1)

If $\nabla$ is c-projectively compact of order $2$ then a non-vanishing
section of $\mathcal{E}(1,1)_{\mathbb{R}}$ which is parallel for $\nabla$
extends by zero to a defining density for $\partial M$.

  2. (2)

If the almost c-projective structure on $M$ determined by $\nabla$ smoothly
extends to $\overline{M}$ and there exists a defining density
$\tau\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$ for $\partial M$ that is
parallel on $M$ for $\nabla$, then $\nabla$ is c-projectively compact of order
2.

Let $(M,\nabla)$ be a smooth manifold equipped with a complex connection. If
there exists a smooth manifold with boundary $\overline{M}$ such that
$\overline{M}=M\cup\partial M$ for which $\nabla$ is c-projectively compact we
will say that $(\overline{M},[\nabla])$ is a c-projective compactification of
$(M,\nabla)$.

###  2.4. Admissible metrics

How do metrics fit into the picture? We discussed earlier that the relevant
pseudo-Riemannian metrics in almost c-projective geometry are those which are
Hermitian with respect to $J$. Minimizing the torsion we come to the class of
connections $\boldsymbol{D}$, which in general have torsion of type $(0,2)$,
and so cannot be the Levi-Civita connection. Fortunately, a minimal complex
connection preserving a pseudo-Riemannian metric, that is Hermitian for $J$,
is uniquely determined. Such a connection need not exist in general.

On an almost complex manifold $(M,J)$ a pseudo-Riemannian metric $g$ that is
Hermitian for $J$ is said to be admissible if and only if there is a minimal
complex affine connection on $(M,J)$ preserving $g$. If such a connection
exists it is termed the canonical connection associated to $g$. By Proposition
4.1 of [12] or Proposition 7 of [15] a pseudo-Riemannian metric on an almost
complex manifold $(M,J)$ that is Hermitian for $J$ is admissible if and only
if it is quasi-Kähler in the sense of Gray-Hervella [42].

In the notation of Gray-Hervella, quasi-Kähler is
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. $\mathcal{W}_{1}$ denotes the class of
nearly Kähler manifolds i.e. $d\omega=3\nabla\omega$ where $\omega$ is the
Kähler form $\omega_{ab}=J_{a}^{c}g_{cb}$ and $\nabla$ is the canonical
connection associated to $g$. $\mathcal{W}_{2}$ denotes the class of almost
Kähler manifolds i.e. $d\omega=0$. If $J$ is integrable, $g$ is admissible if
and only if it is pseudo-Kähler, i.e. $\nabla\omega=0$.

###  2.5. The c-projective Schouten tensor

The curvature tensor, $R\indices{{}_{ab}^{c}{}_{d}}$, of a complex affine
connection, $\nabla$, on an almost complex manifold $(M^{2m},J)$ satisfies
$R\indices{{}_{ab}^{c}{}_{i}}J_{d}^{i}=R\indices{{}_{ab}^{i}{}_{d}}J^{c}_{i}$.
Denoting the Ricci tensor by $R_{ab}=R\indices{{}_{ia}^{i}{}_{b}}$ we define
the Rho tensor or c-projective Schouten tensor by

(9) $\displaystyle
P_{ab}\colonequals\frac{1}{2(m+1)}(R_{ab}+\frac{1}{m-1}(R_{(ab)}-J^{i}_{(a}J^{j}_{b)}R_{ij})).$

Given a complex connection $\nabla$ with Schouten $P$, The Schouten
$\overline{P}$ of the c-projectively related connection
$\tilde{\nabla}=\nabla+\Upsilon$ is given by

(10)
$\displaystyle\tilde{P}_{ab}=P_{ab}-\nabla_{a}\Upsilon_{b}+\Upsilon_{a}\Upsilon_{b}-J_{a}^{i}J_{b}^{j}\Upsilon_{i}\Upsilon_{j}.$

Writing $W\indices{{}_{ab}^{c}{}_{d}}$ for the Weyl curvature we have the
following

(11) $\displaystyle
R\indices{{}_{ab}^{c}{}_{d}}=W\indices{{}_{ab}^{c}{}_{d}}+2\delta^{c}_{[a}P_{b]d}-2P_{[ab]}\delta_{d}^{c}-2J^{i}_{[a}P_{b]i}J^{c}_{d}-2J^{c}_{[a}P_{b]i}J^{i}_{d}.$

Observe that if the Ricci is Hermitian then $P_{ab}=\frac{1}{2(m+1)}R_{ab}$
and if the Ricci is symmetric then the Schouten is symmetric as well.

###  2.6. C-projective tractor bundle

An almost c-projective manifold $(M,J,\boldsymbol{D})$ equipped with a choice
of density bundle $\mathcal{E}(1,0)$ is equivalent to a Cartan geometry
$(\mathcal{P}\twoheadrightarrow M,\omega)$ of type $(G,P)$ where
$G=SL(m+1,\mathbb{C})\cong SL(2m+2,\mathbb{J})$, which we identify with its
standard representation on $\mathbb{C}^{m+1}$, and $P\subseteq G$ is the
isotropy group of a complex line through the origin in $\mathbb{C}^{m+1}$.
Restricting this representation to $P$, call the restricted representation
space $\mathbb{V}$. The corresponding tractor bundle is the standard
c-projective tractor bundle $\mathcal{T}$, i.e.

(12)
$\displaystyle\mathcal{E}^{\mathscr{A}}=\mathcal{T}\colonequals\mathcal{P}\times_{P}\mathbb{V}.$

Its dual, the standard cotractor bundle, is given by

(13)
$\displaystyle\mathcal{E}_{\mathscr{A}}=\mathcal{T}^{*}\colonequals\mathcal{P}\times_{P}\mathbb{V}^{*}.$

We define the standard complex tractor bundle to be the $(1,0)$ component of
the complexification of the real standard tractor bundle

(14)
$\displaystyle\mathcal{E}^{A}=\mathcal{T}^{1,0}\subset\mathbb{C}\mathcal{T},$

We denote it’s conjugate, dual, and dual conjugate by
$\mathcal{E}^{\overline{A}}=\mathcal{T}^{0,1}=\overline{\mathcal{T}^{1,0}}$,
$\mathcal{E}_{A}=(\mathcal{T}^{1,0})^{*}$, and
$\mathcal{E}_{\overline{A}}=(\mathcal{T}^{0,1})^{*}$, respectively.

Recalling the various natural maps denoted by $\Pi$ from the beginning of
Section 2, observe that there are, _mutatis mutandis_ , analogous natural
complex linear inclusions and projections at the tractor level which satisfy
similar identities. For instance,

$\displaystyle(\mathcal{T}^{*})^{(1,0)}$
$\displaystyle\hookrightarrow\mathbb{C}\mathcal{T}^{*}\ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle u_{A}\mapsto\Pi_{\mathscr{A}}^{A}u_{A}$
$\displaystyle(\mathcal{T}^{*})^{(0,1)}$
$\displaystyle\hookrightarrow\mathbb{C}\mathcal{T}^{*}\ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle
u_{\overline{A}}\mapsto\overline{\Pi}_{\mathscr{A}}^{\overline{A}}u_{\overline{A}}.$

Note that we use capital script indices for the real standard (co)tractor
bundle, and its complexification when no confusion can arise. The structure of
the tractor bundles defined above can be described by the following short
exact sequences

$\displaystyle
0\rightarrow\mathcal{E}(-1,0)\xrightarrow{X^{\mathscr{A}}}\mathcal{E}^{\mathscr{A}}\xrightarrow{Z^{a}_{\mathscr{A}}}\mathcal{E}^{a}\otimes_{\mathbb{C}}\mathcal{E}(-1,0)\rightarrow
0$ $\displaystyle
0\rightarrow\mathcal{E}_{a}\otimes_{\mathbb{C}}\mathcal{E}(1,0)\xrightarrow{Z^{a}_{\mathscr{A}}}\mathcal{E}_{\mathscr{A}}\xrightarrow{X^{\mathscr{A}}}\mathcal{E}(1,0)\rightarrow
0$ $\displaystyle
0\rightarrow\mathcal{E}_{\overline{\alpha}}\otimes\mathcal{E}(0,1)\xrightarrow{Z^{\overline{\alpha}}_{\overline{A}}}\mathcal{E}_{\overline{A}}\xrightarrow{X^{\overline{A}}}\mathcal{E}(0,1)\rightarrow
0$ $\displaystyle
0\rightarrow\mathcal{E}_{\alpha}\otimes\mathcal{E}(1,0)\xrightarrow{Z^{{\alpha}}_{{A}}}\mathcal{E}_{{A}}\xrightarrow{X^{{A}}}\mathcal{E}(1,0)\rightarrow
0$ $\displaystyle
0\rightarrow\mathcal{E}(-1,0)\xrightarrow{X^{A}}\mathcal{E}^{A}\xrightarrow{Z_{A}^{\alpha}}\mathcal{E}^{\alpha}\otimes\mathcal{E}(-1,0)\rightarrow
0$ $\displaystyle
0\rightarrow\mathcal{E}(0,-1)\xrightarrow{X^{\overline{A}}}\mathcal{E}^{\overline{A}}\xrightarrow{Z_{\overline{A}}^{\overline{\alpha}}}\mathcal{E}^{\overline{\alpha}}\otimes\mathcal{E}(0,-1)\rightarrow
0.$

A choice of connection $\nabla\in\boldsymbol{D}$ in the c-projective class
determines a Weyl structure which splits these short exact sequences as
follows

$\displaystyle
0\leftarrow\mathcal{E}(-1,0)\xleftarrow{Y_{\mathscr{A}}}\mathcal{E}^{\mathscr{A}}\xleftarrow{W_{a}^{\mathscr{A}}}\mathcal{E}^{a}\otimes_{\mathbb{C}}\mathcal{E}(-1,0)\leftarrow
0$ $\displaystyle
0\leftarrow\mathcal{E}_{a}\otimes_{\mathbb{C}}\mathcal{E}(1,0)\xleftarrow{W_{a}^{\mathscr{A}}}\mathcal{E}_{\mathscr{A}}\xleftarrow{Y_{\mathscr{A}}}\mathcal{E}(1,0)\leftarrow
0$ $\displaystyle
0\leftarrow\mathcal{E}_{\overline{\alpha}}\otimes\mathcal{E}(0,1)\xleftarrow{W_{\overline{\alpha}}^{\overline{A}}}\mathcal{E}_{\overline{A}}\xleftarrow{Y_{\overline{A}}}\mathcal{E}(0,1)\leftarrow
0$ $\displaystyle
0\leftarrow\mathcal{E}_{\alpha}\otimes\mathcal{E}(1,0)\xleftarrow{W_{{\alpha}}^{{A}}}\mathcal{E}_{{A}}\xleftarrow{Y_{{A}}}\mathcal{E}(1,0)\leftarrow
0$ $\displaystyle
0\leftarrow\mathcal{E}(-1,0)\xleftarrow{Y_{A}}\mathcal{E}^{A}\xleftarrow{W^{A}_{\alpha}}\mathcal{E}^{\alpha}\otimes\mathcal{E}(-1,0)\leftarrow
0$ $\displaystyle
0\leftarrow\mathcal{E}(0,-1)\xleftarrow{Y_{\overline{A}}}\mathcal{E}^{\overline{A}}\xleftarrow{W^{\overline{A}}_{\overline{\alpha}}}\mathcal{E}^{\overline{\alpha}}\otimes\mathcal{E}(0,-1)\leftarrow
0.$

These splitting tractors $W$, $X$, $Y$, and $Z$ maps can be viewed as weighted
tractors as follows

$\displaystyle
W_{a}^{\mathscr{A}}\in\Gamma(\mathcal{E}^{\mathscr{A}}\otimes(\mathcal{E}_{a}\otimes_{\mathbb{C}}\mathcal{E}(1,0))\
\ \ \ \ $ $\displaystyle
W_{\overline{\alpha}}^{\overline{A}}\in\Gamma(\mathcal{E}_{\alpha}^{\overline{A}}(0,1))$
$\displaystyle W_{\alpha}^{{A}}\in\Gamma(\mathcal{E}_{\alpha}^{{A}}(1,0)))$
$\displaystyle X^{\mathscr{A}}\in\Gamma(\mathcal{E}^{\mathscr{A}}(1,0))\ \ \ \
\ $ $\displaystyle X^{\overline{A}}\in\Gamma(\mathcal{E}^{\overline{A}}(0,1))$
$\displaystyle X^{{A}}\in\Gamma(\mathcal{E}^{{A}}(1,0))$ $\displaystyle
Y_{\mathscr{A}}\in\Gamma(\mathcal{E}^{\mathscr{A}}(-1,0))\ \ \ \ \ $
$\displaystyle Y_{\overline{A}}\in\Gamma(\mathcal{E}^{\overline{A}}(0,-1))$
$\displaystyle Y_{{A}}\in\Gamma(\mathcal{E}^{{A}}(-1,0))$ $\displaystyle
Z^{a}_{\mathscr{A}}\in\Gamma(\mathcal{E}_{\mathscr{A}}\otimes(\mathcal{E}^{a}\otimes_{\mathbb{C}}\mathcal{E}(-1,0)))\
\ \ \ \ $ $\displaystyle
Z^{\overline{\alpha}}_{\overline{A}}\in\Gamma(\mathcal{E}^{\alpha}_{\overline{A}}(0,-1))$
$\displaystyle Z^{\alpha}_{{A}}\in\Gamma(\mathcal{E}^{\alpha}_{{A}}(-1,0))$

These maps satisfy the obvious relations

Given two splittings (i.e., connections)
$\nabla,\hat{\nabla}\in\boldsymbol{D}$, sections of
$\mathcal{E}^{\mathscr{B}}$ and $\mathcal{E}_{\mathscr{B}}$ change by

(19)
$\displaystyle\widehat{\left(\begin{array}[]{c}\lambda^{b}\otimes_{\mathbb{C}}\sigma\\\
\rho\\\
\end{array}\right)}=\left(\begin{array}[]{c}\lambda^{b}\otimes_{\mathbb{C}}\sigma\\\
\rho-\Upsilon_{b}\lambda^{b}\rho+\Upsilon_{b}J^{b}_{a}\lambda^{a}i\rho\\\
\end{array}\right)$ (24)
$\displaystyle\widehat{\left(\begin{array}[]{c}\upsilon\\\
\nu_{b}\otimes_{\mathbb{C}}\epsilon\\\
\end{array}\right)}=\left(\begin{array}[]{c}\upsilon\\\
\nu_{b}\otimes_{\mathbb{C}}\epsilon+\Upsilon_{b}\otimes_{\mathbb{C}}\epsilon\upsilon+J_{b}^{a}\Upsilon_{a}\otimes_{\mathbb{C}}\epsilon\upsilon
i\\\ \end{array}\right),$

and sections of $\mathcal{E}^{{B}}$ and $\mathcal{E}_{{B}}$ change by

(29) $\displaystyle\widehat{\left(\begin{array}[]{c}\eta^{\beta}\\\ \rho\\\
\end{array}\right)}=\left(\begin{array}[]{c}\eta^{\beta}\\\
\rho-2\Upsilon_{\beta}\eta^{\beta}\\\ \end{array}\right)$ (34)
$\displaystyle\widehat{\left(\begin{array}[]{c}\xi\\\ \mu_{\beta}\\\
\end{array}\right)}=\left(\begin{array}[]{c}\xi\\\
\mu_{\beta}+2\Upsilon_{\beta}\xi\\\ \end{array}\right).$

###  2.7. The c-projective tractor connection

For a choice of splitting, $\nabla\in\boldsymbol{D}$, the tractor connection
on $\mathcal{E}^{\mathscr{B}}$ and $\mathcal{E}_{\mathscr{B}}$ is given by

(39)
$\displaystyle\nabla_{a}{\left(\begin{array}[]{c}\lambda^{b}\otimes_{\mathbb{C}}\sigma\\\
\rho\\\
\end{array}\right)}=\left(\begin{array}[]{c}(\nabla_{a}\lambda^{b})\otimes_{\mathbb{C}}\sigma+\lambda^{b}\otimes_{\mathbb{C}}(\nabla_{a}\sigma)+\delta_{a}^{b}\otimes_{\mathbb{C}}\rho\\\
\nabla_{a}\rho-P_{ab}\lambda^{b}\sigma+P_{ab}J^{b}_{c}\lambda^{c}i\sigma\\\
\end{array}\right)$ (44)
$\displaystyle\nabla_{a}{\left(\begin{array}[]{c}\upsilon\\\
\nu_{b}\otimes_{\mathbb{C}}\epsilon\\\
\end{array}\right)}=\left(\begin{array}[]{c}\nabla_{a}\upsilon-\nu_{a}\epsilon\\\
(\nabla_{a}\nu_{b})\otimes_{\mathbb{C}}\epsilon+\nu_{b}\otimes_{\mathbb{C}}(\nabla_{a}\epsilon)+P_{ab}\otimes_{\mathbb{C}}\epsilon\upsilon-
J_{a}^{c}P_{cb}\otimes_{\mathbb{C}}\epsilon\upsilon i\\\ \end{array}\right),$

and on sections of $\mathcal{E}^{{B}}$ and $\mathcal{E}_{{B}}$ it is given by

(49) $\displaystyle\nabla_{\alpha}{\left(\begin{array}[]{c}\eta^{\beta}\\\
\rho\\\
\end{array}\right)}=\left(\begin{array}[]{c}\nabla_{\alpha}\eta^{\beta}+\delta_{\alpha}^{\beta}\rho\\\
\nabla_{\alpha}\rho-2P_{\alpha\beta}\eta^{\beta}\\\ \end{array}\right)$ (54)
$\displaystyle\nabla_{\alpha}{\left(\begin{array}[]{c}\xi\\\ \mu_{\beta}\\\
\end{array}\right)}=\left(\begin{array}[]{c}\nabla_{\alpha}\xi-\mu_{\alpha}\\\
\nabla_{\alpha}\mu_{\beta}+2P_{\alpha\beta}\xi\\\ \end{array}\right)$ (59)
$\displaystyle\nabla_{\overline{\alpha}}{\left(\begin{array}[]{c}\eta^{\beta}\\\
\rho\\\
\end{array}\right)}=\left(\begin{array}[]{c}\nabla_{\overline{\alpha}}\eta^{\beta}\\\
\nabla_{\overline{\alpha}}\rho-2P_{\overline{\alpha}\beta}\eta^{\beta}\\\
\end{array}\right)$ (64)
$\displaystyle\nabla_{\overline{\alpha}}{\left(\begin{array}[]{c}\xi\\\
\mu_{\beta}\\\
\end{array}\right)}=\left(\begin{array}[]{c}\nabla_{\overline{\alpha}}\xi\\\
\nabla_{\overline{\alpha}}\mu_{\beta}+2P_{\overline{\alpha}\beta}\xi\\\
\end{array}\right),$

where $P_{\alpha\beta}=\Pi^{a}_{\alpha}\Pi^{b}_{\beta}P_{ab}$ and
$P_{\overline{\alpha}\beta}=\overline{\Pi}^{a}_{\overline{\alpha}}\Pi^{b}_{\beta}P_{ab}$.
Using these formulae for the tractor connection a series of straightforward
computations yield the following:

$\displaystyle\nabla_{\gamma}W_{\alpha}^{A}=-2P_{\gamma\alpha}X^{A}$
$\displaystyle\nabla_{\overline{\gamma}}W_{\overline{\alpha}}^{\overline{A}}=-2P_{\overline{\gamma}\overline{\alpha}}X^{\overline{A}}$
$\displaystyle\nabla_{\gamma}W_{\overline{\alpha}}^{\overline{A}}=-2P_{\overline{\gamma}\overline{\alpha}}X^{\overline{A}}$
$\displaystyle\nabla_{\overline{\gamma}}W_{\alpha}^{A}=-2P_{\overline{\gamma}\alpha}X^{A}$
$\displaystyle\nabla_{\gamma}X^{A}=W_{\gamma}^{A}$
$\displaystyle\nabla_{\overline{\gamma}}X^{\overline{A}}=W_{\overline{\gamma}}^{\overline{A}}$
$\displaystyle\nabla_{\gamma}X^{\overline{A}}=0$
$\displaystyle\nabla_{\overline{\gamma}}X^{A}=0$
$\displaystyle\nabla_{\gamma}Y_{A}=2Z_{A}^{\alpha}P_{\gamma\alpha}$
$\displaystyle\nabla_{\overline{\gamma}}Y_{\overline{A}}=2Z_{\overline{A}}^{\overline{\alpha}}P_{\overline{\gamma}\overline{\alpha}}$
$\displaystyle\nabla_{\gamma}Y_{\overline{A}}=2Z_{\overline{A}}^{\overline{\alpha}}P_{\gamma\overline{\alpha}}$
$\displaystyle\nabla_{\overline{\gamma}}Y_{{A}}=2Z_{{A}}^{{\alpha}}P_{\overline{\gamma}{\alpha}}$
$\displaystyle\nabla_{\gamma}Z_{A}^{\alpha}=-\delta^{\alpha}_{\gamma}Y_{A}$
$\displaystyle\nabla_{\overline{\gamma}}Z_{\overline{A}}^{\overline{\alpha}}=-\delta^{\overline{\alpha}}_{\overline{\gamma}}Y_{\overline{A}}$
$\displaystyle\nabla_{\gamma}Z_{\overline{A}}^{\overline{\alpha}}=0$
$\displaystyle\nabla_{\overline{\gamma}}Z_{A}^{\alpha}=0$

as well as

$\displaystyle\nabla_{c}W^{\mathscr{A}}_{a}=-P_{ab}X^{\mathscr{A}}+J^{b}_{c}P_{ba}iX^{A}$
$\displaystyle\nabla_{c}X^{\mathscr{A}}=W^{\mathscr{A}}_{c}$
$\displaystyle\nabla_{c}Y_{\mathscr{A}}=P_{ca}Z_{\mathscr{A}}^{a}-J_{c}^{b}P_{ba}iZ_{\mathscr{A}}^{a}$
$\displaystyle\nabla_{c}Z_{\mathscr{A}}^{a}=-\delta^{a}_{c}Y_{\mathscr{A}}.$

###  2.8. The Thomas D-operator

The Thomas $D$-operator
$D_{\mathscr{A}}:\mathcal{E}^{\bullet}(w,w^{\prime})\rightarrow\mathcal{E}_{\mathscr{A}}^{\bullet}(w-1,w^{\prime})$
is a c-projectively invariant operator. Here $\mathcal{E}^{\bullet}$ denotes
any tractor bundle constructed tensorially out of $\mathcal{E}^{\mathscr{A}}$,
and $\mathcal{E}_{\mathscr{A}}$. For our purposes it will be sufficient to
explicitly describe the action of the Thomas $D$-operator on sections $s$ of
$\mathcal{E}(w,w^{\prime})$. In a splitting the Thomas $D$-operator is given
by

$\displaystyle D_{\mathscr{A}}s\colonequals\left(\begin{array}[]{cc}ws\\\
\nabla_{a}s\\\
\end{array}\right)=Y_{\mathscr{A}}ws+Z^{a}_{\mathscr{A}}\nabla_{a}s.$

###  2.9. The metricity bundle

An almost pseudo-Hermitian manifold is a triple $(M,J,g)$ where $(M,J)$ is an
almost complex manifold and $g$ is a Hermitian metric for $J$, i.e.
$g_{ab}J^{a}_{c}J^{b}_{d}=g_{cd}$. An almost pseudo-Hermitian manifold is
called a pseudo-Hermitain manifold if $J$ is integrable. Recall the Kähler
form of an almost pseudo-Hermitian manifold is the 2-form
$\omega_{ab}=J^{c}_{a}g_{cb}$, which clearly satisfies
$\omega_{ab}J^{a}_{c}J^{b}_{d}=\omega_{cd}$ and
$\omega_{ab}\omega^{bc}=-\delta^{c}_{a}$, where $\Omega^{bc}\colonequals
J^{c}_{a}g^{ba}$ is the Poisson bivector. The almost Hermitian manifold is
said to be almost pseudo-Kähler if the Kähler form, $\omega$, is closed. We
can also view a Hermitian metric $g_{ab}$ as a real non-degenerate section
$g_{\alpha\overline{\beta}}=\Pi_{\alpha}^{a}\overline{\Pi}^{b}_{\overline{\beta}}g_{ab}$
of $\mathcal{E}_{\alpha{\overline{\beta}}}$.

Now we consider the bundle $\mathcal{E}^{A\overline{B}}$ and its real
subbundles $\mathcal{H}^{*}$ (which, following [15], we term the metricity
bundle) and its skew counterpart $\mathcal{P}^{*}$. $\mathcal{H}^{*}$ and
$\mathcal{P}^{*}$ can also be viewed as subbundles of
$\mathcal{E}^{(\mathscr{AB})}$ and $\mathcal{E}^{\mathscr{[AB]}}$,
respectively, whose sections are Hermitian with respect to the almost complex
structure, $J^{\mathscr{A}}_{\mathscr{B}}$, on the tractor bundle
$\mathcal{E}^{\mathscr{A}}$. Note that $J^{\mathscr{A}}_{\mathscr{B}}$ gives
an isomorphism between $\mathcal{H}^{*}$ and $\mathcal{P}^{*}$. In a splitting
we have

$\displaystyle\mathcal{E}^{A\overline{B}}$
$\displaystyle=\mathcal{E}^{\alpha\overline{\beta}}(-1,-1)\oplus{\mathcal{E}^{\alpha}(-1,-1)}\oplus\mathcal{E}^{\overline{\beta}}(-1,-1)\oplus\mathcal{E}(-1,-1)$
$\displaystyle\mathcal{H}^{*}$
$\displaystyle=\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}\oplus\mathcal{E}^{a}(-1,-1)_{\mathbb{R}}\oplus\mathcal{E}(-1,-1)_{\mathbb{R}},$
$\displaystyle\mathcal{P}^{*}$
$\displaystyle=\operatorname{SkewHerm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}\oplus\mathcal{E}^{a}(-1,-1)_{\mathbb{R}}\oplus\mathcal{E}(-1,-1)_{\mathbb{R}}$

where $\operatorname{Herm}(E)$ and $\operatorname{SkewHerm}(E)$ denotes the
bundle of Hermitain metrics and Hermitian forms on a vector bundle $E$,
respectively. We write sections
$h^{A\overline{B}}\in\Gamma(\mathcal{E}^{A\overline{B}})$,
$h^{\mathscr{AB}}\in\Gamma(\mathcal{H}^{*})$, and
$p^{\mathscr{AB}}\in\Gamma(\mathcal{P}^{*})$ as

$\displaystyle
h^{A\overline{B}}=\left(\begin{array}[]{cc}\zeta^{\alpha\overline{\beta}}\\\
\lambda^{\alpha}\ \ \ |\ \ \ \mu^{\overline{\beta}}\\\ \nu\\\
\end{array}\right)\ \ \ \ \ \ \
h^{\mathscr{A}\mathscr{B}}=\left(\begin{array}[]{cc}\zeta^{ab}\\\
\lambda^{c}\\\ \nu\\\ \end{array}\right)\ \ \ \ \ \ \
p^{\mathscr{A}\mathscr{B}}=\left(\begin{array}[]{cc}\pi^{ab}\\\ \iota^{c}\\\
\nu\\\ \end{array}\right)$

where we can identify the slots of $h^{\mathscr{A}\mathscr{B}}$ with real
slots of $\mathcal{E}^{A\overline{B}}$:

$\displaystyle\overline{\zeta^{\overline{\gamma}\beta}}=\zeta^{\overline{\beta}\gamma},\
\ \ \ \ \ \overline{\lambda^{\alpha}}=\mu^{\overline{\alpha}},\ \ \ \ \ \
\operatorname{and}\ \ \ \ \ \ \overline{\nu}=\nu.$

We also see that the slots of $h^{\mathscr{A}\mathscr{B}}$ are related to the
slots of
$p^{\mathscr{A}\mathscr{B}}=J^{\mathscr{B}}_{\mathscr{C}}h^{\mathscr{A}\mathscr{C}}$
by

$\displaystyle\pi^{ab}=J^{b}_{c}\zeta^{ca},\ \ \ \ \ \ \operatorname{and}\ \ \
\ \ \ \iota^{a}=J^{a}_{b}\lambda^{b}.$

We will also need to work with the dual bundles, namely
$\mathcal{E}_{A\overline{B}}$ and its real subbundles $\mathcal{H}$ and
$\mathcal{P}$. In a splitting these decompose into the following direct sums

$\displaystyle\mathcal{E}_{A\overline{B}}$
$\displaystyle=\mathcal{E}_{\alpha\overline{\beta}}(1,1)\oplus{\mathcal{E}_{\alpha}(1,1)}\oplus\mathcal{E}_{\overline{\beta}}(1,1)\oplus\mathcal{E}(1,1)$
$\displaystyle\mathcal{H}$
$\displaystyle=\operatorname{Herm}(TM)\otimes\mathcal{E}(1,1)_{\mathbb{R}}\oplus\mathcal{E}_{a}(1,1)_{\mathbb{R}}\oplus\mathcal{E}(1,1)_{\mathbb{R}},$
$\displaystyle\mathcal{P}$
$\displaystyle=\operatorname{SkewHerm}(TM)\otimes\mathcal{E}(1,1)_{\mathbb{R}}\oplus\mathcal{E}_{a}(1,1)_{\mathbb{R}}\oplus\mathcal{E}(1,1)_{\mathbb{R}}.$

and we write sections
$h_{A\overline{B}}\in\Gamma(\mathcal{E}_{A\overline{B}})$ and
$h_{\mathscr{AB}}\in\Gamma(\mathcal{H}^{*})$ as

$\displaystyle
h_{A\overline{B}}=\left(\begin{array}[]{cc}\varphi_{\alpha\overline{\beta}}\\\
\lambda_{\alpha}\ \ \ |\ \ \ \mu^{\overline{\beta}}\\\ \tau\\\
\end{array}\right)\ \ \ \ \ \ \
h_{\mathscr{A}\mathscr{B}}=\left(\begin{array}[]{cc}\varphi_{ab}\\\
\lambda_{c}\\\ \tau\\\ \end{array}\right)$

The formulae for the tractor connection applied to $\mathcal{H}^{*}$ and
$\mathcal{H}$, respectively, are given by:

$\displaystyle\nabla_{c}^{\mathcal{T}}{h^{\mathscr{A}\mathscr{B}}}$
$\displaystyle=\nabla_{c}^{\mathcal{T}}\left(\begin{array}[]{cc}\zeta^{a{b}}\\\
\lambda^{a}\\\ \nu\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{c}\zeta^{a{b}}+\delta_{c}^{(a}\lambda^{{b)}}+J_{c}^{(a}J_{i}^{b)}\lambda^{i}\\\
\nabla_{c}\lambda^{a}+2\delta^{a}_{c}\nu-2P_{c{b}}\zeta^{a{b}}\\\
\nabla_{c}\nu-P_{cb}\lambda^{b}\\\ \end{array}\right),$
$\displaystyle\nabla_{c}^{\mathcal{T}}{h_{\mathscr{A}\mathscr{B}}}$
$\displaystyle=\nabla_{c}^{\mathcal{T}}\left(\begin{array}[]{cc}\tau\\\
\lambda_{a}\\\ \varphi_{ab}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{c}\tau-2\lambda_{c}\\\
\nabla_{c}\lambda_{a}+P_{ca}\tau-\varphi_{ca}\\\
\nabla_{c}\varphi_{ab}+2P_{c(b}\lambda_{a)}+2P_{ci}\lambda_{j}J^{i}_{(a}J^{j}_{b)}\\\
\end{array}\right).$

We can pass from sections of $h_{\mathscr{AB}}\in\Gamma(\mathcal{H}^{*})$ to
sections of $h_{A\overline{B}}\in\Gamma(\mathcal{E}_{A\overline{B}})$, or vice
versa, via the $\Pi$ and $\overline{\Pi}$ maps discussed earlier. The formulae
for the tractor connection applied to sections of
$\mathcal{E}^{A\overline{B}}(-1,-1)$ are given by:

$\displaystyle\nabla_{\gamma}^{\mathcal{T}}{h^{A\overline{B}}}$
$\displaystyle=\nabla_{\gamma}^{\mathcal{T}}\left(\begin{array}[]{cc}\zeta^{\alpha\overline{\beta}}\\\
\lambda^{\alpha}\ \ \ |\ \ \ \mu^{\overline{\alpha}}\\\ \nu\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{\gamma}\zeta^{\alpha\overline{\beta}}+\delta_{\gamma}^{\alpha}\mu^{\overline{\beta}}\\\
\nabla_{\gamma}\lambda^{\alpha}+2\delta^{\alpha}_{\gamma}\nu-2P_{\gamma\overline{\beta}}\zeta^{\alpha\overline{\beta}}\
\ \ |\ \ \
\nabla_{\gamma}\mu^{\overline{\beta}}-2P_{\gamma\beta}\zeta^{\beta\overline{\alpha}}\\\
\nabla_{\gamma}\nu-
P_{\gamma\overline{\alpha}}\mu^{\overline{\alpha}}-P_{\alpha\beta}\lambda^{\beta}\\\
\end{array}\right),$
$\displaystyle\nabla_{\overline{\gamma}}^{\mathcal{T}}{h^{A\overline{B}}}$
$\displaystyle=\nabla_{\overline{\gamma}}^{\mathcal{T}}\left(\begin{array}[]{cc}\zeta^{\alpha\overline{\beta}}\\\
\lambda^{\alpha}\ \ \ |\ \ \ \mu^{\overline{\alpha}}\\\ \nu\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{\overline{\gamma}}\zeta^{\alpha\overline{\beta}}+\lambda^{{\alpha}}\delta_{\overline{\gamma}}^{\overline{\beta}}\\\
\nabla_{\gamma}\lambda^{\alpha}-2P_{\overline{\gamma}\overline{\beta}}\zeta^{\alpha\overline{\beta}}\
\ \ |\ \ \
\nabla_{\overline{\gamma}}\mu^{\overline{\alpha}}+2\nu\delta_{\overline{\gamma}}^{\overline{\alpha}}-P_{\overline{\gamma}\beta}\zeta^{\beta\overline{\alpha}}\\\
\nabla_{\overline{\gamma}}\nu-2P_{\overline{\gamma}\overline{\alpha}}\mu^{\overline{\alpha}}-2P_{\overline{\gamma}\overline{\beta}}\lambda^{\beta}\\\
\end{array}\right).$

The formulae for the tractor connection applied to sections of
$\mathcal{E}_{A\overline{B}}(1,1)$ are given by:

$\displaystyle\nabla_{\gamma}^{\mathcal{T}}{h_{A\overline{B}}}$
$\displaystyle=\nabla_{\gamma}^{\mathcal{T}}\left(\begin{array}[]{cc}\varphi_{\alpha\overline{\beta}}\\\
\lambda_{\alpha}\ \ \ |\ \ \ \mu_{\overline{\alpha}}\\\ \tau\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{\gamma}\tau-\lambda_{\gamma}\\\
\nabla_{\gamma}\lambda_{\alpha}+2P_{\gamma\alpha}\tau\ \ \ |\ \ \
\nabla_{\gamma}\mu_{\overline{\beta}}+2P_{\gamma\overline{\alpha}}\tau-\varphi_{\gamma\overline{\alpha}}\\\
\nabla_{\gamma}\varphi_{\alpha\overline{\beta}}+P_{\gamma\overline{\beta}}\lambda_{\alpha}+P_{\gamma\alpha}\mu_{\overline{\beta}}\\\
\end{array}\right),$
$\displaystyle\nabla_{\overline{\gamma}}^{\mathcal{T}}{h_{A\overline{B}}}$
$\displaystyle=\nabla_{\overline{\gamma}}^{\mathcal{T}}\left(\begin{array}[]{cc}\tau\\\
\lambda_{\alpha}\ \ \ |\ \ \ \mu_{\overline{\alpha}}\\\
\varphi_{\alpha\overline{\beta}}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{\overline{\gamma}}\tau-\mu_{\overline{\gamma}}\\\
\nabla_{\gamma}\lambda_{\alpha}+2P_{\overline{\gamma}\alpha}\tau-\varphi_{\alpha\overline{\gamma}}\
\ \ |\ \ \
\nabla_{\overline{\gamma}}\mu_{\overline{\alpha}}+2P_{\overline{\alpha}\overline{\gamma}}\tau\\\
\nabla_{\overline{\gamma}}\varphi_{\alpha\overline{\beta}}+P_{\overline{\gamma}\overline{\beta}}\lambda_{\alpha}+P_{\overline{\gamma}\alpha}\mu_{\overline{\beta}}\\\
\end{array}\right).$

###  2.10. C-projective BGG equations

Given a Cartan geometry $(\mathcal{P}\twoheadrightarrow M,\omega)$ of type
$(G,P)$ and a $G$-representation $\mathbb{V}$, we form a tractor bundle
$\mathcal{V}=\mathcal{P}\times_{P}\mathbb{V}$. Then, via the corresponding
tractor connection, we can form the exterior covariant derivative,
$d^{\nabla}$, on $\mathcal{V}$-valued forms to obtain the de Rham sequence
twisted by $\mathcal{V}$.

$0\xrightarrow{}\mathcal{V}\xrightarrow{d^{\nabla}}\mathcal{V}\otimes\mathcal{E}_{a}\xrightarrow{d^{\nabla}}\mathcal{V}\otimes\mathcal{E}_{[ab]}\xrightarrow{d^{\nabla}}...$

Then, via the canonical map

$\dagger:\mathcal{E}_{a}\rightarrow\operatorname{End}(\mathcal{V}),\ \ \ \
\operatorname{given\ explicitly\ by}\ \ \ \ \alpha_{a}\mapsto
X^{\mathscr{B}}Z_{\mathscr{A}}^{a}\alpha_{a}$

in the case when $\mathcal{V}=\mathcal{E}^{\mathscr{A}}$ , one can construct a
special case of the Kostant codifferential $\partial^{*}$, that gives a
complex of natural bundle maps on $\mathcal{V}$-valued differential forms
going in the opposite direction to the twisted de Rham sequence,

$0\xleftarrow{\partial^{*}}\mathcal{V}\xleftarrow{\partial^{*}}\mathcal{V}\otimes\mathcal{E}_{a}\xleftarrow{\partial^{*}}\mathcal{V}\otimes\mathcal{E}_{[ab]}\xleftarrow{\partial^{*}}...$

The homology of this sequence gives natural subquotient bundles

$\displaystyle
H_{k}(M,\mathcal{V})\colonequals\operatorname{ker}(\partial^{*})/\operatorname{im}(\partial^{*}).$

There are natural bundle projections
$\Pi_{k}:\operatorname{ker}(\partial^{*})\subseteq\mathcal{V}\otimes\mathcal{E}_{[ab...c]}\rightarrow
H_{k}(M,\mathcal{V})$, from the indicated $\mathcal{V}$-valued $k$-forms to
the $k$th BGG homology. Given a smooth section $\rho$ of
$H_{k}(M,\mathcal{V})$ there is a unique smooth section $L_{k}(\rho)$ of
$\operatorname{ker}(\partial^{*})\subseteq\mathcal{V}\otimes\mathcal{E}_{[a...b]}$
such that $\Pi_{k}(L_{k}(\rho))=\rho$ and
$\partial^{*}(d^{\nabla^{\mathcal{V}}}(L_{k}(\rho)))$$=0$. This characterizes
a projectively invariant differential operator $L$ called the BGG splitting
operator, or just the splitting operator. We can then define the $k$th BGG
operator $\Theta_{k}:H_{k}(M,\mathcal{V})\rightarrow H_{k+1}(M,\mathcal{V})$
by $\rho\mapsto\Pi_{k+1}(d^{\nabla^{\mathcal{V}}}(L_{k}(\rho)))$. It follows
from these definitions that parallel sections of $\mathcal{V}$ are equivalent
to (via $\Pi_{0}$ and $L_{0}$) a special class of so-called normal solutions
of the first BGG operator $\Theta_{0}:H_{0}(M,\mathcal{V})\rightarrow
H_{1}(M,\mathcal{V})$ associated with $\mathcal{V}$. Equations induced on the
sections of $H_{0}(M,\mathcal{V})$ by the BGG operator $\Theta_{0}$ are known
as (first) BGG equations. Note that the BGG sequence, given by the BGG
operators, is not a complex in general, unless the connection
$\nabla^{\mathcal{V}}$ is flat. Note that a parallel section of a tractor
bundle is necessarily in the image of the splitting operator. Next, we
determine the first BGG equation and splitting operator corresponding to the
c-projective metricity bundle.

######  Proposition 2.2.

Let $(M,J,\boldsymbol{D})$ be an almost c-projective manifold. The first BGG
operator $\Theta_{0}:H_{0}(M,\mathcal{H}^{*})\rightarrow
H_{1}(M,\mathcal{H}^{*})$, induces the following projectively invariant first
order equation on
$\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}$,

(65)
$\displaystyle\nabla_{c}\zeta^{ab}-\frac{1}{m}\delta_{c}^{(a}\nabla_{d}\zeta^{b)d}-\frac{1}{m}J_{c}^{(b}J^{a)}_{e}\nabla_{d}\zeta^{ed}=0.$

###### Proof.

Let $h^{\mathscr{AB}}\in\Gamma(\mathcal{E}^{(\mathscr{AB})})$. Then we compute
$\nabla_{c}^{\mathcal{T}}{h^{\mathscr{AB}}}$.

$\displaystyle\nabla_{c}^{\mathcal{T}}{h^{\mathscr{AB}}}$
$\displaystyle=\nabla_{c}^{\mathcal{T}}\left(\begin{array}[]{cc}\zeta^{ab}\\\
\lambda^{a}\\\ \rho\\\
\end{array}\right)=\left(\begin{array}[]{cc}\nabla_{c}\zeta^{ab}+\delta_{c}^{(a}\lambda^{b)}+J_{c}^{(a}J^{b)}_{i}\lambda^{i}\\\
\nabla_{c}\lambda^{a}+2\delta^{a}_{c}\rho-2P_{cb}\zeta^{ab}\\\ \nabla_{c}\rho-
P_{ca}\lambda^{a}\\\ \end{array}\right).$

Then $\partial^{*}(\nabla_{c}^{\mathcal{T}}h^{\mathscr{AB}})=0$ tells us that
the slots of
$Z_{\mathscr{D}}^{c}X^{(\mathscr{A}}\nabla_{c}^{\mathcal{T}}h^{\mathscr{B})\mathscr{D}}$
are trace-free111See the proof of Proposition 14 of [15] for a more details.,
i.e. we have the following system of equations:

(66) $\displaystyle\operatorname{trace}(\nabla_{c}\zeta^{ab}$
$\displaystyle+\delta_{c}^{(a}\lambda^{b)}+J_{c}^{(a}J^{b)}_{i}\lambda^{i})=0,$
$\displaystyle\operatorname{trace}(\nabla_{c}\lambda^{a}$
$\displaystyle-2P_{cb}\zeta^{ab}+2\delta^{a}_{c}\rho)=0.$

Therefore,

$\displaystyle\lambda^{a}$ $\displaystyle=\frac{-1}{m}\nabla_{b}\zeta^{ab},$
$\displaystyle\rho$
$\displaystyle=\frac{1}{2m}P_{ba}\zeta^{ab}+\frac{1}{4m^{2}}\nabla_{a}\nabla_{b}\zeta^{ab}.$

Thus a Hermitian form, $h$, on the cotractor bundle in the image of the
splitting operator is of the form

$\displaystyle{h^{\mathscr{AB}}}=L(\zeta^{ab})=\left(\begin{array}[]{cc}\zeta^{ab}\\\
\frac{-1}{m}\nabla_{b}\zeta^{ab}\\\
\frac{1}{2m}P_{ba}\zeta^{ab}+\frac{1}{4m^{2}}\nabla_{a}\nabla_{b}\zeta^{ab}\\\
\end{array}\right).$

Substituting gives the following first-order BGG equation on
$\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}$

(67) $\displaystyle\operatorname{trace-
free}(\nabla_{c}\zeta^{ab})=0\Longleftrightarrow\nabla_{c}\zeta^{ab}-\frac{1}{m}\delta_{c}^{(a}\nabla_{d}\zeta^{b)d}-\frac{1}{m}J_{c}^{(b}J^{a)}_{e}\nabla_{d}\zeta^{ed}=0.$

C-projective invariance follows from a straightforward computation.
$H_{0}(M,\mathcal{H}^{*})=\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}$
follows from applications of the general BGG machinery of [22] and this
particular case is treated in Theorem 3.3 of [19]. So we have given the
explicit form of
$\Theta_{0}(\zeta^{ab})\colonequals\Pi_{1}(d^{\nabla}L(\zeta^{ab}))=0$, which
is the c-projective metrizability equation (65). ∎

Applying the procedure above to the bundle $\mathcal{H}$ yields
$H_{0}(M,\mathcal{H})=\mathcal{E}(1,1)$ and also gives an explicit formula for
the splitting operator $L:\mathcal{E}(1,1)\rightarrow\mathcal{H}$. For later
reference, we give the formulae (cf. with Section 3.5 of [15]) for the BGG
splitting operators mapping into $\mathcal{H}^{*}$ and $\mathcal{H}$,
respectively, in the following corollary,

######  Corollary 2.3.

Let $\tau\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$ and
$\zeta\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}})$.
Then their images under their respective splitting operators, both denoted by
$L$, are given by

$\displaystyle L(\tau)$ $\displaystyle=\left(\begin{array}[]{cc}\tau\\\
\frac{1}{2}\nabla_{a}\tau\\\
\frac{1}{2}(\delta^{i}_{(b}\delta^{j}_{c)}+J^{i}_{(b}J^{j}_{c)})(\frac{1}{2}\nabla_{i}\nabla_{j}\tau+P_{ij}\tau)\\\
\end{array}\right)$

and

$\displaystyle L(\zeta^{ab})$
$\displaystyle=\left(\begin{array}[]{cc}\zeta^{ab}\\\
\frac{-1}{m}\nabla_{i}\zeta^{ic}\\\
\frac{1}{2m}P_{ij}\zeta^{ij}+\frac{1}{4m^{2}}\nabla_{i}\nabla_{j}\zeta^{ij}\\\
\end{array}\right).$

###  2.11. Determinants

We now describe several methods of taking determinants which are relevant to
our purposes. Let
$\mathcal{\epsilon}_{\alpha_{1}\cdots\alpha_{m}}\in\Gamma(\mathcal{E}_{[\alpha_{1}\cdots\alpha_{m}]}(m+1,0))$
denote the canonical section giving the identification
$\mathcal{E}^{[\alpha_{1}\cdots\alpha_{m}]}\xrightarrow{\sim}\mathcal{E}(m+1,0)$.
Then we get a well-defined notion of determinant for sections of
$\mathcal{E}^{\alpha\overline{\beta}}(k,k)$ via the map:

$\displaystyle\textbf{det}:\mathcal{E}^{\alpha\overline{\beta}}(k,k)$
$\displaystyle\rightarrow\mathcal{E}(km+m+1,km+m+1)$
$\displaystyle\sigma^{\alpha\overline{\beta}}$
$\displaystyle\mapsto\frac{1}{m!}{\epsilon}_{\alpha_{1}\cdots\alpha_{m}}\epsilon_{\overline{\beta}_{1}\cdots\overline{\beta}_{m}}\sigma^{\alpha_{1}\overline{\beta}_{1}}\cdots\sigma^{\alpha_{m}\overline{\beta}_{m}}.$

For Hermitian sections of $\mathcal{E}^{\alpha\overline{\beta}}(k,k)$, det is
valued in $\mathcal{E}(km+m+1,km+m+1)_{\mathbb{R}}$. The parallel
c-projectively invariant tractor

$\displaystyle\epsilon_{A_{0}\cdots
A_{m}\overline{B}_{0}\cdots\overline{B}_{m}}\colonequals{\epsilon}_{\alpha_{1}\cdots\alpha_{m}}\epsilon_{\overline{\beta}_{1}\cdots\overline{\beta}_{m}}Y_{[A_{0}}Z^{\alpha_{1}}_{A_{1}}\cdots
Z^{\alpha_{m}}_{A_{m}]}Y_{[\overline{B}_{0}}Z^{\overline{\beta}_{1}}_{\overline{B}_{1}}\cdots
Z^{\overline{\beta}_{m}}_{\overline{B}_{m}]},$

which is the (complex) c-projective tractor volume form, provides a method for
taking determinants of sections of $\mathcal{E}^{A\overline{B}}$ as follows,

$\displaystyle\operatorname{det}:\mathcal{E}^{A\overline{B}}$
$\displaystyle\rightarrow\mathcal{E}(0,0)$ $\displaystyle h^{A\overline{B}}$
$\displaystyle\mapsto\frac{1}{(m+1)!}\epsilon_{A_{0}\cdots\
A_{m}\overline{B}_{0}\cdots\ \overline{B}_{m}}h^{A_{0}\overline{B}_{0}}\cdots
h^{A_{m}\overline{B}_{m}}.$

This determinant is real-valued, i.e., valued in
$\mathcal{E}(0,0)_{\mathbb{R}}$, for Hermitian sections of
$\mathcal{E}^{A\overline{B}}$.

Now, letting $\epsilon_{a_{1}\cdots
a_{m}}\colonequals\Pi^{\alpha_{1}}_{a_{1}}\cdots\Pi^{\alpha_{m}}_{a_{m}}\epsilon_{\alpha_{1}\cdots\alpha_{m}}$
and $\overline{\epsilon}_{b_{1}\cdots
b_{m}}\colonequals\overline{\Pi}^{\overline{\beta}_{1}}_{b_{1}}\cdots\overline{\Pi}^{\overline{\beta}_{m}}_{b_{m}}\epsilon_{\overline{\beta}_{1}\cdots\overline{\beta}_{m}}$,
we define

$\epsilon^{2}_{a_{1}\cdots a_{m}b_{1}\cdots
b_{m}}\colonequals{\epsilon}_{[a_{1}\cdots
a_{m}}\overline{\epsilon}_{b_{1}\cdots
b_{m}]}\in\Gamma(\mathcal{E}_{[a_{1}\cdots a_{m}b_{1}\cdots
b_{m}]}\otimes\mathcal{E}(m+1,m+1)_{\mathbb{R}}).$

So $\epsilon^{2}_{a_{1}\cdots a_{m}b_{1}\cdots b_{m}}$ is the canonical
section identifying oriented real line bundles $\mathcal{E}^{[a_{1}\cdots
a_{m}b_{1}\cdots b_{m}]}\xrightarrow{\sim}\mathcal{E}(m+1,m+1)_{\mathbb{R}}$.
Observe that this volume form gives a notion of determinant on sections of
$\operatorname{SkewHerm}(T^{*}M)\otimes\mathcal{E}(k,k)_{\mathbb{R}}$ defined
by,

$\displaystyle\textbf{det}:\operatorname{SkewHerm}(T^{*}M)\otimes\mathcal{E}(k,k)_{\mathbb{R}}$
$\displaystyle\rightarrow\mathcal{E}(km+m+1,km+m+1)_{\mathbb{R}}$
$\displaystyle\pi^{a{b}}$
$\displaystyle\mapsto\frac{1}{m!}\epsilon^{2}_{a_{1}\cdots a_{m}b_{1}\cdots
b_{m}}\pi^{a_{1}{b}_{1}}\cdots\pi^{a_{m}{b}_{m}}.$

Since $J$ identifies $\operatorname{SkewHerm}(T^{*}M)$ with
$\operatorname{Herm}(T^{*}M)$, we can pull the determinant back to
$\operatorname{Herm}(T^{*}M)$. That is, let
$\textbf{det}(\sigma^{ab})\colonequals\textbf{det}(\pi^{ab})$ where
$\pi^{ab}=J^{b}_{c}\sigma^{ac}$. Then, we define the (real) c-projective
tractor volume form by

$\displaystyle\epsilon_{\mathscr{A}_{0}\cdots\
\mathscr{A}_{m}\mathscr{B}_{0}\cdots\
\mathscr{B}_{m}}\colonequals\epsilon^{2}_{a_{1}\cdots a_{m}b_{1}\cdots
b_{m}}\Pi^{A_{0}}_{[\mathscr{A}_{0}}\cdots\overline{\Pi}^{\overline{B}_{m}}_{\mathscr{B}_{m}]}\Pi_{\alpha_{1}}^{a_{1}}\cdots\overline{\Pi}_{\overline{\beta}_{m}}^{b_{m}}Y_{[A_{0}}Z^{\alpha_{1}}_{A_{1}}\cdots
Z^{\alpha_{m}}_{A_{m}]}Y_{[\overline{B}_{0}}Z^{\overline{\beta}_{1}}_{\overline{B}_{1}}\cdots
Z^{\overline{\beta}_{m}}_{\overline{B}_{m}]}.$

Since $Y_{[A_{0}}Z^{\alpha_{1}}_{A_{1}}\cdots
Z^{\alpha_{m}}_{A_{m}]}Y_{[\overline{B}_{0}}Z^{\overline{\beta}_{1}}_{\overline{B}_{1}}\cdots
Z^{\overline{\beta}_{m}}_{\overline{B}_{m}]}$, and hence
$\epsilon_{A_{0}\cdots A_{m}\overline{B}_{0}\cdots\overline{B}_{m}}$, is fixed
under conjugation, it follows that (2.11) is indeed a section of
$(\Lambda_{\mathbb{R}}^{2m+2}\mathcal{T}^{*})$. Thus it provides a notion of
determinant on the real subbundle $\mathcal{P}^{*}$ of
$\mathcal{E}^{A\overline{B}}$ via

$\displaystyle\operatorname{det}:\mathcal{P^{*}}$
$\displaystyle\rightarrow\mathcal{E}(0,0)_{\mathbb{R}}$ $\displaystyle
p^{\mathscr{AB}}$
$\displaystyle\mapsto\frac{1}{(m+1)!}\epsilon_{\mathscr{A}_{0}\cdots\
\mathscr{A}_{m}\mathscr{B}_{0}\cdots\
\mathscr{B}_{m}}p^{\mathscr{A}_{0}\mathscr{B}_{0}}\cdots
p^{\mathscr{A}_{m}\mathscr{B}_{m}}.$

Since $J$ (viewed as a complex structure at the tractor bundle level)
identifies $\mathcal{P}^{*}$ with $\mathcal{H}^{*}$, we pull the determinant
back to $\mathcal{H}^{*}$. That is, we let
$\operatorname{det}(h^{\mathscr{AB}})\colonequals\operatorname{det}(p^{\mathscr{AB}})$
where $p^{\mathscr{AB}}=J^{\mathscr{B}}_{\mathscr{C}}h^{\mathscr{AC}}$.

###  2.12. Scalar curvature

Let $(M,J)$ be an almost complex manifold equipped with an admissible
Hermitian pseudo-Riemannian metric $g$. The volume form for $g$,
$\operatorname{vol}_{g}\in\Gamma(\mathcal{E}(-2m-2))$, is parallel for any
affine connection $\nabla$ preserving $g$ and hence any root of
$\operatorname{vol}_{g}$ is parallel for $\nabla$ as well. In particular,
$\tau\colonequals\operatorname{vol}_{g}^{-\frac{1}{m+1}}\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$
is parallel for the canonical connection $\nabla^{g}$ of $g$. It follows that
$\zeta^{ab}\colonequals\tau^{-1}g^{ab}\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-2))$
is a solution to the metrizability equation. Further, in the splitting
determined by the canonical connection for $g$ we see that

$\displaystyle L(\zeta^{ab})=\left(\begin{array}[]{cc}\zeta^{ab}\\\ 0\\\
\frac{1}{2m}\zeta^{ij}P_{ij}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\tau^{-1}g^{ab}\\\ 0\\\
\frac{\tau^{-1}}{2m}g^{ij}P_{ij}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\tau^{-1}g^{ab}\\\ 0\\\
\frac{\tau^{-1}R^{g}}{4m(m+1)}\\\ \end{array}\right),$

where $R^{g}$ denotes the scalar curvature of $g$. Then, up to a constant
multiple, the determinant of $h^{\mathscr{AB}}=L(\tau^{-}1g^{ab})$ agrees with
the scalar curvature $R^{g}$ of $g$, as was observed in Proposition 15 of [15]
and (in the special case of parallel sections of
$\mathcal{E}^{A\overline{B}}$) in Proposition 4.8 of [12]. If $(M,J)$ can be
realized as the interior of manifold with boundary $\overline{M}$ such that
the c-projective structure of $\nabla^{g}$ admits a smooth extension to
$\overline{M}$, then $\tau^{-1}g^{ab}$ and $R^{g}$ can be extended from the
interior to $\overline{M}$, for details see Corollary 16 of [15]. This is
closely related to the questions we consider in section 3.

##  3\. Induced stratifications

We will show that given nondegeneracy of $L(\tau)$ or $L(\zeta)$ where
$\tau\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$ and
$\zeta\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}})$
is a solution to the metrizability equation (65) induces a stratification of
the underlying almost c-projective manifold in analogous fashion to the
projective cases considered in [31]. The following theorems can be viewed as
generalizations of curved orbit decomposition result of Theorem 3.3 in [19]
where we are primarily using the more hands-on machinery developed in [15].

######  Theorem 3.1.

Let $(M,J,\boldsymbol{D})$ be an almost c-projective manifold with real
dimension $2n$ equipped with a real density
$\tau\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$ such that
$L(\tau)\in\Gamma(\mathcal{H})$ is non-degenerate as a Hermitian form on the
tractor bundle. If $L(\tau)$ is definite then the zero locus
$\mathcal{Z}(\tau)$ is empty and $(M,J,\boldsymbol{D},g)$ is Hermitian with
metric $g_{bc}=(\delta^{i}_{(b}\delta^{j}_{c)}+J^{i}_{(b}J^{j}_{c)})P_{ij}$,
which is not, in general, admissible. If $L(\tau)$ has indefinite signature
then $\mathcal{Z}(\tau)$ is either empty or it is a smoothly embedded
separating real hypersurface such that the following hold:

  1. (i)

$M$ is stratified by the strict sign of $\tau$ with curved orbit decomposition
given by

$M=\coprod\limits_{i\in\\{+,0,-\\}}M_{i}$

where $\tau$ is positive, zero ,and negative on $M_{+}$, $M_{0}$, and $M_{-}$,
respectively.

  2. (ii)

If $M$ is closed, then the open components $(M\backslash
M_{\mp},J,\boldsymbol{D})$ are c-projective compactifications of
$(M_{\pm},J,\nabla^{\tau})$, with boundary $M_{0}$.

  3. (iii)

The open components $(M\backslash M_{\mp},J,g)$ are pseudo-Hermitian with
metric $g_{bc}=(\delta^{i}_{(b}\delta^{j}_{c)}+J^{i}_{(b}J^{j}_{c)})P_{ij}$.
The metric $g$ is not admissible for $\boldsymbol{D}$ in general, but if
$L(\tau)$ is parallel, then it is admissible and further $g$ is Kähler-
Einstein.

  4. (iv)

$M_{0}$ inherits a (possibly degenerate) almost CR structure of hypersurface
type.

###### Proof.

(i) Since $L(\tau)$ is non-degenerate observe that
$\mathcal{Z}(\tau)\cap\mathcal{Z}(\nabla\tau)=\varnothing$. The implicit
function theorem implies then that $\mathcal{Z}(\tau)$ is a smoothly embedded
real hypersurface, which is necessarily separating since $\nabla\tau\neq 0$ on
$\mathcal{Z}(\tau)$.
(ii) Since $\tau$ is a defining density for $M_{0}$ it follows from
Proposition 2.1 that $(M\backslash M_{\mp},J,\boldsymbol{D})$ are c-projective
compactifications of $(M_{\pm},J,\nabla^{\tau})$, with boundary $M_{0}$.
(iii) Away from $\mathcal{Z}(\tau)$, in the splitting $\nabla^{\tau}$, we see
that
$L(\tau)=(\tau,0,(\delta^{i}_{(b}\delta^{j}_{c)}+J^{i}_{(b}J^{j}_{c)})P_{ij})$.
Nondegeneracy of $L(\tau)$ implies that the Hermitian form
$g_{bc}=(\delta^{i}_{(b}\delta^{j}_{c)}+J^{i}_{(b}J^{j}_{c)})P_{ij}$ is itself
non-degenerate, and hence a Hermitian metric, away from $\mathcal{Z}(\tau)$.
If the bottom slot of $\nabla L(\tau)$ vanishes in the splitting
$\nabla^{\tau}$, then $g$ is necessarily admissible and hence quasi-Kähler by
our discussion in Section 2.3222 For details see Proposition 4.1 of [12] or
Proposition 7 of [15].. In particular, if $L(\tau)$ is parallel, then $g$ is
admissible.
(iv) Observe that, for $x\in M_{0}$, $H_{x}\colonequals T_{x}M_{0}\cap
J(T_{x}M_{0})$ defines a corank one smooth distribution $H\subset TM_{0}$ and
the pullback $i^{*}J$ of the almost complex structure along the inclusion
$i:M_{0}\hookrightarrow M$ defines an almost complex structure on $H$. Thus
$(M_{0}^{2n-1},H,i^{*}J)$ is a (possibly degenerate) almost CR structure of
hypersurface type. If $J$ is integrable then $i^{*}J$ is integrable. ∎

Next we examine an analogous result in the dual case.

###  3.1. Degenerate solutions of the c-projective metrizability equation: the
order 2 c-projective compactification case

######  Theorem 3.2.

Let $(M,J,\boldsymbol{D})$ be a almost c-projective manifold with real
dimension $2m$ equipped with a solution
$\zeta\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}})$
of the metrizability equation such that $L(\zeta)\in\Gamma(\mathcal{H}^{*})$
is non-degenerate as a pseudo-Hermitian form on the cotractor bundle. If
$L(\zeta)$ is definite then the degeneracy locus $\mathcal{D}(\zeta)$ is empty
and $(M,J,\boldsymbol{D},\zeta)$ is a quasi-Kähler manifold with inverse
Hermitian metric $g^{-1}=\operatorname{sgn}(\tau)\tau\zeta$ where
$\tau={\textbf{det}}(\zeta)$. If $L(\zeta)$ has signature $(p+1,q+1)$, with
$p,q\geq 0$, then $\mathcal{D}(\zeta)$ is either empty or it is a smoothly
embedded separating real hypersurface such that the following hold:

  1. (i)

$M$ is stratified by the strict signature of $\zeta$ as a (density weighted)
Hermitian form on $T^{*}M$ with curved orbit decomposition given by

$M=\coprod\limits_{i\in\\{+,0,-\\}}M_{i}$

where $\zeta$ has signature $(p+1,q)$, $(p,q+1)$,and $(p,q,1)$ on $TM$
restricted to $M_{+}$, $M_{-}$, and $M_{0}$, respectively.

  2. (ii)

On $M_{\pm}$, $\zeta$ induces a quasi-Kähler metric $g_{\pm}$ with
nonvanishing scalar curvature $R^{g_{\pm}}$, with the same signature as
$\zeta$, with inverse
$g^{-1}_{\pm}=\operatorname{sgn}(\tau)\tau\zeta|_{M_{\pm}}$ where
$\tau={\textbf{det}}(\zeta)$.

  3. (iii)

If $M$ is closed, then the components $(M\backslash M_{\mp},J,\boldsymbol{D})$
are c-projective compactifications of $(M_{\pm},J,\nabla^{\zeta})$, with
boundary $M_{0}$.

  4. (iv)

$M_{0}$ inherits a signature $(p,q)$ almost CR structure of hypersurface type.

###### Proof.

(i) Let $\Phi_{\mathscr{AB}}\colonequals(h^{\mathscr{AB}})^{-1}$ denote the
pointwise inverse of $h^{\mathscr{AB}}=L(\zeta^{ab})$. Given a splitting, say
$\nabla\in\boldsymbol{D}$, we write $h$ and $\Phi$ as

(74) $\displaystyle h^{\mathscr{AB}}=\left(\begin{array}[]{cc}\zeta^{ab}\\\
\lambda^{a}\\\ \rho\\\ \end{array}\right)\ \ \ \text{and}\ \ \
{\Phi_{\mathscr{AB}}}=\left(\begin{array}[]{cc}\hat{\tau}\\\ \hat{\eta}_{a}\\\
\hat{\varphi}_{ab}\\\ \end{array}\right),$

for smooth sections $\rho\in\Gamma(\mathcal{E}(-1,-1)_{\mathbb{R}})$,
$\lambda^{a}\in\Gamma(\mathcal{E}^{a}(-1,-1)_{\mathbb{R}})$,
$\zeta^{ab}\in\Gamma(\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}})$,
$\hat{\tau}\in\Gamma(\mathcal{E}(1,1)_{\mathbb{R}})$,
$\hat{\eta}_{a}\in\Gamma(\mathcal{E}_{a}(1,1)_{\mathbb{R}})$, and
$\hat{\varphi}_{ab}\in\Gamma(\operatorname{Herm}(TM)\otimes\mathcal{E}(1,1)_{\mathbb{R}})$.
Observe that, up to a non-zero constant,
$\hat{\tau}=\frac{\operatorname{\textbf{det}}(\zeta^{ab})}{\operatorname{det}(h^{\mathscr{AB}})}$.
By definition we have

(75)
$\displaystyle\Phi_{\mathscr{AC}}h^{\mathscr{CB}}=\delta^{\mathscr{B}}_{\mathscr{A}}.$

Applying the tractor connection, $\nabla^{\mathcal{T}}_{i}$, to both sides
gives

$\displaystyle-(\nabla^{\mathcal{T}}_{i}\Phi_{\mathscr{AC}})h^{\mathscr{CB}}=\Phi_{\mathscr{AC}}\nabla^{\mathcal{T}}_{i}h^{\mathscr{CB}}.$

Applying $\Phi_{\mathscr{BD}}$ to each side gives

(76)
$\displaystyle-\nabla^{\mathcal{T}}_{i}\Phi_{\mathscr{AD}}=\Phi_{\mathscr{AC}}(\nabla^{\mathcal{T}}_{i}h^{\mathscr{CB}})\Phi_{\mathscr{BD}}.$

Computing, the top two slots of (76) are given by

$\displaystyle 2\hat{\eta}_{i}-\nabla_{i}\hat{\tau}$
$\displaystyle=\hat{\tau}^{2}\beta_{i}-2\hat{\tau}\hat{\eta}_{b}\alpha^{b}_{i}$
$\displaystyle\hat{\varphi}_{ic}-\nabla_{i}\hat{\eta}_{c}-P_{ic}\hat{\tau}$
$\displaystyle=\hat{\tau}\hat{\eta}_{c}\beta_{i}+\hat{\tau}\hat{\varphi}_{jc}\alpha_{i}^{j}+\hat{\eta}_{c}\alpha_{i}^{j}\hat{\eta}_{j}$

where
$\nabla_{i}^{\mathcal{T}}h^{\mathscr{CB}}=(0,\alpha_{i}^{b},\beta_{i})^{t}$.
On $\mathcal{D}(\zeta)=\mathcal{Z}(\hat{\tau})$ these reduce to

(77) $\displaystyle\nabla_{i}\hat{\tau}$ $\displaystyle=2\hat{\eta}_{i}$ (78)
$\displaystyle\hat{\varphi}_{ic}$
$\displaystyle=\hat{\eta}_{c}\alpha_{i}^{j}\hat{\eta}_{j}+\nabla_{i}\hat{\eta}_{c}.$

It follows that, on $\mathcal{D}(\zeta)$, $\Phi$ has the form

(82)
$\displaystyle{\Phi_{\mathscr{AB}}}=\left(\begin{array}[]{cc}\hat{\tau}\\\
\frac{1}{2}\nabla_{c}\hat{\tau}\\\
\frac{1}{2}\nabla_{a}\nabla_{b}\hat{\tau}+\frac{1}{4}(\nabla_{b}\hat{\tau})(\nabla_{i}\hat{\tau})\alpha_{a}^{i}\\\
\end{array}\right).$

Nondegeneracy of $\Phi$ implies that
$\mathcal{Z}(\hat{\tau})\cap\mathcal{Z}(\nabla\hat{\tau})=\varnothing$ whence
we conclude that $\mathcal{Z}(\hat{\tau})$ is a smoothly embedded real
hypersurface that is necessarily separating.

(ii) On the open orbits $M_{\pm}$ the complex connection
$\nabla^{\zeta}\in\boldsymbol{D}$ preserving $\zeta$ necessarily preserves
$\operatorname{\textbf{det}}(\zeta)$, and hence preserves the pseudo-
Riemannian metric $g^{-1}_{\pm}=\operatorname{sgn}(\tau)\tau\zeta|_{M_{\pm}}$.
Since $\zeta$ is Hermitian it follows that $g_{\pm}$ is Hermitian and hence
admissible. As observed in Section 2.4, a Hermitian pseudo-Riemannian metric,
on an almost complex manifold, is admissible if and only if it is quasi-
Kähler.
On $M_{\pm}$ in the splitting $\nabla^{\zeta}$ we see that
$L(\zeta)=(\operatorname{sgn}(\tau)\tau^{-1}g^{-1}_{\pm},0,\frac{1}{4m(m+1)}\operatorname{sgn}(\tau)\tau^{-1}R^{g_{\pm}})^{t}$.
Nondegeneracy of $L(\zeta)$ implies that the scalar curvature $R^{g_{\pm}}$ is
nonvanishing.

(iii) Since the $(1,1)_{\mathbb{R}}$-density
$\tau=\operatorname{\textbf{det}}(\zeta)$ is a defining density for $M_{0}$
and it is necessarily preserved by $\nabla^{\zeta}$, it follows from
Proposition 2.1 that the components $(M\backslash M_{\mp},J,\boldsymbol{D})$
are c-projective compactifications of $(M_{\pm},J,\nabla^{\zeta})$, with
boundary $M_{0}$.

(iv) Observe that, for $x\in M_{0}$, $H_{x}\colonequals T_{x}M_{0}\cap
J(T_{x}M_{0})$ defines a corank one smooth distribution $H\subset TM_{0}$ and
the pullback $i^{*}J$ of the almost complex structure along the inclusion
$i:M_{0}\hookrightarrow M$ defines an almost complex structure on $H$. Thus
$(M_{0}^{2n-1},H,i^{*}J)$ is a (possibly degenerate) almost CR structure of
hypersurface type. If $J$ is integrable then $i^{*}J$ is integrable.

Since $\nabla_{a}\tau$ is conormal to $TM_{0}\subset TM$ it follows that
$J^{a}_{b}\nabla_{a}\tau$ is conormal to $J(TM_{0})\subset TM$ so that
$\nabla_{a}\tau\perp H$ and $J^{a}_{b}\nabla_{a}\tau\perp H$. Now we show that
when $\zeta$ degenerates, its nullity is pointwise spanned by $\nabla_{a}\tau$
and $J_{b}^{a}\nabla_{a}\tau$. Since $\zeta$ is Hermitian we need only show
$\nabla_{a}\tau$ is in the nullity of $\zeta$ and it will follow that
$J_{b}^{a}\nabla_{a}\tau$ is in the nullity as well.

Now we show that there exists a scale such that $Y$ is null, i.e.
$H^{\mathscr{A}\mathscr{B}}Y_{\mathscr{A}}Y_{\mathscr{B}}=0$, along $M_{0}$.
Given $f$ a non-vanishing section of $\mathcal{E}(1,1)$, then in the scale
corresponding to $f$, we have that

$\displaystyle
f^{-1}D_{\mathscr{A}}f=f^{-1}Y_{\mathscr{A}}f+f^{-1}Z^{a}_{\mathscr{A}}\nabla_{a}f=Y_{\mathscr{A}}.$

Observing that, on $\mathcal{D}(\zeta)$,

$\displaystyle
D_{\mathscr{A}}\hat{\tau}=Y_{\mathscr{A}}\hat{\tau}+Z^{a}_{\mathscr{A}}\nabla_{a}\hat{\tau}=2\Phi_{\mathscr{A}\mathscr{B}}X^{\mathscr{B}},$

This implies that along $\mathcal{D}(\zeta)$

$\displaystyle
h^{\mathscr{A}\mathscr{B}}(D_{\mathscr{A}}\hat{\tau})(D_{\mathscr{B}}\hat{\tau})=4h^{\mathscr{A}\mathscr{B}}\Phi_{\mathscr{A}\mathscr{D}}X^{\mathscr{D}}\Phi_{\mathscr{B}\mathscr{C}}X^{\mathscr{C}}=4\Phi_{\mathscr{A}\mathscr{D}}X^{\mathscr{D}}X^{\mathscr{A}}=\hat{\tau}=0.$

Also, observe that

$\displaystyle\frac{1}{2}h^{\mathscr{A}\mathscr{C}}D_{\mathscr{A}}\hat{\tau}=X^{\mathscr{C}}.$

Define $\xi\colonequals-\frac{1}{4}f\rho$ where $f$ is an arbitrary non-
vanishing section of $\mathcal{E}(1,1)$ and $\rho\colonequals
Y_{\mathscr{A}}Y_{\mathscr{B}}h^{\mathscr{A}\mathscr{B}}$. Now let
$\gamma\colonequals f+\xi\hat{\tau}$. Then along $\mathcal{D}(\zeta)$ we have

$\displaystyle
h^{\mathscr{A}\mathscr{B}}Y^{\gamma}_{\mathscr{A}}Y^{\gamma}_{\mathscr{B}}=$
$\displaystyle\
\gamma^{-2}(D_{\mathscr{A}}\gamma)({D_{\mathscr{B}}\gamma})h^{\mathscr{A}\mathscr{B}}$
$\displaystyle=$ $\displaystyle\ f^{-2}(D_{\mathscr{A}}f+\xi
D_{\mathscr{A}}\hat{\tau})({D_{\mathscr{B}}f}+\xi{D_{\mathscr{B}}\hat{\tau}})h^{\mathscr{A}\mathscr{B}}$
$\displaystyle=$ $\displaystyle\
f^{-2}(D_{\mathscr{A}}f)({D_{\mathscr{B}}f})h^{\mathscr{AB}}+f^{-2}\xi(D_{\mathscr{A}}\hat{\tau})({D_{\mathscr{B}}f})h^{\mathscr{AB}}+f^{-2}\xi(D_{\mathscr{A}}f)({D_{\mathscr{B}}\hat{\tau}})h^{\mathscr{AB}}$
$\displaystyle\
+f^{-2}\xi^{2}(D_{\mathscr{A}}\hat{\tau})({D_{\mathscr{B}}\hat{\tau}})h^{\mathscr{AB}}$
$\displaystyle=$ $\displaystyle\
Y_{\mathscr{A}}Y_{\mathscr{B}}h^{\mathscr{A}\mathscr{B}}+2f^{-2}\xi
X^{\mathscr{B}}({D_{\mathscr{B}}f})+2f^{-2}\xi
X^{\mathscr{A}}(D_{\mathscr{A}}f)+0$ $\displaystyle=$ $\displaystyle\
\rho+4f^{-1}\xi$ $\displaystyle=$ $\displaystyle\ \rho-\rho$ $\displaystyle=$
$\displaystyle\ 0.$

A scale $\nabla^{\gamma}$ preserving $\gamma$ such that $Y^{\gamma}$ is null
along along $\mathcal{D}(\zeta)$ will be known as a special boundary
scale.333This term and the method of constructing a special boundary scale was
first done the setting of projective differential geometry by Sam Porath in
[40]. In such a scale, $\rho$ necessarily vanishes along $\mathcal{D}(\zeta)$.

Computing the slots of 75 along $\mathcal{D}(\zeta)$ in a special boundary
scale yields the following system of equations

(83) $\displaystyle\frac{1}{2}(\nabla_{a}\hat{\tau})\zeta^{ab}=0$ (84)
$\displaystyle\frac{1}{2}(\nabla_{c}\nabla_{a}\hat{\tau})\nabla_{j}\zeta^{jc}+\frac{1}{4}(\nabla_{a}\hat{\tau})(\nabla_{i}\hat{\tau})\alpha_{c}^{i}\nabla_{j}\zeta^{jc}=0$
(85) $\displaystyle\frac{1}{2m}(\nabla_{c}\hat{\tau})(\nabla_{i}\zeta^{ic})=1$
(86)
$\displaystyle\frac{1}{2m}(\nabla_{a}\hat{\tau})(\nabla_{i}\zeta^{ib})+\frac{1}{2}(\nabla_{c}\nabla_{a}\hat{\tau})\zeta^{cb}+\frac{1}{4}(\nabla_{a}\hat{\tau})(\nabla_{i}\hat{\tau})\alpha_{c}^{i}\nabla_{j}\zeta^{jc}=\delta^{b}_{a}.$

It follows from (83) that the kernel of $\zeta$ on $\mathcal{D}(\zeta)$ is
spanned by $\nabla_{a}\hat{\tau}$ and $J_{b}^{a}\nabla_{a}\hat{\tau}$. So
$\zeta$ induces a pseudo Hermitian metric with signature $(p,q)$ on the
distribution $H$. Contracting $\xi^{a}\in\Gamma(H)$ into (84) gives us

(87)
$\displaystyle\frac{1}{2}(\nabla_{c}\nabla_{a}\hat{\tau})(\nabla_{j}\zeta^{jc})\xi^{a}=0.$

The restriction of $\theta_{a}\colonequals J_{a}^{i}\nabla_{i}\hat{\tau}$ to
$TM_{0}$ is a weighted contact form for the CR structure induced on
$\mathcal{D}(\zeta)$. Then (87) together with (85) this implies that
$T^{a}\colonequals-\frac{1}{2m}J^{a}_{j}\nabla_{i}\zeta^{ij}$ is a candidate
for the Reeb vector field since $\theta_{a}T^{a}=1$.

Contracting $\xi^{a}\in\Gamma(H)$ into (86) gives us

$\displaystyle\frac{1}{2}(\nabla_{c}\nabla_{a}\hat{\tau})\zeta^{cb}\xi^{a}=\delta^{b}_{a}\xi^{a}=\xi^{b},$

whence $(\zeta|_{H})^{-1}=\frac{1}{2}\nabla\nabla\hat{\tau}$. Since and the
real part of the CR Levi form corresponding to $\theta_{a}$ is the restriction
of $(d\theta)_{ab}=2\nabla_{[a}J_{b]}^{i}\nabla_{i}\hat{\tau}$ to $H$, it
follows immediately that the real part of the CR Levi form is
$(J^{a}_{b}\check{\zeta}^{bc})^{-1}$, where
$\check{\zeta}\colonequals\zeta|_{H}$. Thus the CR structure is Levi-non-
degenerate with signature $(p,q)$ and, since $T^{a}(d\theta)_{ab}=0$, we see
that $T^{a}$ is indeed the Reeb. ∎

######  Remark 3.3.

If the original almost complex structure is integrable then the open orbits
are in fact pseudo-Kähler and the c-projective Schouten tensor $P_{ab}$ is
symmetric and Hermitian, so by Theorem 23 of [15], $g_{\pm}$ satisfies an
asymptotic version of the Einstein equation. If $L(\zeta)$ is parallel then
the vanishing of the middle slot of $\nabla L(\zeta)$ implies that the quasi-
Kähler metrics on the open orbits are Einstein. If $J$ is integrable and
$L(\zeta)$ is parallel then the open orbits are pseudo-Kähler-Einstein.

######  Corollary 3.4.

Let $(\overline{M},J)$ be a connected almost complex manifold with boundary
$\partial M$ and interior $M$, equipped with a pseudo-quasi-Kähler metric $g$
on $M$ which is Hermitian for $J$, has nonvanishing scalar curvature, and such
that the minimal complex connection $\nabla^{g}$ preserving $g$ does not
extend to any neighborhood of a boundary point, but the almost c-projective
structure $\boldsymbol{D}\colonequals[\nabla^{g}]$ does extend to the
boundary. Let $\tau\colonequals\operatorname{vol}(g)^{-\frac{1}{m+1}}$. Then
$\zeta^{ab}\colonequals\tau^{-1}g^{ab}$ extends to the boundary. If
$L(\zeta^{ab})$ is non-degenerate on $\overline{M}$, then $(M,J,\nabla^{g})$
is c-projectively compact.

###### Proof.

$L(\zeta)$ is defined on the interior $M$. It extends via parallel transport
for the prolongation connection444The prolongation connection is a natural
modification of the tractor connection. See, e.g., [HSSS] for a construction
for general BGG operators to a parallel (for the prolongation connection)
tractor on $\overline{M}$. Projection to the quotient bundle
$\operatorname{Herm}(T^{*}M)\otimes\mathcal{E}(-1,-1)_{\mathbb{R}}$ gives a
smooth extension of $\zeta$ to all of $\overline{M}$ (cf. Corollary 16 of
[15]). The degeneracy locus of $\zeta$ is precisely $\partial M$, otherwise it
would contradict our assumption that $\nabla^{g}$ does not extend to any
neighborhood of a boundary point. Then the result follows from Theorem 3.2. ∎

###  3.2. The Model

We briefly discuss here the model for the structures considered in Theorem
3.2. The standard homogeneous model for c-projective geometry is the complex
projective space arising as the complex scalar projectivization,
$\mathbb{CP}^{m}=\mathbb{P}(\mathbb{C}^{m+1})$, of $\mathbb{C}^{m+1}$. The
J-planar curves in $\mathbb{CP}^{m}$ are the smooth (real) curves lying in
linearly embedded complex curves
$\mathbb{CP}^{1}\hookrightarrow\mathbb{CP}^{m}$. On $\mathbb{CP}^{m}$ the
group $G=SL(\mathbb{C}^{m+1})$ acts transitively. On this c-projective
structure it is well known that the tractor connection is induced by the
trivial connection on $\mathbb{C}^{m+1}$. Now suppose we fix, on
$\mathbb{C}^{m+1}$, a non-degenerate symmetric bilinear form $h$ of signature
$(p+1,q+1)$. This may be identified with a corresponding parallel tractor. In
$G$ consider the subgroup $H:=SU(h)\cong SU(p+1,q+1)$ fixing $h$ (so
$p+q=m-1$). This acts on the complex projective space $\mathbb{CP}^{m}$ but
now with orbits parametrized by the strict sign of $h(X,X)$ where $X$ denotes
the homogeneous coördinates of a given point on $\mathbb{CP}^{m}$. Complex
projective space $\mathbb{CP}^{m}$ equipped with this action of $H$ and
accompanying orbit decomposition is the model for the structure discussed in
Theorem 3.2. This follows easily from the tractor approach that we use with
the interpretation of the tractor bundles over the homogeneous space $G/P$. So
the Theorem also reveals, for this model, the general features of the orbits
and the geometries thereon. In fact, $h^{-1}=L(\zeta)$ where $\zeta$ is the
corresponding solution of (65) and, in the language of [21], this is a
holonomy reduction of a flat Cartan geometry (namely
$G\rightarrow\mathbb{CP}^{m}$). Turning this around, we see that Theorem 3.2
shows that solutions $\zeta$ of equation (65), satisfying that
$\det(L(\zeta))$ is nowhere zero, provide well behaved curved generalizations
of this model even though $\zeta$ is not required to be normal (i.e.
$L(\zeta)$ is not required to be parallel).

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