# On the regularity of conical Calabi-Yau potentials

Tran-Trung Nghiem

###### Abstract.

Using pluripotential theory on degenerate Sasakian manifolds, we show that a
locally bounded conical Calabi-Yau potential on a Fano cone is actually smooth
on the regular locus. This work is motivated by a similar result obtained by
R. Berman in the case where the cone is toric. Our proof is purely
pluripotential and independent of any extra symmetry imposed on the cone.

##  1\. Introduction

###  1.1. Background and motivation

The problem of finding Kähler-Einstein metrics has been central in the
development of Kähler geometry, leading to the solution by Chen-Donaldson-Sun
of the celebrated Yau-Tian-Donaldson conjecture [CDS15a, CDS15b, CDS15c].
While the problem is well understood on compact Kähler manifolds, or more
generally compact Kähler varieties [EGZ09, Li22], the non-compact case is
still relatively open. In the pioneering work of Martelli-Sparks-Yau [MSY08],
the existence of conical Calabi-Yau metrics (alias Ricci-flat Kähler cone
metrics) on toric varieties with an isolated singularity is shown to be
equivalent to a volume minimization principle for Euclidean convex cones. This
principle still holds for mildly singular toric varieties as proved by Berman
[Ber20]. A more systematic study of polarized affine varieties with an
isolated singularity was done by Collins and Székelyhidi [CS19], generalizing
the work of Chen-Donaldson-Sun to the context of Kähler cones, or
equivalently, Sasakian manifolds.

A Sasakian manifold is a compact Riemannian manifold such that the metric cone
over it is Kähler. Sasakian manifolds can be viewed as odd-dimensional analogs
of compact Kähler manifolds since they have a natural transverse Kähler
structure on an intrinsic horizontal distribution. The existence of Ricci-flat
Kähler cone metrics on a Kähler cone is in fact equivalent to the existence of
Sasaki-Einstein metrics on the link, which boils down to a Kähler-Einstein-
like problem on the transverse structure.

The existence of a (singular) Kähler-Einstein metrics is equivalent to solving
a (degenerate) complex Monge-Ampère equation. An interesting problem to ask is
the regularity of a singular Kähler-Einstein metric on the smooth locus. In
the present paper, we are concerned with the regularity problem on a class of
mildly singular affine varieties called Fano cones.

In order to state the main result, let us first give some preliminaries on
Fano cones and conical Calabi-Yau potentials. Recall that a normal variety is
called $\mathbb{Q}$-Gorenstein if a multiple of its canonical line bundle is
Cartier. The action of a complex torus $T$ on $Y$ is said to be good if it is
effective and has a unique fixed point contained in any orbit closure.

######  Definition 1.1.

A _cone_ $Y$ is a normal affine variety endowed with the good action of a
complex torus $T\simeq(\mathbb{C}^{*})^{k}$. We say that $Y$ is a _Fano cone_
if it is $\mathbb{Q}$-Gorenstein with klt singularities. The unique fixed
point of $Y$, denoted by $0_{Y}$, is called the _vertex_ of $Y$.

Let $\mathcal{M}:=\operatorname{Hom}(T,\mathbb{C}^{*})\simeq\mathbb{Z}^{k}$ be
the weight lattice and
$\mathcal{N}:=\mathcal{M}^{*}=\operatorname{Hom}(\mathbb{C}^{*},T)$ the
coweight lattice. The ring of regular functions of $Y$ admits a decomposition
into $T$-modules

$\mathbb{C}[Y]=\oplus_{\alpha\in\Gamma}R_{\alpha},\quad\Gamma:=\left\\{\alpha\in\mathcal{M},R_{\alpha}\neq
0\right\\}$

where $R_{\alpha}$ is the $T$-module with weight $\alpha$. Let
$\mathcal{M}_{\mathbb{R}}:=\mathcal{M}\otimes\mathbb{R}$ and
$\mathcal{N}_{\mathbb{R}}:=\mathcal{N}\otimes\mathbb{R}$. The set $\Gamma$ is
an affine semi-group of finite type which generates a strictly convex
polyhedral cone $\sigma^{\vee}\subset\mathcal{M}_{\mathbb{R}}$. Equivalently,
the dual cone $\sigma$ in $\mathcal{N}_{\mathbb{R}}$ is polyhedral of maximal
dimension $k$. This results from the assumption that $Y$ has a unique fixed
point lying in the closure of every $T$-orbit (cf. [AH06]). The interior of
$\sigma$ is then non-empty and coincides with its relative interior:

$\text{Int}(\sigma)=\left\\{\xi\in\mathcal{N}_{\mathbb{R}},\left<\alpha,\xi\right>>0,\forall\alpha\in\Gamma\right\\}$

######  Definition 1.2.

The interior of the cone $\sigma$ is called the _Reeb cone_ of $Y$. An element
$\xi\in\text{Int}(\sigma)$ is called a _Reeb vector_. A Fano cone decorated
with a Reeb vector $(Y,\xi)$ is said to be a _polarized Fano cone_. We say
that $(Y,\xi)$ is _quasi-regular_ if $\xi\in\mathcal{N}_{\mathbb{Q}}$, and
otherwise _irregular_ if $\xi\notin\mathcal{N}_{\mathbb{Q}}$.

The closure inside $\operatorname{Aut}(Y)$ of the one-parameter subgroup
generated by the infinitesimal action of $\xi$ is a compact torus
$T_{\xi}\subset T_{c}$, where $T_{c}\simeq(\mathbb{S}^{1})^{k}$ is a maximal
compact subtorus of $T$. If $\xi$ is quasi-regular then
$T_{\xi}\simeq\mathbb{S}^{1}$, but if it is irregular then
$T_{\xi}\simeq(\mathbb{S}^{1})^{m}$, $k\geq m>1$. Equivalently, in the quasi-
regular (resp. irregular) case, the holomorphic vector field associated to
$\xi$ generates an action of $\mathbb{C}^{*}$ (resp. $(\mathbb{C}^{*})^{k}$).
It can be shown that in the quasi-regular case, the quotient
$(Y\backslash\left\\{0_{Y}\right\\})/\mathbb{C}^{*}$ is a Fano orbifold (see
[Kol04, Paragraph 42]). Note however that in the irregular case, the quotient
by $(\mathbb{C}^{*})^{k}$ is only well-defined as an algebraic space (cf.
[Kol97] ). For more details on Fano cones, the reader may consult for example
[LLX20], [DS17] and references therein.

Given a Fano cone $(Y,T)$, by Sumihiro’s theorem (see [Sum74, Theorem 1, Lemma
8]), there exists an embedding $Y\subset\mathbb{C}^{N}$ such that $T$
corresponds to a diagonal subgroup of $GL_{N}(\mathbb{C}^{N})$ acting
linearly. Given an embedding $Y\subset\mathbb{C}^{N}$, we say that a function
$f$ is plurisubharmonic (psh for short) on $Y$ if it is locally the
restriction to $Y$ of a psh function on the ambient space $\mathbb{C}^{N}$.

######  Definition 1.3.

A _$\xi$ -radial function_ (or _$\xi$ -conical potential_)
$r^{2}:Y\to\mathbb{R}_{>0}$ is a psh function on $Y$ that is invariant under
the action of $\xi$ and 2-homogeneous under $-J\xi$, namely

$\mathcal{L}_{\xi}r^{2}=0,\quad\mathcal{L}_{-J\xi}r^{2}=2r^{2}$

on $Y_{\text{reg}}$.

If $Y$ is a $\mathbb{Q}$-Gorenstein cone, then for $m>0$ large enough,
$mK_{Y}$ is a Cartier divisor and naturally linearized by the $T$-action.
Moreover, there exists a $T$-invariant non-vanishing holomorphic section $s\in
mK_{Y}$ and a volume form $dV_{Y}$ such that

$dV_{Y}=\left(i^{(n+1)^{2}m}s\wedge\overline{s}\right)^{1/m}$

where $n+1=\dim_{\mathbb{C}}Y$. To simplify the notation, by an abuse of
language we will sometimes say that $s$ is a “multivalued” section of $K_{Y}$
and simply write $dV_{Y}=i^{(n+1)^{2}}s\wedge\overline{s}$.

A canonical volume form $dV_{Y}$ on $Y$ is a volume form that is
$(2n+2)$-homogeneous under the action of $r\partial r$, namely

$\mathcal{L}_{r\partial r}dV_{Y}=2(n+1)dV_{Y}$

on $Y_{\text{reg}}$.

The $\mathbb{Q}$-Gorenstein and klt singularities assumptions on $Y$ guarantee
that there exists a unique canonical volume form on $Y$ up to a constant, see
[MSY08], [CS19].

A $(1,1)$-Kähler current $\omega$ on a polarized Fano cone $(Y,\xi)$ is said
to be a $\xi$-Kähler cone current if there exists a locally bounded
$\xi$-radial function such that

$\omega=dd^{c}r^{2}$

This is well-defined thanks to the local theory of Bedford-Taylor [BT76]. If
moreover the function $r^{2}$ satisfies the Calabi-Yau condition

(1) $\omega^{n+1}=(dd^{c}r^{2})^{n+1}=dV_{Y}$

in the pluripotential sense, then $r^{2}$ is said to be a (singular) conical
Calabi-Yau potential.

######  Definition 1.4.

We say that a Kähler cone current $\omega=dd^{c}r^{2}$ is a conical Calabi-Yau
metric if the function $r^{2}$ is a singular conical Calabi-Yau potential
which is smooth on the regular locus of $Y$.

The motivation for studying these metrics on Fano cones actually has its
origin in the compact Fano case. Concretely, Fano cones arise as metric
tangent cones of the Gromov-Hausdorff limit of a Fano manifolds sequence
[DS17]. If each term of the sequence is moreover Kähler-Einstein, then the
Fano cone admits conical Calabi-Yau metrics. As discussed in [Ber20, Section
4] (see also Remark 4.10), it is expected that a singular conical Calabi-Yau
potential restricts to a smooth function on the regular locus of $Y$. Our goal
in this article is to give an affirmative answer to this problem.

######  Theorem 1.5.

Let $(Y,\xi)$ be a polarized Fano cone and $r^{2}$ be a singular $\xi$-conical
Calabi-Yau potential on $Y$. Then $r^{2}$ is smooth on the regular locus of
$Y$. In particular, the curvature form of $r^{2}$ is a well-defined conical
Calabi-Yau metric.

Such smoothness result is well-known for singular Kähler-Einstein metrics on
compact Kähler varieties [EGZ09], [BEGZ10], [BBE+19, Lemma 3.6]. In the non-
compact setting, when the cone has a unique singularity at the vertex, the
Sasakian link is smooth, so the conical metric is automatically smooth outside
the vertex. For toric Fano cones with non-isolated singularities, a regularity
property was obtained by Berman [Ber20] by using the toric symmetry to
reformulate the problem in terms of real Monge-Ampère equations. As discussed
in [Ber20, Remark 4.10], the only places where the toric structure was used
were the $L^{\infty}$-estimate and uniqueness of the Monge-Ampère equation.
Although it is possible to generalize the same approach to a larger class of
highly symmetric varieties, such as horospherical varieties, we provide a
proof closer to the pluripotential spirit and independent of any symmetry
other than the given effective torus action. It is an interesting problem to
ask if we can weaken the regularity assumption of the solution.

###  1.2. Organization

The organization of the article is as follows.

  * •

In Section 2, we give a quick review of the structure of degenerate Sasakian
manifolds. We then gather results in pluripotential theory on these manifolds
based the on the work of Guedj-Zeriahi [GZ05] and He-Li [HL21]. We also
introduce extremal functions associated to a Reeb-invariant Borel set on a
degenerate Sasakian manifold, which seems to be new in the literature. These
objects were not studied in [HL21] in all generality (but see [HL21, Prop.
3.17, Thm. 3.1] for results concerning weighted global extremal functions).
The capacity-extremal function comparison is crucial in the proof of the
uniform estimate.

  * •

Section 3 is devoted to the proof of our main result. The general strategy is
based on [EGZ09], [BEGZ10], [BBE+19] and [Ber20]. Let us give a brief
explanation. After taking a resolution of singularities, the conical Calabi-
Yau problem is translated by pullback to a Calabi-Yau problem on a degenerate
Sasakian manifold.

Our key theorem is the uniform $L^{\infty}$-estimate of a family of solutions,
which relies on a domination-by-capacity property (cf. Prop. 3.7). This,
combined with a transverse Yau-Aubin inequality, allows us to obtain a
Laplacian estimate of the family, which implies regularity of the solution.

  * •

In Section 4, we provide a proof for the transverse version of Yau-Aubin
inequality, which is used in the Laplacian estimate.

Acknowledgements. This article is part of a thesis supervised by Thibaut
Delcroix and Marc Herzlich. I wish to thank Vincent Guedj, Eleonora Di Nezza,
and Tat-Dat To for their generosity as well as many helpful discussions and
remarks. Thanks are also due to the hospitality of the Vietnam Institute for
Advanced Study in Mathematics (VIASM), where this work first begun.

##  2\. Pluripotential theory on Sasakian manifolds

###  2.1. Structure of Sasakian manifolds

In this section, we introduce the notion of degenerate Sasakian manifolds.
These are compact manifolds having all the essential properties of a Sasakian
manifold, except that the form $d\eta$ is not positive-definite, hence does
not define a transverse Kähler structure. Still, we assume that a degenerate
Sasakian manifold has a transverse Kähler structure, but that the basic Kähler
form is not induced by the contact form.

Degenerate Sasakian manifolds arise as the link of the resolution of Fano
cones (see Lem. 3.1). The reader should compare this setting to the Kähler
situation: a resolution of a Kähler space is still Kähler, but the Kähler
structure of the resolution is not the pullback of the Kähler structure on the
base.

We refer the reader to [BG08] for a detailed treatment of almost contact
structures and Sasakian manifolds.

Let $S$ be a compact differentiable manifold of dimension $2n+1$. A contact
structure on $S$ is the data of a $1$-form $\eta$ on $S$ such that
$\eta\wedge(d\eta)^{n}\neq 0$. The manifold $S$ is then said to be a contact
manifold. On a contact manifold, there exists a unique vector field $\xi$,
called the Reeb vector field, such that $\eta(\xi)=1,\mathcal{L}_{\xi}\eta=0$.
The distribution $\mathcal{D}:=\text{ker}(\eta)$ is called the horizontal
distribution of $S$.

######  Definition 2.1.

An almost contact structure is given by $(S,\xi,\eta,\Phi)$, where $\eta$ is a
contact form, $\xi$ the corresponding Reeb vector field, and $\Phi$ a
$(1,1)$-tensor of $TS$ such that:

$\Phi^{2}=-Id+\xi\otimes\eta,\quad d\eta(\Phi.,\Phi.)=d\eta,\quad
d\eta(.,\Phi.)>0$

In particular, $\Phi|_{\mathcal{D}}$ is an almost complex structure.

A degenerate almost contact structure is the same as an almost contact
structure, except that $d\eta(.,\Phi.)$ is only semipositive, i.e.
$d\eta(.,\Phi,.)\geq 0$.

######  Definition 2.2.

A degenerate metric contact structure is a degenerate almost contact structure
$(S,\xi,\eta,\Phi)$ endowed with a Riemannian metric $g$ satisfying

$g(\Phi.,\Phi.)=g(.,.)-\eta\otimes\eta$

Such a metric is said to be compatible.

A (degenerate) almost contact structure is said to be normal if the horizontal
distribution $\mathcal{D}$ is integrable. A form $\alpha$ on $S$ is said to be
basic if

$\mathcal{L}_{\xi}\alpha=i_{\xi}\alpha=0$

######  Definition 2.3.

A degenerate Sasakian manifold $(S,\xi,\eta,\omega_{B})$ is a normal
degenerate contact structure with a transverse Kähler metric defined by a
basic positive-definite $(1,1)$-form $\omega_{B}$.

Let $g_{B}$ be the Riemannian metric associated to $\omega_{B}$. A degenerate
Sasakian manifold admits a Riemannian metric, defined by

$g_{S}:=\eta\otimes\eta+g_{B},$

which restricts to a transverse Kähler metric on $\mathcal{D}$, but the latter
is in general different from the semipositive form induced by the contact
form. In particular, a degenerate Sasakian manifold has a degenerate metric
contact structure.

######  Remark.

In [BG08], a Sasakian manifold is defined as a normal metric contact
structure. In our paper, one should distinguish between a metric contact
structure and an degenerate metric contact structure. Both are almost contact
structures with a compatible metric, but the metric of the former is exactly
$d\eta(Id\otimes\Phi)$, while the latter has a compatible metric $g_{S}\neq
d\eta(Id\otimes\Phi)$.

Many properties of Sasakian manifolds still hold on their degenerate
counterparts. For example, on a degenerate Sasakian manifold, we still have a
cover by local foliation charts, coming from the foliation $\mathcal{F}_{\xi}$
by the Reeb vector field $\xi$ on $S$.

######  Definition 2.4.

The _foliation atlas_ on a degenerate Sasakian manifold is defined as a
collection of charts $(U_{\alpha},\Phi_{\alpha})$ covering $S$ with
diffeomorphisms:

$\displaystyle\Phi_{\alpha}:W_{\alpha}$ $\displaystyle\times]-t,t[\to
U_{\alpha}$
$\displaystyle(z,x)\longrightarrow(\varphi_{\alpha}(z),\tau_{\alpha}(z,x))$

such that:

  * •

The open interval $]-t,t[\subset\mathbb{R}$ has coordinate $x$. Here, $t$ can
be taken to be independent of $\alpha$.

  * •

For all $\alpha$, $W_{\alpha}\simeq B_{\delta}(0)$ is the ball of radius
$\delta>0$ centered in $0\in\mathbb{C}^{n}$ with coordinates
$z=(z_{1},\dots,z_{n})$. Moreover, the transition map
$\varphi_{\alpha\beta}:=\varphi_{\alpha}\circ\varphi_{\beta}^{-1}$ from
$W_{\alpha}\cap W_{\beta}$ to itself is holomorphic. In pratice, we usually
take $\delta=1$.

Each chart $(U_{\alpha},\Phi_{\alpha})$ is called a _foliation chart_ , and
each $W_{\alpha}$ is said to be a _transverse chart_ (or _transverse
neighborhood_).

In a foliation chart $U_{\alpha}$, we may identify $\xi$ with $\partial_{x}$
and a point $p\in S$ can be written as $p=(z_{1},\dots,z_{n},x)$.

Let $\Omega_{B}^{k}$ be the sheaf of basic $k$-forms on $S$. Since the
exterior differential $d$ on $S$ preserves basic forms, it descends to the
basic exterior differential $d_{B}:=d|_{\Omega_{B}^{k}}$. We then have a
subcomplex $\Omega_{B}^{.}(\mathcal{F}_{\xi})$ of the de Rham complex, and the
corresponding basic cohomology $H_{B}^{*}$. The integrable complex structure
on $\mathcal{D}$ leads to the decompositions

$d_{B}=\partial_{B}+\overline{\partial}_{B},\quad\Omega_{B}^{k}=\bigoplus_{p+q=k}\Omega_{B}^{p,q}$

as well as the basic Dolbeault complex and the corresponding cohomologies
$H_{B}^{p,q}$. We then say that a basic function is  transversely holomorphic
if it vanishes under $\overline{\partial}_{B}$. The Kähler structure on
$\mathcal{D}$ induces the decomposition in basic cohomologies as in the
classic Hodge theory:

$H^{k}_{B}=\bigoplus_{p+q=k}H^{p,q}_{B}$

In short, usual Kähler properties still hold for a Kähler leaf space. We refer
the reader to [EKA90] for proofs.

###  2.2. Quasipsh functions and capacities

We present here some results concerning intrinsic capacities on degenerate
Sasakian manifolds, following the lines of Guedj-Zeriahi [GZ05], slightly
generalizing the work of He-Li [HL21]. Apart from a subtlety in the definition
of capacity, there are generally no supplementary difficulties compared to the
case of a classic Sasakian manifold studied by He and Li.

Let $(S,\xi,\eta,\omega_{B})$ be a degenerate Sasakian manifold of dimension
$(2n+1)$, where $\omega_{B}$ a basic Kähler form on $S$, while $\theta:=d\eta$
is smooth, semipositive and big; the latter meaning:

$0<\text{vol}_{\theta}(S):=\int_{S}\theta^{n}\wedge\eta<+\infty$

Let $g_{S}:=\eta\otimes\eta+g_{B}$ be the corresponding Riemannian metric on
$S$. We denote by

$\mu_{\omega_{B}}:=\omega_{B}^{n}\wedge\eta$

the volume form on $S$ associated to $g_{S}$.

######  Definition 2.5.

By a _$\xi$ -invariant object_ (function, set, etc.), we mean that the object
is invariant under the action of the compact torus $T_{\xi}$ generated by
$\xi$.

By a function in $L^{1}(S)$, we mean a function being $L^{1}$ with respect to
the measure $\mu_{\omega_{B}}$ on $S$.

A $(p,q)$-transverse current is a collection
$\left\\{(W_{\alpha},T_{\alpha})\right\\}$ where $W_{\alpha}$ is a transverse
neighborhood and $T_{\alpha}$ a current of bidegre $(p,q)$ on $W_{\alpha}$
such that

$\varphi_{\alpha\beta}^{*}T_{\beta}|_{W_{\alpha}\cap
W_{\beta}}=T_{\alpha}|_{W_{\alpha}\cap W_{\beta}}$

The current $T$ is said to be closed (resp. positive) if each $T_{\alpha}$ is
closed (resp. positive) on $W_{\alpha}$. Recall that a basic function on $S$
is a $\xi$-invariant function. A basic psh function $u$ on $U_{\alpha}$ is a
basic, upper-semicontinuous function on $U_{\alpha}$ such that
$u|_{W_{\alpha}}$ is a classical psh function. In particular, $u$ is locally
integrable.

######  Definition 2.6.

We say that a function $u$: $S\to\mathbb{R}\cup\left\\{-\infty\right\\}$ is
basic $\theta$-psh if $u$ is locally the sum of a basic smooth function and a
basic psh function, such that

$(\theta+d_{B}d_{B}^{c}u)|_{\mathcal{D}}\geq 0$

in the sense of transverse currents. We will denote by $PSH(S,\xi,\theta)$ the
set of basic $\theta$-psh functions. If $u\in PSH(S,\xi,\theta)$, we put
$\theta_{u}:=\theta+dd^{c}u$.

In particular, a $\theta$-psh function is $\xi$-invariant, upper-
semicontinuous and $L^{1}(S)$. A Sasakian analogue of the Bedford-Taylor
theory was developed by van Coevering [vC18] in the case where $\theta$ is
Kähler and $u$ is a $\theta$-psh bounded function on $S$. Let us give some
details of the construction.

Let $u\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$ and $T$ a transverse closed
positive current on $S$. Since $\theta$ is a closed and basic $(1,1)$-form,
$\theta_{u}$ defines a transverse $(1,1)$-current. After perharps resizing the
transverse neighborhood $W_{\alpha}$, there exists a local $\xi$-invariant
potential $v$ such that $\theta=dd^{c}v$. We then define on each $W_{\alpha}$

$\theta_{u}\wedge T:=dd^{c}((v+u).T)$

This allows one to define inductively $\theta_{u}^{k}\wedge T$ on each
$W_{\alpha}$. Passing to the foliation chart
$U_{\alpha}=W_{\alpha}\times]-t,t[$, the Monge-Ampère operator of $u$ is
defined as

$\theta_{u}^{n}\wedge dx$

where we identify the contact form $\eta$ with $dx$ in the local coordinate of
$]-t,t[$. One can check that this definition is independent of the foliation
chart. We will denote the (sasakian) Monge-Ampère measure of $u$ by

$\text{MA}_{\theta}(u):=\theta_{u}^{n}\wedge\eta$

In particular, $\text{MA}_{\theta}(u)$ is a $\xi$-invariant Radon measure,
which has the following continuity property.

######  Proposition 2.7.

[vC18, Theorem 2.3.1] The sasakian Monge-Ampère operator is continuous for
monotone convergence. In other words, if $(u_{k})_{k\in\mathbb{N}}\subset
PSH(S,\xi,\theta)^{\mathbb{N}}\cap L^{\infty}(S)$ increases (or decreases)
towards $u$, then $\text{MA}_{\theta}(u_{k})\to\text{MA}_{\theta}(u)$ in the
sense of measures.

If $u$ is bounded, then by supposing $u\geq 0$ and noting that $u^{2}$ is
basic and psh, one can define the transverse closed positive current:

$du\wedge d^{c}u\wedge T:=\frac{1}{2}dd^{c}u^{2}\wedge T-udd^{c}\wedge T$

As in the (transverse) Kähler case, we have for all $u\in
PSH(S,\xi,\theta)\cap L^{\infty}(S)$

$\int_{S}\theta_{u}^{n}\wedge\eta=\text{vol}_{\theta}(S)$

i.e. a locally bounded $\theta$-psh function is of full mass.

We record the following regularization property for a later use:

######  Lemma 2.8.

Given $u\in PSH(S,\xi,\omega_{B})$, there exists a sequence
$(u_{k})_{k\in\mathbb{N}}\subset PSH(S,\xi,\omega_{B})\cap C^{\infty}(S)$
decreasing to $u$.

###### Proof.

We use the regularization procedure as in [Ber19, Theorem 3.3]. First, for a
smooth basic function $f$ and $\beta>0$, consider the basic Calabi-Yau-type
problem on $S$:

$(\omega_{B}+d_{B}d_{B}\varphi_{\beta})^{n}\wedge\eta=e^{\beta(\varphi_{\beta}-f)}\omega_{B}^{n}\wedge\eta$

A solution $\varphi_{\beta}$ verifying $\sup\varphi_{\beta}=0$ exists and is
unique (cf. [EKA90, 3.5.5]). We will denote by $P_{\beta}(f)$, $\beta>0$ the
unique solution.

Now let

$P_{\omega_{B}}(f)(p):=\sup\left\\{\varphi(p),\varphi\leq f,\varphi\in
PSH(S,\xi,\omega_{B})\right\\},$

This function belongs to $PSH(S,\xi,\omega_{B})$ (cf. [HL21, Proposition
3.17]). Consider

$P^{{}^{\prime}}_{\omega_{B}}(f)(p):=\sup\left\\{\varphi(p),\varphi\leq
f,\varphi\in PSH(S,\xi,\omega_{B})\cap C^{\infty}(S)\right\\}$

Since $u$ is u.s.c. and basic, it is a decreasing limit of a sequence of
smooth basic functions $(f_{j})$. We assert that the sequence
$(v_{j})_{j\in\mathbb{N}}:=(P^{{}^{\prime}}_{\omega_{B}}(f_{j}))_{j\in\mathbb{N}}$,
which consists of basic functions, decreases to $u$. Indeed, since
$P^{{}^{\prime}}_{\omega_{B}}$ is a decreasing operator, $(v_{j})$ is a
decreasing sequence and $f_{j}\geq v_{j}\geq u$ by construction. Since
$f_{j}\searrow u$, for all $x$ and $\varepsilon>0$, there exists $j_{0}$ such
that for all $j\geq j_{0}$:

$u(x)\leq v_{j}(x)\leq f_{j}(x)\leq u(x)+\varepsilon$

hence $v_{j}(x)$ decreases to $u(x)$.

Arguing as in [Ber19, Proposition 2.3], one can show that the sequence of
basic $\omega_{B}$-psh functions $v_{j,\beta}:=P_{\beta}(f_{j})$ converges
uniformly to $v_{j}$ as $\beta\to\infty$, hence for appropriate
$\varepsilon_{j}\to 0$, the sequence

$u_{j}:=v_{j,\beta(j)}+\varepsilon_{j}$

which consists of smooth basic $\omega_{B}$-psh functions, decreases to $u$. ∎

We also have the comparison principle for $\theta$-psh functions in the
degenerate Sasakian context.

######  Proposition 2.9.

For all $u,v\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$,

$\int_{\left\\{v<u\right\\}}\text{MA}_{\theta}(u)\leq\int_{\left\\{v<u\right\\}}\text{MA}_{\theta}(u)$

###### Proof.

We first prove the following maximum principle:

$1_{\left\\{v<u\right\\}}\text{MA}_{\theta}(\max(u,v))=1_{\left\\{v<u\right\\}}\text{MA}_{\theta}(u)$

It is enough to prove the equality on a foliation chart $U_{\alpha}$. First
remark that since $u,v$ are both basic, on $U_{\alpha}$ they depend only on
the $z$-coordinates, hence
$U_{\alpha}\cap\left\\{v<u\right\\}=]-t,t[\times\left\\{z\in
W_{\alpha},v<u\right\\}$. Since $\text{MA}_{\theta}(u)$ is $\xi$-invariant, it
restricts to $\theta_{u}^{n}\wedge dx$ on $U_{\alpha}$. The equality is then
equivalent to

$1_{]-t,t[\times\left\\{z\in
W_{\alpha},v<u\right\\}}\theta_{\max(u,v)}^{n}\wedge
dx=1_{]-t,t[\times\left\\{z\in W_{\alpha},v<u\right\\}}\theta_{u}^{n}\wedge
dx$

on each foliation chart. By contracting with $\xi=\partial_{x}$, this is
exactly the classical local maximum principle for $\theta$-psh functions.

It follows from the maximum principle that

$\displaystyle\int_{\left\\{v<u\right\\}}\text{MA}_{\theta}(u)$
$\displaystyle=\int_{S}1_{\left\\{v<u\right\\}}\text{MA}_{\theta}(\max(u,v))$
$\displaystyle=\text{vol}_{\theta}(S)-\int_{\left\\{v\geq
u\right\\}}\text{MA}_{\theta}(\max(u,v))$
$\displaystyle\leq\int_{S}\text{MA}_{\theta}(v)-\int_{\left\\{v>u\right\\}}\text{MA}_{\theta}(\max(u,v))=\int_{\left\\{v\leq
u\right\\}}\text{MA}_{\theta}(v)$

By arguing the same way with $u-\varepsilon$ and $v$, we obtain

$\int_{\left\\{v<u-\varepsilon\right\\}}\text{MA}_{\theta}(u)\leq\int_{\left\\{v\leq
u-\varepsilon\right\\}}\text{MA}_{\theta}(v)\leq\int_{\left\\{v<u\right\\}}\text{MA}(v)$

The proof is now concluded by remarking that
$\left\\{v<u-\varepsilon\right\\}$ increases to $\left\\{v<u\right\\}$. ∎

We record the following result for a later use.

######  Proposition 2.10.

Let $U=B_{1}(0)\times]-t,t[$ be a foliation chart on $S$. For every
$\varphi\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$, there exists a unique
$\widetilde{\varphi}\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$ such that

$\text{MA}_{\theta}(\widetilde{\varphi})=0\;\text{on}\;U,\;\widetilde{\varphi}=\varphi\;\text{on}\;S\backslash
U,\;\widetilde{\varphi}\geq\varphi\;\text{on}\;S$

Moreover, if $\varphi_{1}\leq\varphi_{2}$, then
$\widetilde{\varphi}_{1}\leq\widetilde{\varphi}_{2}$.

###### Proof.

The proof is a direct consequence of the local Dirichlet problem on a
degenerate Sasakian manifold. The problem can be solved in exactly the same
way as in the classical case by remarking that for a basic function $u$ in a
foliation chart $(z_{1},\dots,z_{n},x)$.

$(d_{B}d_{B}^{c}u)^{n}\wedge\eta=\det\left(\frac{\partial^{2}u}{\partial
z_{i}\partial\overline{z}_{j}}\right)\bigwedge_{k=1}^{n}\frac{i}{2}dz_{k}\wedge
d\overline{z}_{k}\wedge dx=0\iff\det(u_{i\overline{j}})=0$

Hence the local Dirichlet problem on a degenerate Sasakian manifold becomes
the classical Dirichlet problem (see [BT76], [BT82] for a proof). ∎

######  Proposition 2.11.

Let $(\varphi_{j})_{j\in\mathbb{N}}\subset PSH(S,\xi,\theta)^{\mathbb{N}}$.

  1. 1.

There exists a constant $C=C(\mu_{\omega_{B}},\theta)$ such that for all $u\in
PSH(S,\xi,\theta)$:

$-C+\sup_{S}u\leq\int_{S}ud\mu_{\omega_{B}}\leq\text{vol}_{\omega_{B}}(S)\sup_{S}u$

  2. 2.

If $(\varphi_{j})$ is uniformly bounded on $S$, then either $(\varphi_{j})$
converges locally uniformly to $-\infty$, or $(\varphi_{j})$ is relatively
compact in $L^{1}(S)$.

  3. 3.

If $\varphi_{j}\to\varphi$ in $L^{1}(S)$, then $\varphi$ coincides almost-
everywhere with a function $\varphi^{*}\in PSH(S,\xi,\theta)$. Moreover,

$\sup_{S}\varphi^{*}=\lim_{j\to+\infty}\sup_{S}\varphi_{j}$

  4. 4.

The family

$\mathcal{F}_{0}:=\left\\{\varphi\in PSH(S,\xi,\theta),\sup\varphi=0\right\\}$

is a compact subset of $PSH(S,\xi,\theta)$.

###### Proof.

For $1)$, we can adapt the strategy in [HL21, Prop. 3.3] to the non-degenerate
Sasakian case. Let us sketch the arguments. We only need to prove the first
inequality in the statement (the second one is trivial). Assuming without loss
of generality that $\sup_{S}u=0$, the inequality then reduces to

$\int_{S}ud\mu_{\omega_{B}}\geq-C$

There exists two finite covering of $S$ by foliation charts $V_{\alpha}\subset
U_{\alpha}$ such that $V_{\alpha}\simeq B_{1}(0)\times]-t,t[$ is relatively
compact in $U_{\alpha}\simeq B_{4}(0)\times]-2t,2t[$. To prove the desired
result, it is enough to show that

$\int_{V_{\alpha}}ud\mu_{\omega_{B}}\geq-C_{\alpha}$

where $C_{\alpha}=C_{\alpha}(\theta)$. But on $V_{\alpha}$, this is equivalent
to

$\int_{B_{1}(0)\times]-t,t[}ud\mu_{z,x}=2t\int_{B_{1}(0)}u(z)d\mu_{z}\geq-
C_{\alpha}$

where $d\mu_{z,x}$ and $d\mu_{z}$ are respectively the measures
$\omega_{B}^{n}\wedge\eta$ and $\omega_{B}^{n}$ on $V_{\alpha}$ and
$B_{1}(0)$. Let $\varphi_{\alpha}$ be a local potential of $\theta$ on
$B_{4}(0)$ ($\varphi_{\alpha}$ exists by the
$\partial_{B}\overline{\partial}_{B}$-lemma). The function
$\varphi_{\alpha}+u$ is independent of $x$ and psh in $B_{4}(0)$. By upper-
semicontinuity, $u$ attains its local supremum $u(p_{1})=0$ at
$p_{1}=(z_{1},0)\in B_{4}(0)$ . By the submean inequality on
$B_{2}(z_{1})\subset B_{4}(0)$,

$(\varphi_{\alpha}+u)(z_{1},0)=\varphi_{\alpha}(z_{1},0)\leq\frac{1}{\mu_{z}(B_{2}(z_{1}))}\int_{B_{2}(z_{1})}(\varphi_{\alpha}+u)(z,0)d\mu_{z}$

Since $u\leq 0$ and $B_{1}(0)\subset B_{2}(z_{1})$, this completes our proof.

$2)$ is a consequence of $1)$ (cf. [HL21, Proposition 3.4]).

$3)$ is a consequence of the local result for psh functions (see e.g. [GZ17,
Theorem 1.46 (2)]). Indeed, by assumption, on each foliation chart
$U_{\alpha}\simeq B_{1}(0)\times]-t,t[$, we have $\varphi_{j}\to\varphi$ in
$L^{1}_{\text{loc}}(U_{\alpha})$. In particular, $\varphi_{j}\to\varphi$ in
$L^{1}_{\text{loc}}(B_{1}(0))$ as psh functions.

$4)$ is a direct consequence of $2)$ and $3)$. ∎

The following is a Chern-Levine-Nirenberg-type inequality.

######  Lemma 2.12.

Let $v,u\in PSH(S,\xi,\theta)$ such that $0\leq u\leq 1$. Then

$0\leq\int_{S}\left|v\right|\theta_{u}^{n}\wedge\eta\leq\int_{S}\left|v\right|\theta^{n}\wedge\eta+n(1+2\sup
v)\text{vol}_{\theta}(S)$

###### Proof.

We first suppose that $v\leq 0$. It is enough to establish the equality for
$v_{k}:=\max\left\\{v,-k\right\\}$. Indeed, the sequence $-v_{k}$ increases to
$-v$, which allows us to conclude by monotone convergence theorem. Now let us
prove the desired result for $v_{k}$. It is clear that $v_{k}$ is
$\theta$-psh. We then have the following chain of inequalities:

$\displaystyle\int_{S}(-v_{k})\theta_{u}^{n}\wedge\eta$
$\displaystyle=\int_{S}(-v_{k})\theta_{u}^{n-1}\wedge(\theta+\sqrt{-1}\partial_{B}\overline{\partial}_{B}u)\wedge\eta$
$\displaystyle=\int_{S}(-v_{k})\theta_{u}^{n-1}\wedge\theta\wedge\eta+\int_{S}(-v_{k})\theta_{u}^{n-1}\wedge\sqrt{-1}\partial_{B}\overline{\partial}_{B}u\wedge\eta$
$\displaystyle=\int_{S}(-v_{k})\theta_{u}^{n-1}\wedge\theta\wedge\eta+\int_{S}u\theta_{u}^{n-1}\wedge(-\sqrt{-1}\partial_{B}\overline{\partial}_{B}v_{k})\wedge\eta$
$\displaystyle\leq\int_{S}(-v_{k})\theta_{u}^{n-1}\wedge\theta\wedge\eta+\int_{S}\theta_{u}^{n-1}\wedge\theta\wedge\eta$

A simple induction allows us to conclude for the case $v\leq 0$. The general
case follows by considering $v^{\prime}:=v-\sup_{S}v$. ∎

######  Definition 2.13.

The capacity of a Borel set $E\subset S$ is defined as:

$Cap_{\theta}(E):=\sup\left\\{\int_{E}\text{MA}_{\theta}(u),u\in
PSH(S,\xi,\theta),0\leq u\leq 1\right\\}$

This definition makes sense since $\theta$ is supposed to be big (otherwise
$Cap$ would be identically zero). It is clear by definition that
$Cap_{\theta}(.)\geq 0$.

Now let $PSH^{-}(S,\xi,\theta)$ be the set of negative, basic $\theta$-psh
functions.

######  Proposition 2.14.

  * 1)

If $\theta_{1}\leq\theta_{2}$ are two basic semipositive $(1,1)$-forms on $S$,
then $Cap_{\theta_{1}}(.)\leq Cap_{\theta_{2}}(.)$. Moreover, for all
$\delta\geq 1$,

$Cap_{\theta}(.)\leq Cap_{\delta\theta}(.)\leq\delta^{n}Cap_{\theta}(.)$

For every Borel set $K\subset E$, we have

$0\leq Cap_{\theta}(K)\leq Cap_{\theta}(E)\leq
Cap_{\theta}(X)=\text{vol}_{\theta}(X)$

  * 2)

For all $v\in PSH^{-}(S,\xi,\theta)$, there exists a constant
$C=C(S,\theta)>0$ such that:

$Cap_{\theta}(v<-t)\leq\frac{C}{t}$

for all $t>0$. In particular, $\lim_{t\to+\infty}Cap_{\theta}(v<-t)=0$.

###### Proof.

1) It is clear that if $\theta_{1}\leq\theta_{2}$ then
$\text{MA}_{\theta_{1}}(.)\leq\text{MA}_{\theta_{2}}(.)$ by a property of the
complex Hessian in local coordinates. Moreover, if $\theta_{1}\leq\theta_{2}$,
then $PSH(S,\xi,\theta_{1})\subset PSH(S,\xi,\theta_{2})$, so
$Cap_{\theta_{1}}\leq Cap_{\theta_{2}}$. For all $\delta\geq 1$ and $u\in
PSH(S,\xi,\delta\theta)$, $0\leq u\leq 1$, we have $u\in PSH(S,\xi,\theta)$
and:

$0\leq(u/\delta)\leq(1/\delta)\leq
1,\quad(\delta\theta+d_{B}d_{B}^{c}u)^{n}=\delta^{n}\left(\theta+\frac{d_{B}d_{B}^{c}u}{\delta}\right)^{n}$

Therefore $Cap_{\delta\theta}(.)\leq\delta^{n}Cap_{\theta}(.)$ by definition.

For all $K\subset E$ and all candidate function $u$ in the definition of
$Cap$, $\int_{K}\text{MA}_{\theta}(u)\leq\int_{E}\text{MA}_{\theta}(u)$, hence
$Cap_{\theta}(K)\leq Cap_{\theta}(E)\leq Cap_{\theta}(X)$. Finally,
$Cap_{\theta}(X)=\text{vol}_{\theta}(X)$ since a locally bounded function has
full mass.

2) By the Chern-Levine-Nirenberg inequality in Lem. 2.12, for a $\theta$-psh
function $u$ such that $0\leq u\leq 1$ and $v\in PSH(S,\xi,\theta),v\leq 0$,
we have:

(2)
$\int_{S}(-v)\theta_{u}^{n}\wedge\eta\leq\int_{S}(-v)\theta^{n}\wedge\eta+n\text{vol}_{\theta}(S)$

This inequality allows us to complete the proof. Indeed, for all $u\in
PSH(S,\xi,\theta)$ such that $0\leq u\leq 1$:

$\displaystyle\int_{\left\\{v<-t\right\\}}\theta_{u}^{n}\wedge\eta$
$\displaystyle\leq\frac{1}{t}\int_{S}(-v)\theta_{u}^{n}\wedge\eta$
$\displaystyle\leq\frac{1}{t}\left(\int_{S}(-v)\theta^{n}\wedge\eta+n\text{vol}_{\theta}(S)\right)$
$\displaystyle\leq\frac{1}{t}\left(C(S,\theta)+n\text{vol}_{\theta}(S)\right)(\text{by
Prop. \ref{compacity_properties}})$

We conclude then by the definition of capacity. ∎

The following uniqueness result still holds in the context of degenerate
Sasakian manifolds.

######  Proposition 2.15.

Let $u,v\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$. If

$\text{MA}_{\theta}(u)=\text{MA}_{\theta}(v)$

then $u=v+cst$.

###### Proof.

We borrow the proof from [GZ07, Theorem 3.3] (see also [HL21, Theorem 6.4]),
which still applies when $\theta$ is only semipositive. Let $f=(u-v)/2$ and
$h=(u+v)/2$. We can assume that $u,v\geq-C_{\theta}$ so that
$\int_{S}(-h)\theta_{h}^{n}\wedge\eta\geq 1$. The key idea is to obtain the
following inequalities:

(3) $\displaystyle\int_{S}d_{B}f\wedge
d_{B}^{c}f\wedge\theta^{n-1}_{h}\wedge\eta$
$\displaystyle\leq\int_{S}\frac{f}{2}(\theta_{u}^{n}-\theta_{v}^{n})\wedge\eta$
(4) $\displaystyle\frac{\int_{S}d_{B}f\wedge
d_{B}^{c}f\wedge\theta^{n-1}\wedge\eta}{\int_{S}(-h)\theta_{h}^{n}\wedge\eta}$
$\displaystyle\leq 3^{n}\left(\int_{S}d_{B}f\wedge
d_{B}^{c}f\wedge\theta_{h}^{n-1}\wedge\eta\right)^{1/2^{n-1}}$

As a consequence, if $\theta_{u}^{n}\wedge\eta=\theta_{v}^{n}\wedge\eta$, then
combining (3) and (4) yields $\nabla f=0$, hence $u=v+cst$ as desired. We give
a quick proof of (3).

Note that the current under integration on the lhs of (3) is well-defined
since $u$ and $v$ are supposed to be bounded. A direct calculation yields

$\displaystyle\int_{S}d_{B}f\wedge d_{B}^{c}f\wedge\theta^{n-1}_{h}\wedge\eta$
$\displaystyle\leq\sum_{k=1}^{n-1}\int_{S}d_{B}f\wedge
d_{B}^{c}f\wedge\theta_{u}^{k}\wedge\theta_{v}^{n-1-k}\wedge\eta$
$\displaystyle=\sum\int_{S}f(d_{B}d^{c}_{B}f)\wedge\theta_{u}^{k}\wedge\theta_{v}^{n-1-k}\wedge\eta$
$\displaystyle=\int_{S}\frac{f}{2}(\theta_{u}^{n}-\theta_{v}^{n})\wedge\eta$

The first inequality follows from $C^{k}_{n-1}\leq 2^{n-1}$, the second one
from Stokes’ theorem, and the third from the fact that
$2d_{B}d_{B}^{c}f=\theta_{u}-\theta_{v}$.

The proof of (4) still goes through unchanged. It consists of proving
inductively that for $T=\theta_{h}^{l}\wedge\theta^{n-2-l}\wedge\eta$,
$l=n-2,\dots,0$, we have

$\frac{\int_{S}df\wedge d^{c}f\wedge\theta\wedge
T}{\left(\int_{S}(-h)\theta_{h}^{2}\wedge T\right)^{1/2}}\leq
3\left(\int_{S}df\wedge d^{c}f\wedge\theta_{h}\wedge T\right)^{1/2}$

using an integration by parts and Cauchy-Schwartz inequality. ∎

###  2.3. Extremal functions

Motivated by extremal functions in pluripotential theory, we introduce the
following counterpart in the Sasakian setting.

######  Definition 2.16.

Let $K\subset S$ be a ${\xi}$-invariant Borel subset. The extremal function
associated to $\theta$ and $K$ is defined as:

$V_{K,\theta}(p):=\sup\left\\{\varphi(p),\varphi\in
PSH(S,\xi,\theta),\varphi\leq 0\;\text{on}\;K\right\\}$

Let $V^{*}_{K,\theta}$ be the u.s.c. regularization of $V_{K,\theta}$. We say
that a $\xi$-invariant Borel set $K\subset S$ is $PSH(S,\xi,\theta)$\-
pluripolar if $K$ belongs to the $-\infty$ locus of a basic $\theta$-psh
function. Clearly $\left\\{u=-\infty\right\\}$ is $\xi$-invariant if $u$ is
basic $\theta$-psh. Here we impose the symmetry by $\xi$ on $K$ so that there
is no inherent contradiction in the definition of pluripolarity. The
pluripolarity of $K$ is determined by its extremal function, as the following
lemma shows.

######  Lemma 2.17.

Let $K\subset S$ be a $\xi$-invariant Borel set.

  * 1)

$K$ is $PSH(S,\xi,\theta)$-pluripolar $\iff V^{*}_{K,\theta}=+\infty\iff\sup
V^{*}_{K,\theta}=+\infty$.

  * 2)

If $K$ is not $PSH(S,\xi,\theta)$-pluripolar, then $V^{*}_{K,\theta}\in
PSH(S,\xi,\theta)$ and $V^{*}_{K,\theta}=0$ on $\text{Int}(K)$. Moreover,

$\int_{\overline{K}}\text{MA}_{\theta}(V^{*}_{K,\theta})=\int_{\overline{K}}(V^{*}_{K,\theta})^{n}\wedge\eta=\text{vol}_{\theta}(S),\quad\int_{S\backslash\overline{K}}\text{MA}_{\theta}(V^{*}_{K,\theta})=0$

###### Proof.

1) Suppose that $\sup_{S}V^{*}_{K,\theta}=+\infty$. By Choquet’s lemma, there
exists an increasing sequence of functions $\varphi_{j}\in PSH(S,\xi,\theta)$
such that $\varphi_{j}=0$ on $K$ and
$V^{*}_{K,\theta}=(\lim\nearrow\varphi_{j})^{*}$. Up to extracting a
subsequence, we can assume that $\sup_{S}\varphi_{j}\geq 2^{j}$. Define
$\psi_{j}:=\varphi_{j}-\sup_{S}\varphi_{j}$. The sequence
$\left\\{\psi_{j}\right\\}_{j\in\mathbb{N}}\subset PSH(S,\xi,\theta)$ is
compact and satisfies
$\int_{S}\psi_{j}d\mu_{\omega_{B}}\geq-C(\mu_{\omega_{B}})$ (cf. Lem. 2.11).
Let

$\psi:=\sum_{j\geq 1}2^{-j}\psi_{j}$

The function $\psi$ is basic $\theta$-psh as a limit of basic $\theta$-psh
functions, and satisfies $\int\psi d\mu_{\omega_{B}}\geq-C(\mu_{\omega_{B}})$.
It is clear that $\psi_{j}(x)=-\sup_{S}\varphi_{j}$, $\forall x\in K$, hence
$K\subset\left\\{\psi=-\infty\right\\}$.

Now suppose that $K\subset\left\\{\psi=-\infty\right\\}$ where $\psi\in
PSH(S,\xi,\theta)$. For all $c\in\mathbb{R}$, $\psi+c\in PSH(S,\xi,\theta)$
and $\psi+c\leq 0$ on $K$. It follows that $V_{K,\theta}^{*}\geq\psi+c$, hence
$V_{K,\theta}^{*}=+\infty$ on $S\backslash\left\\{\psi=-\infty\right\\}$.
Finally, $V_{K,\theta}^{*}=+\infty$ on $S$ since
$\left\\{\psi=-\infty\right\\}$ has zero mass with respect to
$\mu_{\omega_{B}}=\omega_{B}^{n}\wedge\eta$.

2) Clearly $V_{K,\theta}^{*}=0$ in $\text{Int}(K)$ by definition. The function
$V_{K,\theta}$ is basic as the sup-envelope of basic functions, hence its
u.s.c. regularization $V^{*}_{K,\theta}$ is also basic. The fact that
$V^{*}_{K,\theta}$ is $\theta$-psh follows from (3) of Prop. 2.11. Since a
locally bounded $\theta$-psh function has full mass, we have

$\int_{\overline{K}}\text{MA}_{\theta}(V^{*}_{K,\theta})=\int_{S}\text{MA}_{\theta}(V^{*}_{K,\theta})=\int_{S}(\theta+d_{B}d_{B}^{c}V^{*}_{K,\theta})^{n}\wedge\eta=\text{vol}_{\theta}(S)$

It only remains to show that $\text{MA}_{\theta}(V^{*}_{K,\theta})=0$ on
$S\backslash\overline{K}$, which is equivalent to showing

$\int_{U_{\alpha}}\text{MA}_{\theta}(V^{*}_{K,\theta})=0$

on each foliation chart $U_{\alpha}=B_{1}(0)\times]-t,t[\subset
S\backslash\overline{K}$. By Choquet’s lemma, there exists an increasing
sequence of functions $\varphi_{j}\in PSH(S,\xi,\theta)$ such that
$\varphi_{j}=0$ on $K$ and $V^{*}_{K,\theta}=(\lim\nearrow\varphi_{j})^{*}$.
Let $\widetilde{\varphi}_{j}$ the unique solution of local Dirichlet problem
with initial datum $\varphi_{j}$ (which exists by Prop. 2.10). In particular,

$\text{MA}_{\theta}(\widetilde{\varphi}_{j})=0\;\text{on}\;U_{\alpha}$

Moreover, the sequecne $(\widetilde{\varphi}_{j})$ is increasing and
$\widetilde{\varphi}_{j}=\varphi_{j}$ on $S\backslash U_{\alpha}$, hence
$\widetilde{\varphi}_{j}=0$ on $K$. This shows that
$\widetilde{\varphi}_{j}\leq V_{K,\theta}^{*}$ , therefore
$\widetilde{\varphi}_{j}\nearrow V_{K,\theta}^{*}$. By continuity of the
Monge-Ampère operator along a monotone sequence (cf. Thm 2.7),
$\text{MA}_{\theta}(V^{*}_{K,\theta})=0$ on $U_{\alpha}$. ∎

Let us now state an important comparison theorem between capacity and extremal
functions.

######  Lemma 2.18.

Let $M_{K,\theta}:=\sup_{S}V^{*}_{K,\theta}$. For all compact non-pluripolar
and $\xi$-invariant $K\subset S$ we have:

$1\leq\text{vol}_{\theta}(S)^{1/n}Cap_{\theta}(K)^{-1/n}\leq\max(1,M_{K,\theta})$

###### Proof.

The inequality on the left is clear by Prop. 2.14. First suppose that
$M_{K,\theta}\leq 1$, then $V_{K,\theta}^{*}$ is bounded. Since $K$ is non-
pluripolar, $V^{*}_{K,\theta}\in PSH(S,\xi,\theta)$. Moreover,
$\text{MA}_{\theta}(V^{*}_{K,\theta})$ is supported in $K$ (cf. Lem. 2.17),
hence

$Cap_{\theta}(K)\geq\int_{K}\text{MA}_{\theta}(V^{*}_{K,\theta})=\int_{S}\text{MA}_{\theta}(V^{*}_{K,\theta})=\text{vol}_{\theta}(S),$

which completes the proof in the $M_{K,\theta}\leq 1$ case.

Assume now that $M:=M_{K,\theta}\geq 1$. Since the function
$V_{K,\theta}^{*}/M$ is a candidate in the definition of $Cap_{\theta}$, it
follows that

$\displaystyle Cap_{\theta}(K)$
$\displaystyle\geq\int_{K}\text{MA}_{\theta}(M^{-1}V^{*}_{K,\theta})$
$\displaystyle=\int_{S}\text{MA}_{\theta}(M^{-1}V^{*}_{K,\theta})\;(\text{by
Lem. \ref{extremal_fucntion_properties}})$ $\displaystyle\geq
M^{-n}\int_{S}\text{MA}_{\theta}(V^{*}_{K,\theta})=M^{-n}\text{vol}_{\theta}(S)$

This allows us to conclude. ∎

###  2.4. Lelong number and integrability

We define the Lelong number of a basic psh function $u$ on a foliation chart
$U_{\alpha}$ at a point $p$ with coordinates $(z,x)$ by

$\nu(u,p):=\lim_{r\to
0^{+}}\frac{1}{\log(r)\text{vol}(B(z,r))}\int_{B(z,r)}u(z)\omega_{B}^{n}$

This number does not depend on the foliation chart since the transition maps
restrict to biholomorphisms on transverse neighborhoods and that the right-
hand side is invariant under biholomorphisms by a theorem of Siu.

It is clear by our definition that the Lelong number is $\xi$-invariant.
Moreover, in a foliation chart $B_{1}(0)\times]-t,t[$, the function
$x\in]-t,t[\to\nu(u,(z,x))$ is constant for all $z\in B_{1}(0)$. The Lelong
number at a point $p$ on a Sasakian manifold therefore equals to its value at
the projection of $p$ to the transverse holomorphic ball of a foliation chart.
Local properties of Lelong number can be translated word by word to the
Sasakian setting.

######  Proposition 2.19.

The number

$\nu(\left\\{\theta\right\\}):=\sup\left\\{\nu(\varphi,x),(\varphi,x)\in
PSH(S,\xi,\theta)\times S\right\\}$

is finite and depends only on the basic cohomology class of $\theta$.

###### Proof.

Since $S$ is compact, there exists a basic Kähler form $\theta^{\prime}$ such
that $\theta^{\prime}\geq\theta$, hence $PSH(S,\xi,\theta^{\prime})\supset
PSH(S,\xi,\theta)$, so
$\nu(\left\\{\theta^{\prime}\right\\})\geq\nu(\left\\{\theta\right\\})$. It is
then enough to prove the assertion when $\theta$ is transverse Kähler.

For $p\in S$, we define $\chi$ to be a smooth function equals to $1$ in a
neighborhood of $p$ and $0$ outside a larger neighborhood. Let

$g_{p}(.):=\chi(.)\log d(.,p)$

where $d$ is the Riemannian distance associated to $\theta$. It is clear that
$g_{p}$ is smooth on $S\backslash\left\\{p\right\\}$ and psh on a neighborhood
of $p$, hence $A\theta$-psh for $A>0$. Since $S$ is compact, we can choose a
uniform constant $A=A(\theta)$ such that for all $p\in S$:

$dd^{c}g_{p}\geq-A\theta$

By taking average with respect to the action of the compact torus generated by
$\xi$, we can suppose that $g_{p}$ is $\xi$-invariant, hence $g_{p}\in
PSH(S,\xi,A\theta)$.

A basic psh function $\varphi$ in a foliation chart $B_{1}(0)\times]-t,t[$
restricts to a psh function on the ball $B_{1}(0)$, so we have

$\nu(\varphi,0)=\int_{\left\\{0_{z}\right\\}}d_{B}d_{B}^{c}\varphi\wedge(d_{B}d_{B}^{c}\log\left|z\right|)^{n-1}$

with $0_{z}$ being the center of $B_{1}(0)$ (see e.g. [GZ17, Lemma 2.46] for a
proof). It follows from this local result that for any $a=(z,x)$

(5)
$\nu(\varphi,a)=\int_{\left\\{z\right\\}}\theta_{\varphi}\wedge(A\theta+d_{B}d_{B}^{c}g_{a})^{n-1}$

The right-hand side is bounded by
$\int_{S}A^{n}\theta^{n}\wedge\eta=A^{n}\text{vol}_{\theta}(S)$. This
completes our proof. ∎

######  Theorem 2.20.

Let $\mathcal{F}_{0}:=\left\\{\varphi\in
PSH(S,\xi,\theta),\sup_{S}\varphi=0\right\\}$. If

$A<2\nu(\left\\{\theta\right\\})^{-1},$

then

$\sup_{\varphi\in\mathcal{F}_{0}}\left\\{\int_{S}e^{-A\varphi}\omega_{B}^{n}\wedge\eta\right\\}\leq
C$

for a constant $C$ depending only on $\omega_{B}$ and $\theta$.

###### Proof.

We will reduce the problem to the classic Skoda’s integrability theorem. First
remark that there exist two covers of $S$ by a finite number of foliations
charts $(V_{j})_{1\leq j\leq N}$ and $(U_{j})_{1\leq j\leq N}$, where
$U_{j}=B_{1}(0)\times]-t,t[$, such that $\overline{V}_{j}\subset U_{j}$. We
need to show that on each foliation chart $U_{j}$, there exists a constant
$C_{j}=C(V_{j},\mathcal{F}_{0},A)$ satisfying

$\int_{U_{j}}e^{-A\varphi}\omega_{B}^{n}\wedge\eta\leq C_{j}$

But since on $U_{j}$, $\varphi$ depends only on the $z$ coordinates and $\eta$
coincides with $dx$, it is enough to show that

$\int_{U_{j}}e^{-A\varphi}\omega_{B}^{n}\wedge\eta=2t\int_{B_{1}(0)}e^{-A\varphi}\omega_{B}^{n}\leq
C_{j}$

This follows from the local Skoda’s integrability theorem since the family
$\mathcal{F}_{0}$ is compact (cf. [GZ17, Theorem 2.50] for a proof). ∎

##  3\. Regularity of the potential

This part is dedicated to the proof of our main theorem. Let us first give
some preliminaries and outline the arguments of the proof. Consider a Fano
cone $Y$ of complex dimension $n+1$ with a good action by
$T\simeq(\mathbb{C}^{*})^{k}$. Let $T_{c}\simeq(\mathbb{S}^{1})^{k}$ be the
maximal compact subtorus of $T$.

Consider a $T_{c}$-equivariant embedding of $Y$ into $\mathbb{C}^{N}$ such
that $T_{c}$ corresponds to a diagonal group acting linearly on
$\mathbb{C}^{N}$. Recall that $\xi$ generates the action of a compact torus
$T_{\xi}\subset T_{c}$. Now fix a locally bounded conical Calabi-Yau potential
$r^{2}$ and a Reeb vector $\xi$ on $Y$, whose action by $T_{\xi}$ extends to
$\mathbb{C}^{N}$ through the embedding. Let $r^{2}_{\xi}$ be the radial
function on $\mathbb{C}^{N}$ associated to $\xi$ with conical metric
$\omega_{\xi}=dd^{c}r^{2}_{\xi}$. Then $r_{\xi}^{2}$ restricts to a
$\xi$-conical potential on $Y$. The link of $Y$ is homeomorphic to the set
$Y\cap\left\\{r_{\xi}^{2}=1\right\\}$. Now let

$\pi:X\to Y$

be a $T$-equivariant resolution of $Y$ (which exists by Lem. 3.1). Let

$\mathcal{U}:=\pi^{-1}(Y_{\text{reg}})$

be the open Zariski subset of $X$ isomorphic to $Y_{\text{reg}}$. Consider the
following submanifold of $X$:

$(S=\pi^{-1}(Y\cap\left\\{r_{\xi}^{2}=1\right\\}),\xi,\eta,\omega_{B})$

where by an abuse of notation $\xi$ still denotes the pullback of the given
Reeb field on $\mathbb{C}^{N}$, $\omega_{B}$ is a transverse Kähler form on
$S$ (cf. Lem. 3.3), and $\eta=2\pi^{*}d^{c}\log r_{\xi}^{2}$ the contact form
on $S$, which is pullback of the contact form associated to $\xi$ on
$\mathbb{C}^{N}$. Since $d\theta$ is only semipositive, $S$ is degenerate
Sasakian. One can show (see Prop. 3.5) that the conical Calabi-Yau equation

$(dd^{c}r)^{n+1}=dV_{Y}$

is in fact equivalent to the following transverse equation on $\mathcal{U}\cap
S$:

$(\theta_{X}+d_{B}d_{B}^{c}\varphi_{X})^{n}\wedge\eta=e^{-(n+1)\varphi_{X}}e^{(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}\wedge\eta$

Here

  * •

$\theta_{X}:=d\eta$,

  * •

$\varphi_{X}:=\pi^{*}\varphi,\;\varphi:=\log(r^{2}/r_{\xi}^{2})$,

  * •

$\Psi_{\pm}$ are basic $A\omega_{B}$-quasi-psh on $S$ for $A>0$ large enough,

Remark that $\theta_{X}$ is a semipostive and big form on $S$. By
construction, $\varphi_{X}$ is invariant under the induced actions of $\xi$
and $-J\xi$ on $X$. In a foliation chart $(z_{1},\dots,z_{n},x)$ of $S$, the
equation can be written as:

$\det\left(\theta_{X,i\overline{j}}+\frac{\partial^{2}\varphi_{X}}{\partial
z_{i}\partial\overline{z}_{j}}\right)=e^{-(n+1)\varphi_{X}(z)}e^{(n+1)(\Psi_{+}(z)-\Psi_{-}(z))}\det(\omega_{B,i\overline{j}})$

The smoothness of $r^{2}=r_{\xi}^{2}e^{\varphi}$ on $Y_{\text{reg}}$ is then
equivalent to the regularity of $\varphi_{X}:=\pi^{*}\varphi$ on
$S\cap\mathcal{U}$. Consider the family of equations:

$(\theta_{X}+\varepsilon\omega_{B}+d_{B}d_{B}^{c}\varphi_{j,\varepsilon})^{n}=e^{(n+1)(\psi_{+,j}-\psi_{-,j})}\omega_{B}^{n}$

where $\psi_{\pm,j}$ are two sequences of basic $A\omega_{B}$-qpsh functions
decreasing to $\psi_{+}:=\Psi_{+}$ and $\psi_{-}:=\Psi_{-}+\varphi_{X}$ for
$A>0$ large enough. The existence of a unique $\varphi_{j,\varepsilon}$
verifying $\sup\varphi_{j,\varepsilon}=0$ is guaranteed by the transverse
Calabi-Yau theorem of [EKA90]. Finally, to obtain the regularity of
$\varphi_{X}$, we proceed by the following classic steps:

  * 1)

Uniform estimate: The functions $\varphi_{j,\varepsilon}$ are uniformly
bounded, i.e. there exists a constant $C$ independent of $j$ and
$\varepsilon$, such that:

$\left\|\varphi_{j,\varepsilon}\right\|_{L^{\infty}(S)}\leq C$

  * 2)

Laplacian uniform estimate: Using the uniform estimate of the previous step,
one can show that there exists $C^{\prime}$ such that for all $j,\varepsilon$,

$\sup_{S\cap\mathcal{U}}\left|\Delta_{\omega_{B}}\varphi_{j,\varepsilon}\right|\leq
C^{\prime}$

where

$\operatorname{Tr}_{\omega_{B}}f:=n\frac{d_{B}d_{B}^{c}f\wedge\omega_{B}^{n-1}}{\omega_{B}^{n}}$

  * 3)

By the complex Evans-Krylov theory, we obtain the following uniform estimate:

$\left\|\varphi_{j,\varepsilon}\right\|_{C^{2,\beta}(S)}\leq
C^{\prime\prime},$

which implies $C^{k+2,\beta}$-estimates for all $k>0$ by Schauder estimate and
a bootstrapping argument.

The last step is classic and well-known in the literature (cf. [Blo12]). Our
focus will be mostly on the first and second steps (see Prop. 3.10 and Prop.
3.12).

###  3.1. Transverse Kähler form

Let $V$ be an irreducible projective variety. Following [Kol07, Paragraph 3],
by a strong resolution we mean a proper morphism $\pi:V^{\prime}\to V$ such
that

  * •

$V^{\prime}$ is smooth and $\pi$ is birational.

  * •

$\pi:\pi^{-1}(V_{\text{reg}})\to V_{\text{reg}}$ is a biholomorphism.

  * •

$\pi^{-1}(V_{\text{sing}})$ is a divisor with simple normal crossings (s.n.c).

In the sense of [Kol07, Paragraph 4], we say that a resolution is functorial
if for any varieties $V,W$ with resolutions $\pi_{V}:V^{\prime}\to V$,
$\pi_{W}:W^{\prime}\to W$, every smooth morphism $\varphi:V\to W$ can be
lifted to a smooth morphism $\varphi^{\prime}:V^{\prime}\to W^{\prime}$ such
that $\pi_{W}\circ\varphi^{\prime}=\varphi\circ\pi_{V}$.

######  Lemma 3.1.

There exists a smooth $T$-equivariant resolution of singularities $\pi:X\to
Y$.

###### Proof.

Let us embed $Y$ in a $T$-equivariant manner into $\mathbb{C}^{N}$ such that
$T$ is identified with a diagonal group. Let
$\overline{Y}\subset\mathbb{P}^{N}$ be the closure of $Y$ in $\mathbb{P}^{N}$.
There exists a $T$-equivariant resolution $\pi:\overline{X}\to\overline{Y}$.
Indeed, it is enough to take a $\pi$ as a strong and functorial resolution in
the sense of Kollar as recalled above (see [Kol07, Theorem 36] for a proof of
existence).

The functoriality of the resolution implies that the action of all algebraic
group on $\overline{Y}$ lifts on $\overline{X}$ such that $\pi$ is equivariant
(see [Kol07, Paragraph 9]). We conclude that
$\pi:X:=\overline{X}\cap\mathbb{C}^{N}\to Y$ is a $T$-equivariant resolution
of $Y$. ∎

Now let $(X,\pi)$ be the resolution of $Y$, constructed in the previous lemma.
Let $E_{0}:=\pi^{-1}(0_{Y})$ be the “vertex exceptional divisor”. Since $\pi$
is equivariant, the vector fields $\xi$ and $-J\xi$ induce by pullback the
respective actions on $X$ (still denoted by $\xi$ and $-J\xi$). The action
generated by $-J\xi$ is an action of $\mathbb{R}^{*}_{+}$.

The pullback by $\pi$ of the holomorphic vector field
$v_{\xi}:=(-J\xi-\sqrt{-1}\xi)/2$ defines a holomorphic foliation
$\mathcal{F}_{v_{\xi}}$ on $X\backslash E_{0}$. At every point $p\in
X\backslash E_{0}$, there exists a transverse holomorphic coordinates
$(z_{1},\dots,z_{n},w)$ such that

$v_{\xi}.z_{j}=0,\;\frac{\partial}{\partial\Im
w}=\xi,\;\frac{\partial}{\partial\Re w}=(-J\xi)$

which restrict to the coordinate $(z,x)$ on $S$. In other words,
$w=\pi^{*}\log r_{\xi}+\sqrt{-1}x$. A form $\alpha$ on $X\backslash E_{0}$ is
said to be basic if

$\mathcal{L}_{V}\alpha=i_{V}\alpha=0,\;\forall
V\in\mathbb{R}\left\\{\xi,-J\xi\right\\}$

The restriction map allows us to identify basic forms on $X\backslash E_{0}$
and basic forms on $S$.

######  Lemma 3.2.

There exists a $T_{c}$-invariant Kähler form $\omega$ on $X$ and a global
smooth function $\Phi_{\omega}$ defined on $U$ such that

$dd^{c}\Phi_{\omega}=\omega,\;\Phi_{\omega}\to-\infty\;\text{near}\;\partial\mathcal{U}$

###### Proof.

Let $\pi:\overline{X}\to\overline{Y}\subset\mathbb{P}^{N}$ be the resolution
as in the previous lemma. Let $\mathcal{O}(1)$ be the $T$-linearized
hyperplane line bundle of $\mathbb{P}^{N}$ and $E$ the exceptional divisor of
$(\overline{X},\pi)$. Since $\pi$ is relatively ample, there exists an ample
line bundle $A$ on $\overline{X}$ such that:

$\pi^{*}\mathcal{O}(1)=A+E$

Now let $\left\|.\right\|_{E}$ a $T_{c}$-invariant metric on
$E=\left\\{s_{E}=0\right\\}$ and
$\varphi_{E}:=-\log\left\|s_{E}\right\|^{2}_{E}$ its potential. Let $h_{A}$ be
a $T_{c}$-invariant metric of strictly positive curvature on $A$. Let $h$ be
the $T_{c}$-invariant metric $h:=h_{A}e^{-\varphi_{E}}$ on
$\pi^{*}\mathcal{O}(1)$ and $\Phi_{\omega}:=-\log h$ the potential of $h$.
Since $X$ is contained in an open affine set $\simeq\mathbb{C}^{N}$ of
$\mathbb{P}^{N}$, there exists a global trivializing $T_{c}$-invariant section
$s$ of the line bundle $\pi^{*}\mathcal{O}(1)|_{X}$. The global form

$\omega:=dd^{c}\Phi_{\omega}|_{X}=-dd^{c}\log h(s)$

is clearly closed and positive definite. Indeed, since $s\neq 0$ in $X$, we
have

$-dd^{c}\log h(s)=-dd^{c}\log h_{A}|_{X}>0$

Finally, since $s_{E}\to 0$ near $\partial\mathcal{U}$, we have
$\Phi_{\omega}=-\log h_{A}+\varphi_{E}\to-\infty$ near $\partial\mathcal{U}$.
∎

######  Lemma 3.3.

[Ber20, Prop. 4.3] There exists a global smooth function $\Phi_{B}$ on
$\mathcal{U}$ satisfying

$\mathcal{L}_{\xi}\Phi_{B}=0,\mathcal{L}_{-J\xi}\Phi_{B}=2,\Phi_{B}\to-\infty\;\text{near}\;\partial\mathcal{U}$

and a transverse basic Kähler form $\omega_{B}$ on $X\backslash E_{0}$ such
that $dd^{c}\Phi_{B}=\omega_{B}$ on $\mathcal{U}$.

######  Remark.

The information on the behavior of $\Phi_{B}$ near the border of $\mathcal{U}$
is crucial in the Laplacian estimate of the potential $\varphi_{X}$.

###### Proof.

The proof in [Ber20] is an adaptation of the construction of reduced Kähler
metrics on a symplectic quotient (see e.g. [BG04, Formulae 4.5, 4.6]). We
provide here the details for the reader’s convenience.

Remark however that in our case, the symplectic quotient is not well defined
since the action generated by $\xi$ on the level set of the hamiltonian is not
free in general. However, the construction still applies since it is local in
nature.

Let $\omega$ be the $T_{c}$-invariant Kähler form on $X$, constructed in Lem.
3.2. Remark that the action generated by $\xi$ is hamiltonian with respect to
$\omega$ (since by the embedding of $Y$ into $\mathbb{C}^{N}$, $\xi$ is
identified with a hamiltonian action on $\mathbb{C}^{N}$). It follows that
there exists a smooth function $\mathcal{H}:X\to\mathbb{R}$ such that:

$d\mathcal{H}(.)=-\omega(\xi,.)=g_{\omega}(-J\xi,.)$

where $g_{\omega}$ is the metric associated to $\omega$. In particular,
$d\mathcal{H}(-J\xi)>0$, so $d_{x}\mathcal{H}$ is surjective for $x\notin
E_{0}$. It follows that $\mathcal{H}$ is a submersion for $x\notin E_{0}$;
hence for $\lambda$ positive, sufficiently large,

$S_{\lambda}=\left\\{\mathcal{H}=\lambda\right\\}$

is a compact submanifold of $X\backslash E_{0}$, diffeomorphic to
$(X\backslash E_{0})/\mathbb{R}^{*}_{+}$. Now let

$\pi_{\lambda}:X\backslash E_{0}\to S_{\lambda},\quad
i_{\lambda}:S_{\lambda}\to X\backslash E_{0}$

be the natural projection and inclusion. Let $\Phi_{\omega}$ be the global
potential on $\mathcal{U}$ constructed in Lemma 3.2. Let $V_{p}$ be the
neighborhood of a point $p\in X\backslash E_{0}$ with local transverse
coordinates $(z,w)$. Consider the following $\xi$-invariant function on
$S_{\lambda}\cap\mathcal{U}$:

$\Phi_{\lambda}:=i^{*}_{\lambda}(\Phi_{\omega}-\lambda\Im w)$

The function

(6) $\Psi_{B}=\pi^{*}_{\lambda}\Phi_{\lambda}+\lambda\Im
w=\pi^{*}_{\lambda}\Phi_{\omega}|_{S_{\lambda}}+\lambda(\Im
w-i^{*}_{\lambda}\Im w)$

is then $\xi$-invariant on $V_{p}$ and well-defined on $V_{p}$. Indeed, let
$V_{p^{\prime}}$ be another local transverse neighborhood of a point
$p^{\prime}\in S_{\lambda}\cap V_{p}$. By the definition of $w$,
$v_{\xi}(w-w^{\prime})=0$, so there exists a basic transversely holomorphic
function $f(z)$ on $V_{p}\cap V_{p^{\prime}}$ such that $w-w^{\prime}=f(z)$.
It follows that:

$\Im(w-w^{\prime})|_{V_{p}\cap
V_{p^{\prime}}}=\Im(w-w^{\prime})|_{S_{\lambda}\cap V_{p}\cap
V_{p^{\prime}}}=i^{*}_{\lambda}\Im(w-w^{\prime})|_{V_{p}\cap V_{p^{\prime}}}$

By construction, we have $\mathcal{L}_{-J\xi}\Psi_{B}=\lambda$, hence
$\Psi_{B}$ extends uniquely to a smooth function on $\mathcal{U}$. The
function

$\Phi_{B}:=2(\Psi_{B}/\lambda)$

satisfies $\mathcal{L}_{\xi}\Phi_{B}=0,\;\mathcal{L}_{-J\xi}\Phi_{B}=2$. We
assert that the following global form on $\mathcal{U}$

$\omega_{B}:=dd^{c}\Phi_{B}$

defines a transverse Kähler metric on $\mathcal{U}$. By a direct computation
from the equation (6) as in [BG04, Section 9], $2\lambda^{-1}\omega$ is
exactly $\omega_{B}$ on $S_{\lambda}$. After replacing $\Phi_{B}$ with
$2\lambda^{-1}\Phi_{\omega}$ on each $V_{p}$, we see that $\omega_{B}$ extends
to a transverse Kähler metric on $X\backslash E_{0}$.

It remains to show that $\Phi_{B}\to-\infty$ on $\partial\mathcal{U}$. Indeed,
on $\mathcal{U}\cap V_{p}$, $\Phi_{B}-2\lambda^{-1}\Phi_{\omega}=\Im
w-i^{*}_{\lambda}(\Im w)$ for all $p\in S_{\lambda}$. It follows that
$\Phi_{B}-2\lambda^{-1}\Phi_{\omega}$ is bounded on
$S_{\lambda}\cap\mathcal{U}$, so
$\Phi_{B}=(\Phi_{B}-2\lambda^{-1}\Phi_{\omega})+2\lambda^{-1}\Phi_{\omega}\to-\infty$
near $\partial\mathcal{U}$ since $\Phi_{\omega}\to-\infty$ near
$\partial\mathcal{U}$. ∎

Since $X$ is a $T_{c}$-invariant resolution of $Y$ and that $Y$ has klt
singularities, there exists a $T_{c}$-invariant divisor $D$ such that:

$\pi^{*}K_{Y}=K_{X}+D,\;D=\sum_{a_{j}>-1}a_{j}D_{j},$

We have moreover a decomposition $D=D_{+}-D_{-}$ where:

$D_{+}:=\sum_{a_{j}>0}D_{j},\;D_{-}:=\sum_{a_{j}<0}(-a_{j})D_{j}$

are two effective $T_{c}$-invariant $\mathbb{Q}$-divisors. There exists then a
$T_{c}$-invariant volume form $dV_{X}$ on $X$, two multivalued sections
$s_{\pm}$ and hermitian $T_{c}$-invariant metrics $h^{\pm}$ on $D_{\pm}$, such
that:

(7)
$\pi^{*}dV_{Y}=\left\|s_{+}\right\|^{2}_{h^{+}}\left\|s_{-}\right\|^{-2}_{h^{-}}dV_{X}$

To be precise, we may choose:

$\left\|s_{+}\right\|^{2}_{h^{+}}:=\prod_{a_{j}>0}\left|s_{j}\right|^{2a_{j}}_{h_{j}},\quad\left\|s_{-}\right\|^{2}_{h^{-}}:=\prod_{a_{j}<0}\left|s_{j}\right|^{-2a_{j}}_{h_{j}}$

where $h_{j}$ are $T_{c}$-invariant hermitian metrics of the fiber
$\mathcal{O}_{X}(D_{j})$. Up to a positive constant, we have the following
volume form on $S$:

$dV_{X}(-J\xi,.)=\omega_{B}^{n}\wedge\eta$

######  Lemma 3.4.

There exist two basic psh $T_{c}$-invariant functions $\Psi_{\pm}$ on $S$,
smooth on $\mathcal{U}$ and a constant $A>0$ such that on $S$,

$\pi^{*}dV_{Y}(-J\xi,.)=e^{(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}\wedge\eta,\quad\frac{i}{2\pi}\partial_{B}\overline{\partial}_{B}\Psi_{\pm}\geq-A\omega_{B}$

Moreover, $e^{-\Psi_{-}}\in L^{p}(S),p>1$.

###### Proof.

Assume that there exists a positive constant $C>0$ satisfying:

(8)
$\frac{i}{2\pi}\partial_{B}\overline{\partial}_{B}\log\left\|s_{\pm}\right\|^{2}_{h^{\pm}|_{S}}\geq-C\omega_{B}$

Then by choosing $\Psi_{\pm}$ such that:

$(n+1)\Psi_{\pm}=\log\left\|s_{\pm}\right\|^{2}_{h^{\pm}|_{S}}\in
C_{B}^{\infty}(S)$

we obtain the equality between volume forms from (7) and the estimate of
$\partial_{B}\overline{\partial}_{B}\Psi_{\pm}$ follows immediately.

It remains to prove (8). By definition of $s_{\pm}$ and
$\left\|.\right\|_{h^{\pm}}$, in a transverse holomorphic chart of
$X\backslash E_{0}$ with coordinates $(z,w)$, there exist $T_{c}$-invariant
local potentials $\varphi_{\pm}$ and holomorphic $T_{c}$-semi-invariant local
functions $f_{\pm}$ such that:

$\left\|s_{\pm}\right\|_{h^{\pm}}=\left|f_{\pm}(z,w)\right|e^{-\varphi_{\pm}(z,w)}$

In particular, there exist $\lambda_{\pm}\in\mathbb{R}$ satisfying:

$\frac{\partial}{\partial\Im w}f_{\pm}=i\lambda_{\pm}f$

After replacing $f_{\pm}$ by $f_{\pm}e^{-\lambda_{\pm}w}$, one can suppose
that $f_{\pm}$ are $\xi$-invariant (hence basic), so
$\overline{\partial}_{B}f_{\pm}=0$. It follows that $f_{\pm}$ are transversely
holomorphic, hence $d_{B}d^{c}_{B}\log\left|f_{\pm}(z,w)\right|^{2}\geq 0$, so
locally:

$d_{B}d^{c}_{B}\log\left\|s_{\pm}\right\|^{2}_{h^{\pm}|_{S}}\geq-
Cd_{B}d_{B}^{c}\varphi_{\pm}$

for some constant $C$ depending only on the local open set. Moreover, since
$\omega_{B}$ is Kähler, one can find in a transverse neighborhood a constant
$A>0$ (which depends only on the neighborhood) such that

$d_{B}d^{c}_{B}\varphi_{\pm}\leq A\omega_{B}$

The compacity of $S$ then completes the proof of (8). Finally, since $Y$ has
klt singularities, $D_{j}$ are normal crossing divisors, hence there exists
$p>1$ such that $pa_{j}>-1$ for all $j$, so $e^{-\Psi^{-}}\in L^{p}(S)$ for
some $p>1$. ∎

###  3.2. Transverse Monge-Ampère equation

######  Proposition 3.5.

The conical potential $r$ is a solution in the pluripotential sense of the
equation:

(9) $(dd^{c}r^{2})^{n+1}=dV_{Y}$

on $Y_{\text{reg}}$ if and only if $\varphi_{X}$ satisfies the following
equation on $S\cap\mathcal{U}$:

(10)
$(\theta_{X}+d_{B}d_{B}^{c}\varphi_{X})^{n}\wedge\eta=e^{-(n+1)\varphi_{X}}e^{(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}\wedge\eta$

In particular, in a transverse holomorphic neighborhood $S\cap\mathcal{U}$,

$(\theta_{X}+d_{B}d^{c}_{B}\varphi_{X})^{n}=e^{-(n+1)\varphi_{X}}e^{-(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}$

###### Proof.

By definition $\Phi=\log r^{2}$, hence:

$dd^{c}r^{2}=e^{\Phi}(dd^{c}\Phi+d\Phi\wedge
d^{c}\Phi)=r^{2}(dd^{c}\Phi+d\Phi\wedge d^{c}\Phi)$

in the current sense. We have

$(dd^{c}\Phi+d\Phi\wedge d^{c}\Phi)^{n+1}=\sum
c_{k,n}(dd^{c}\Phi)^{k}\wedge(d\Phi\wedge
d^{c}\Phi)^{n-k}=(dd^{c}\Phi)^{n}\wedge d\Phi\wedge d^{c}\Phi$

Indeed, in the transverse coordinates $(z,w)$ on $X\backslash E_{0}$,

$\frac{\partial\Phi}{\partial w}=\frac{\partial\Phi}{\partial\overline{w}}=1$

hence $(dd^{c}\Phi)^{n+1}=0$. It follows that

$\displaystyle(dd^{c}r^{2})^{n+1}=dV_{Y}\iff r^{2n+2}(dd^{c}\Phi)^{n}\wedge
d\Phi\wedge d^{c}\Phi=dV_{Y}$

Since

$\mathcal{L}_{\xi}\Phi=0,$

the restriction of $\Phi$ in $S$ is basic. It follows that

$\displaystyle(dd^{c}\Phi)^{n}\wedge d\Phi\wedge d^{c}\Phi$
$\displaystyle=\det\left(\frac{\partial^{2}\Phi}{\partial
z_{l}\partial\overline{z}_{m}}\right)\bigwedge(i/2)dz_{k}\wedge
d\overline{z}_{k}\wedge d\Phi\wedge d^{c}\Phi$
$\displaystyle=(dd^{c}\Phi)^{n}\wedge(dw+d\overline{w})\wedge(d^{c}w+d^{c}\overline{w})$
$\displaystyle=(dd^{c}\Phi)^{n}\wedge 2d\Re w\wedge 2d^{c}\Re w$

The conical Calabi-Yau equation then becomes

$r^{2n+2}(dd^{c}\Phi)^{n}\wedge 2d\Re w\wedge 2d^{c}\Re w=dV_{Y}$

By contracting the equality with $-J\xi$, and using $2d\Re w(-J\xi)=1$, we
have:

$r^{2n+2}(dd^{c}\Phi)^{n}\wedge 2d^{c}\Re
w=dV_{Y}(-J\xi)=\omega_{B}^{n}\wedge\eta$

By using $dd^{c}\Phi=\theta+dd^{c}\varphi=\theta+d_{B}d_{B}^{c}\varphi$,
$2d^{c}\Re w=\eta$, the previous lemma and the fact that
$S=X\cap\pi^{-1}(\left\\{r_{\xi}^{2}=1\right\\})$, we obtain by pullback the
following equation on $S\cap\mathcal{U}$:

$(\theta_{X}+d_{B}d^{c}_{B}\varphi_{X})^{n}\wedge\eta=e^{-(n+1)\varphi_{X}}e^{-(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}\wedge\eta$

Finally by applying $i_{\xi}$ and using that $\eta(\xi)=1$, the equation on
$S\cap\mathcal{U}$ becomes

$(\theta_{X}+d_{B}d^{c}_{B}\varphi_{X})^{n}=e^{-(n+1)\varphi_{X}}e^{-(n+1)(\Psi_{+}-\Psi_{-})}\omega_{B}^{n}$

The converse is proved in the same manner. ∎

###  3.3. Uniform estimate

Let $\psi_{\pm,j}$ be two sequences of smooth basic quasi-psh functions which
decrease to

$\psi_{+}:=\Psi_{+},\quad\psi_{-}:=\Psi_{-}+\varphi_{X},$

such that :

(11) $\quad d_{B}d^{c}_{B}\psi_{\pm,j}\geq-C\omega_{B}$

for a uniform constant $C$ independent of $j$. Such a sequence exists by
virtue of Lem. 2.8.

Let $\varepsilon>0$. Recall that the form $\theta_{X}=\pi^{*}dd^{c}\log
r_{\xi}^{2}$ is semi-positive, big and basic, hence
$\theta_{X}+\varepsilon\omega_{B}$ is a transverse Kähler form. Consider the
following equation on $S$ for a smooth basic
$(\theta_{X}+\varepsilon\omega_{B})$-psh function $\varphi_{j,\varepsilon}$:

(12)
$\left(\theta_{X}+\varepsilon\omega_{B}+d_{B}d^{c}_{B}\varphi_{j,\varepsilon}\right)^{n}\wedge\eta=e^{(n+1)(\psi_{+,j}-\psi_{-,j})}\omega_{B}^{n}\wedge\eta$

By the transverse Calabi-Yau theorem of El-Kacimi Alaoui [EKA90, 3.5.5], for
all $j,\varepsilon$, there exists a unique basic solution satisfying:

$\sup\varphi_{j,\varepsilon}=0$

Now let $\mu_{j}$ be the smooth volume form
$e^{(n+1)(\psi_{+,j}-\psi_{-,j})}\omega_{B}^{n}\wedge\eta$ on $S$. The
following lemma is elementary:

######  Lemma 3.6.

Let $\mu$ be an inner-regular positive Borel measure on $S$. Then for all
$\xi$-invariant Borel set $E\subset S$,

$\mu(E)=\sup\left\\{\mu(K),K\subset
E\;\text{compact},\xi-\text{invariant}\right\\}$

In particular, $\mu_{j}$ satisfies this property.

###### Proof.

It is enough to show that for all $j\in\mathbb{N}^{*}$, there exists a compact
$\xi$-invariant $K_{j}$ such that:

$\mu(E)\leq\mu(K_{j})+\frac{1}{j}$

By inner regularity of $E$, there exists a compact $C_{j}\subset E$ such that:

$\mu(E)\leq\mu(C_{j})+1/j$

The idea is to average $C_{j}$ by the action of $T_{\xi}$. We define

$K_{j}:=\cup_{g\in T_{\xi}}g.C_{j}=T_{\xi}.C_{j}$

For each $j$, the set $K_{j}$ is compact and $\xi$-invariant by construction.
Moreover, $K_{j}\subset E$ since $g.C_{j}\subset g.E\subset E$. Finally, the
fact that $C_{j}\subset K_{j}$ implies $\mu(C_{j})\leq\mu(K_{j})$. This
completes our proof. ∎

We also have the important domination by capacity property of the measures
$\mu_{j}$.

######  Proposition 3.7.

The measures $\mu_{j}$ satisfy the $\mathcal{H}(\alpha,A,\theta)$ condition
for all $\alpha$. Namely, for all $\alpha>0$, there exists a constant $A$
independent of $j$ such that:

$\mu_{j}(E)\leq ACap_{\theta}(E)^{1+\alpha}$

for all $\xi$-invariant Borel subset $E\subset S$.

###### Proof.

By inner regularity of $\mu_{j}$, it is enough to establish the lemma for a
compact $\xi$-invariant $K\subset S$. Indeed, suppose that the inequality is
true for all such $K$, then for all Borel $\xi$-invariant set $E$,

$\displaystyle\mu_{j}(E)$ $\displaystyle=\sup\left\\{\mu_{j}(K),K\subset
E\;\text{compact},\xi-\text{invariant}\right\\}$ $\displaystyle\leq
A\sup\left\\{Cap_{\theta}(K)^{1+\alpha},K\subset
E\;\text{compact},\xi-\text{invariant}\right\\}$ $\displaystyle\leq
ACap_{\theta}(E)^{1+\alpha}\;(\text{by Prop. \ref{capacity_properties}(1))}$

We can suppose furthermore that $K$ is non-pluripolar (otherwise
$\mu_{j}(K)=0$ and the inequality is then trivial).

Now let $K$ be a compact ${\xi}$-invariant and non-pluripolar. Let $p>1$ be as
in Lemma 3.4. By Hölder inequality, we have:

$0\leq\mu_{j}(K)\leq\left\|f_{j}\right\|_{L^{p}(\omega_{B}^{n}\wedge\eta)}\text{vol}_{\omega_{B}}(K)^{1/q}$

where $1/p+1/q=1$. Since $\psi_{+,j}\leq\psi_{+,1}$ and that
$\psi_{j}\geq\psi_{-}$, the function $e^{(n+1)(\psi_{+,j}-\psi_{-,j})}$ is
bounded in $L^{p}$ by $e^{(n+1)(C-\psi_{-})}$, where $C:=\sup_{S}\psi_{+,1}$.
It follows that the norm
$\left\|f_{j}\right\|_{L^{p}(\omega_{B}^{n}\wedge\eta)}$ is uniformly bounded,
therefore it is enough to show that

$\text{vol}_{\omega_{B}}(K)\leq
C\exp\left(-\gamma(Cap_{\theta}(K))^{-1/n}\right))$

where $C=C(\theta,\omega_{B}),\gamma=\gamma(\theta)$ are constants independent
of $j$. The conclusion then follows from the elementary equality
$\exp(-x^{\beta})\leq A_{\alpha}x^{\alpha}$, for all $x\in[0,1],\alpha>0$.

By Theorem 2.20, for $\gamma:=2/(\nu(\left\\{\theta\right\\})+1)$, there
exists a constant $C=C(\theta,\omega_{B})$ such that:

$\sup_{\psi\in\mathcal{F}_{0}}\int_{S}\exp(-\gamma\psi)\omega_{B}^{n}\wedge\eta\leq
C$

In particular, for $\psi:=V^{*}_{K,\theta}-M_{K,\theta}$ (recall that
$M_{K,\theta}=\sup V^{*}_{K,\theta}$), we obtain

$\int_{S}\exp(-\gamma V^{*}_{K,\theta})\omega_{B}^{n}\wedge\eta\leq
C\exp(-\gamma M_{K,\theta})$

Note that $V^{*}_{K,\theta}$ is well defined thanks to the $\xi$-invariance of
$K$. Finally, since $V^{*}_{K,\theta}\leq 0$ $\mu_{\omega_{B}}$-a.e. on $K$,
we have

$\text{vol}_{\omega_{B}}(K)\leq C\exp(-\gamma M_{K,\theta})$

An application of Lemma 2.18 then completes our proof. ∎

Let us first establish some more useful lemmas before proving the uniform
estimate.

######  Lemma 3.8.

Let $u\in PSH(S,\xi,\theta)\cap L^{\infty}(S)$ be a negative function. For all
$s\geq 0$, $0\leq t\leq 1$,

$t^{n}Cap_{\theta}(u<-s-t)\leq\int_{\left\\{u<-s\right\\}}\theta_{u}^{n}\wedge\eta$

###### Proof.

Let $v\in PSH(S,\xi,\theta)$, $0\leq v\leq 1$. Then

$\left\\{u<-s-t\right\\}\subset\left\\{u\leq tv-
s-t\right\\}\subset\left\\{u<-s\right\\}$

By definition of the Monge-Ampère operator

$\int_{\left\\{u<-s-t\right\\}}\text{MA}_{\theta}(v)\leq\int_{\left\\{u\leq
tv-s-t\right\\}}\text{MA}_{\theta}(v)\leq t^{-n}\int_{\left\\{u\leq tv-
s-t\right\\}}\text{MA}_{\theta}(tv)$

Applying the comparison princple 2.9 to the functions $u+s+t$ and $tv$,

$t^{-n}\int_{\left\\{u\leq tv-s-t\right\\}}\text{MA}_{\theta}(tv)\leq
t^{-n}\int_{\left\\{u\leq tv-
s-t\right\\}}\text{MA}_{\theta}(u)\leq\int_{\left\\{u<-s\right\\}}\text{MA}_{\theta}(u)$

which terminates our proof. ∎

######  Lemma 3.9.

[EGZ09, Lem. 2.4] Let $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a right-
continuous decreasing function such that $\lim_{s\to+\infty}f(s)=0$. If $f$
satisfies the condition

$H(\alpha,B),\quad tf(s+t)\leq Bf(s)^{1+\alpha},\;\forall s\geq 0,0\leq t\leq
1$

then there exists $s_{0}=s_{0}(\alpha,B)$ such that $f(s)=0$, $\forall s\geq
s_{0}$.

######  Proposition 3.10.

There exists a uniform constant $C$ such that:

$\left\|\varphi_{j,\varepsilon}\right\|_{L^{\infty}(S)}\leq C$

###### Proof.

Let $f(s):=Cap_{\theta}(\varphi_{j,\varepsilon}<-s)^{1/n}$. It is clear that
$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is right-continuous, and
$\lim_{s\to+\infty}f(s)=0$ (cf. Prop. 2.14). Moreover, $f$ is decreasing: for
all $t>s$,
$\left\\{\varphi<-t\right\\}\subset\left\\{\varphi<-s\right\\},\;\forall t>s$,
hence $f(t)\leq f(s)$. Following Lem. 3.8 and the fact that $\mu_{j}$ satisfy
$\mathcal{H}(\alpha,A,\theta)$, $f$ satisfies the condition $H(\alpha,B)$ with
$B=A^{1/n}$. Indeed,

$\displaystyle t^{n}f(s+t)^{n}$ $\displaystyle\leq
t^{n}Cap_{\theta+\varepsilon\omega_{B}}(\varphi_{j,\varepsilon}<-s-t)$
$\displaystyle\leq\int_{\left\\{\varphi_{j,\varepsilon}<-s\right\\}}(\theta+\varepsilon\omega_{B}+d_{B}d_{B}^{c}\varphi_{j,\varepsilon})^{n}\wedge\eta$
$\displaystyle=\int_{\left\\{\varphi_{j,\varepsilon}<-s\right\\}}\mu_{j}\leq
ACap_{\theta}(\varphi_{j,\varepsilon}<-s)^{1+\alpha}=Af(s)^{n(1+\alpha)}$

The first inequality follows from Lem. 2.14, the second is direct from Lem.
3.8, while the fourth is a consequence of Lem. 3.7. Now let
$\omega_{\varepsilon}:=\theta_{X}+\varepsilon\omega_{B}$. For $\varepsilon$
sufficiently small and $\delta$ large enough, there exists
$\delta=\delta(S)\geq 1$ such that $\omega_{\varepsilon}\leq\delta\omega_{B}$.
In particular, $\varphi_{j,\varepsilon}\in PSH^{-}(S,\xi,\delta\omega_{B})$.
Again by Lem. 2.14,

$\displaystyle f(s)^{n}$ $\displaystyle\leq
Cap_{\delta\omega_{B}}(\varphi_{j,\varepsilon}<-s)$
$\displaystyle\leq\frac{\delta^{n}}{s}\left(\int_{S}(-\varphi_{j,\varepsilon})\omega_{B}^{n}\wedge\eta+n\text{vol}_{\omega_{B}}(S)\right)$

But by (1) of Lem. 2.11:

$\int_{S}-\varphi_{j,\varepsilon}d\mu_{\omega_{B}}\leq-\sup\varphi_{j,\varepsilon}+C(\omega_{B})=C(\omega_{B})$

Therefore, $f(s)\leq(C_{1}/s^{1/n})$, where
$C_{1}=C_{1}(\omega_{B},\theta_{X})$. We can then apply Lem. 3.9 to select
$s_{0}=s_{0}(n,\alpha,A,\omega_{B},\theta_{X})$ as in [EGZ09, Lemma 2.3,
Theorem 2.1] such that:

$Cap_{\theta_{X}}(\varphi_{j,\varepsilon}<-s)=0,\;\forall s\geq s_{0}$

In particular, $\mu_{j}(\varphi_{j,\varepsilon}<-s_{0})=0$ by Lem. 3.7. Hence
$\varphi_{j,\varepsilon}\geq s_{0}$ on $S$, so there exists
$C=C(n,\alpha,A,\omega_{B},\theta_{X})$ such that:

$\left\|\varphi_{j,\varepsilon}\right\|_{L^{\infty}(S)}\leq C$

∎

###  3.4. Laplacian estimate

We will need the transverse version of the Yau-Aubin inequality, obtained by
Siu for two cohomologous forms [Siu87], but the proof can be generalized to
any couple of Kähler forms. Let

$\Delta_{\omega^{\prime}_{B}}:=\operatorname{Tr}_{\omega^{\prime}_{B}}d_{B}d_{B}^{c}$

be the Laplacian associated to the transverse Kähler form
$\omega_{B}^{\prime}$.

######  Lemma 3.11.

For all transverse Kähler form $\omega_{B}^{\prime}$, there exists a constant
$\kappa$ depending only on the transverse bisectional curvature of
$\omega_{B}$ such that:

$\Delta_{\omega^{\prime}_{B}}\log\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}\geq-\kappa\operatorname{Tr}_{\omega_{B}^{\prime}}\omega_{B}-\frac{\operatorname{Tr}_{\omega_{B}}\text{Ric}(\omega_{B}^{\prime})}{\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}$

where $\text{Ric}(\omega^{\prime}_{B})$ is the transverse Ricci curvature.

###### Proof.

On each foliation chart, the transverse Kähler forms depend only on the
$z$-coordinates. The inequality thus follows from the purely local proof in
the compact Kähler case. The reader may consult Appendice 4 for a proof. ∎

The following proposition gives a a priori Laplacian estimate of the solution
$\varphi_{j,\varepsilon}$ of equation (12). We follow the arguments of
[BBE+19, Appendice B]. In this section, by a uniform constant, we mean a
constant independent of the $j,\varepsilon$ parameters.

######  Proposition 3.12.

Let $\psi:=\Phi_{B}-r_{\xi}^{2}$ and
$\omega_{\varepsilon}=\theta_{X}+\varepsilon\omega_{B},\;\omega^{\prime}_{\varepsilon}:=\omega_{\varepsilon}+d_{B}d_{B}^{c}\varphi_{j,\varepsilon}$.
There exist uniform constants $C_{1},C_{2}$ such that:

$\sup_{S\cap\mathcal{U}}\operatorname{Tr}_{\omega_{\varepsilon}}\omega^{\prime}_{\varepsilon}\leq
C_{2}e^{-C_{1}\psi-\psi_{-,j}}\leq C_{2}e^{-C_{1}\psi-\psi_{-}}$

In particular, there exists a uniform constant $C_{3}$ such that

$\sup_{S\cap\mathcal{U}}\left|\Delta_{\omega_{B}}\varphi_{j,\varepsilon}\right|\leq
C_{3}e^{-C_{1}\psi-\psi_{-}}$

###### Proof.

The function $\psi$ is clearly basic $\theta_{X}$-psh and $\psi\to-\infty$
near $\partial\mathcal{U}$ by the construction of $\Phi_{B}$ in Prop. 3.3.
Moreover, $\omega_{B}|_{\mathcal{U}}=(\theta_{X}+dd^{c}\psi)|_{\mathcal{U}}$
is the restriction into $\mathcal{U}$ of the transverse Kähler form
$\omega_{B}$, constructed on $X\backslash E_{0}$.

Consider the following smooth function on $S\cap\mathcal{U}$:

$h:=\log(\operatorname{Tr}_{\omega_{\varepsilon}}\omega_{\varepsilon}^{\prime})+n\psi_{-,j}-A_{1}(\varphi_{j,\varepsilon}-\psi)$

where $A_{1}:=A_{1}(\kappa)$ is a constant sufficiently large and depends on
$\kappa$. The compacity of $S$, the $L^{\infty}$-estimate in Prop. 3.10,
combined with transverse Yau-Aubin inequality in Lem. 3.11 are all the
ingredients we need to repeat the arguments of [BBE+19, Appendice B] to
conclude.

For the reader’s convenience, we provide here some details of the proof. By
the transverse Yau-Aubin inequality, we have on $S\cap\mathcal{U}$:

$\Delta_{\omega^{\prime}_{\varepsilon}}h\geq\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}(\omega_{\varepsilon})-A_{2}$

where $A_{2}$ depends only on $A_{1}$ and $n$. Since $\varphi_{j,\varepsilon}$
is uniformly bounded and that $\psi\to-\infty$ near
$\partial(S\cap\mathcal{U})$, $h$ attains its maximum at $x_{0}\in
S\cap\mathcal{U}$. It follows from the maximum principle that

$0\geq\Delta_{\omega^{\prime}_{\varepsilon}}h(x_{0})\geq\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}(\omega_{\varepsilon})(x_{0})-A_{2}$

By local elementary reasonings as in the compact Kähler case, we obtain the
following inequality for two transverse Kähler forms:

$\operatorname{Tr}_{\omega_{\varepsilon}}(\omega^{\prime}_{\varepsilon})\leq
n\frac{(\omega^{\prime}_{\varepsilon})^{n}}{\omega_{\varepsilon}^{n}}(\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}(\omega_{\varepsilon}))^{n}=(n+1)e^{\psi_{+,j}-\psi_{-,j}}(\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}(\omega_{\varepsilon}))^{n}$

Taking log on both sides gives us

$\log(\operatorname{Tr}_{\omega_{\varepsilon}}\omega^{\prime}_{\varepsilon})\leq\log(n)+(n+1)(\psi_{+,j}-\psi_{-,j})+n\log(\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}\omega_{\varepsilon})$

hence by definition of $h$,

$h\leq\log(n)+(n+1)\psi_{+,j}+n\log(\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}\omega_{\varepsilon})-A_{1}(\varphi_{j,\varepsilon}-\psi)$

Therefore

$\sup_{S\cap\mathcal{U}}h\leq h(x_{0})\leq
A_{3}-A_{1}\inf_{S\cap\mathcal{U}}(\varphi_{j,\varepsilon}-\psi)\leq
A_{3}-A_{1}\inf_{S\cap\mathcal{U}}\varphi_{j,\varepsilon}$

where $A_{3}$ is a uniform constant since $\psi_{+,j}$ and
$\operatorname{Tr}_{\omega^{\prime}_{\varepsilon}}\omega_{\varepsilon}(x_{0})$
are both uniformly bounded. As a consequence, there exists a uniform constant
$A_{4}$ such that:

$h:=\log(\operatorname{Tr}_{\omega_{\varepsilon}}\omega_{\varepsilon}^{\prime})+(n+1)\psi_{-,j}-A_{1}(\varphi_{j,\varepsilon}-\psi)\leq
A_{4}$

which leads to:

$\operatorname{Tr}_{\omega_{\varepsilon}}\omega_{\varepsilon}^{\prime}\leq
e^{-(n+1)\psi_{-,j}}e^{A_{1}(\varphi_{j,\varepsilon}-\psi)}e^{A_{4}}$

hence the existence of uniform constants $A_{1},A_{5}$, depending only on $C$
in inequality (11), $\kappa$, and the bound of the $L^{\infty}$-estimate 3.10
such that:

$\sup_{S\cap\mathcal{U}}\operatorname{Tr}_{\omega_{\varepsilon}}\omega^{\prime}_{\varepsilon}\leq
A_{5}e^{-A_{1}\psi-\psi_{-,j}}\leq A_{4}e^{-A_{1}\psi-\psi_{-}}$

For the estimate of $\Delta_{\omega_{B}}\varphi_{j,\varepsilon}$, we make the
following remark. By compacity of $S$, there exists a uniform constant
$\delta$ sufficiently large such that

$\omega_{\varepsilon}=\theta+\varepsilon\omega_{B}\leq\delta\omega_{B}$

hence

$\operatorname{Tr}_{\omega_{B}}(.)\leq\delta^{-1}\operatorname{Tr}_{\omega_{\varepsilon}}(.)$

But since

$\sup_{S\cap\mathcal{U}}\operatorname{Tr}_{\omega_{\varepsilon}}(\omega_{\varepsilon}+dd^{c}\varphi_{j,\varepsilon})=n+\sup_{S\cap\mathcal{U}}\Delta_{\omega_{\varepsilon}}\varphi_{j,\varepsilon}\leq
A_{4}e^{-A_{1}\psi-\psi_{-}},$

this completes our proof. ∎

###  3.5. Conclusion

###### Proof of the main Theorem.

By using the $L^{\infty}$-estimate in Lem. 3.10 and the transverse Yau-Aubin
inequality 3.11, we obtained in Lem. 3.12 the estimate of
$\Delta_{\omega_{B}}\varphi_{j,\varepsilon}$. As a consequence,
$\Delta_{\omega_{B}}\varphi_{j,\varepsilon}$ is locally uniformly bounded on
$S\cap\mathcal{U}$ since $\psi_{-}:=\Psi_{-}+\varphi_{X}$ is locally bounded
by our assumption. It follows that there exists a subsequence
$\varphi_{j,\varepsilon(j)}$ which is $C^{1}$-convergent on $S\cap\mathcal{U}$
to

$\varphi_{0}\in L^{\infty}(S\cap\mathcal{U}),\Delta_{\omega_{B}}\varphi_{0}\in
L^{\infty}_{\text{loc}}(S\cap\mathcal{U})$

which is a solution of

(13)
$(\theta+d_{B}d_{B}^{c}\varphi_{0})^{n}\wedge\eta=e^{-(n+1)\varphi_{X}}\pi^{*}dV_{Y}(-J\xi,.)$

on $S\cap\mathcal{U}$. The equation admits a unique solution up to constant
(cf. Prop. 2.15), hence:

$\varphi_{0}=\varphi_{X}+c$

which implies that $\Delta_{\omega_{B}}\varphi_{X}$ is locally bounded. This
allows us to obtain a $C^{2,\alpha}$-estimate of $\varphi_{X}$, as well as
higher order estimates using Schauder’s estimate and complex Evans-Krylov
theory as in [Blo12, 5.3, p.210], hence the smoothness of $\varphi_{X}$ on
$S\cap\mathcal{U}$.

By definition, $r^{2}=r_{\xi}^{2}e^{\varphi}$ and
$\varphi_{X}=\pi^{*}\varphi$. Using symmetry by $\mathbb{R}_{>0}$-action
generated by $-J\xi$, we conclude that $\varphi_{X}=\varphi\circ\pi$ is
actually smooth on $\mathcal{U}$, hence $\varphi$ is smooth on
$Y_{\text{reg}}$. In particular, $r^{2}$ is smooth on $Y_{\text{reg}}$. ∎

##  4\. Appendice : Transverse Yau-Aubin inequality

In the sequel, we will use the summation convention. Let
$\omega_{B},\omega_{B}^{\prime}$ be two transverse Kähler forms on $S$. Let
$(z,x)$ be the coordinates on a foliation chart of $S$ such that:

$\omega_{B}=g_{j\overline{k}}\sqrt{-1}dz^{j}\wedge
d\overline{z}^{k},\quad\omega^{\prime}_{B}=g^{\prime}_{j\overline{k}}\sqrt{-1}dz^{j}\wedge
d\overline{z}^{k}$

After choosing a normal transverse holomorphic chart, one can suppose that
$g_{j\overline{k}}=\delta_{jk}$ and that $\omega^{\prime}_{B}$ is diagonal.
Let $(g^{j\overline{k}})$ denote the inverse of $(g_{j\overline{k}})$. We
have:

$\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}=g^{j\overline{j}}g^{\prime}_{j\overline{j}}=\sum_{j}g^{\prime}_{j\overline{j}},\quad\operatorname{Tr}_{\omega_{B}^{\prime}}\omega_{B}=g^{\prime
j\overline{j}}g_{j\overline{j}}=\sum_{j}g^{\prime j\overline{j}}$

Denote

$\partial_{j}:=\frac{\partial}{\partial
z_{j}},\;\overline{\partial}_{k}:=\frac{\partial}{\partial\overline{z}_{k}},\;\partial_{j}\overline{\partial}_{k}:=\frac{\partial^{2}}{\partial
z_{j}\partial\overline{z}_{k}}$

######  Lemma 4.1.

We have the following inequality:

$g^{\prime
p\overline{p}}(\partial_{p}g^{\prime}_{a\overline{a}})(\overline{\partial}_{p}g^{\prime}_{b\overline{b}})\leq(\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime})\sum_{p,a,j}g^{\prime
p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{j}}\right|^{2}$

###### Proof.

The lemma follows from repeated applications of Cauchy-Schwartz inequality:

$\displaystyle\sum_{p,a,b}g^{p\overline{p}}(\partial_{p}g^{\prime}_{a\overline{a}})(\overline{\partial}_{p}g^{\prime}_{b\overline{b}})$
$\displaystyle\leq\sum_{a,b}(g^{p\overline{p}}\left|\partial_{p}g^{\prime}_{a\overline{a}}\right|^{2})^{1/2}(g^{p\overline{p}}\left|\overline{\partial}_{p}g^{\prime}_{b\overline{b}}\right|^{2})^{1/2}$
$\displaystyle=(\sum_{a}(\sum_{p}g^{p\overline{p}}\left|\partial_{p}g^{\prime}_{a\overline{a}}\right|^{2})^{1/2})^{2}$
$\displaystyle=(\sum_{a}\sqrt{g_{a\overline{a}}^{\prime}}(\sum_{p}g^{p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{a}}\right|^{2})^{1/2})^{2}$
$\displaystyle\leq(\sum_{a}g^{\prime}_{a\overline{a}})(\sum_{p,a}g^{p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{a}}\right|^{2})$
$\displaystyle\leq(\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime})(\sum_{p,a,j}g^{p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{j}}\right|^{2})$

∎

Recall the statement of the transverse Yau-Aubin inequality:

######  Lemma 4.2.

$\Delta_{\omega^{\prime}_{B}}\log\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}\geq-\kappa\operatorname{Tr}_{\omega_{B}^{\prime}}\omega_{B}-\frac{\operatorname{Tr}_{\omega_{B}}\text{Ric}(\omega_{B}^{\prime})}{\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}$

###### Proof.

We have:

$\displaystyle\Delta_{\omega^{\prime}_{B}}\log\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B}$
$\displaystyle=\frac{\Delta_{\omega^{\prime}_{B}}\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}{\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B}}-g^{p\overline{q}}\frac{(\overline{\partial}_{q}\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B})(\partial_{p}\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B})}{(\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime})^{2}}$
$\displaystyle=\frac{\Delta_{\omega^{\prime}_{B}}\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}{\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B}}-\frac{g^{p\overline{p}}(\partial_{p}g^{\prime}_{a\overline{a}})(\overline{\partial}_{p}g^{\prime}_{b\overline{b}})}{(\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime})^{2}}$

By definition,

$\displaystyle\Delta_{\omega^{\prime}_{B}}\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}$
$\displaystyle=g^{\prime
p\overline{q}}(\partial_{p}\overline{\partial}_{q}g^{j\overline{k}})g^{\prime}_{j\overline{k}}+g^{\prime
p\overline{q}}g^{j\overline{k}}\partial_{p}\overline{\partial}_{q}g^{\prime}_{j\overline{k}}$
$\displaystyle=g^{\prime
p\overline{q}}(\partial_{p}\overline{\partial}_{q}g^{j\overline{k}})g^{\prime}_{j\overline{k}}-g^{\prime
p\overline{q}}g^{j\overline{k}}R^{\prime}_{j\overline{k}p\overline{q}}+g^{\prime
p\overline{q}}g^{j\overline{k}}g^{\prime
a\overline{b}}(\partial_{p}g^{\prime}_{j\overline{b}})(\overline{\partial}_{q}g^{\prime}_{a\overline{k}})$

where $R^{\prime}_{j\overline{k}p\overline{q}}$ is the local expression of the
transverse curvature form of $\omega_{B}^{\prime}$. Let us estimate the three
terms of the expression above.

  * •

Since $\omega_{B}$ and $\omega_{B}^{\prime}$ are diagonal, we have for the
first term:

$g^{\prime
p\overline{q}}(\partial_{p}\overline{\partial}_{q}g^{j\overline{k}})g^{\prime}_{j\overline{k}}=g^{\prime
p\overline{p}}(\partial_{p}\overline{\partial}_{p}g^{j\overline{j}})g^{\prime}_{j\overline{j}}\geq-\kappa(\operatorname{Tr}_{\omega_{B}}{\omega^{\prime}_{B}})(\operatorname{Tr}_{\omega_{B}^{\prime}}\omega_{B})$

where $\kappa$ is the infimum of the transverse sectional curvature (which
exists since $S$ is compact).

  * •

In the second term, $g^{\prime
p\overline{q}}R_{j\overline{k}p\overline{q}}=R^{\prime}_{j\overline{k}}$,
where $R^{\prime}_{j\overline{k}}$ is the local expression of the transverse
Ricci-form $\text{Ric}(\omega^{\prime}_{B})$.

  * •

For the third term, we have:

$g^{\prime p\overline{q}}g^{j\overline{k}}g^{\prime
a\overline{b}}(\partial_{p}g^{\prime}_{j\overline{b}})(\overline{\partial}_{q}g^{\prime}_{a\overline{k}})=g^{\prime
p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{j}}\right|^{2}$

It follows that

$\Delta_{\omega^{\prime}_{B}}\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}\geq-\kappa\operatorname{Tr}_{\omega_{B}}{\omega^{\prime}_{B}}\operatorname{Tr}_{\omega_{B}^{\prime}}\omega_{B}-g^{j\overline{k}}R^{\prime}_{j\overline{k}}+\sum_{p,a,j}g^{\prime
p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{j}}\right|^{2}$

hence

$\displaystyle\Delta_{\omega^{\prime}_{B}}\log\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B}$
$\displaystyle\geq-\kappa\operatorname{Tr}_{\omega^{\prime}_{B}}\omega_{B}-\frac{\operatorname{Tr}_{\omega_{B}}\text{Ric}(\omega_{B}^{\prime})}{\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}$
$\displaystyle+\frac{\sum_{p,a,j}g^{\prime p\overline{p}}g^{\prime
a\overline{a}}\left|\partial_{p}g^{\prime}_{a\overline{j}}\right|^{2}}{\operatorname{Tr}_{\omega_{B}}\omega^{\prime}_{B}}-\frac{g^{p\overline{p}}(\partial_{p}g^{\prime}_{a\overline{a}})(\overline{\partial}_{p}g^{\prime}_{b\overline{b}})}{(\operatorname{Tr}_{\omega_{B}}\omega_{B})^{2}}$
$\displaystyle\geq-\kappa\operatorname{Tr}_{\omega^{\prime}_{B}}\omega_{B}-\frac{\operatorname{Tr}_{\omega_{B}}(\text{Ric}(\omega_{B}^{\prime})}{\operatorname{Tr}_{\omega_{B}}\omega_{B}^{\prime}}$

by the previous lemma. ∎

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