# Pluricomplex Green’s functions and Fano manifolds

Nicholas McCleerey and Valentino Tosatti

###### Abstract

We show that if a Fano manifold does not admit Kähler-Einstein metrics then
the Kähler potentials along the continuity method subconverge to a function
with analytic singularities along a subvariety which solves the homogeneous
complex Monge-Ampère equation on its complement, confirming an expectation of
Tian-Yau.

  *     * Keywords. Fano manifold; pluricomplex Green function; algebraic Kähler potentials

2010 Mathematics Subject Classification. 32W20 (primary); 32U35, 14J45, 32Q20,
53C25 (secondary)

[Français]

Titre. Fonctions de Green pluricomplexes et variétés de Fano Résumé. Nous
montrons que si une variété de Fano n’admet aucune métrique de Kähler-Einstein
alors, suivant la méthode de continuité, les potentiels kählériens sous-
convergent vers une fonction à singularités analytiques le long d’une sous-
variété, sur le complémentaire de laquelle la fonction est solution de
l’équation de Monge-Ampère complexe homogène. Cela confirme une attente de
Tian-Yau.

###### Contents

  1. 1\. Introduction
  2. 2\. The partial $C^{0}$ estimate
  3. 3\. The singularity model function
  4. 4\. Understanding the class $\mathcal{C}$
  5. 5\. The pluricomplex Green’s function
  6. References

##  1\. Introduction

Let $X^{n}$ be a Fano manifold, i.e. a compact complex manifold with
$c_{1}(X)>0$. A Kähler-Einstein metric on $X$ is a Kähler metric $\omega$
which satisfies

$\mathrm{Ric}(\omega)=\omega.$

This implies that $[\omega]=c_{1}(X)$. We assume throughout this paper that
$X$ does not admit a Kähler-Einstein metric. This is known to be equivalent to
K-unstability by [13] (see also [40]), but we will not use this fact. We fix a
Kähler metric $\omega$ with $[\omega]=c_{1}(X)$, with Ricci potential $F$
defined by $\mathrm{Ric}(\omega)=\omega+\sqrt{-1}\partial\overline{\partial}F$
(normalized by $\int_{X}(e^{F}-1)\omega^{n}=0$). We consider Kähler metrics
$\omega_{t}$ with $[\omega_{t}]=c_{1}(X)$ which satisfy

$\mathrm{Ric}(\omega_{t})=t\omega_{t}+(1-t)\omega.$

We can write
$\omega_{t}=\omega+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$ and the
functions $\varphi_{t}$ solve the complex Monge-Ampère equation [46]

$\omega_{t}^{n}=e^{F-t\varphi_{t}}\omega^{n}.$ (1.1)

A solution $\varphi_{t}$ exists on $[0,R(X))$ where $R(X)\leqslant 1$ is the
greatest lower bound for the Ricci curvature of Kähler metrics in $c_{1}(X)$
[36]. It is known [35, 38] that since $X$ does not admit Kähler-Einstein
metrics, we must have that $\lim_{t\to R(X)}\sup_{X}\varphi_{t}=+\infty$. We
fix a sequence $t_{i}\to R(X)$ and write $\varphi_{i}:=\varphi_{t_{i}}$ and
$\omega_{i}:=\omega_{t_{i}}$. Using this result, together with multiplier
ideal sheaves, Nadel [29, Proposition 4.1] proved that (up to passing to a
subsequence) the measures $\omega_{i}^{n}$ converge to zero (as measures) on
compact sets of $X\backslash V$ for some proper analytic subvariety $V\subset
X$, and in [44] the second-named author improved this to uniform convergence.
By weak compactness of closed positive currents in a fixed cohomology class,
up to subsequences we can extract a limit $\rho$ of
$\varphi_{i}-\sup_{X}\varphi_{i}$ (which may depend on the subsequence), which
is an unbounded $\omega$-psh function, and the convergence happens in the
$L^{1}$ topology. In their work [41, p.178], Tian-Yau expressed the
expectation that $\rho$ should have logarithmic poles along a proper analytic
subvariety $V\subset X$, and that it should satisfy
$(\omega+\sqrt{-1}\partial\overline{\partial}\rho)^{n}=0$ on $X\backslash V$,
so that $\rho$ could be thought of as a kind of pluricomplex Green’s function
(see also [38, p.238] and [39, p.109]). In this note we confirm Tian-Yau’s
expectation:

###### Theorem 1.1

Let $X$ be a Fano manifold without a Kähler-Einstein metric, and let
$\omega_{t}=\omega+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$ be the
solutions of the continuity method (1.1). Given any sequence
$t_{i}\in[0,R(X))$ with $t_{i}\to R(X)$, choose a subsequence such that
$\varphi_{t_{i}}-\sup_{X}\varphi_{t_{i}}$ converge in $L^{1}(X)$ to an
$\omega$-psh function $\rho$. Then we can find $m\geqslant 1$ and an
$\omega$-psh function $\psi$ on $X$ with analytic singularities

$\psi=\frac{1}{m}\log\sum_{j=1}^{p}\lambda_{j}^{2}|S_{j}|^{2}_{h^{m}},$ (1.2)

for some $\lambda_{j}\in(0,1]$ and some sections $S_{j}\in
H^{0}(X,K_{X}^{-m})$, with nonempty common zero locus $V\subset X$ such that
$\rho-\psi$ is bounded on $X$, and on $X\backslash V$ we have

$(\omega+\sqrt{-1}\partial\overline{\partial}\rho)^{n}=0,$ (1.3)

where the Monge-Ampère product is in the sense of Bedford-Taylor [6].

In particular, Theorem 1.1 implies that the non-pluripolar Monge-Ampère
operator of $\rho$ (defined in [12]) vanishes identically on $X$. On the other
hand, there is another meaningful Monge-Ampère operator that can be applied to
$\rho$. Indeed, the fact that $\rho-\psi\in L^{\infty}(X)$ implies that $\rho$
itself has analytic singularities. In [3] Andersson-Błocki-Wulcan defined a
Monge-Ampère operator for $\omega$-psh functions with analytic singularities
(generalizing earlier work of Andersson-Wulcan [4] in the local setting). In
general, applying this Monge-Ampère operator to $\rho$ will produce a Radon
measure $\mu$ on $X$ (which may be identically zero in some cases), which by
Theorem 1.1 is supported on the analytic set $V$, thus providing geometrically
interesting examples of unbounded quasi-psh functions on compact Kähler
manifolds with Monge-Ampère operator concentrated on a subvariety (see also
[1, 2] for related results in the local setting). In particular, this answers
[11, Question 1 (c)], an open problem raised at the AIM workshop “The complex
Monge-Ampère equation” in August 2016 (cf. the related [23, Question 12]).
Note that in general a formula for the total mass of $\mu$ is proved in [3,
Theorem 1.2], and it satisfies

$\int_{X}\mu\leqslant\int_{X}\omega^{n},$

with strict inequality in general (but it is not hard to see that if $\dim
X=2$ and $V$ is a finite set then equality holds). Therefore, the measure
$\mu$ is in general different from the measures that one obtains as weak
limits of $(\omega+\sqrt{-1}\partial\overline{\partial}\varphi_{i})^{n}$ (up
to subsequences), whose total mass is always equal to $\int_{X}\omega^{n}$.

###### Remark 1.2

[Remark added in proof] After this work was posted on the arXiv, and partly
prompted by it, Błocki [10] modified the definition of the Monge-Ampère
operator for $\omega$-psh functions with analytic singularities of [4, 3], and
with his definition the total mass is always equal to $\int_{X}\omega^{n}$. It
is an interesting question to determine whether this Monge-Ampère operator
equals the weak limit of
$(\omega+\sqrt{-1}\partial\overline{\partial}\varphi_{i})^{n}$.

###### Remark 1.3

As in the second-named author’s previous work [44], Theorem 1.1 has a direct
counterpart for solutions of the normalized Kähler-Ricci flow, instead of the
continuity method (1.1). The statement is identical to Theorem 1.1, except
that now the sequence $t_{i}$ goes to $+\infty$. The proof is also almost
verbatim the same, and the partial $C^{0}$ estimate along the flow is proved
in [14, 15] (see also [5]). All other ingredients used also have well-known
counterparts for the flow (see [44]). We leave the simple details to the
interested reader.

###### Remark 1.4

The behavior of the solutions $\omega_{t}$ of (1.1) as $t\to R(X)$ has been
investigated in the past. If the manifold is K-stable, [20] show that
$\omega_{t}$ converge smoothly to a Kähler-Einstein metric. If on the other
hand no such metric exists, the blowup behavior of $\omega_{t}$ has been
investigated in [29, 44] in the setting of this paper, and also in [20, 26,
33] by allowing reparametrizations of the metrics by diffeomorphisms.

The proof of Theorem 1.1 relies on the partial $C^{0}$ estimate for solutions
of (1.1) which was established by Székelyhidi [37]. We recall this in section
2, together with a well-known reformulation of this estimate (Proposition
2.1). In section 3 we observe that this gives us the singularity model
function $\psi$ in (1.2), and it also implies that $\rho$ has the same
singularity type as $\psi$. In section 4 we show the general fact that every
$\omega$-psh function on $X$ with the same singularity type as $\psi$ has
vanishing Monge-Ampère operator outside $V$, thus proving Theorem 1.1. This
relies on a geometric understanding of the rational map defined by the
sections $\\{S_{j}\\}$ as in Theorem 1.1. Lastly, in section 5 we discuss the
pluricomplex Green’s function with the same singularity type as $\psi$.

Acknowledgments.  We thank Z. Błocki, T.C. Collins, S. Kołodziej and D.H.
Phong for discussions on this topic at the AIM workshop “The complex Monge-
Ampère equation” in August 2016, E. Wulcan for her interest in this work and
A. Rashkovskii and the referee for useful comments. The second-named author is
also grateful to S.-T. Yau for related discussions over the years. This work
was completed during the second-named author’s visit to the Institut Henri
Poincaré in Paris (supported by a Chaire Poincaré funded by the Clay
Mathematics Institute) which he would like to thank for the hospitality and
support.

##  2\. The partial $C^{0}$ estimate

To start we fix some notation. We choose a Hermitian metric $h$ on
$K_{X}^{-1}$ with curvature $R_{h}=\omega$ (such $h$ is unique up to scaling),
and let $h^{m}$ be the induced metric on $K_{X}^{-m}$, for all $m\geqslant 1$.
Let $N_{m}=\dim H^{0}(X,K_{X}^{-m})$, and for any $m\geqslant 1$ define the
density of states function

$\rho_{m}(\omega)=\sum_{j=1}^{N_{m}}|S_{j}|^{2}_{h^{m}},$

where $S_{1},\dots,S_{N_{m}}$ are a basis of $H^{0}(X,K_{X}^{-m})$ which is
orthonormal with respect to the $L^{2}$ inner product $\int_{X}\langle
S_{1},S_{2}\rangle_{h^{m}}\omega^{n}$. Clearly $\rho_{m}(\omega)$ is
independent of the choice of basis, and is also unchanged if we scale $h$ by a
constant. The integral $\int_{X}\rho_{m}(\omega)\omega^{n}$ equals $N_{m}$,
and if $m$ is sufficiently large so that $K_{X}^{-m}$ is very ample, then
$\rho_{m}(\omega)$ is strictly positive on $X$. If we apply this same
construction to the metrics $\omega_{t}$ and Hermitian metrics
$h_{t}=he^{-\varphi_{t}}$ we get a density of states function
$\rho_{m}(\omega_{t}).$ Following [39], we say that a “partial $C^{0}$
estimate” holds if there exist $m\geqslant 1$ and a constant $C>0$ such that

$\inf_{X}\rho_{m}(\omega_{t})\geqslant C^{-1},$ (2.1)

holds for all $t\in[0,R(X)).$ The reason for this name is explained by the
following proposition, which is essentially well-known (see [39, Lemma 2.2]
and [43, Proposition 5.1]), but we provide the details for convenience:

###### Proposition 2.1

If a partial $C^{0}$ estimate holds then there exists $m\geqslant 1$, such
that for all $\varepsilon>0$ we can find a constant $C>0$ so that for all
$t\in[\varepsilon,R(X))$ we can find real numbers
$1=\lambda_{1}(t)\geqslant\dots\geqslant\lambda_{N_{m}}(t)>0$ and a basis
$\\{S_{j}(t)\\}_{1\leqslant j\leqslant N_{m}}$ of $H^{0}(X,K_{X}^{-m})$,
orthonormal with respect to the $L^{2}$ inner product of $\omega,h^{m}$, such
that for all $t\in[0,R(X))$ we have

$\sup_{X}\left|\varphi_{t}-\sup_{X}\varphi_{t}-\frac{1}{m}\log\sum_{j=1}^{N_{m}}\lambda_{j}(t)^{2}|S_{j}(t)|^{2}_{h^{m}}\right|\leqslant
C.$ (2.2)

In the rest of the paper we will fix a value of $\varepsilon>0$ once and for
all, for example $\varepsilon=R(X)/2$. The precise choice is irrelevant, since
we are only interested in the behavior as $t\to R(X)$.

  * Proof.

First, it is well-known that for all $m\geqslant 1$ and $\varepsilon>0$ there
is a constant $C$ such that for all $t\in[\varepsilon,R(X))$ we have

$\rho_{m}(\omega_{t})\leqslant C.$ (2.3)

To see this, first observe for every $S\in H^{0}(X,K_{X}^{-m})$ we have

$\Delta_{\omega_{t}}|S|^{2}_{h_{t}^{m}}=|\nabla
S|^{2}_{t}-2m|S|^{2}_{h_{t}^{m}}\geqslant-2m|S|^{2}_{h_{t}^{m}},$ (2.4)

and that since $\mathrm{Ric}(\omega_{t})\geqslant
t\omega_{t}\geqslant\varepsilon\omega_{t}$, Myers’ Theorem gives a uniform
upper bound for $\mathrm{diam}(X,\omega_{t})$ and then Croke [19] and Li [27]
show that the Sobolev constant of $(X,\omega_{t})$ has a uniform upper bound.
We can then apply Moser iteration to (2.4) to get

$\sup_{X}|S|^{2}_{h_{t}^{m}}\leqslant
C\int_{X}|S|^{2}_{h_{t}^{m}}\omega_{t}^{n}\leqslant C,$ (2.5)

provided we assume that $\int_{X}|S|^{2}_{h_{t}^{m}}\omega_{t}^{n}=1$. Taking
now an orthonormal basis of sections and summing we obtain (2.3). Thanks to
(2.3) we know that for $t\in[\varepsilon,R(X))$ a partial $C^{0}$ estimate is
equivalent to

$\sup_{X}|\log\rho_{m}(\omega_{t})|\leqslant C.$ (2.6)

We now take a basis $\\{\tilde{S}_{j}(t)\\}_{1\leqslant j\leqslant N_{m}}$ of
$H^{0}(X,K_{X}^{-m})$ orthonormal with respect to the $L^{2}$ inner product of
$\omega_{t},h_{t}^{m}$ and notice that since
$h_{t}^{m}=e^{-m\varphi_{t}}h^{m}$ we clearly have

$\varphi_{t}=\frac{1}{m}\log\frac{\sum_{j=1}^{N_{m}}|\tilde{S}_{j}(t)|^{2}_{h^{m}}}{\sum_{j=1}^{N_{m}}|\tilde{S}_{j}(t)|^{2}_{h^{m}_{t}}},$

which is equivalent to

$\varphi_{t}-\frac{1}{m}\log\sum_{j=1}^{N_{m}}|\tilde{S}_{j}(t)|^{2}_{h^{m}}=-\frac{1}{m}\log\rho_{m}(\omega_{t}).$
(2.7)

It follows from (2.6) and (2.7) that that for $t\in[\varepsilon,R(X))$ a
partial $C^{0}$ estimate is equivalent to an estimate

$\sup_{X}\left|\varphi_{t}-\frac{1}{m}\log\sum_{j=1}^{N_{m}}|\tilde{S}_{j}(t)|^{2}_{h^{m}}\right|\leqslant
C.$

We now choose another basis $\\{S_{j}\\}_{1\leqslant j\leqslant N_{m}}$ of
$H^{0}(X,K_{X}^{-m})$ orthonormal with respect to the $L^{2}$ inner product of
$\omega,h^{m}$. After modifying $S_{j}$ and $\tilde{S}_{j}(t)$ by
$t$-dependent unitary transformations, we obtain orthonormal bases
$\\{S_{j}(t)\\}_{1\leqslant j\leqslant N_{m}}$ with respect to $\omega,h^{m}$,
and $\\{\tilde{S}_{j}(t)\\}_{1\leqslant j\leqslant N_{m}}$ with respect to
$\omega_{t},h_{t}^{m}$ such that

$\tilde{S}_{j}(t)=\mu_{j}(t)S_{j}(t),$

for some positive real numbers $\mu_{j}(t)$, with
$\mu_{1}(t)\geqslant\dots\geqslant\mu_{N_{m}}(t)>0$. We then let
$\lambda_{j}(t)=\mu_{j}(t)/\mu_{1}(t)$ and we see that a partial $C^{0}$
estimate is equivalent to

$\sup_{X}\left|\varphi_{t}-\frac{2}{m}\log\mu_{1}(t)-\frac{1}{m}\log\sum_{j=1}^{N_{m}}\lambda_{j}(t)^{2}|S_{j}(t)|^{2}_{h^{m}}\right|\leqslant
C.$ (2.8)

We now claim that if a partial $C^{0}$ estimate holds, then for all
$t\in[\varepsilon,R(X))$ we also have

$\left|\frac{2}{m}\log\mu_{1}(t)-\sup_{X}\varphi_{t}\right|\leqslant C.$ (2.9)

Once this is proved, combining (2.8) and (2.9) we get (2.2). To prove (2.9),
first use (2.5) to get

$C\geqslant\sup_{X}|\tilde{S}_{1}(t)|^{2}_{h_{t}^{m}}\geqslant\mu_{1}(t)^{2}\sup_{X}|S_{1}(t)|^{2}_{h^{m}}e^{-m\sup_{X}\varphi_{t}},$

and the fact $\int_{X}|S_{1}(t)|^{2}_{h^{m}}\omega^{n}=1$ implies that
$\sup_{X}|S_{1}(t)|^{2}_{h^{m}}\geqslant 1/\mathrm{Vol}(X,\omega),$ and so

$\left(\frac{2}{m}\log\mu_{1}(t)-\sup_{X}\varphi_{t}\right)\leqslant C.$

On the other hand the partial $C^{0}$ estimate (2.1) implies that

$C^{-1}\leqslant\rho_{m}(\omega_{t})=\sum_{j=1}^{N_{m}}|\tilde{S}_{j}(t)|^{2}_{h_{t}^{m}}\leqslant\mu_{1}(t)^{2}\sum_{j=1}^{N_{m}}|S_{j}(t)|^{2}_{h^{m}}e^{-m\varphi_{t}},$
(2.10)

and we clearly have that

$\sup_{j}\sup_{X}|S_{j}(t)|^{2}_{h^{m}}\leqslant C,$ (2.11)

since the sections $\\{S_{j}(t)\\}$ are just varying in a compact unitary
group (or one can also repeat the Moser iteration argument of (2.3) for the
fixed metric $\omega$). This together with (2.10), evaluated at the point
where $\varphi_{t}$ achieves its maximum, gives the reverse inequality

$\left(\sup_{X}\varphi_{t}-\frac{2}{m}\log\mu_{1}(t)\right)\leqslant C,$

which completes the proof of (2.9). $\Box$

##  3\. The singularity model function

The next goal is to use the partial $C^{0}$ estimate in Proposition 2.1 to
construct a singular $\omega$-psh function $\psi$ which will have the same
singularity type of any weak limit of the normalized solutions
$\varphi_{i}-\sup_{X}\varphi_{i}$ of the continuity method. Let the notation
be as in Proposition 2.1, and in particular we fix once and for all a value of
$m\geqslant 1$ given there. We can find a sequence $t_{i}\to R(X)$ and an
$\omega$-psh function $\rho$ with $\sup_{X}\rho=0$ such that
$\varphi_{i}-\sup_{X}\varphi_{i}\to\rho$ in $L^{1}(X)$, and pointwise a.e.
Passing to a subsequence, we can find a basis $\\{S_{j}\\}_{1\leqslant
j\leqslant N_{m}}$ of $H^{0}(X,K_{X}^{-m})$ orthonormal with respect to the
$L^{2}$ inner product of $\omega,h^{m}$, such that $S_{j}(t_{i})\to S_{j}$
smoothly as $i\to\infty$, for all $1\leqslant j\leqslant N_{m}$. The change of
basis matrix from $\\{S_{j}\\}_{1\leqslant j\leqslant N_{m}}$ to
$\\{S_{j}(t)\\}_{1\leqslant j\leqslant N_{m}}$ induces an automorphism
$\sigma(t)$ of $\mathbb{CP}^{N_{m}-1}$, such that
$\sigma(t_{i})\to\mathrm{Id}$ smoothly as $i\to\infty$. For ease of notation,
write

$\psi_{t}=\frac{1}{m}\log\sum_{j=1}^{N_{m}}\lambda_{j}(t)^{2}|S_{j}(t)|^{2}_{h^{m}}.$

These functions are Kähler potentials for $\omega$ since

$\omega+\sqrt{-1}\partial\overline{\partial}\psi_{t}=\frac{\iota^{*}\sigma(t)^{*}\tau(t)^{*}\omega_{FS}}{m}>0,$
(3.1)

where $\iota:X\hookrightarrow\mathbb{CP}^{N_{m}-1}$ is the Kodaira embedding
map given by the sections $\\{S_{j}\\}_{1\leqslant j\leqslant N_{m}},$ the map
$\tau(t)$ is the automorphism of $\mathbb{CP}^{N_{m}-1}$ induced by the
diagonal matrix with entries $\\{\lambda_{j}(t)\\}_{1\leqslant j\leqslant
N_{m}}$, and $\omega_{FS}$ is the Fubini-Study metric on
$\mathbb{CP}^{N_{m}-1}$. The identity in (3.1) follows directly from the
definition of the Fubini-Study metric $\omega_{FS}$ on
$\mathbb{CP}^{N_{m}-1}$, which on $\mathbb{C}^{N_{m}}\backslash\\{0\\}$ is
given explicitly by
$\omega_{FS}=\sqrt{-1}\partial\overline{\partial}\log\sum_{j=1}^{N_{m}}|z_{j}|^{2},$
and from the fact that the curvature of $h$ is $\omega$. Up to passing to a
subsequence of $t_{i}$, we may assume that $\lambda_{j}(t_{i})\to\lambda_{j}$
as $i\to\infty$ for all $j$, and we have

$1=\lambda_{1}\geqslant\dots\geqslant\lambda_{p}>0=\lambda_{p+1}=\dots=\lambda_{N_{m}},$

for some $1\leqslant p<N_{m}$. The case $p=N_{m}$ is impossible because by
(2.2) it would imply a uniform $L^{\infty}$ bound for $\varphi_{t}$ and so $X$
would admit a Kähler-Einstein metric. For the same reason, the set
$V:=\\{S_{1}=\cdots=S_{p}=0\\}$ must be a nonempty proper analytic subvariety
of $X$. Note that thanks to (2.2) we can write

$\omega_{t}^{n}=e^{F-t(\varphi_{t}-\sup_{X}\varphi_{t})}e^{-t\sup_{X}\varphi_{t}}\omega^{n}\leqslant
Ce^{t\psi_{t}}e^{-t\sup_{X}\varphi_{t}}\omega^{n},$

and since the term $e^{t\psi_{t}}$ is uniformly bounded on compact sets of
$X\backslash V$, we see immediately that

$\omega_{t}^{n}\to 0,$ (3.2)

uniformly on compact sets of $X\backslash V$ (this result was proved in [44]
without using the partial $C^{0}$ estimate, which was not available at the
time, with weaker results established earlier in [29]). Let then

$\psi=\frac{1}{m}\log\sum_{j=1}^{p}\lambda_{j}^{2}|S_{j}|^{2}_{h^{m}},$

which is a smooth function on $X\backslash V$ which approaches $-\infty$
uniformly on $V$. Since $e^{m\psi_{t}}\to e^{m\psi}$ smoothly on $X$, and
since $\psi_{t}$ are smooth and $\omega$-psh, it follows that $\psi$ is
$\omega$-psh. This will be our singularity model function in the rest of the
argument, as we now explain:

###### Lemma 3.1

Define the class

$\mathcal{C}=\\{\eta\in PSH(X,\omega)\ |\ \eta-\psi\in L^{\infty}(X)\\},$

of $\omega$-psh functions with the same singularity type as $\psi$. Then we
have that $\rho\in\mathcal{C}$.

  * Proof.

Recall that we have $\varphi_{i}-\sup_{X}\varphi_{i}\to\rho$ a.e. on $X$.
Thanks to (2.2), the function $\rho$ satisfies

$|\rho-\psi|\leqslant C,$ (3.3)

a.e. on $X$, which implies the same inequality on all of $X$ by elementary
properties of psh functions (cf. [25, Theorem K.15]), thus showing that
$\rho\in\mathcal{C}$. $\Box$

##  4\. Understanding the class $\mathcal{C}$

We now exploit the geometry of our setting to gain a better understanding of
the class of functions $\mathcal{C}$. The sections
$\\{\lambda_{j}S_{j}\\}_{1\leqslant j\leqslant p}$ define a rational map
$\Phi:X\dashrightarrow\mathbb{CP}^{p-1}$, with indeterminacy locus $Z\subset
V$ (this inclusion is in general strict, since $\mathrm{codim}Z\geqslant 2$
while $V$ may contain divisorial components). Let $Y$ be the image of $\Phi$,
i.e. the closure of $\Phi(X\backslash Z)$ in $\mathbb{CP}^{p-1}$, which is an
irreducible projective variety. By resolving the indeterminacies of $\Phi$ we
get a modification $\mu:\tilde{X}\to X$, obtained as a sequence of blowups
with smooth centers, and a holomorphic map $\Psi:\tilde{X}\to Y$ such that
$\Psi=\Phi\circ\mu$ holds on $\tilde{X}\backslash\mu^{-1}(Z)$. We may also
assume without loss of generality that $\mu$ principalizes the ideal sheaf
generated by $\\{S_{j}\\}_{1\leqslant j\leqslant p}$, so that we have

$\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\psi)=\theta+[E],$

where $E$ is an effective $\mathbb{R}$-divisor with $\mu(E)\subset V$, and
$\theta$ is a smooth closed semipositive $(1,1)$ form on $\tilde{X}$. We will
denote by $\omega_{FS,p}$ the Fubini-Study metric on $\mathbb{CP}^{p-1}$. To
identify $\theta$, note that on $X\backslash V$ we have by definition
$\omega+\sqrt{-1}\partial\overline{\partial}\psi=\frac{\Phi^{*}\omega_{FS,p}}{m},$
and so on $\tilde{X}\backslash\mu^{-1}(V)$ we have

$\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\psi)=\frac{\mu^{*}\Phi^{*}\omega_{FS,p}}{m}=\frac{\Psi^{*}\omega_{FS,p}}{m},$

and so $\theta=\frac{\Psi^{*}\omega_{FS,p}}{m}$ on
$\tilde{X}\backslash\mu^{-1}(V)$, and hence everywhere since both sides of
this equality are smooth forms on all of $\tilde{X}$. This proves the key
relation

$\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\psi)=\frac{\Psi^{*}\omega_{FS,p}}{m}+[E].$
(4.1)

Let $\tilde{X}\overset{\nu}{\to}\tilde{Y}\overset{q}{\to}Y$ be the Stein
factorization of $\Psi$, where $\tilde{Y}$ is an irreducible projective
variety, the map $\nu$ has connected fibers, and $q$ is a finite morphism. We
have that $q^{*}\omega_{FS,p}$ is a smooth semipositive $(1,1)$ form on
$\tilde{Y}$, in the sense of analytic spaces. Since $\nu$ has compact
connected fibers, a standard argument shows that the set of
$\frac{\Psi^{*}\omega_{FS,p}}{m}$-psh functions on $\tilde{X}$ can be
identified with the set of (weakly) $\frac{q^{*}\omega_{FS,p}}{m}$-psh
functions on $\tilde{Y}$ via $\nu^{*}$ (indeed the restriction of every
$\frac{\Psi^{*}\omega_{FS,p}}{m}$-psh function to any fiber of $\nu$ is
plurisubharmonic and hence constant on that fiber). We will use this standard
argument several other times in the following. Here and in the following, as
in [21], a weakly quasi-psh function on a compact analytic space means a
quasi-psh function on its regular part which is locally bounded above near the
singular set. As shown in [21, §1], weakly quasi-psh functions are the same as
usual quasi-psh functions if the analytic space is normal, and otherwise they
can be identified with quasi-psh functions on its normalization.

###### Proposition 4.1

Given any function $\eta\in\mathcal{C}$, there is a unique bounded weakly
$\frac{q^{*}\omega_{FS,p}}{m}$-psh function $u$ on $\tilde{Y}$ such that

$\mu^{*}\eta=\mu^{*}\psi+\nu^{*}u.$ (4.2)

Conversely, given any bounded weakly $\frac{q^{*}\omega_{FS,p}}{m}$-psh
function $u$ on $\tilde{Y}$ there is a unique function $\eta\in\mathcal{C}$
such that (4.2) holds.

The relation in (4.2) thus allows us to identify the class $\mathcal{C}$ with
the class of bounded weakly $\frac{q^{*}\omega_{FS,p}}{m}$-psh functions on
$\tilde{Y}$. Next, we observe that

###### Proposition 4.2

We have that

$\dim Y<\dim X.$

This is a consequence of our assumption that $X$ does not admit a Kähler-
Einstein metric. Lastly, every function $\eta\in\mathcal{C}$ belongs to
$L^{\infty}_{\rm loc}(X\backslash V)$, and so its Monge-Ampère operator
$(\omega+\sqrt{-1}\partial\overline{\partial}\eta)^{n}$ is well-defined on
$X\backslash V$ thanks to Bedford-Taylor [6]. Combining the results in
Propositions 4.1 and 4.2 we will obtain:

###### Theorem 4.3

For every $\eta\in\mathcal{C}$ we have that

$(\omega+\sqrt{-1}\partial\overline{\partial}\eta)^{n}=0,$

on $X\backslash V$.

In particular, this holds for the function $\rho$, thanks to Lemma 3.1, and
Theorem 1.1 thus follows from these.

  * Proof of Proposition 4.1.

If $\eta$ is an $\omega$-psh function on $X$ with $\eta-\psi\in
L^{\infty}(X)$, i.e. $\eta$ is an element of $\mathcal{C}$, then using (4.1)
we can write

$\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\eta)=\frac{\Psi^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}\mu^{*}(\eta-\psi)+[E],$

where $E$ is as in (4.1) and $\mu^{*}(\eta-\psi)\in L^{\infty}(\tilde{X})$.
Applying the Siu decomposition, we see that

$\frac{\Psi^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}\mu^{*}(\eta-\psi)\geqslant
0,$

weakly, and so

$\mu^{*}(\eta-\psi)=\nu^{*}u_{\eta},$

for a bounded weakly $\frac{q^{*}\omega_{FS,p}}{m}$-psh functions $u_{\eta}$
on $\tilde{Y}$, which is uniquely determined by $\eta$ (and $\psi$, which we
view as fixed). Conversely, given a bounded weakly
$\frac{q^{*}\omega_{FS,p}}{m}$-psh function $u$ on $\tilde{Y}$, we have that
$\nu^{*}u$ is $\frac{\Psi^{*}\omega_{FS,p}}{m}$-psh and bounded on $\tilde{X}$
and so

$0\leqslant\frac{\Psi^{*}\omega_{FS,p}}{m}+[E]+\sqrt{-1}\partial\overline{\partial}\nu^{*}u=\mu^{*}\omega+\sqrt{-1}\partial\overline{\partial}(\mu^{*}\psi+\nu^{*}u),$

and so $\mu^{*}\psi+\nu^{*}u$ descends to an $\omega$-psh function $\eta_{u}$
on $X$ with $\eta_{u}-\psi\in L^{\infty}(X)$, i.e. $\eta_{u}\in\mathcal{C}$.
These two constructions are inverses to each other, and so we obtain the
desired bijective correspondence between functions in $\mathcal{C}$ and
bounded weakly $\frac{q^{*}\omega_{FS,p}}{m}$-psh functions on $\tilde{Y}$.
$\Box$

  * Proof of Proposition 4.2.

On $X$ we have the estimate

$\omega_{t}\geqslant
C^{-1}\frac{\iota^{*}\sigma(t)^{*}\tau(t)^{*}\omega_{FS}}{m},$ (4.3)

which is a direct consequence of the partial $C^{0}$ estimate (see e.g. [24,
Lemma 4.2]). We can also give a direct proof by calculating

$\Delta_{\omega_{t}}\left(\log\textrm{tr}_{\omega_{t}}{\left(\frac{\iota^{*}\sigma(t)^{*}\tau(t)^{*}\omega_{FS}}{m}\right)}-A(\varphi_{t}-\sup_{X}\varphi_{t}-\psi_{t})\right)\geqslant\textrm{tr}_{\omega_{t}}{\left(\frac{\iota^{*}\sigma(t)^{*}\tau(t)^{*}\omega_{FS}}{m}\right)}-C,$

if $A$ is sufficiently large, and applying the maximum principle together with
the partial $C^{0}$ estimate (2.2) (for this calculation we used that the
bisectional curvature of the metrics
$\frac{\iota^{*}\sigma(t)^{*}\tau(t)^{*}\omega_{FS}}{m}$ have a uniform upper
bound independent of $t$). If we had $\dim Y=\dim X$ then the rational map
$\Phi$ would be generically finite, so there would be a nonempty open subset
$U\Subset X\backslash V$ such that $\Phi|_{U}$ is a biholomorphism with its
image. Recall that $\Phi$ is the rational map defined by the sections
$\\{\lambda_{j}S_{j}\\}_{1\leqslant j\leqslant p}$, while
$\iota:X\hookrightarrow\mathbb{CP}^{N_{m}-1}$ is the embedding defined by the
sections $\\{S_{j}\\}_{1\leqslant j\leqslant N_{m}}$, and so
$\Phi=\tilde{\tau}\circ P\circ\iota$ where
$P:\mathbb{CP}^{N_{m}-1}\dashrightarrow\mathbb{CP}^{p-1}$ is the linear
projection given by $[z_{1}:\cdots:z_{N_{m}}]\mapsto[z_{1}:\cdots:z_{p}]$ and
$\tilde{\tau}:\mathbb{CP}^{p-1}\to\mathbb{CP}^{p-1}$ is the automorphism given
by

$[z_{1}:\cdots:z_{p}]\mapsto[\lambda_{1}z_{1}:\cdots:\lambda_{p}z_{p}].$

In particular, on the embedded open $n$-fold $\iota(U)$, we have that
$P|_{\iota(U)}$ is also a biholomorphism with its image. The automorphisms
$\tau(t_{i})$ descend to automorphisms $\tilde{\tau}(t_{i})$ on
$\mathbb{CP}^{p-1}$, and now as $i\to\infty$ these converge smoothly to the
automorphism $\tilde{\tau}$. Thus
$P\circ\tau(t_{i})\circ\sigma(t_{i})\circ\iota=\tilde{\tau}(t_{i})\circ
P\circ\sigma(t_{i})\circ\iota$, which converge smoothly as maps to
$\tilde{\tau}\circ P\circ\iota=\Phi$ on $U$ as $i\to\infty$. Since $\Phi$ is
an isomorphism on $U$, smooth convergence gives us that
$P\circ\tau(t_{i})\circ\sigma(t_{i})\circ\iota$ is a local isomorphism. Thus,
after possibly shrinking $U$,

$P:(\tau(t_{i})\circ\sigma(t_{i})\circ\iota)(U)\to(P\circ\tau(t_{i})\circ\sigma(t_{i})\circ\iota)(U)=(\tilde{\tau}(t_{i})\circ
P\circ\sigma(t_{i})\circ\iota)(U)$

is an isomorphism, and for $i$ large the open sets $(\tilde{\tau}(t_{i})\circ
P\circ\sigma(t_{i})\circ\iota)(U)\subset\mathbb{CP}^{p-1}$ converge to the
open set $(\tilde{\tau}\circ P\circ\iota)(U)$ in the Hausdorff sense. Up to
shrinking $U$, there is an open subset $V\subset\mathbb{CP}^{p-1}$ that
contains $(\tilde{\tau}(t_{i})\circ P\circ\sigma(t_{i})\circ\iota)(U)$ for all
$i$ large, and still $P^{-1}$ is well-defined on $V$ (and $P:P^{-1}(V)\to V$
is a biholomorphism), so that $P^{-1}(V)$ contains
$(\tau(t_{i})\circ\sigma(t_{i})\circ\iota)(U)$ for all $i$ large, and on
$P^{-1}(V)$ we have

$P^{*}\omega_{FS,p}\leqslant C\omega_{FS},$ (4.4)

On $U$ we also have that
$\frac{\iota^{*}\sigma(t_{i})^{*}\tau(t_{i})^{*}P^{*}\omega_{FS,p}}{m}$
converges smoothly to $\frac{\Phi^{*}\omega_{FS,p}}{m}$, which is a Kähler
metric on $U$. Thanks to (4.3) and (4.4), on $U$ we have

$\omega_{i}\geqslant
C^{-1}\frac{\iota^{*}\sigma(t_{i})^{*}\tau(t_{i})^{*}\omega_{FS}}{m}\geqslant
C^{-1}\frac{\iota^{*}\sigma(t_{i})^{*}\tau(t_{i})^{*}P^{*}\omega_{FS,p}}{m}\geqslant
C^{-1}\frac{\Phi^{*}\omega_{FS,p}}{m},$

for all $i$ large, which implies that $\int_{U}\omega_{i}^{n}\geqslant
C^{-1}$, which is absurd thanks to (3.2). $\Box$

###### Remark 4.4

In particular we see that if $\dim Y=0$ (i.e. $Y$ is a point) then we have
$\mathcal{C}=\\{\psi+s\\}_{s\in\mathbb{R}}$. On the other hand as long as
$\dim Y>0$ the class $\mathcal{C}$ is always rather large.

  * Proof of Theorem 4.3.

Thanks to Proposition 4.1, every $\eta\in\mathcal{C}$ satisfies
$\mu^{*}\eta=\mu^{*}\psi+\nu^{*}u$ for some bounded weakly
$\frac{q^{*}\omega_{FS,p}}{m}$-psh function $u$ on $\tilde{Y}$. Then using
(4.1) we have

$\begin{split}\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\eta)&=\frac{\Psi^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}\nu^{*}u+[E]\\\
&=\nu^{*}\left(\frac{q^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}u\right)+[E],\end{split}$

and so if $K$ is any compact subset of $X\backslash V$, since $\mu$ is an
isomorphism on $\mu^{-1}(K)$, we get

$\begin{split}\int_{K}(\omega+\sqrt{-1}\partial\overline{\partial}\eta)^{n}&=\int_{\mu^{-1}(K)}\mu^{*}(\omega+\sqrt{-1}\partial\overline{\partial}\eta)^{n}\\\
&=\int_{\mu^{-1}(K)}\nu^{*}\left(\frac{q^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}u\right)^{n}=0,\end{split}$

since $\dim\tilde{Y}=\dim Y<\dim X$ by Proposition 4.2. $\Box$

##  5\. The pluricomplex Green’s function

We can also consider the pluricomplex Green’s function with singularity type
determined by $\psi$, namely

$G=\sup\\{u\ |\ u\in PSH(X,\omega),u\leqslant 0,u\leqslant\psi+O(1)\\}^{*},$
(5.1)

which is the compact manifold analog of the construction in [31], and has been
studied in detail in [18, 30, 32] and references therein. In particular, since
$\psi$ has analytic singularities, it follows from [31, 32] that
$G\in\mathcal{C}$. Thanks to Proposition 4.1 we can write

$\mu^{*}G=\mu^{*}\psi+\nu^{*}F,$ (5.2)

for a bounded weakly $\frac{q^{*}\omega_{FS,p}}{m}$-psh function $F$ on
$\tilde{Y}$. The function $F$ is itself given by a suitable envelope.

###### Proposition 5.1

The pluricomplex Green’s function $G$ satisfies (5.2) where $F$ is the
envelope on $\tilde{Y}$ given by

$F=\sup\\{w\ |\ w\in
PSH(\tilde{Y},q^{*}\omega_{FS,p}/m),w\leqslant-\nu_{*}\mu^{*}\psi\\}^{*},$
(5.3)

and where we are writing

$\nu_{*}(f)(y)=\sup_{x\in\nu^{-1}(y)}f(x),$

for any function $f$ on $\tilde{X},y\in\tilde{Y}$.

In other words, $F$ is given by a quasi-psh envelope with obstacle
$-\nu_{*}\mu^{*}\psi$ on $\tilde{Y}$.

  * Proof.

Write $E=\sum_{i}\lambda_{i}E_{i}$ for $E_{i}$ prime divisors and
$\lambda_{i}\in\mathbb{R}_{>0}$, and for each $i$ fix a defining section
$s_{i}$ of $\mathcal{O}(E_{i})$ and a smooth metric $h_{i}$ on
$\mathcal{O}(E_{i})$ with curvature form $R_{i}$. For brevity, we will write
$|s|^{2}_{h}=\prod_{i}|s_{i}|^{2\lambda_{i}}_{h_{i}}$ and
$R_{h}=\sum_{i}\lambda_{i}R_{i}$. Then the Poincaré-Lelong formula gives

$[E]=\sqrt{-1}\partial\overline{\partial}\log|s|^{2}_{h}+R_{h},$

and we obtain that $\mu^{*}\omega-R_{h}$ is cohomologous to
$\frac{\Psi^{*}\omega_{FS,p}}{m}$ and

$\mu^{*}\omega-
R_{h}=\frac{\Psi^{*}\omega_{FS,p}}{m}+\sqrt{-1}\partial\overline{\partial}(\log|s|^{2}_{h}-\mu^{*}\psi),$

and $\mu^{*}\psi-\log|s|^{2}_{h}$ is smooth on all of $\tilde{X}$. Note that
if we denote by

$\tilde{G}=\sup\\{u\ |\ u\in PSH(\tilde{X},\mu^{*}\omega),u\leqslant
0,u\leqslant\log|s|^{2}_{h}+O(1)\\}^{*},$

then we have that $\tilde{G}=\mu^{*}G$ (this is again because every
$\mu^{*}\omega$-psh function on $\tilde{X}$ is in fact the pullback of an
$\omega$-psh function on $X$). As in [28], we use a trick from [8, Section 4]
(see also [31]), to show that

$\tilde{G}=\log|s|^{2}_{h}+\sup\\{v\ |\ v\in PSH(\tilde{X},\mu^{*}\omega-
R_{h}),v\leqslant-\log|s|^{2}_{h}\\}^{*}.$

For the reader’s convenience, we supply the simple proof. Denote the right
hand side by $\hat{G}$. For one direction, if $v$ is $(\mu^{*}\omega-
R_{h})$-psh and satisfies $v\leqslant-\log|s|^{2}_{h}$, then
$u:=v+\log|s|^{2}_{h}$ satisfies $u\leqslant 0$ but also since $v\leqslant C$
on $\tilde{X}$, we see that $u\leqslant\log|s|^{2}_{h}+C$, and also

$\begin{split}\mu^{*}\omega+\sqrt{-1}\partial\overline{\partial}u&=\mu^{*}\omega+\sqrt{-1}\partial\overline{\partial}\log|s|^{2}_{h}+\sqrt{-1}\partial\overline{\partial}v\\\
&=\mu^{*}\omega-R_{h}+[E]+\sqrt{-1}\partial\overline{\partial}v\\\
&\geqslant\mu^{*}\omega-R_{h}+\sqrt{-1}\partial\overline{\partial}v\geqslant
0,\end{split}$

and so $\hat{G}\leqslant\tilde{G}$. Conversely, if $u$ is $\mu^{*}\omega$-psh
and satisfies $u\leqslant 0$ and $u\leqslant\log|s|^{2}_{h}+C$ for some $C$,
then the Siu decomposition of
$\mu^{*}\omega+\sqrt{-1}\partial\overline{\partial}u$ contains $[E]$ and so

$0\leqslant\mu^{*}\omega+\sqrt{-1}\partial\overline{\partial}u-[E]=\mu^{*}\omega-
R_{h}+\sqrt{-1}\partial\overline{\partial}(u-\log|s|^{2}_{h}),$

and so $v:=u-\log|s|^{2}_{h}$ is $(\mu^{*}\omega-R_{h})$-psh and satisfies
$v\leqslant-\log|s|^{2}_{h},$ and it follows that $\tilde{G}\leqslant\hat{G}$,
which proves our claim. But finally note that for all $x\in\tilde{X}$ we have

$\begin{split}&\log|s|^{2}_{h}(x)+\sup\\{v(x)\ |\ v\in
PSH(\tilde{X},\mu^{*}\omega-R_{h}),v\leqslant-\log|s|^{2}_{h}\\}\\\
&=\mu^{*}\psi(x)+\sup\\{v(x)\ |\ v\in
PSH(\tilde{X},\Psi^{*}\omega_{FS,p}/m),v\leqslant-\mu^{*}\psi\\}\\\
&=\mu^{*}\psi(x)+\sup\\{w(\nu(x))\ |\ w\in
PSH(\tilde{Y},q^{*}\omega_{FS,p}/m),w\leqslant-\nu_{*}\mu^{*}\psi\\}\end{split}$

and taking the upper-semicontinuous regularization and using the claim above
gives $\mu^{*}G=\mu^{*}\psi+\nu^{*}F,$ which completes the proof. $\Box$

Using Proposition 5.1 we can see that $F$ is continuous on a Zariski open
subset of $\tilde{Y}$, using the following argument. Let
$g:Y^{\prime}\rightarrow\tilde{Y}$ be a resolution of the singularities of
$\tilde{Y}$. Then we have:

$g^{*}F=\sup\\{w\ |\ w\in
PSH(Y^{\prime},g^{*}q^{*}\omega_{FS,p}/m),w\leqslant-g^{*}\nu_{*}\mu^{*}\psi\\}^{*}.$

Note that $g^{*}q^{*}\omega_{FS,p}/m$ is semi-positive and big, and that
$-g^{*}\nu_{*}\mu^{*}\psi$ is continuous off of
$g^{-1}(\nu(\mu^{-1}(\psi^{-1}(-\infty))))$, where it is unbounded. Using the
trick in [28], we can replace the obstacle $-g^{*}\nu_{*}\mu^{*}\psi$ with a
globally continuous obstacle $h$ without changing $g^{*}F$. Now, approximate
$h$ uniformly by smooth functions $h_{j}$. It is easy to see that the
envelopes:

$F_{j}:=\sup\\{w\ |\ w\in PSH(Y^{\prime},g^{*}q^{*}\omega_{FS,p}/m),w\leqslant
h_{j}\\}^{*}.$

converge uniformly to $g^{*}F$. But then by [8], the $F_{j}$ are continuous
away from the non-Kähler locus of $g^{*}q^{*}\omega_{FS,p}/m$ (a proper
Zariski closed subset, see e.g. [12]), so we are done.

###### Remark 5.2

One is naturally led to wonder about what the optimal regularity of $G$ is.
The sharp $C^{1,1}$ regularity (on a Zariski open subset) of envelopes of the
form (5.3) has been recently obtained in [17, 45] in Kähler classes and in
[16] in nef and big classes (see also [7, 8, 9]) when the obstacle is smooth
(or at least $C^{1,1}$), but in our case the regularity of
$-\nu_{*}\mu^{*}\psi$ does not seem to be very good, especially near the
points where $\nu$ is not a submersion. On the other hand, the first-named
author [28] has very recently obtained $C^{1,1}$ regularity (on a Zariski open
subset) of envelopes with prescribed analytic singularities, which include
those of the form (5.1), generalizing results in [32] in the case of line
bundles. In our situation, the results of [28, 32] do not apply since in (5.1)
the functions $u$ and $\psi$ are both $\omega$-psh (while for these results
one would need them to be quasi-psh with respect to two different
$(1,1)$-forms such that the cohomology class of their difference is big).
Moreover, the main result of [28] also allows for $u$ and $\psi$ being both
$\omega$-psh, but then needs the condition that the total mass of the non-
pluripolar Monge-Ampère operator of $\psi$ be strictly positive. This is
obviously not the case in our situation however, by Theorem 4.3.

###### Remark 5.3

One possibly interesting approach to studying higher regularity of functions
$v\in\mathcal{C}$ which are already continuous on $X\setminus V$ is the
following. Suppose $\sup_{X}v=0$. Fix an $M>0$ and let $\Omega$ be the open
set $\Omega:=\\{v<-M\\}$. Then one can easily show using the comparison
principle and Theorem 4.3 that we have:

$\max\\{v,-M\\}=V_{\Omega}-M,$

where here $V_{\Omega}$ is the global (Siciak) extremal function for $\Omega$.
In particular, one sees that $\Omega$ is regular. There is then a well-
developed theory about Hölder continuous regularity for such functions (the so
called HCP property), see e.g. [34]. It may be possible to use this theory to
study $G$, if one can first show that it is continuous in at least a
neighborhood of $V$. Another possibility may be to study regularity of the
boundary of $\Omega$ – see the very end of [28].

###### Remark 5.4

On can also naturally ask whether the function $\rho$ (and therefore also its
singularity type $\psi$) in Theorem 1.1 is actually independent of the choice
of subsequence $t_{i}$, and also how regular $\rho$ is on $X\backslash V$. Our
guess is that $\rho$ is indeed uniquely determined, and is smooth on
$X\backslash V$. These properties would both follow if one could show that the
map $\Phi:X\dashrightarrow Y$ is independent of the chosen subsequence, and
that the corresponding function $u$ on $\tilde{Y}$ given by Lemma 3.1 and
Proposition 4.1 which satisfies

$\mu^{*}\rho=\mu^{*}\psi+\nu^{*}u,$

actually solves a suitable complex Monge-Ampère equation on $\tilde{Y}$. In a
related setting of Calabi-Yau manifolds fibered over lower-dimensional spaces,
such a limiting equation after collapsing the fibers was obtained by the
second-named author in [42, Theorem 4.1].

###### Remark 5.5

Lastly, we can also ask whether the limit $\rho$ (if it is unique) is
necessarily equal to the pluricomplex Green’s function $G$ up to addition of a
constant. By remark 4.4 this is the case if the rational map $\Phi$ is
constant, so that $Y$ is a point. In general though this seems rather likely
false.

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