Introduction
Let
$Y$
be a compact Kähler manifold and
$\unicode[STIX]{x1D6FC}_{Y}\in H^{1,1}(Y,\mathbb{R})$
a Kähler class. The space
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
of Kähler metrics
$\unicode[STIX]{x1D714}_{Y}$
in
$\unicode[STIX]{x1D6FC}_{Y}$
can be seen as an infinite dimensional Riemannian manifold, whose tangent spaces
$T_{\unicode[STIX]{x1D714}_{Y}}{\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
can all be identified with
${\mathcal{C}}^{\infty }(Y,\mathbb{R})$
. Mabuchi has introduced in [Reference MabuchiMab87] an
$L^{2}$
-metric on
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
, by setting
$$\begin{eqnarray}\langle f,g\rangle _{\unicode[STIX]{x1D714}_{Y}}:=\int _{Y}f\,g\,\frac{{\unicode[STIX]{x1D714}_{Y}}^{n}}{V_{\unicode[STIX]{x1D6FC}_{Y}}},\end{eqnarray}$$
where
$n=\dim _{\mathbb{C}}Y$
and
$V_{\unicode[STIX]{x1D6FC}_{Y}}=\int _{Y}{\unicode[STIX]{x1D714}_{Y}}^{n}=\unicode[STIX]{x1D6FC}_{Y}^{n}$
denotes the volume of
$\unicode[STIX]{x1D6FC}_{Y}$
.
Mabuchi studied the corresponding geometry of
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
, showing, in particular, that it can formally be seen as a locally symmetric space of non-positive curvature. Semmes [Reference SemmesSem92] re-interpreted the geodesic equation as a complex homogeneous equation, while Donaldson [Reference DonaldsonDon99] strongly motivated the search for smooth geodesics through its connection with the uniqueness of constant scalar curvature Kähler metrics.
In a series of remarkable works [Reference ChenChe00, Reference Calabi and ChenCC02, Reference Chen and TianCT08, Reference ChenChe09, Reference Chen and SunCS12], Chen and his collaborators have studied the metric and geometric properties of the space
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
, showing in particular that it is a path metric space (a non-trivial assertion in this infinite-dimensional setting). A key step from [Reference ChenChe00] has been the production of
${\mathcal{C}}^{1,\overline{1}}$
-geodesics, which turn out to minimize the intrinsic distance
$d$
. Very recently, such a regularity result was improved by Chu et al. [Reference Chu, Tosatti and WeinkoveCTW17]: they showed that geodesics are
$C^{1,1}$
. It follows from the work of Lempert and Vivas [Reference Lempert and VivasLV13], Darvas and Lempert [Reference Darvas and LempertDL12], and Ross and Witt-Nyström [Reference Ross and Witt-NyströmRW15] that one cannot expect better regularity, but for the toric setting.
The metric study of the space
$({\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}},d)$
has been recently pushed further by Darvas [Reference DarvasDar17b, Reference DarvasDar17c, Reference DarvasDar15]. He characterized there the metric completion of
$({\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}},d)$
and showed that such a completion is non-positively curved in the sense of Alexandrov. He also introduced several Finsler-type metrics on
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
, which turn out to be quite useful (see [Reference Darvas and RubinsteinDR17, Reference Berman, Boucksom and JonssonBBJ15]). For each
$p\geqslant 1$
, we set
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}):=\inf \{\ell _{p}(\unicode[STIX]{x1D719})\mid \unicode[STIX]{x1D719}~\text{is a path joining}~\unicode[STIX]{x1D719}_{0}~\text{to}~\unicode[STIX]{x1D719}_{1}\},\quad \forall \unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}},\end{eqnarray}$$
where
$$\begin{eqnarray}\ell _{p}(\unicode[STIX]{x1D719}):=\int _{0}^{1}|\dot{\unicode[STIX]{x1D719}_{t}}|_{p}\,dt=\int _{0}^{1}\biggl(\int _{Y}|\dot{\unicode[STIX]{x1D719}}_{t}|^{p}\operatorname{MA}(\unicode[STIX]{x1D719}_{t})\biggr)^{1/p}dt,\end{eqnarray}$$
and
$\operatorname{MA}(\unicode[STIX]{x1D719}_{t}):=(\unicode[STIX]{x1D714}_{Y}+dd^{c}\unicode[STIX]{x1D719}_{t})^{n}/V_{\unicode[STIX]{x1D6FC}_{Y}}$
. The goal of this article is to extend these studies to the case when the underlying space has singularities.
From now on, let
$Y$
be a compact Kähler normal space and
$\unicode[STIX]{x1D6FC}_{Y}\in H_{\mathit{BC}}^{1,1}(Y)$
a Kähler class, where
$H_{\mathit{BC}}^{1,1}(Y)$
denotes the Bott–Chern cohomology space. We fix a base point
$\unicode[STIX]{x1D714}_{Y}$
representing
$\unicode[STIX]{x1D6FC}_{Y}$
and work with the space of Kähler potentials
Our first main result extends the main results of [Reference ChenChe00] and [Reference DarvasDar15, Theorem 1], as follows.
Theorem A.
(i)
$({\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}},d_{p})$
is a metric space.(ii)
$d_{p}(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1})=(\int _{Y}|\dot{\unicode[STIX]{x1D719}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D719}_{0}))^{1/p}=(\int _{Y}|\dot{\unicode[STIX]{x1D719}}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D719}_{1}))^{1/p}$
,
$\forall \unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
.
As we are going to discuss, in Remark 1.11, the singularities of
$Y$
prevent us from defining the distance
$d_{p}$
as in (1). We instead work on a resolution of
$Y$
and there define
$d_{p}$
as a limit of path length metrics. We refer to Definition 1.10 and Remark 1.14 for the precise definition of
$d_{p}$
.
Following [Reference DarvasDar17c, Reference DarvasDar15] we then study the metric completion of the space
$({\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}},d_{p})$
and establish the following generalization of [Reference DarvasDar15, Theorem 2].
Theorem B. Let
$Y$
be a projective normal variety and assume that
$\unicode[STIX]{x1D714}_{Y}$
is a Hodge form. The metric completion of
$({\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}},d_{p})$
is a geodesic metric space, which can be identified with the finite-energy class
$({\mathcal{E}}^{p}(Y,\unicode[STIX]{x1D714}_{Y}),I_{p})$
.
Finite-energy classes have been introduced in [Reference Guedj and ZeriahiGZ07] and further studied in [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13]; we recall their definition in § 2. The Mabuchi geodesics can be extended to finite-energy geodesics, which are still metric geodesics. A key technical tool here is Theorem 3.6, which compares
$d_{p}$
and
$I_{p}$
, where
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}):=\bigg(\int _{Y}|\unicode[STIX]{x1D719}_{0}-\unicode[STIX]{x1D719}_{1}|^{p}\bigg[\frac{\operatorname{MA}(\unicode[STIX]{x1D719}_{0})+\operatorname{MA}(\unicode[STIX]{x1D719}_{1})}{2}\bigg]\bigg)^{1/p}.\end{eqnarray}$$
This is a natural quantity which allows one to define the ‘strong topology’ on
${\mathcal{E}}^{p}(Y,\unicode[STIX]{x1D714}_{Y})$
.
The metric completion of
$({\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}},d)$
has been considered by Streets in his study of the Calabi flow [Reference StreetsStr16] and also plays an important role in recent works by Berman et al. [Reference Berman, Boucksom and JonssonBBJ15] and Berman et al. [Reference Berman, Darvas and LuBDL16]. There is no doubt that the extension to the singular setting will play a leading role in subsequent applications. We illustrate this here by generalizing Tian’s analytic criterion [Reference TianTia97, Reference Phong, Song, Sturm and WeinkovePSSW08], using results in [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ] and an idea in [Reference Darvas and RubinsteinDR17].
Theorem C. Let
$(Y,D)$
be a log Fano pair. It admits a unique Kähler–Einstein metric if and only if there exists
$\unicode[STIX]{x1D700},M>0$
, such that, for all
$\unicode[STIX]{x1D719}\in {\mathcal{H}}_{\text{norm}}$
,
$$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D719})\leqslant -\unicode[STIX]{x1D700}d_{1}(0,\unicode[STIX]{x1D719})+M.\end{eqnarray}$$
Here,
${\mathcal{F}}$
is a functional whose critical points are Kähler–Einstein potentials (§ 5) and
${\mathcal{H}}_{\text{norm}}$
is the set of potentials in
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
normalized such that the supremum is 0. This result has been independently obtained by Darvas [Reference DarvasDar17a] using a different approach.
Our results should also be useful in analyzing more generally constant scalar curvature Kähler (cscK) metrics on mildly singular varieties (see, for example, the recent construction by Arezzo and Spotti of cscK metrics on crepant resolutions of Calabi–Yau varieties with non-orbifold singularities [Reference Arezzo and SpottiAS16]).
A way to establish these results is to consider a resolution of singularities
$\unicode[STIX]{x1D70B}:X\rightarrow Y$
and to work with the space
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
of potentials associated with the form
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{Y}$
. All these results actually hold in the more general setting when
$\unicode[STIX]{x1D714}$
is merely a semi-positive and big form (i.e.
$\int _{X}\unicode[STIX]{x1D714}^{n}>0$
). We approximate
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
by spaces of Kähler potentials
${\mathcal{H}}_{\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}}$
and show that the most important metric properties of
$({\mathcal{H}}_{\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}},d_{\unicode[STIX]{x1D700}})$
pass to the limit.
The organization of the paper is as follows. Section 1 starts with a recap of Mabuchi geodesics and metrics. Theorem A is proved in § 1.2, where we develop a low-regularity approach for understanding geodesics by approximation. We introduce in § 2 classes of finite-energy currents and compare their natural topologies with the one induced by the Mabuchi distances in § 3. We study finite-energy geodesics in § 4 and prove Theorem B. We finally prove Theorem C in § 5.
1 The space of Kähler currents
Let
$(Y,\unicode[STIX]{x1D714}_{Y})$
be a compact Kähler normal space of dimension
$n$
. It follows from the definition of
$H_{\mathit{BC}}^{1,1}(Y)$
(see, for example, [Reference Boucksom, Eyssidieux and GuedjBEG13, Definition 4.6.2]) that any other Kähler metric on
$Y$
in the same Bott–Chern cohomology class of
$\unicode[STIX]{x1D714}_{Y}$
can be written as
$$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D714}_{Y}+dd^{c}\unicode[STIX]{x1D719},\end{eqnarray}$$
where
$d=\unicode[STIX]{x2202}+\overline{\unicode[STIX]{x2202}}$
and
$d^{c}=(1/2i\unicode[STIX]{x1D70B})(\unicode[STIX]{x2202}-\overline{\unicode[STIX]{x2202}})$
. Let
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
be the space of Kähler potentials
$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}=\{\unicode[STIX]{x1D719}\in C^{\infty }(Y,\mathbb{R});\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D719}>0\}.\end{eqnarray}$$
This is a convex open subset of the Fréchet vector space
$C^{\infty }(Y):=C^{\infty }(Y,\mathbb{R})$
and thus itself a Fréchet manifold, which is, moreover, parallelizable:
$$\begin{eqnarray}T{\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}={\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}\times C^{\infty }(Y).\end{eqnarray}$$
For any
$\unicode[STIX]{x1D719}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
, each tangent space
$T_{\unicode[STIX]{x1D719}}{\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
is identified with
$C^{\infty }(Y)$
.
As two Kähler potentials define the same metric when (and only when) they differ by an additive constant, we set
$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}={\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}/\mathbb{R},\end{eqnarray}$$
where
$\mathbb{R}$
acts on
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
by addition. The set
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}$
is therefore the space of Kähler metrics on
$Y$
in the cohomology class
$\unicode[STIX]{x1D6FC}_{Y}:=\{\unicode[STIX]{x1D714}_{Y}\}\in H_{\mathit{BC}}^{1,1}(Y)$
.
In the whole article we fix
$\unicode[STIX]{x1D70B}:X\rightarrow Y$
a resolution of singularities and set
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{Y}$
,
$\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D6FC}_{Y}$
. Since
$\unicode[STIX]{x1D6FC}$
is no longer Kähler, we fix
$\unicode[STIX]{x1D714}_{X}$
a Kähler form on
$X$
and set
$$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}\end{eqnarray}$$
for
$\unicode[STIX]{x1D700}>0$
. We will study the geometry and the topology of the spaces
$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D70B}^{\ast }{\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{Y}}\quad \text{and}\quad {\mathcal{H}}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70B}^{\ast }{\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}\end{eqnarray}$$
by approximating them by the spaces
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}},{\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
, where
$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}:=\{\unicode[STIX]{x1D711}\in C^{\infty }(X,\mathbb{R});\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}>0\}\quad \text{and}\quad \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D700}}:=\{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}\}.\end{eqnarray}$$
All the properties that we are going to establish actually hold for cohomology classes
$\unicode[STIX]{x1D6FC}$
that are merely semi-positive and big (not necessarily the pull-back of a Kähler class under a desingularization).
Our analysis will focus on the ample locus of
$\unicode[STIX]{x1D6FC}$
.
Definition 1.1. The ample locus
$\operatorname{Amp}(\unicode[STIX]{x1D6FC})$
of
$\unicode[STIX]{x1D6FC}$
is the Zariski open set of those points
$x\in X$
, such that
$\unicode[STIX]{x1D6FC}$
can be represented by a positive closed
$(1,1)$
-current that is a smooth positive form near
$x$
.
We then let
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
denote the space of potentials
$\unicode[STIX]{x1D711}\in C^{\infty }(X,\mathbb{R})$
such that
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}$
is a Kähler form in
$\operatorname{Amp}(\unicode[STIX]{x1D6FC})$
. In our main case of interest, i.e. when
$\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D6FC}_{Y}$
for some Kähler class
$\unicode[STIX]{x1D6FC}_{Y}$
on a normal space
$Y$
, the ample locus
$$\begin{eqnarray}\operatorname{Amp}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D70B}^{-1}(Y^{\text{reg}})\end{eqnarray}$$
is the preimage of the set of regular points of
$Y$
.
1.1 The Riemannian structure
1.1.1 Mabuchi geodesics
Definition 1.2 [Reference MabuchiMab87].
The Mabuchi metric is the
$L^{2}$
Riemannian metric on
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. It is defined by
$$\begin{eqnarray}\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}\rangle _{\unicode[STIX]{x1D711}}=\int _{X}\unicode[STIX]{x1D713}_{1}\unicode[STIX]{x1D713}_{2}\frac{(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}}{V_{\unicode[STIX]{x1D6FC}}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D711}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
,
$\unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}\in C^{\infty }(X)$
, and
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}/V_{\unicode[STIX]{x1D6FC}}$
is the volume element, normalized so that it is a probability measure. Here,
$V_{\unicode[STIX]{x1D6FC}}:=\unicode[STIX]{x1D6FC}^{n}=\int _{X}\unicode[STIX]{x1D714}^{n}$
.
In the following, we shall also use the notation
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}:=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}$
and
$$\begin{eqnarray}\operatorname{MA}(\unicode[STIX]{x1D711}):=V_{\unicode[STIX]{x1D6FC}}^{-1}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n}.\end{eqnarray}$$
Geodesics between two points
$\unicode[STIX]{x1D711}_{0}$
,
$\unicode[STIX]{x1D711}_{1}$
in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
correspond to the extremals of the energy functional
$$\begin{eqnarray}\unicode[STIX]{x1D711}\mapsto H(\unicode[STIX]{x1D711})=\frac{1}{2}\int _{0}^{1}\int _{X}(\dot{\unicode[STIX]{x1D711}_{t}})^{2}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})\,dt,\end{eqnarray}$$
where
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{t}$
is a smooth path in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
joining
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
. The geodesic equation is formally obtained by computing the Euler–Lagrange equation for this energy functional (with fixed end points). It is given by
$$\begin{eqnarray}\ddot{\unicode[STIX]{x1D711}}\operatorname{MA}(\unicode[STIX]{x1D711})=\frac{n}{V_{\unicode[STIX]{x1D6FC}}}\,d\dot{\unicode[STIX]{x1D711}}\wedge d^{c}\dot{\unicode[STIX]{x1D711}}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n-1}.\end{eqnarray}$$
We are interested in the boundary value problem for the geodesic equation: given
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
, two distinct points in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, can one find a path
$(\unicode[STIX]{x1D711}(t))_{0\leqslant t\leqslant 1}$
in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
which is a solution of (2) with end points
$\unicode[STIX]{x1D711}(0)=\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}(1)=\unicode[STIX]{x1D711}_{1}$
?
For each path
$(\unicode[STIX]{x1D711}_{t})_{t\in [0,1]}$
in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, we set
$$\begin{eqnarray}\unicode[STIX]{x1D711}(x,t+is)=\unicode[STIX]{x1D711}_{t}(x),\quad x\in X,\quad t+is\in S=\{z\in \mathbb{C}:0<\Re (z)<1\};\end{eqnarray}$$
i.e. we associate with each path
$(\unicode[STIX]{x1D711}_{t})$
a function
$\unicode[STIX]{x1D711}$
on the complex manifold
$M=X\times S$
, which only depends on the real part of the strip coordinate: we consider
$S$
as a Riemann surface with boundary and use the complex coordinate
$z=t+is$
to parametrize the strip
$S$
. Set
$\unicode[STIX]{x1D714}(x,z):=\unicode[STIX]{x1D714}(x)$
.
Semmes observed [Reference SemmesSem92] that the path
$\unicode[STIX]{x1D711}_{t}$
is a geodesic in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
if and only if the associated function
$\unicode[STIX]{x1D711}$
on
$X\times S$
is a
$\unicode[STIX]{x1D714}$
-psh solution of the homogeneous complex Monge–Ampère equation
$$\begin{eqnarray}(\unicode[STIX]{x1D714}+dd_{x,z}^{c}\unicode[STIX]{x1D711})^{n+1}=0.\end{eqnarray}$$
This motivates the following.
Definition 1.3. The function
$$\begin{eqnarray}\unicode[STIX]{x1D711}=\sup \{u\,;\,u\in \text{PSH}(M,\unicode[STIX]{x1D714})~\text{and}~u\leqslant \unicode[STIX]{x1D711}_{0,1}~\text{on}~\unicode[STIX]{x2202}M\}\end{eqnarray}$$
is the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
.
Here
$\text{PSH}(M,\unicode[STIX]{x1D714})$
denotes the set of
$\unicode[STIX]{x1D714}$
-psh functions on
$M$
: these are functions
$u:M\rightarrow \mathbb{R}\cap \{-\infty \}$
that are locally the sum of a plurisubharmonic and a smooth function, such that
$\unicode[STIX]{x1D714}+dd_{x,z}^{c}u\geqslant 0$
in the sense of currents (see § 2.1.1 for more details).
Proposition 1.4. Let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
be the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
. Then:
(i)
$\unicode[STIX]{x1D711}\in \text{PSH}(M,\unicode[STIX]{x1D714})$
is uniformly bounded on
$M$
and continuous on
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})\times \bar{S}$
;(ii)
$|\unicode[STIX]{x1D711}(x,z)-\unicode[STIX]{x1D711}(x,z^{\prime })|\leqslant A|\Re (z)-\Re (z^{\prime })|$
with
$A=\Vert \unicode[STIX]{x1D711}_{0}-\unicode[STIX]{x1D711}_{1}\Vert _{L^{\infty }(X)}$
;(iii)
$\unicode[STIX]{x1D711}_{|\{\Re (z)=0\}}=\unicode[STIX]{x1D711}_{0}$
,
$\unicode[STIX]{x1D711}_{|\{\Re (z)=1\}}=\unicode[STIX]{x1D711}_{1}$
and
$(\unicode[STIX]{x1D714}+dd_{x,z}^{c}\unicode[STIX]{x1D711})^{n+1}=0$
.
It is, moreover, the unique bounded
$\unicode[STIX]{x1D714}$
-psh solution to this Dirichlet problem.
We thank Hoang Chinh Lu for sharing his ideas on the continuity of
$\unicode[STIX]{x1D711}$
.
Proof. The proof follows from a classical balayage technique together with a barrier argument, as noted by Berndtsson [Reference BerndtssonBer15]. Set
$A=\Vert \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\Vert _{L^{\infty }(X)}$
.
Observe that the function
$\unicode[STIX]{x1D711}_{0}-At$
, with
$t=\Re (z)$
, is
$\unicode[STIX]{x1D714}$
-psh on
$M$
and
$\unicode[STIX]{x1D711}_{0}-At|_{\unicode[STIX]{x2202}M}\leqslant \unicode[STIX]{x1D711}_{0,1}$
. Hence, it belongs to the family
${\mathcal{F}}$
defining the upper envelope
$\unicode[STIX]{x1D711}$
, so
$\unicode[STIX]{x1D711}_{0}-At\leqslant \unicode[STIX]{x1D711}_{t}$
.
Similarly,
$\unicode[STIX]{x1D711}_{0}+At$
is a
$\unicode[STIX]{x1D714}$
-psh function on
$M$
and
$\unicode[STIX]{x1D711}_{0}+At|_{\unicode[STIX]{x2202}M}\geqslant \unicode[STIX]{x1D711}_{0,1}$
. Since
$(\unicode[STIX]{x1D714}+dd_{x,z}^{c}(\unicode[STIX]{x1D711}_{0}+At))^{n+1}=0$
, it follows from the maximum principle that
$u\leqslant \unicode[STIX]{x1D711}_{0}+At$
, for any
$u\in {\mathcal{F}}$
in the family. Therefore,
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}-At\leqslant \unicode[STIX]{x1D711}_{t}\leqslant \unicode[STIX]{x1D711}_{0}+At.\end{eqnarray}$$
Similar arguments show that
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{1}+A(t-1)\leqslant \unicode[STIX]{x1D711}_{t}\leqslant \unicode[STIX]{x1D711}_{1}-A(t-1).\end{eqnarray}$$
The upper semi-continuous regularization
$\unicode[STIX]{x1D711}^{\ast }$
of
$\unicode[STIX]{x1D711}$
satisfies the same estimates, showing, in particular, that
$\unicode[STIX]{x1D711}^{\ast }|_{\unicode[STIX]{x2202}M}=\unicode[STIX]{x1D711}_{0,1}$
. Since
$\unicode[STIX]{x1D711}^{\ast }$
is
$\unicode[STIX]{x1D714}$
-psh, we infer
$\unicode[STIX]{x1D711}^{\ast }\in {\mathcal{F}}$
; hence,
$\unicode[STIX]{x1D711}^{\ast }=\unicode[STIX]{x1D711}$
. Thus
$\unicode[STIX]{x1D711}$
is
$\unicode[STIX]{x1D714}$
-psh and uniformly bounded, proving the first statement in part (i). Classical balayage arguments show that
$(\unicode[STIX]{x1D714}+dd_{x,z}^{c}\unicode[STIX]{x1D711})^{n+1}=0$
, proving part (iii).
We now prove part (ii). Consider the function
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{t}(x)=\max \{\unicode[STIX]{x1D711}_{0}(x)-A\,\text{log}|z|,\unicode[STIX]{x1D711}_{1}(x)+A(\log |z|-1)\}\end{eqnarray}$$
and note that it belongs to
${\mathcal{F}}$
and has the right boundary values.
Since
$\unicode[STIX]{x1D712}_{-}=\unicode[STIX]{x1D711}_{0}(x)-At\leqslant \unicode[STIX]{x1D711}$
with equality at
$t=0$
, we infer, for all
$x$
,
$$\begin{eqnarray}-A=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{-}}{\unicode[STIX]{x2202}t}_{|t=0}\leqslant \dot{\unicode[STIX]{x1D711}}_{0}(x).\end{eqnarray}$$
Similarly
$\unicode[STIX]{x1D712}_{+}=\unicode[STIX]{x1D711}_{1}(x)+A(t-1)\leqslant \unicode[STIX]{x1D711}$
with equality at
$t=1$
yields, for all
$x$
,
$\dot{\unicode[STIX]{x1D711}}_{1}(x)\leqslant +A=(\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{+}/\unicode[STIX]{x2202}t)_{|t=1}$
. Since
$t\mapsto \unicode[STIX]{x1D711}_{t}(x)$
is convex (by subharmonicity in
$z$
), we infer that for a.e.
$t,x$
,
$-A\leqslant \dot{\unicode[STIX]{x1D711}}_{0}(x)\leqslant \dot{\unicode[STIX]{x1D711}}_{t}(x)\leqslant \dot{\unicode[STIX]{x1D711}}_{1}(x)\leqslant +A$
.
It remains to show that
$\unicode[STIX]{x1D711}$
is continuous on
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})\times \bar{S}$
. We can assume, without loss of generality, that
$\unicode[STIX]{x1D711}_{0}<\unicode[STIX]{x1D711}_{1}$
. Indeed, given any
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, there exists
$C>0$
, such that
$\unicode[STIX]{x1D711}_{0}<\unicode[STIX]{x1D711}_{1}+C$
. By Lemma 1.8, the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}+C$
is
$\unicode[STIX]{x1D713}_{t}=\unicode[STIX]{x1D711}_{t}+Ct$
,
$t\in [0,1]$
. The continuity of
$(x,t)\rightarrow \unicode[STIX]{x1D713}_{t}(x)$
will then imply the continuity of
$(x,t)\rightarrow \unicode[STIX]{x1D711}_{t}(x)$
.
We change notation slightly, replacing the strip
$S$
by the annulus
$D:=\{z=e^{t+is}\in \mathbb{C}:1\leqslant |w|\leqslant e\}$
. We are going to express the function
$\unicode[STIX]{x1D711}$
as a global
$\unicode[STIX]{x1D6E9}$
-psh envelope on the compact manifold
$X\times \mathbb{P}^{1}$
, where we view the annulus
$D$
as a subset of the Riemann sphere,
$\mathbb{C}\subset \mathbb{P}^{1}=\mathbb{C}\cup \{\infty \}$
. The form
$\unicode[STIX]{x1D6E9}(x,z)=\unicode[STIX]{x1D714}(x)+A\unicode[STIX]{x1D714}_{\text{FS}}(z)$
is a semi-positive and big form on the compact Kähler manifold
$\widetilde{M}:=X\times \mathbb{P}^{1}$
, so the viscosity approach of [Reference Eyssidieux, Guedj and ZeriahiEGZ17] can be applied, showing that the envelope
$\unicode[STIX]{x1D711}$
is continuous on
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})\times \bar{S}$
. Here,
$\unicode[STIX]{x1D714}_{\text{FS}}$
denotes the Fubini–Study metric on
$\mathbb{P}^{1}$
and
$A>0$
is a constant to be chosen next.
Consider
$U=\max (U_{0},U_{1})$
, where
$U_{0}(x,z):=\unicode[STIX]{x1D711}_{0}(x)$
and
$$\begin{eqnarray}U_{1}(x,z):=\unicode[STIX]{x1D711}_{1}(x)+A(\log |z|^{2}-\log (|z|^{2}+1)+\log (e^{2}+1)-2).\end{eqnarray}$$
We choose
$A>0$
so large that
$U(x,1)\equiv \unicode[STIX]{x1D711}_{0}(x)$
. Note that
$U(x,e)\equiv \unicode[STIX]{x1D711}_{1}(x)$
since
$\unicode[STIX]{x1D711}_{0}<\unicode[STIX]{x1D711}_{1}$
. Both
$U_{0}$
and
$U_{1}$
are
$\unicode[STIX]{x1D6E9}$
-psh on
$\widetilde{M}$
, hence so is
$U$
.
Fix
$\unicode[STIX]{x1D70C}$
a local potential of
$A\unicode[STIX]{x1D714}_{\text{FS}}$
in
$D$
, such that
$\unicode[STIX]{x1D70C}|_{\unicode[STIX]{x2202}D}=0$
and let
$F$
be a continuous
$S^{1}$
-invariant function on
$\widetilde{M}$
, such that:
(a)
$F=\unicode[STIX]{x1D711}_{0,1}$
on
$X\times \unicode[STIX]{x2202}D$
;(b)
$F(x,z)\geqslant U(x,z)\geqslant \unicode[STIX]{x1D711}_{0}(x)$
;(c)
$F(x,z)+\unicode[STIX]{x1D70C}(z)>\unicode[STIX]{x1D711}_{t}(x)$
in
$X\times D$
, with
$t=\log |z|$
.
We let the reader check that the function
$F=U$
in
$\widetilde{M}\setminus X\times D$
and
$$\begin{eqnarray}F(x,z):=(1-\log |z|)\unicode[STIX]{x1D711}_{0}(x)+(\log |z|)\unicode[STIX]{x1D711}_{1}(x)-\unicode[STIX]{x1D70C}(z)+(\log |z|)(1-\log |z|),\end{eqnarray}$$
for
$(x,z)\in X\times D$
, does the job.
We claim that for all
$(x,z)\in X\times D$
,
$$\begin{eqnarray}P_{\unicode[STIX]{x1D6E9}}(F)(x,z)+\unicode[STIX]{x1D70C}(z)=\unicode[STIX]{x1D711}_{\log |z|}(x)\,,\end{eqnarray}$$
where
$$\begin{eqnarray}P_{\unicode[STIX]{x1D6E9}}(F):=\sup \{v:v\in \operatorname{PSH}(\widetilde{M},\unicode[STIX]{x1D6E9})~\text{and}~v\leqslant F\}.\end{eqnarray}$$
Indeed
$P_{\unicode[STIX]{x1D6E9}}(F)+\unicode[STIX]{x1D70C}$
is
$\unicode[STIX]{x1D714}$
-psh in
$X\times D$
and has boundary values
${\leqslant}\unicode[STIX]{x1D711}_{0,1}$
. It follows from the definition of the geodesic that
$P_{\unicode[STIX]{x1D6E9}}(F)+\unicode[STIX]{x1D70C}\leqslant \unicode[STIX]{x1D711}_{t}$
. Conversely,
$F+\unicode[STIX]{x1D70C}\geqslant U+\unicode[STIX]{x1D70C}\in \operatorname{PSH}(X\times D,\unicode[STIX]{x1D714})$
and
$U=\unicode[STIX]{x1D711}_{0,1}$
on
$\unicode[STIX]{x2202}M$
, thus
$P_{\unicode[STIX]{x1D6E9}}(F)+\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D711}_{0,1}$
on
$\unicode[STIX]{x2202}M$
. Condition (c) ensures that
$M=X\times D$
does not meet the contact set
$\{P_{\unicode[STIX]{x1D6E9}}(F)=F\}$
, since
$F+\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D711}_{t}\geqslant P_{\unicode[STIX]{x1D6E9}}(F)+\unicode[STIX]{x1D70C}$
. It thus follows from a balayage argument [Reference Bedford and TaylorBT82] that
$(\unicode[STIX]{x1D6E9}+dd^{c}P_{\unicode[STIX]{x1D6E9}}(F))^{n+1}=0$
in
$M$
, and the maximum principle yields
$$\begin{eqnarray}P_{\unicode[STIX]{x1D6E9}}(F)+\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D711}_{t}.\end{eqnarray}$$
The continuity of
$\unicode[STIX]{x1D711}$
on
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})\times \bar{S}$
now follows from [Reference Eyssidieux, Guedj and ZeriahiEGZ17], together with the following easy observation: the arguments in [Reference Eyssidieux, Guedj and ZeriahiEGZ17, § 2.2] ensure that if
$F$
is a smooth function on
$\widetilde{M}$
,
$P_{\unicode[STIX]{x1D6E9}}(F)$
is a
$\unicode[STIX]{x1D6E9}$
-psh function, continuous on
$\operatorname{Amp}(\{\unicode[STIX]{x1D6E9}\})$
. The same result holds if
$F$
is merely continuous. Indeed, let
$F_{j}$
be a sequence of smooth functions on
$\widetilde{M}$
converging uniformly to
$F$
. Taking the envelope at both sides of the inequality
$F_{j}\leqslant F+\Vert F_{j}-F\Vert _{L^{\infty }(X)}$
, we get
$P_{\unicode[STIX]{x1D6E9}}(F_{j})\leqslant P_{\unicode[STIX]{x1D6E9}}(F)+\Vert F_{j}-F\Vert _{L^{\infty }(X)}$
. Hence,
$\Vert P_{\unicode[STIX]{x1D6E9}}(F_{j})-P_{\unicode[STIX]{x1D6E9}}(F)\Vert _{L^{\infty }(X)}\leqslant \Vert F_{j}-F\Vert _{L^{\infty }(X)}$
. Thus,
$P_{\unicode[STIX]{x1D6E9}}(F_{j})$
converges uniformly to
$P_{\unicode[STIX]{x1D6E9}}(F)$
, and so
$P_{\unicode[STIX]{x1D6E9}}(F)$
is a
$\unicode[STIX]{x1D6E9}$
-psh function that is continuous on
$\operatorname{Amp}(\{\unicode[STIX]{x1D6E9}\})=\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})\times \bar{S}$
.◻
Remark 1.5. If one could choose
$F$
smooth in this proof, it would follow from [Reference Berman and DemaillyBD12] (or [Reference BermanBer13, Theorem 1.2]) that
$\unicode[STIX]{x1D711}\in {\mathcal{C}}^{1,\bar{1}}(\operatorname{Amp}(\unicode[STIX]{x1D6FC})\times S)$
. This would also provide a compact proof of Chen’s regularity result.
We now observe that geodesics in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
are projections of those in
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
,
Proposition 1.6. Let
$\unicode[STIX]{x1D711}$
denote the geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
and let
$\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
denote the corresponding geodesic in the space
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
. The map
$\unicode[STIX]{x1D700}\mapsto \unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
is increasing and
$\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
decreases to
$\unicode[STIX]{x1D711}$
as
$\unicode[STIX]{x1D700}$
decreases to zero. Moreover,
$$\begin{eqnarray}\unicode[STIX]{x1D711}=P(\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}),\end{eqnarray}$$
where
$P$
denotes the projection operator onto the space
$\text{PSH}(M,\unicode[STIX]{x1D714})$
.
Recall that, for an upper semi-continuous function
$u:M\rightarrow \mathbb{R}$
, its projection
$P(u)$
is defined by
$$\begin{eqnarray}P(u):=\sup \{v\in \text{PSH}(M,\unicode[STIX]{x1D714});v\leqslant u\}.\end{eqnarray}$$
The function
$P(u)$
is either identical to
$-\infty$
or belongs to
$\text{PSH}(M,\unicode[STIX]{x1D714})$
. It is the greatest
$\unicode[STIX]{x1D714}$
-psh function on
$M$
that lies below
$u$
.
Proof. Set
$\unicode[STIX]{x1D713}:=P(\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}})$
. Since
$\unicode[STIX]{x1D714}\leqslant \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
, it follows from the envelope point of view that
$\unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
. Thus,
$\unicode[STIX]{x1D711}=P(\unicode[STIX]{x1D711})\leqslant P(\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}})=\unicode[STIX]{x1D713}$
and
$\unicode[STIX]{x1D713}\in \text{PSH}(M,\unicode[STIX]{x1D714})$
. Now
$\unicode[STIX]{x1D713}\leqslant \unicode[STIX]{x1D711}$
, since
$\unicode[STIX]{x1D713}\leqslant \unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
on
$\unicode[STIX]{x2202}M$
and
$\unicode[STIX]{x1D713}\in \text{PSH}(M,\unicode[STIX]{x1D714})$
. Thus,
$\unicode[STIX]{x1D713}=P(\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}})=\unicode[STIX]{x1D711}$
.
Fix
$\unicode[STIX]{x1D700}^{\prime }\leqslant \unicode[STIX]{x1D700}$
. The inclusion
$\text{PSH}(M,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}^{\prime }})\subset \text{PSH}(M,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
implies similarly that
$\unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}^{\prime }}\leqslant \unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
. The decreasing limit
$v$
of
$\unicode[STIX]{x1D711}^{\unicode[STIX]{x1D700}}$
, as
$\unicode[STIX]{x1D700}$
decreases to zero, satisfies both
$\unicode[STIX]{x1D711}\leqslant v$
and
$v\in \text{PSH}(M,\unicode[STIX]{x1D714})$
with boundary values
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
, thus
$v=\unicode[STIX]{x1D711}$
.◻
It will also be interesting to consider subgeodesics.
Definition 1.7. A subgeodesic is a path
$(\unicode[STIX]{x1D711}_{t})$
of functions in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
(or in larger classes of
$\unicode[STIX]{x1D714}$
-psh functions) such that the associated function is a
$\unicode[STIX]{x1D714}$
-psh function on
$X\times S$
.
We shall soon need the following simple observation.
Lemma 1.8. Fix
$c\in \mathbb{R}$
,
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
and let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
denote the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D713}$
. Then
$\unicode[STIX]{x1D713}_{t}(x):=\unicode[STIX]{x1D711}_{t}(x)-ct,0\leqslant t\leqslant 1,x\in X,$
is the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}$
to
$\unicode[STIX]{x1D713}-c$
.
Proof. The proof follows from Definition 1.3 and the definition of envelopes, since
$\sup \{v;v\in \text{PSH}(M,\unicode[STIX]{x1D714})~\text{and}~v\leqslant \unicode[STIX]{x1D711},v\leqslant \unicode[STIX]{x1D713}-c~\text{on}~\unicode[STIX]{x2202}M\}=\unicode[STIX]{x1D711}_{t}-ct$
.◻
1.1.2 Mabuchi and other Finsler distances
When
$\unicode[STIX]{x1D714}$
is Kähler, the length of a smooth path
$(\unicode[STIX]{x1D711}_{t})_{t\in [0,1]}$
in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
is defined in a standard way,
$$\begin{eqnarray}\ell (\unicode[STIX]{x1D711}):=\int _{0}^{1}|\dot{\unicode[STIX]{x1D711}_{t}}|\,dt=\int _{0}^{1}\sqrt{\int _{X}\dot{\unicode[STIX]{x1D711}}_{t}^{2}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})}\,dt.\end{eqnarray}$$
The distance between two points in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
is then
$$\begin{eqnarray}d(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}):=\inf \{\ell (\unicode[STIX]{x1D711})\mid \unicode[STIX]{x1D711}~\text{is a smooth path joining}~\unicode[STIX]{x1D711}_{0}~\text{to}~\unicode[STIX]{x1D711}_{1}\}.\end{eqnarray}$$
It is easy to verify that
$d$
defines a semi-distance (i.e. non-negative, symmetric, and satisfying the triangle inequality). It is, however, non-trivial to check that
$d$
is non-degenerate (see [Reference Michor and MumfordMM05] for a striking example).
Observe that
$d$
induces a distance on
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$
(that we abusively still denote
$d$
) compatible with the Riemannian splitting
${\mathcal{H}}_{\unicode[STIX]{x1D714}}={\mathcal{H}}_{\unicode[STIX]{x1D6FC}}\times \mathbb{R}$
, by setting
$$\begin{eqnarray}d(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}):=d(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\end{eqnarray}$$
whenever the potentials
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
of
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$
are normalized by
$E(\unicode[STIX]{x1D711})=E(\unicode[STIX]{x1D713})=0$
(see § 2.2.1 for the definition of the functional
$E$
).
It is rather easy to check that
$({\mathcal{H}}_{\unicode[STIX]{x1D6FC}},d)$
is not a complete metric space. We shall describe the metric completion
$(\overline{{\mathcal{H}}}_{\unicode[STIX]{x1D6FC}},d)$
in § 4. Following Darvas [Reference DarvasDar15], we introduce a family of distances that generalize
$d$
:
Definition 1.9. For
$p\geqslant 1$
and
$\unicode[STIX]{x1D714}$
Kähler, we set
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}):=\inf \{\ell _{p}(\unicode[STIX]{x1D711})\mid \unicode[STIX]{x1D711}~\text{is a smooth path joining}~\unicode[STIX]{x1D711}_{0}~\text{to}~\unicode[STIX]{x1D711}_{1}\},\end{eqnarray}$$
where
$\ell _{p}(\unicode[STIX]{x1D711}):=\int _{0}^{1}|\dot{\unicode[STIX]{x1D711}_{t}}|_{p}\,dt=\int _{0}^{1}(\int _{X}|\dot{\unicode[STIX]{x1D711}}_{t}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{t}))^{1/p}\,dt$
.
Note that
$d_{2}=d$
is the Mabuchi distance. Mabuchi geodesics have constant speed with respect to all the Finsler structures
$\ell _{p}$
, as was observed by Berndtsson [Reference BerndtssonBer09, Lemma 2.1]: for any
${\mathcal{C}}^{1}$
-function
$\unicode[STIX]{x1D712}$
,
$$\begin{eqnarray}t\mapsto \int _{X}\unicode[STIX]{x1D712}(\dot{\unicode[STIX]{x1D711}}_{t})\operatorname{MA}(\unicode[STIX]{x1D711}_{t})\end{eqnarray}$$
is constant along a geodesic. Indeed
$$\begin{eqnarray}\displaystyle \frac{d}{dt}\int _{X}\unicode[STIX]{x1D712}(\dot{\unicode[STIX]{x1D711}}_{t})\operatorname{MA}(\unicode[STIX]{x1D711}_{t}) & = & \displaystyle \int _{X}\unicode[STIX]{x1D712}^{\prime }(\dot{\unicode[STIX]{x1D711}}_{t})\ddot{\unicode[STIX]{x1D711}}_{t}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})+\frac{n}{V_{\unicode[STIX]{x1D6FC}}}\int _{X}\unicode[STIX]{x1D712}(\dot{\unicode[STIX]{x1D711}}_{t})\,dd^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}_{t}}^{n-1}\nonumber\\ \displaystyle & = & \displaystyle \int _{X}\unicode[STIX]{x1D712}^{\prime }(\dot{\unicode[STIX]{x1D711}}_{t})\biggl\{\ddot{\unicode[STIX]{x1D711}}_{t}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})-\frac{n}{V_{\unicode[STIX]{x1D6FC}}}d\dot{\unicode[STIX]{x1D711}}_{t}\wedge d^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}_{t}}^{n-1}\biggr\}=0\nonumber\end{eqnarray}$$
since
$\ddot{\unicode[STIX]{x1D711}}_{t}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})-(n/V_{\unicode[STIX]{x1D6FC}})\,d\dot{\unicode[STIX]{x1D711}}_{t}\wedge d^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}_{t}}^{n-1}=0$
. Applying this observation to
$\unicode[STIX]{x1D712}(t)=t^{p}$
shows that Mabuchi geodesics have constant
$\ell _{p}$
-speed.
When
$\unicode[STIX]{x1D714}$
is merely semi-positive, there are fewer smooth paths within
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. It is natural to consider smooth paths in
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
and pass to the limit in the previous definitions.
Definition 1.10. Assume
$\unicode[STIX]{x1D714}$
is semi-positive and big. Let
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. We define the Mabuchi distance between
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
as
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}):=\liminf _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
where
$d_{p,\unicode[STIX]{x1D700}}$
is the distance with respect to the Kähler form
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
.
We will show in Theorem 1.13 that it is a distance, which moreover does not depend on the way we approximate
$\unicode[STIX]{x1D714}$
by Kähler classes.
Remark 1.11. For any smooth path
$\unicode[STIX]{x1D713}:[0,1]\rightarrow {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, we can still define
$$\begin{eqnarray}\ell _{p}(\unicode[STIX]{x1D713}):=\int _{0}^{1}\biggl(\frac{1}{V}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{t}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713}_{t})^{n}\biggr)^{1/p}\,dt\end{eqnarray}$$
when
$\unicode[STIX]{x1D714}$
is merely semi-positive. Since
$\text{PSH}(M,\unicode[STIX]{x1D714})\subset \text{PSH}(M,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
,
$\unicode[STIX]{x1D713}_{t}$
is both in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
and
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
. Observe that
$$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D700}}^{-1}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{t}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D713}_{t})^{n} & = & \displaystyle V_{\unicode[STIX]{x1D700}}^{-1}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{t}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713}_{t}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X})^{n}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle V^{-1}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{t}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713}_{t})^{n}+A\unicode[STIX]{x1D700},\nonumber\end{eqnarray}$$
hence
$$\begin{eqnarray}\ell _{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D713})\leqslant \ell _{p}(\unicode[STIX]{x1D713})+A^{\prime }\unicode[STIX]{x1D700}\,,\end{eqnarray}$$
where
$\ell _{p,\unicode[STIX]{x1D700}}$
denotes the length in
${\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
. We infer
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant \inf \{\ell _{p}(\unicode[STIX]{x1D713})~\unicode[STIX]{x1D713}~\text{smooth path joining}~\unicode[STIX]{x1D711}_{0}~\text{and}~\unicode[STIX]{x1D711}_{1}~\text{in}~{\mathcal{H}}_{\unicode[STIX]{x1D714}}\}.\end{eqnarray}$$
The converse inequality is, however, unclear, owing to the lack of positivity of
$\unicode[STIX]{x1D714}$
: it is difficult to smooth out
$\unicode[STIX]{x1D714}$
-psh functions if
$\unicode[STIX]{x1D714}$
is not Kähler. This partially explains Definition 1.10.
1.2 Approximation by Kähler classes
Fix
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. We let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
denote the Mabuchi geodesic in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
.
Definition 1.12. For
$t=0,1$
we set
$$\begin{eqnarray}I(t):=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{t}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{t}).\end{eqnarray}$$
Theorem 1.13. Set
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
,
$\unicode[STIX]{x1D700}>0$
. Then
$\lim _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
exists and is independent of
$\unicode[STIX]{x1D714}_{X}$
. More precisely,
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\rightarrow I(0)=I(1).\end{eqnarray}$$
In particular,
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=I(0)^{1/p}=I(1)^{1/p}$
defines a distance on
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
.
In the definition of
$I(0),I(1)$
, the time derivatives
$\dot{\unicode[STIX]{x1D711}}_{0}=\dot{\unicode[STIX]{x1D711}}_{0}^{+}$
,
$\dot{\unicode[STIX]{x1D711}}_{1}=\dot{\unicode[STIX]{x1D711}}_{1}^{-}$
denote the right and left derivatives, respectively.
Remark 1.14. When
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{Y}$
, for some Kähler form
$\unicode[STIX]{x1D714}_{Y}$
on a compact normal space
$Y$
, for each
$p\geqslant 1$
and
$\forall \unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{Y}}$
, we define
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1}):=d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\quad \text{where}~\unicode[STIX]{x1D711}_{0}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D719}_{1}.\end{eqnarray}$$
This definition does not depend on the choice of resolution. Indeed, let
$\unicode[STIX]{x1D70B}^{\prime }:X^{\prime }\rightarrow Y$
be another resolution of
$Y$
that dominates
$X$
, i.e. there exists a holomorphic and bimeromorphic map
$f:X^{\prime }\rightarrow X$
, such that
$\unicode[STIX]{x1D70B}^{\prime }=\unicode[STIX]{x1D70B}\circ f$
. Set
$\unicode[STIX]{x1D714}^{\prime }:={\unicode[STIX]{x1D70B}^{\prime }}^{\ast }\unicode[STIX]{x1D714}_{Y}=f^{\ast }\unicode[STIX]{x1D714}$
. We need to show that
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=d_{p,f^{\ast }\unicode[STIX]{x1D714}}(f^{\ast }\unicode[STIX]{x1D711}_{0},f^{\ast }\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Denote by
$\unicode[STIX]{x1D713}_{t}$
the
$f^{\ast }\unicode[STIX]{x1D714}$
-geodesic joining
$f^{\ast }\unicode[STIX]{x1D711}_{0}$
and
$f^{\ast }\unicode[STIX]{x1D711}_{1}$
. We claim that
$\unicode[STIX]{x1D713}_{t}=f^{\ast }\unicode[STIX]{x1D711}_{t}$
. We first observe that, since
$\unicode[STIX]{x1D713}_{t}$
is a
$f^{\ast }\unicode[STIX]{x1D714}$
-psh function for each fixed
$t$
,
$\unicode[STIX]{x1D713}_{t}=f^{\ast }\unicode[STIX]{x1D6FE}_{t}$
, where
$\unicode[STIX]{x1D6FE}_{t}$
is a
$\unicode[STIX]{x1D714}$
-psh function on
$X$
. Set
$M^{\prime }:=X^{\prime }\times S$
,
$\unicode[STIX]{x1D713}(x^{\prime },t):=\unicode[STIX]{x1D713}_{t}(x^{\prime })$
, and
$\unicode[STIX]{x1D6FE}(x^{\prime },t):=\unicode[STIX]{x1D6FE}_{t}(x^{\prime })$
for each
$(x^{\prime },t)\in M^{\prime }$
. By construction, we have that
$$\begin{eqnarray}f^{\ast }(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D6FE})^{n+1}=(f^{\ast }\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713})^{n+1}=0\quad \text{on}~M^{\prime }:=X^{\prime }\times S,\quad \unicode[STIX]{x1D713}|_{\unicode[STIX]{x2202}M^{\prime }}=f^{\ast }\unicode[STIX]{x1D711}_{0,1}.\end{eqnarray}$$
The claim follows from the uniqueness of the solution of the Dirichlet problem in Proposition 1.4. The invariance of the non-pluripolar Monge–Ampère measure under bimeromorphic maps [Reference Di NezzaDiN15], together with the fact that
$V:=\int _{X}\unicode[STIX]{x1D714}^{n}=\int _{X^{\prime }}f^{\ast }\unicode[STIX]{x1D714}$
, give
$$\begin{eqnarray}\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\frac{(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}}{V}=\int _{X^{\prime }}|\dot{f^{\ast }\unicode[STIX]{x1D711}}_{0}|^{p}\frac{(f^{\ast }\unicode[STIX]{x1D714}+dd^{c}f^{\ast }\unicode[STIX]{x1D711}_{0})^{n}}{V}=\int _{X^{\prime }}|\dot{\unicode[STIX]{x1D713}}_{0}|^{p}\frac{(\unicode[STIX]{x1D714}^{\prime }+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}}{V}.\end{eqnarray}$$
The conclusion then follows from Theorem 1.13.
Proof. Observe that
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
and let
$\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}$
be the corresponding geodesic. It follows from [Reference DarvasDar15, Theorem 3.5] that
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=V_{\unicode[STIX]{x1D700}}^{-1}\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}.\end{eqnarray}$$
Now observe that
$$\begin{eqnarray}\dot{\unicode[STIX]{x1D711}}_{0}^{+}\leqslant \dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}\leqslant \frac{\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}-\unicode[STIX]{x1D711}_{0}}{t}\quad \forall t\in (0,1),\end{eqnarray}$$
where the first inequality follows from the fact that
$\unicode[STIX]{x1D700}\rightarrow \unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}$
is decreasing (Proposition 1.6), while the second uses the convexity of
$t\mapsto \unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}$
. Thus,
$$\begin{eqnarray}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}-\dot{\unicode[STIX]{x1D711}}_{0}^{+}|\leqslant \biggl|\frac{\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}-\unicode[STIX]{x1D711}_{0}}{t}-\dot{\unicode[STIX]{x1D711}}_{0}^{+}\biggr|.\end{eqnarray}$$
Letting
$\unicode[STIX]{x1D700}{\searrow}0$
and then
$t\rightarrow 0$
shows that
$|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}-\dot{\unicode[STIX]{x1D711}}_{0}^{+}|$
converges pointwise to zero. Moreover,
$(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}=f_{\unicode[STIX]{x1D700}}\,dV$
where
$dV$
is the Lebesgue measure and
$f_{\unicode[STIX]{x1D700}}>0$
are smooth densities, which converge locally uniformly to
$f\geqslant 0$
with
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}=f\,dV$
. The dominated convergence theorem thus yields
$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=V^{-1}\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}^{+}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}=I(0).\end{eqnarray}$$
The argument for
$I(1)$
is similar.
This shows, in particular, that
$d_{p}$
is a distance on
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
: if
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=0$
, then
$I(0)=I(1)=0$
, hence,
$\dot{\unicode[STIX]{x1D711}}_{0}(x)=\dot{\unicode[STIX]{x1D711}}_{1}(x)=0$
for a.e.
$x\in X$
, which implies
$\dot{\unicode[STIX]{x1D711}}_{t}(x)=0$
for a.e.
$x\in X$
, by convexity of
$t\mapsto \unicode[STIX]{x1D711}_{t}(x)$
. Thus,
$\unicode[STIX]{x1D711}_{0}(x)=\unicode[STIX]{x1D711}_{1}(x)$
for a.e.
$x\in X$
.◻
We now extend the definition of the distance
$d_{p}$
for bounded
$\unicode[STIX]{x1D714}$
-psh potentials.
Definition 1.15. Let
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in \operatorname{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)$
; then
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}):=\liminf _{\unicode[STIX]{x1D700}\rightarrow 0}\liminf _{j,k\rightarrow +\infty }d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{k})=\liminf _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{k}$
are smooth sequences of
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-psh functions decreasing to
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
, respectively.
Observe that
$d_{p,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
is well defined for potentials in
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
[Reference DarvasDar15], and so, in particular, for bounded
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-psh functions.
Proposition 1.16. Let
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in \operatorname{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)$
. The limit of
$d_{p,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
as
$\unicode[STIX]{x1D700}$
goes to zero exists and does not depend on the choice of
$\unicode[STIX]{x1D714}_{X}$
.
Proof. First, observe that since
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
are bounded, they belong to
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
for any
$0\leqslant \unicode[STIX]{x1D700}\leqslant 1$
. By [Reference DarvasDar15, Corollary 4.14] we know that the Pythagorean formula holds true, i.e.
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})+d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D713}:=\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}$
is the greatest
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-psh function that lies below
$\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
. Fix
$\unicode[STIX]{x1D700}\leqslant \unicode[STIX]{x1D700}^{\prime }$
. We claim that
$$\begin{eqnarray}V_{\unicode[STIX]{x1D700}}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D713})\leqslant V_{\unicode[STIX]{x1D700}^{\prime }}d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D713})\quad \text{and}\quad V_{\unicode[STIX]{x1D700}}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{1})\leqslant V_{\unicode[STIX]{x1D700}^{\prime }}d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Let
$\unicode[STIX]{x1D713}_{t}^{\unicode[STIX]{x1D700}}$
,
$\unicode[STIX]{x1D713}_{t}^{\unicode[STIX]{x1D700}^{\prime }}$
denote the
$\unicode[STIX]{x1D700}$
-geodesic and the
$\unicode[STIX]{x1D700}^{\prime }$
-geodesic, both joining
$\unicode[STIX]{x1D713}$
and
$\unicode[STIX]{x1D711}_{0}$
. Since
$\unicode[STIX]{x1D700}\rightarrow \unicode[STIX]{x1D713}_{t}^{\unicode[STIX]{x1D700}}$
is increasing (Proposition 1.6), we have that, for any
$t\in (0,1)$
$$\begin{eqnarray}\frac{\unicode[STIX]{x1D713}_{t}^{\unicode[STIX]{x1D700}}-\unicode[STIX]{x1D713}}{t}\leqslant \frac{\unicode[STIX]{x1D713}_{t}^{\unicode[STIX]{x1D700}^{\prime }}-\unicode[STIX]{x1D713}}{t}\,,\end{eqnarray}$$
which implies
$\dot{\unicode[STIX]{x1D713}}_{0}^{\unicode[STIX]{x1D700}}\leqslant \dot{\unicode[STIX]{x1D713}}_{0}^{\unicode[STIX]{x1D700}^{\prime }}$
. Moreover, observe that, since
$\unicode[STIX]{x1D711}_{0}(x)\geqslant \unicode[STIX]{x1D713}(x)$
for all
$x\in X$
, Lemma 3.3 yields
$\dot{\unicode[STIX]{x1D713}}_{0}^{\unicode[STIX]{x1D700}}(x)\geqslant 0$
for all
$x\in X$
. It then follows that
$$\begin{eqnarray}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D713})^{n}\leqslant \int _{X}|\dot{\unicode[STIX]{x1D713}}_{0}^{\unicode[STIX]{x1D700}^{\prime }}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}^{\prime }}+dd^{c}\unicode[STIX]{x1D713})^{n},\end{eqnarray}$$
hence the claim. The same type of arguments give
$V_{\unicode[STIX]{x1D700}}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{1})\leqslant V_{\unicode[STIX]{x1D700}^{\prime }}d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{1})$
. Hence,
$$\begin{eqnarray}V_{\unicode[STIX]{x1D700}}V_{\unicode[STIX]{x1D700}^{\prime }}^{-1}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})+d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Using again [Reference DarvasDar15, Corollary 4.14] and the triangle inequality we get
$$\begin{eqnarray}V_{\unicode[STIX]{x1D700}}V_{\unicode[STIX]{x1D700}^{\prime }}^{-1}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})+2d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}^{\prime }}\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Moreover, since
$\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}^{\prime }}\unicode[STIX]{x1D711}_{1}\geqslant \unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}$
, [Reference DarvasDar15, Lemma 5.1] yields
$$\begin{eqnarray}\displaystyle d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}^{\prime }}\unicode[STIX]{x1D711}_{1}) & {\leqslant} & \displaystyle \frac{1}{V_{\unicode[STIX]{x1D700}^{\prime }}}\int _{X}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}^{\prime }}\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})^{p}\,(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}^{\prime }}+dd^{c}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}))^{n}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{V_{\unicode[STIX]{x1D700}^{\prime }}}\int _{X}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}^{\prime }}\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})^{p}\,(\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}(\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}))^{n}\nonumber\\ \displaystyle & := & \displaystyle V_{\unicode[STIX]{x1D700}^{\prime }}^{-1}\unicode[STIX]{x1D702}(\unicode[STIX]{x1D700},\unicode[STIX]{x1D700}^{\prime }).\nonumber\end{eqnarray}$$
Observe that
$\unicode[STIX]{x1D702}(\unicode[STIX]{x1D700},\unicode[STIX]{x1D700}^{\prime })$
converges to 0 as
$\unicode[STIX]{x1D700}^{\prime }$
goes to 0. From above, we have
$$\begin{eqnarray}V_{\unicode[STIX]{x1D700}}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant V_{\unicode[STIX]{x1D700}^{\prime }}d_{p,\unicode[STIX]{x1D700}^{\prime }}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})+\unicode[STIX]{x1D702}(\unicode[STIX]{x1D700},\unicode[STIX]{x1D700}^{\prime }).\end{eqnarray}$$
Hence, the limit exists.
Now, let
$\unicode[STIX]{x1D714}_{X},\widetilde{\unicode[STIX]{x1D714}}_{X}$
be two Kähler metrics on
$X$
, such that
$$\begin{eqnarray}\unicode[STIX]{x1D714}_{X}\leqslant \widetilde{\unicode[STIX]{x1D714}}_{X}\leqslant C\unicode[STIX]{x1D714}_{X}\end{eqnarray}$$
for some
$C>0$
. Assume first
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
. Set
$\widetilde{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\widetilde{\unicode[STIX]{x1D714}}_{X}$
and observe that
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}\leqslant \widetilde{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D700}}\leqslant \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}^{\prime }}$
, where
$\unicode[STIX]{x1D700}^{\prime }=\unicode[STIX]{x1D700}C$
. Let
$\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}},\tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D700}}$
be the geodesic with respect to
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
and
$\widetilde{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D700}}$
, respectively, and observe that
$\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}\leqslant \tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D700}}\leqslant \unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}^{\prime }}$
. The same arguments as before give
$$\begin{eqnarray}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}\leqslant |\dot{\tilde{\unicode[STIX]{x1D711}}}_{0}^{\unicode[STIX]{x1D700}}|^{p}\leqslant |\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}^{\prime }}|^{p}\,,\end{eqnarray}$$
hence
$$\begin{eqnarray}\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}\leqslant \int _{X}|\dot{\tilde{\unicode[STIX]{x1D711}}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\tilde{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}\leqslant \int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}^{\prime }}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}^{\prime }}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}.\end{eqnarray}$$
The latter tells us that the limit does not depend on
$\unicode[STIX]{x1D714}_{X}$
. To get rid of the assumption
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
, one can use the Pythagorean formula, as before.◻
An adaptation of the classical Perron envelope technique yields the following result of Berndtsson [Reference BerndtssonBer15].
Proposition 1.17. Assume that
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
are bounded
$\unicode[STIX]{x1D714}$
-psh functions. Then
$$\begin{eqnarray}\unicode[STIX]{x1D711}(x,z):=\sup \Bigl\{u(x,z)\mid u\in \text{PSH}(X\times S,\unicode[STIX]{x1D714})~\text{with}~\lim _{t\rightarrow 0,1}u\leqslant \unicode[STIX]{x1D711}_{0,1}\Bigr\}\end{eqnarray}$$
is the unique bounded
$\unicode[STIX]{x1D714}$
-psh function on
$X\times S$
, which is the solution of the Dirichlet problem
$\unicode[STIX]{x1D711}_{|X\times \unicode[STIX]{x2202}S}=\unicode[STIX]{x1D711}_{0,1}$
, with
$$\begin{eqnarray}(\unicode[STIX]{x1D714}+dd_{x,z}^{c}\unicode[STIX]{x1D711})^{n+1}=0\quad \text{in}~X\times S.\end{eqnarray}$$
Moreover
$\unicode[STIX]{x1D711}(x,z)=\unicode[STIX]{x1D711}(x,t)$
only depends on
$\Re (z)$
and
$|\dot{\unicode[STIX]{x1D711}}|\leqslant \Vert \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\Vert _{L^{\infty }(X)}$
.
The proof goes exactly as that of Proposition 1.4. The function
$\unicode[STIX]{x1D711}$
(or rather the path
$\unicode[STIX]{x1D711}_{t}\subset \text{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)$
) is called a bounded geodesic in [Reference BerndtssonBer15]. We use the same terminology here, as it turns out that bounded geodesics are geodesics in the metric sense.
Proposition 1.18. Bounded geodesics are metric geodesics. More precisely, if
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
are bounded
$\unicode[STIX]{x1D714}$
-psh functions and
$\unicode[STIX]{x1D711}(x,z)=\unicode[STIX]{x1D711}_{t}(x)$
is the bounded geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
, then for all
$t,s\in [0,1]$
,
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{t},\unicode[STIX]{x1D711}_{s})=|t-s|\,d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. Let
$\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{k}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
be sequences decreasing, respectively, to
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
. It follows from the comparison principle and the uniqueness in Proposition 1.17 that
$\unicode[STIX]{x1D711}_{t,j}$
decreases to
$\unicode[STIX]{x1D711}_{t}$
as
$j$
increases to
$+\infty$
. From Definition 1.15, Proposition 1.16 and the fact that the identity in the statement holds in the Kähler setting for
$d_{\unicode[STIX]{x1D700}}$
we obtain
$$\begin{eqnarray}\displaystyle d_{p}(\unicode[STIX]{x1D711}_{t},\unicode[STIX]{x1D711}_{s}) & = & \displaystyle \liminf _{\unicode[STIX]{x1D700}\rightarrow 0}\liminf _{j,k\rightarrow +\infty }d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{t,j},\unicode[STIX]{x1D711}_{s,k})\nonumber\\ \displaystyle & = & \displaystyle |t-s|\liminf _{\unicode[STIX]{x1D700}\rightarrow 0}\liminf _{j,k\rightarrow +\infty }d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{k})=|t-s|d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\Box \nonumber\end{eqnarray}$$
Remark 1.19. One can no longer expect that
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{t}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})$
for a.e.
$t\in [0,1]$
, as simple examples show. One can, e.g., take
$\unicode[STIX]{x1D711}_{0}\equiv 0$
and
$\unicode[STIX]{x1D711}_{1}=\max (u,0)$
, where
$u$
takes positive values, has isolated singularities, and solves
$\operatorname{MA}(u)=\text{Dirac mass}$
at some point: in this case
$\operatorname{MA}(\unicode[STIX]{x1D711}_{1})$
is concentrated on the contact set
$(u=0)$
while
$\dot{\unicode[STIX]{x1D711}}_{1}\equiv 0$
on this set, hence
$\int _{X}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1})=0$
. We thank Darvas for pointing this out to us.
As this remark points out, we do not have that
$d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=I(0)=I(1)$
when
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
are just bounded
$\unicode[STIX]{x1D714}$
-psh functions. Nevertheless, we can still recover the formula in some special cases.
We start by recalling the following.
Theorem 1.20. Let
$f$
be a continuous function, such that
$dd^{c}f\leqslant C\unicode[STIX]{x1D714}_{X}$
on
$X$
, for some
$C>0$
. Then
$P(f)$
has bounded Laplacian on
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})$
and
$$\begin{eqnarray}(\unicode[STIX]{x1D714}+dd^{c}P_{\unicode[STIX]{x1D714}}(f))^{n}=\unicode[STIX]{x1D7D9}_{\{P_{\unicode[STIX]{x1D714}}(f)=f\}}(\unicode[STIX]{x1D714}+dd^{c}f)^{n}.\end{eqnarray}$$
The fact that
$P(f)$
has a locally bounded Laplacian in
$\operatorname{Amp}(\{\unicode[STIX]{x1D714}\})$
is essentially [Reference BermanBer13, Theorem 1.2]. We do not assume here that
$f$
is smooth but one can check that the upper bound on
$dd^{c}f$
is the only estimate needed to pursue Berman’s approach. One can then argue as in [Reference Guedj and ZeriahiGZ17, Theorem 9.25] to get (4).
Set
$$\begin{eqnarray}{\mathcal{H}}_{bd}:=\{\unicode[STIX]{x1D711}\in \operatorname{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X),\unicode[STIX]{x1D711}=P_{\unicode[STIX]{x1D714}}(f)~\text{for some}~f\in C^{0}(X)~\text{with}~dd^{c}f\leqslant C\unicode[STIX]{x1D714}_{X},C>0\}.\end{eqnarray}$$
Theorem 1.21. Assume that
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
. Let
$\unicode[STIX]{x1D711}_{t}$
be the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
. Then
$$\begin{eqnarray}d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. Set
$\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}}:=P_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(f_{0})$
and
$\unicode[STIX]{x1D711}_{1,\unicode[STIX]{x1D700}}:=P_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(f_{1})$
. Clearly,
$\unicode[STIX]{x1D711}_{i,\unicode[STIX]{x1D700}}$
decreases pointwise to
$\unicode[STIX]{x1D711}_{i}$
,
$i=1,2$
. Let
$\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}$
be the
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-geodesic joining
$\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}}$
and
$\unicode[STIX]{x1D711}_{1,\unicode[STIX]{x1D700}}$
. Combining [Reference DarvasDar15, Theorem 3.5] with (4), we get
$$\begin{eqnarray}V_{\unicode[STIX]{x1D700}}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}},\unicode[STIX]{x1D711}_{1,\unicode[STIX]{x1D700}})=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}})^{n}=\int _{\{\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}}=f_{0}\}}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}f_{0})^{n}.\end{eqnarray}$$
Set
$D_{\unicode[STIX]{x1D700}}:=\{\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}}=f_{0}\}$
,
$D_{0}:=\{\unicode[STIX]{x1D711}_{0}=f_{0}\}$
, and observe that
$D_{0}\subseteq D_{\unicode[STIX]{x1D700}}$
. Since
$\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}}=P_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(f)$
and
$\unicode[STIX]{x1D711}_{0}=P_{\unicode[STIX]{x1D714}}(f)$
, Theorem 1.20 ensures that
$(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}})^{n}=g_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D714}_{X}^{n}$
and
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}=g_{0}\unicode[STIX]{x1D714}_{X}^{n}$
where
$g_{\unicode[STIX]{x1D700}},g_{0}$
are defined as
$$\begin{eqnarray}g_{\unicode[STIX]{x1D700}}:=\left\{\begin{array}{@{}ll@{}}0,\quad & x\notin D_{\unicode[STIX]{x1D700}},\\ \displaystyle \frac{(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}f_{0})^{n}}{\unicode[STIX]{x1D714}_{X}^{n}},\quad & x\in D_{\unicode[STIX]{x1D700}},\end{array}\right.\quad g_{0}:=\left\{\begin{array}{@{}ll@{}}0,\quad & x\notin D_{0},\\ \displaystyle \frac{(\unicode[STIX]{x1D714}+dd^{c}f_{0})^{n}}{\unicode[STIX]{x1D714}_{X}^{n}},\quad & x\in D_{0}.\end{array}\right.\end{eqnarray}$$
We claim that
$g_{\unicode[STIX]{x1D700}}$
converges pointwise to
$g_{0}$
. Indeed, when
$x\in D_{0}\subseteq D_{\unicode[STIX]{x1D700}}$
, then
$g_{\unicode[STIX]{x1D700}}(x)=((\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}f_{0})^{n}/\unicode[STIX]{x1D714}_{X}^{n})(x)$
converges to
$((\unicode[STIX]{x1D714}+dd^{c}f_{0})^{n}/\unicode[STIX]{x1D714}_{X}^{n})(x)=g_{0}(x)$
as
$\unicode[STIX]{x1D700}$
goes to 0. In the case when
$x\notin D_{0}$
, i.e.
$\unicode[STIX]{x1D711}_{0}(x)<f_{0}(x)$
, since
$\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700}}(x)$
decreases to
$\unicode[STIX]{x1D711}_{0}(x)$
as
$\unicode[STIX]{x1D700}$
goes to zero, we can infer that, for
$\unicode[STIX]{x1D700}$
sufficiently small, we still have
$\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700}}(x)<f_{0}(x)$
, which means
$x\notin D_{\unicode[STIX]{x1D700}}$
. Hence,
$g_{\unicode[STIX]{x1D700}}(x)=0=g_{0}(x)$
. The claim is then proved.
Since
$\unicode[STIX]{x1D7D9}_{D_{\unicode[STIX]{x1D700}}}\unicode[STIX]{x1D711}_{0}^{\unicode[STIX]{x1D700}}=f_{0}=\unicode[STIX]{x1D7D9}_{D_{0}}\unicode[STIX]{x1D711}_{0}$
, the same arguments in Theorem 1.13 show that
$|\unicode[STIX]{x1D7D9}_{D_{\unicode[STIX]{x1D700}}}\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}-\unicode[STIX]{x1D7D9}_{D_{0}}\dot{\unicode[STIX]{x1D711}}_{0}|$
converges pointwise to 0 as
$\unicode[STIX]{x1D700}$
goes to zero.
We thus infer that
$\unicode[STIX]{x1D7D9}_{D_{\unicode[STIX]{x1D700}}}|\dot{\unicode[STIX]{x1D711}_{0}^{\unicode[STIX]{x1D700}}}|^{p}g_{\unicode[STIX]{x1D700}}$
converges pointwise to
$\unicode[STIX]{x1D7D9}_{D_{0}}|\dot{\unicode[STIX]{x1D711}_{0}}|^{p}g_{0}$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
. The dominated convergence theorem yields
$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D700}}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=\lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{X}\unicode[STIX]{x1D7D9}_{D_{\unicode[STIX]{x1D700}}}|\dot{\unicode[STIX]{x1D711}}_{0}^{\unicode[STIX]{x1D700}}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{0,\unicode[STIX]{x1D700}})^{n}=\int _{X}\unicode[STIX]{x1D7D9}_{D_{0}}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n},\end{eqnarray}$$
hence the conclusion. ◻
Observe that if
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, then
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
. Indeed, since
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
are smooth, the functions
$-\unicode[STIX]{x1D711}_{0},-\unicode[STIX]{x1D711}_{1}$
are quasi-plurisubharmonic, i.e. there exists
$C>0$
such that
$dd^{c}(-\unicode[STIX]{x1D711}_{i})\geqslant -C\unicode[STIX]{x1D714}_{X}$
for any
$i=1,2$
. Thus,
$\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=-\text{max}(-\unicode[STIX]{x1D711}_{0},-\unicode[STIX]{x1D711}_{1})$
is such that
$$\begin{eqnarray}dd^{c}\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=-dd^{c}\max (-\unicode[STIX]{x1D711}_{0},-\unicode[STIX]{x1D711}_{1})\leqslant C\unicode[STIX]{x1D714}_{X}.\end{eqnarray}$$
In particular, (5) holds for
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})$
and
$d_{p}(\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})$
.
2 Finite-energy classes
We define in this section the set
${\mathcal{E}}(\unicode[STIX]{x1D6FC})$
(respectively
${\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC})$
) of positive closed currents
$T=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}$
with full Monge–Ampère mass (respectively finite weighted energy) in
$\unicode[STIX]{x1D6FC}$
, by defining the corresponding class
${\mathcal{E}}(X,\unicode[STIX]{x1D714})$
(respectively
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
) of finite-energy potentials
$\unicode[STIX]{x1D711}$
.
2.1 The space
${\mathcal{E}}(\unicode[STIX]{x1D6FC})$
2.1.1 Quasi-plurisubharmonic functions
Recall that a function is quasi-plurisubharmonic if it is locally given as the sum of a smooth and a psh function. In particular quasi-psh (qpsh for short) functions are upper semi-continuous and integrable.
Definition 2.1. We let
$\text{PSH}(X,\unicode[STIX]{x1D714})$
denote the set of all
$\unicode[STIX]{x1D714}$
-plurisubharmonic functions. These are quasi-psh functions
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty \}$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}\geqslant 0\end{eqnarray}$$
in the weak sense of currents.
The set
$\text{PSH}(X,\unicode[STIX]{x1D714})$
is a closed subset of
$L^{1}(X)$
, for the
$L^{1}$
-topology.
2.1.2 The class
${\mathcal{E}}(X,\unicode[STIX]{x1D714})$
Given
$\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714})$
, we consider
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}:=\max (\unicode[STIX]{x1D711},-j)\in \text{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X).\end{eqnarray}$$
It follows from the Bedford–Taylor theory [Reference Bedford and TaylorBT82] that the
$\operatorname{MA}(\unicode[STIX]{x1D711}_{j})$
are well-defined probability measures. Moreover, the sequence
$\unicode[STIX]{x1D707}_{j}:=\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}>-j\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{j})$
is increasing [Reference Guedj and ZeriahiGZ07, p. 445]. Since the
$\unicode[STIX]{x1D707}_{j}$
all have total mass bounded from above by
$1$
, we consider
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D711}}:=\lim _{j\rightarrow +\infty }\unicode[STIX]{x1D707}_{j},\end{eqnarray}$$
which is a positive Borel measure on
$X$
, with total mass
${\leqslant}1$
.
Definition 2.2. We set
$$\begin{eqnarray}{\mathcal{E}}(X,\unicode[STIX]{x1D714}):=\{\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714})\mid \unicode[STIX]{x1D707}_{\unicode[STIX]{x1D711}}(X)=1\}.\end{eqnarray}$$
For
$\unicode[STIX]{x1D711}\in {\mathcal{E}}(X,\unicode[STIX]{x1D714})$
, we set
$\operatorname{MA}(\unicode[STIX]{x1D711}):=\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D711}}$
.
The latter can be characterized as the largest class for which the complex Monge–Ampère mass is well defined and the maximum principle holds [Reference Guedj and ZeriahiGZ07, Theorem 1.5]. We further note that the domination principle holds ([Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Corollary 2.5], [Reference Darvas, Di Nezza and LuDDL18, Proposition 2.4]).
Proposition 2.3. If
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D714})$
are such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(x)\leqslant \unicode[STIX]{x1D713}(x)\quad \text{for}~\text{MA}(\unicode[STIX]{x1D713})\text{-}\text{a.e.}~x,\end{eqnarray}$$
then
$\unicode[STIX]{x1D711}(x)\leqslant \unicode[STIX]{x1D713}(x)$
for all
$x\in X$
.
It follows from the
$\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$
-lemma that any positive closed current
$T\in \unicode[STIX]{x1D6FC}$
can be written
$T=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}$
for some function
$\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714})$
that is unique up to an additive constant.
Definition 2.4. We let
${\mathcal{E}}(\unicode[STIX]{x1D6FC})$
denote the set of all positive currents in
$\unicode[STIX]{x1D6FC}$
,
$T=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}$
, with
$\unicode[STIX]{x1D711}\in {\mathcal{E}}(X,\unicode[STIX]{x1D714})$
.
Note that this definition does not depend on the choice of
$\unicode[STIX]{x1D714}$
, nor does it depend on the choice of
$\unicode[STIX]{x1D711}$
.
2.2 The class
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
2.2.1 The Aubin–Mabuchi functional
Each tangent space
$T_{\unicode[STIX]{x1D711}}{\mathcal{H}}_{\unicode[STIX]{x1D714}}$
admits the following orthogonal decomposition
$$\begin{eqnarray}T_{\unicode[STIX]{x1D711}}{\mathcal{H}}_{\unicode[STIX]{x1D714}}=\{\unicode[STIX]{x1D713}\in C^{\infty }(X);\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D713})=0\}\oplus \mathbb{R},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FD}=\operatorname{MA}$
is the 1-form defined on
${\mathcal{H}}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D713})=\int _{X}\unicode[STIX]{x1D713}\operatorname{MA}(\unicode[STIX]{x1D711}).\end{eqnarray}$$
It is a classical observation of Mabuchi that the 1-form
$\unicode[STIX]{x1D6FD}$
is closed. Therefore, there exists a unique function
$E$
defined on the convex open set
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
, such that
$\unicode[STIX]{x1D6FD}=dE$
and
$E(0)=0$
. It is often called the Aubin–Mabuchi functional and can be expressed (after integration along affine paths) by
$$\begin{eqnarray}E(\unicode[STIX]{x1D711})=\frac{1}{(n+1)V_{\unicode[STIX]{x1D6FC}}}\mathop{\sum }_{j=0}^{n}\int _{X}\unicode[STIX]{x1D711}\,(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{j}\wedge \unicode[STIX]{x1D714}^{n-j}.\end{eqnarray}$$
Lemma 2.5. The Aubin–Mabuchi functional
$E$
is concave along Euclidean segments, non-decreasing, and satisfies the cocycle condition
$$\begin{eqnarray}E(\unicode[STIX]{x1D711})-E(\unicode[STIX]{x1D713})=\frac{1}{(n+1)V_{\unicode[STIX]{x1D6FC}}}\mathop{\sum }_{j=0}^{n}\int _{X}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\,(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{j}\wedge (\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713})^{n-j}\,.\end{eqnarray}$$
It is affine along geodesics and convex along subgeodesics in
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
.
Proof. These properties are well known when
$\unicode[STIX]{x1D714}$
is in a Kähler class.
The monotonicity property follows from the definition since the first derivative of
$E$
is
$dE=\unicode[STIX]{x1D6FD}=\operatorname{MA}\geqslant 0$
, a probability measure: if
$\unicode[STIX]{x1D711}_{t}$
is an arbitrary path, then
$$\begin{eqnarray}\frac{d}{dt}E(\unicode[STIX]{x1D711}_{t})=\int _{X}\dot{\unicode[STIX]{x1D711}}_{t}\operatorname{MA}(\unicode[STIX]{x1D711}_{t}).\end{eqnarray}$$
It follows from Stokes theorem that
$$\begin{eqnarray}\displaystyle \frac{d^{2}}{dt^{2}}E(\unicode[STIX]{x1D711}_{t}) & = & \displaystyle \int _{X}\ddot{\unicode[STIX]{x1D711}_{t}}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})+\frac{n}{V_{\unicode[STIX]{x1D6FC}}}\int _{X}\dot{\unicode[STIX]{x1D711}_{t}}\,dd^{c}\dot{\unicode[STIX]{x1D711}_{t}}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n-1}\nonumber\\ \displaystyle & = & \displaystyle \int _{X}\biggl\{\ddot{\unicode[STIX]{x1D711}_{t}}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})-\frac{n}{V_{\unicode[STIX]{x1D6FC}}}\,d\dot{\unicode[STIX]{x1D711}_{t}}\wedge d^{c}\dot{\unicode[STIX]{x1D711}_{t}}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}_{t}}^{n-1}\biggr\}.\nonumber\end{eqnarray}$$
Thus,
$E$
is concave along Euclidean segments (
$\ddot{\unicode[STIX]{x1D711}}_{t}=0$
), affine along Mabuchi geodesics, and convex along Mabuchi subgeodesics. The cocycle condition follows by differentiating
$E(t\unicode[STIX]{x1D711}+(1-t)\unicode[STIX]{x1D713})$
.
These computations are merely heuristic as
$t\rightarrow \unicode[STIX]{x1D711}_{t}(x)$
is poorly regular when
$\unicode[STIX]{x1D711}_{t}$
is a geodesic or subgeodesic. We can, however, approximate
$\unicode[STIX]{x1D714}$
by
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
; consider
$(\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}})$
the corresponding geodesic
$$\begin{eqnarray}E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}})=\frac{1}{(n+1)V_{\unicode[STIX]{x1D700}}}\mathop{\sum }_{j=0}^{n}\int _{X}\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}})\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}^{n-j}.\end{eqnarray}$$
It follows from Proposition 1.6 that
$\unicode[STIX]{x1D700}\mapsto \unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}}$
decreases to
$\unicode[STIX]{x1D711}_{t}$
; hence,
$t\mapsto E(\unicode[STIX]{x1D711}_{t})$
is affine, where the limit of the affine maps
$t\mapsto E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}_{t}^{\unicode[STIX]{x1D700}})$
.
For subgeodesics, we again approximate
$\unicode[STIX]{x1D714}$
by
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
and we proceed as in the Kähler case.◻
Observe that
$E(\unicode[STIX]{x1D711}+t)=E(\unicode[STIX]{x1D711})+t$
. Given
$\unicode[STIX]{x1D711}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
there exists a unique
$c\in \mathbb{R}$
such that
$E(\unicode[STIX]{x1D711}+c)=0$
. The restriction of the Mabuchi metric to the fiber
$E^{-1}(0)$
induces a Riemannian structure on the quotient space
${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}={\mathcal{H}}_{\unicode[STIX]{x1D714}}/\mathbb{R}$
and allows decomposition of
${\mathcal{H}}_{\unicode[STIX]{x1D714}}={\mathcal{H}}_{\unicode[STIX]{x1D6FC}}\times \mathbb{R}$
as a product of Riemannian manifolds.
Definition 2.6. For
$\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714})$
, we set
$$\begin{eqnarray}E(\unicode[STIX]{x1D711}):=\inf \{E(\unicode[STIX]{x1D713});\unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x1D713}~\text{and}~\unicode[STIX]{x1D713}\in \text{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)\}\in [-\infty ,+\infty \
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714}):=\{\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714});E(\unicode[STIX]{x1D711})>-\infty \}$
.
2.2.2 Strong topology on
${\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC})$
Set
$$\begin{eqnarray}I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=\int _{X}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})(\operatorname{MA}(\unicode[STIX]{x1D713})-\operatorname{MA}(\unicode[STIX]{x1D711})).\end{eqnarray}$$
It has been shown in [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ] that
$I$
defines a complete metrizable uniform structure on
${\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC})$
. More precisely, we identify
${\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC})$
with the set
$$\begin{eqnarray}{\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})=\Big\{\unicode[STIX]{x1D711}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})\mid \sup _{X}\unicode[STIX]{x1D711}=0\Big\}\end{eqnarray}$$
of normalized potentials. Then:
(a)
$I$
is symmetric and positive on
${\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})^{2}\setminus \{\text{diagonal}\}$
;(b)
$I$
satisfies a quasi-triangle inequality [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorem 1.8];(c)
$I$
induces a uniform structure that is metrizable [Reference BourbakiBou07];(d) the metric space
$({\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC}),d_{I})$
is complete [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Proposition 2.4], where
$d_{I}$
denotes one of the distances induced by the uniform structure
$I$
.
Definition 2.7. The strong topology on
${\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC})$
is the metrizable topology defined by
$I$
.
The corresponding notion of convergence is the convergence in energy previously introduced in [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13] (see [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Proposition 2.3]). It is the coarsest refinement of the weak topology such that
$E$
becomes continuous. In particular, if
$T_{j}\longrightarrow T$
in
$({\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC}),d_{I})$
, then
$$\begin{eqnarray}T_{j}\longrightarrow T\quad \text{weakly and}\quad T_{j}^{n}\longrightarrow T^{n}\end{eqnarray}$$
in the weak sense of Radon measures, while the Monge–Ampère operator is usually discontinuous for the weak topology of currents.
2.2.3 Yet another distance
To fit in with the notation of the next section, we introduce yet another notion of convergence in
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
. We set
$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}):=\int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|\biggl[\frac{\operatorname{MA}(\unicode[STIX]{x1D711})+\operatorname{MA}(\unicode[STIX]{x1D713})}{2}\biggr].\end{eqnarray}$$
This symmetric quantity is non-negative. It follows from Proposition 2.3 that it only vanishes on the diagonal of
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})^{2}$
, while Theorem 3.6 will insure that it satisfies a quasi-triangle inequality. Hence,
$I_{1}$
induces a uniform structure, which is metrizable [Reference BourbakiBou07].
For
$C>0$
, we set
$$\begin{eqnarray}{\mathcal{E}}_{C}^{1}(X,\unicode[STIX]{x1D714}):=\{\unicode[STIX]{x1D711}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714});E(\unicode[STIX]{x1D711})\geqslant -C~\text{and}~\unicode[STIX]{x1D711}\leqslant C\}.\end{eqnarray}$$
It follows from Hartogs’ lemma, the upper semi-continuity, and the concavity of
$E$
along Euclidean segments (Lemma 2.5) that this set is a compact and convex subset of
$\text{PSH}(X,\unicode[STIX]{x1D714})$
, when endowed with the
$L^{1}$
-topology (see [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Lemma 2.6]).
Proposition 2.8. For all
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
,
$I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\leqslant 2I_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})$
. Conversely, for each
$C>0$
, there exists
$A>0$
such that, for all
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}_{C}^{1}(X,\unicode[STIX]{x1D714})$
,
$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\leqslant \int _{X}[2\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})-(\unicode[STIX]{x1D711}+\unicode[STIX]{x1D713})]\operatorname{MA}(0)+A\,I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})^{1/2^{n}}.\end{eqnarray}$$
In particular, the topologies induced by
$I,I_{1}$
on
${\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})$
are the same.
Observe that
$I_{1}$
induces a distance on
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
, but
$I$
is merely defined on
${\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})$
, as
$I(\unicode[STIX]{x1D711}+c,\unicode[STIX]{x1D713}+c^{\prime })=I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})$
, for any
$c,c^{\prime }\in \mathbb{R}$
.
Proof. The first inequality is obvious, as
$$\begin{eqnarray}I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=\int _{X}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})(\operatorname{MA}(\unicode[STIX]{x1D713})-\operatorname{MA}(\unicode[STIX]{x1D711}))\leqslant \int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|(\operatorname{MA}(\unicode[STIX]{x1D713})+\operatorname{MA}(\unicode[STIX]{x1D711})).\end{eqnarray}$$
It follows from Proposition 2.13 that
$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=I_{1}(\unicode[STIX]{x1D711},\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))+I_{1}(\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}),\unicode[STIX]{x1D713}),\end{eqnarray}$$
hence it suffices to establish the second inequality when
$\unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x1D713}$
. In this case
$$\begin{eqnarray}I_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\leqslant \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})\operatorname{MA}(\unicode[STIX]{x1D711}),\end{eqnarray}$$
by Lemma 2.12, while the Cauchy–Schwarz inequality yields
$$\begin{eqnarray}\displaystyle \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})\operatorname{MA}(\unicode[STIX]{x1D711}) & = & \displaystyle \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})\operatorname{MA}(0)+\int _{X}d(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge d^{c}\unicode[STIX]{x1D711}\wedge S_{\unicode[STIX]{x1D711}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})\operatorname{MA}(0)+I(\unicode[STIX]{x1D711},0)^{1/2}\biggl(\int _{X}d(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge d^{c}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge S_{\unicode[STIX]{x1D711}}\biggr)^{1/2},\nonumber\end{eqnarray}$$
where we have set
$S_{\unicode[STIX]{x1D711}}:=\sum _{j=0}^{n-1}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{j}\wedge \unicode[STIX]{x1D714}^{n-1-j}$
. Observing that
$S_{\unicode[STIX]{x1D711}}\leqslant 2^{n-1}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}/2}^{n-1}$
, we can invoke [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Lemma 1.9] to obtain
$$\begin{eqnarray}\int _{X}d(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge d^{c}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge S_{\unicode[STIX]{x1D711}}\leqslant c_{n}I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})^{1/2^{n-1}}\biggl\{I\biggl(\unicode[STIX]{x1D711},\frac{\unicode[STIX]{x1D711}}{2}\biggr)^{1-1/2^{n-1}}+I\biggl(\unicode[STIX]{x1D713},\frac{\unicode[STIX]{x1D711}}{2}\biggr)^{1-1/2^{n-1}}\biggr\}.\end{eqnarray}$$
Now
$I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D711}/2)\leqslant a_{n}I(\unicode[STIX]{x1D711},0)\leqslant C^{\prime }$
and [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorem 1.3] yields
$$\begin{eqnarray}I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}/2)\leqslant b_{n}\{I(\unicode[STIX]{x1D713},0)+I(\unicode[STIX]{x1D711}/2,0)\}\leqslant b_{n}^{\prime }\{I(\unicode[STIX]{x1D713},0)+I(\unicode[STIX]{x1D711},0)\}\leqslant C^{\prime \prime }.\end{eqnarray}$$
We thus get (7).
To prove the last statement, we need to show that, given a sequence
$\unicode[STIX]{x1D711}_{j}\in {\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})$
converging to
$\unicode[STIX]{x1D713}$
with respect to
$I$
, then it also converges to
$\unicode[STIX]{x1D713}$
with respect to
$I_{1}$
, and vice versa. We first note that the
$I$
-convergence implies the
$L^{1}$
-convergence of the potentials [Reference Guedj and ZeriahiGZ17, Theorem 10.37]. This ensures that
$$\begin{eqnarray}\int _{X}[2\max (\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713})-(\unicode[STIX]{x1D711}_{j}+\unicode[STIX]{x1D713})]\operatorname{MA}(0)\rightarrow 0\quad \text{as}~j\rightarrow +\infty ,\end{eqnarray}$$
and moreover we have that
$\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}\in {\mathcal{E}}_{C}^{1}(X,\unicode[STIX]{x1D714})$
for some
$C>0$
[Reference Guedj and ZeriahiGZ17, Lemma 10.33 and Definition 10.34]. The
$I_{1}$
-convergence would then follow from (7). Moreover, since
$I(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713})\leqslant 2I_{1}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713})$
, we conclude that the
$I_{1}$
-convergence implies the
$I$
-convergence.◻
2.3 The complete metric spaces
${\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC})$
Fix
$p\geqslant 1$
. Following [Reference Guedj and ZeriahiGZ07, Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10], we consider the following finite-energy classes.
Definition 2.9. We set
$$\begin{eqnarray}{\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}):=\{\unicode[STIX]{x1D711}\in {\mathcal{E}}(X,\unicode[STIX]{x1D714})/|\unicode[STIX]{x1D711}|^{p}\in L^{1}(\operatorname{MA}(\unicode[STIX]{x1D711}))\}\end{eqnarray}$$
and let
${\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC})=\{T=\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}\mid \unicode[STIX]{x1D711}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})\}$
denote the corresponding sets of finite-energy currents.
On the class
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
,
$p\geqslant 1$
, we define
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}):=\biggl(\int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}\biggl[\frac{\operatorname{MA}(\unicode[STIX]{x1D711})+\operatorname{MA}(\unicode[STIX]{x1D713})}{2}\biggr]\biggr)^{1/p}\,.\end{eqnarray}$$
This quantity is well defined by [Reference Guedj and ZeriahiGZ07, Proposition 3.6]. It is obviously non-negative and symmetric. It follows from the domination principle (Proposition 2.3) that
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=0\Longrightarrow \unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}.\end{eqnarray}$$
Moreover, it will follow from Theorem 3.6 (which shows in particular that
$I_{p}$
satisfies a quasi-triangle inequality) that
$I_{p}$
induces a uniform structure. We can then define the following:
Definition 2.10. The strong topology on
${\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC})$
is the one induced by
$I_{p}$
.
By [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 2.17], a decreasing sequence converges strongly. We also have good convergence properties if we approximate by slightly larger finite-energy classes
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
.
Proposition 2.11. Fix
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
,
$\unicode[STIX]{x1D700}>0$
. If
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)$
, then
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})\cap L^{\infty }(X)$
and
$I_{p,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\rightarrow I_{p,\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
.
Moreover, if
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
and
$\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j}$
are sequences of smooth
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}$
-psh functions decreasing to
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
with
$\unicode[STIX]{x1D700}_{j}\rightarrow 0$
, then
$$\begin{eqnarray}I_{p,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j})\rightarrow I_{p,\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\end{eqnarray}$$
as
$j$
goes to
$+\infty$
.
Proof. Note that
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
belong to any energy class with respect to any Kähler form since they are bounded. In particular,
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}})$
. The first assertion follows from the fact that
$(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711})^{n}$
and
$(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D713})^{n}$
converge weakly to
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}$
and
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D713})^{n}$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
, respectively. For the second statement, we observe that, by symmetry, it suffices to prove that
$$\begin{eqnarray}\int _{X}|\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D713}_{j}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\rightarrow \int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}\quad \text{as}~j\rightarrow +\infty .\end{eqnarray}$$
Given a bounded function
$f$
on
$X$
, we set
$$\begin{eqnarray}|f|_{p}:=\biggl(\int _{X}|f|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\biggr)^{1/p}.\end{eqnarray}$$
The triangle inequality yields
$$\begin{eqnarray}|\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D713}_{j}|_{p}\leqslant |\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|_{p}+|(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})|+|(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{j})|_{p}\end{eqnarray}$$
and similarly
$$\begin{eqnarray}|\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D713}_{j}|_{p}\geqslant |\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|_{p}-|(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})|-|(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{j})|_{p}.\end{eqnarray}$$
Since
$\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}$
is a positive quasi-continuous uniformly bounded function on
$X$
, it follows from [Reference Guedj and ZeriahiGZ17, Theorem 4.26] that
$$\begin{eqnarray}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|_{p}^{p}=\int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\rightarrow \int _{X}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}\end{eqnarray}$$
as
$j\rightarrow +\infty$
. Moreover, we claim that the terms
$|(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})|_{p}$
and
$|(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{j})|_{p}$
go to 0 as
$j\rightarrow +\infty$
. Lemma 2.12, together with the fact that
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}\leqslant \unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}$
, yields
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\leqslant \int _{X}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})^{p}(\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}\unicode[STIX]{x1D711})^{n}.\end{eqnarray}$$
Note that
$\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D711}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X})$
(since they are bounded). Hence [Reference Guedj and ZeriahiGZ07, Theorem 3.8] ensures that the integral at the right-hand side of this inequality is finite.
Since
$\unicode[STIX]{x1D711}_{j}$
is decreasing to
$\unicode[STIX]{x1D711}$
, it then follows from the dominated convergence theorem that
$|(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})|_{p}^{p}\rightarrow 0$
as
$j\rightarrow +\infty$
. Fix
$j_{0}<j$
. Then
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D713}_{j}-\unicode[STIX]{x1D713})^{p}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}_{j}}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\leqslant \int _{X}(\unicode[STIX]{x1D713}_{j_{0}}-\unicode[STIX]{x1D713})^{p}(\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}.\end{eqnarray}$$
It follows again from the continuity of the Monge–Ampère operator along the decreasing sequence, [Reference KolodziejKol05, Corollary 1.14], and the dominated convergence theorem that letting
$j\rightarrow +\infty$
and then
$j_{0}\rightarrow +\infty$
we get
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D713}_{j_{0}}-\unicode[STIX]{x1D713})^{p}(\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}\unicode[STIX]{x1D711}_{j})^{n}\rightarrow 0.\end{eqnarray}$$
Thus,
$|(\unicode[STIX]{x1D713}_{j}-\unicode[STIX]{x1D713})|_{p}^{p}\rightarrow 0$
as
$j\rightarrow +\infty$
. Hence the conclusion.◻
It follows from Hölder’s inequality that the strong topology on
${\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC})$
is stronger than the one on
${\mathcal{E}}^{1}(\unicode[STIX]{x1D6FC})$
: if a sequence
$(\unicode[STIX]{x1D711}_{j})\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
is a Cauchy sequence for
$I_{p}$
, then it is a Cauchy sequence in
$({\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714}),d_{I})$
, since
$$\begin{eqnarray}0\leqslant I(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=\int _{X}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})[\operatorname{MA}(\unicode[STIX]{x1D713})-\operatorname{MA}(\unicode[STIX]{x1D711})]\leqslant 2^{1/p}I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}).\end{eqnarray}$$
Since
$({\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714}),d_{I})$
is complete, there is
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
such that
$d_{I}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D711})\rightarrow 0$
. Now
$I_{p}(\unicode[STIX]{x1D711}_{j},0)$
is bounded and
$\operatorname{MA}(\unicode[STIX]{x1D711}_{j})$
converges to
$\operatorname{MA}(\unicode[STIX]{x1D711})$
(by [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Proposition 5.6]). Thus,
$\unicode[STIX]{x1D711}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
by Fatou’s and Hartogs’ lemmas.
One would now like to prove that
$I_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D711})\rightarrow 0$
and conclude that the space
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),I_{p})$
is complete, arguing as in [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Proposition 2.4]. We refer the reader to Theorem 4.2 for a neat treatment.
Lemma 2.12. Let
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
be bounded
$\unicode[STIX]{x1D714}$
-psh functions and
$S$
be a positive closed current of bidimension
$(1,1)$
on
$X$
. If
$\unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x1D713}$
, then
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}\wedge S\leqslant \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}\wedge S.\end{eqnarray}$$
In particular,
$V_{\unicode[STIX]{x1D6FC}}^{-1}\int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}^{j}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n-j}\leqslant \int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\operatorname{MA}(\unicode[STIX]{x1D711})$
.
Proof. By Stokes’ theorem,
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}\wedge S-\int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}\wedge S=p\int _{X}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})^{p-1}d(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge d^{c}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713})\wedge S\end{eqnarray}$$
is non-negative if
$(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D711})\geqslant 0$
.
The second assertion follows by applying the first one inductively. ◻
We now establish a few useful properties of
$I_{p}$
that will notably allow us to compare
$I_{p}$
with
$d_{p}$
in the next section.
Proposition 2.13. For
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
,
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})^{p}=I_{p}(\unicode[STIX]{x1D711},\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))^{p}+I_{p}(\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}),\unicode[STIX]{x1D713})^{p}.\end{eqnarray}$$
Proof. Recall that the maximum principle ensures that
$$\begin{eqnarray}\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}<\unicode[STIX]{x1D713}\}}\operatorname{MA}(\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))=\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}<\unicode[STIX]{x1D713}\}}\operatorname{MA}(\unicode[STIX]{x1D713}),\end{eqnarray}$$
while
$(\unicode[STIX]{x1D711}-\text{max}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))^{p}=0$
on
$(\unicode[STIX]{x1D711}\geqslant \unicode[STIX]{x1D713})$
; thus,
$$\begin{eqnarray}2I_{p}(\unicode[STIX]{x1D711},\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))^{p}=\int _{\{\unicode[STIX]{x1D711}<\unicode[STIX]{x1D713}\}}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}[\operatorname{MA}(\unicode[STIX]{x1D711})+\operatorname{MA}(\unicode[STIX]{x1D713})].\end{eqnarray}$$
Similarly,
$2I_{p}(\unicode[STIX]{x1D713},\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))^{p}=\int _{\{\unicode[STIX]{x1D711}>\unicode[STIX]{x1D713}\}}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}[\operatorname{MA}(\unicode[STIX]{x1D711})+\operatorname{MA}(\unicode[STIX]{x1D713})]$
and the result follows, since
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})^{p}=\frac{1}{2}\int _{\{\unicode[STIX]{x1D711}\neq \unicode[STIX]{x1D713}\}}|\unicode[STIX]{x1D711}-\unicode[STIX]{x1D713}|^{p}[\operatorname{MA}(\unicode[STIX]{x1D711})+\operatorname{MA}(\unicode[STIX]{x1D713})].\square\end{eqnarray}$$
Corollary 2.14. For all
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
,
$$\begin{eqnarray}I_{p}\biggl(\frac{\unicode[STIX]{x1D711}+\unicode[STIX]{x1D713}}{2},\unicode[STIX]{x1D713}\biggr)\leqslant I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}).\end{eqnarray}$$
Proof. By approximating
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
from the above by a decreasing sequence, it suffices to treat the case when
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. Changing
$\unicode[STIX]{x1D714}$
in
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$
, we can further assume that
$\unicode[STIX]{x1D713}=0$
. It follows from Proposition 2.13 that
$$\begin{eqnarray}I_{p}(0,\unicode[STIX]{x1D711}/2)^{p}=I_{p}(0,\max (0,\unicode[STIX]{x1D711}/2))^{p}+I_{p}(\max (0,\unicode[STIX]{x1D711}/2),\unicode[STIX]{x1D711}/2)^{p}.\end{eqnarray}$$
It follows from Lemma 2.12 that
$$\begin{eqnarray}\displaystyle I_{p}(0,\max (0,\unicode[STIX]{x1D711}/2))^{p} & {\leqslant} & \displaystyle \int _{X}\max (0,\unicode[STIX]{x1D711}/2)^{p}\operatorname{MA}(0)\nonumber\\ \displaystyle & = & \displaystyle 2^{-p}\int _{X}\max (0,\unicode[STIX]{x1D711})^{p}\operatorname{MA}(0)\leqslant I_{p}(0,\max (0,\unicode[STIX]{x1D711}))^{p}.\nonumber\end{eqnarray}$$
We claim that, for all
$0\leqslant j\leqslant n$
,
$$\begin{eqnarray}\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{j}\wedge \unicode[STIX]{x1D714}^{n-j}\leqslant \int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n}.\end{eqnarray}$$
Assuming this for the moment, it follows again from Lemma 2.12 that
$$\begin{eqnarray}\displaystyle I_{p}(\max (0,\unicode[STIX]{x1D711}/2),\unicode[STIX]{x1D711}/2)^{p} & {\leqslant} & \displaystyle \int _{X}(\max (0,\unicode[STIX]{x1D711}/2)-\unicode[STIX]{x1D711}/2)^{p}\operatorname{MA}(\unicode[STIX]{x1D711}/2)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2^{n+p}V_{\unicode[STIX]{x1D6FC}}}\mathop{\sum }_{j=0}^{n}C_{n}^{j}\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{j}\wedge \unicode[STIX]{x1D714}^{n-j}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{2}\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\operatorname{MA}(\unicode[STIX]{x1D711})\leqslant I_{p}(\unicode[STIX]{x1D711},\max (0,\unicode[STIX]{x1D711}))^{p}.\nonumber\end{eqnarray}$$
We infer
$$\begin{eqnarray}I_{p}(0,\unicode[STIX]{x1D711}/2)^{p}\leqslant I_{p}(0,\max (0,\unicode[STIX]{x1D711}))^{p}+I_{p}(\max (0,\unicode[STIX]{x1D711}),\unicode[STIX]{x1D711})^{p}=I_{p}(0,\unicode[STIX]{x1D711})^{p},\end{eqnarray}$$
by using Proposition 2.13 again.
It remains to justify our claim. Set
$S=\unicode[STIX]{x1D714}^{j-1}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}^{n-j}$
. It suffices, by induction, to establish the following inequality:
$$\begin{eqnarray}\displaystyle \int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}\wedge S & = & \displaystyle \int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}\wedge S-\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\,dd^{c}\unicode[STIX]{x1D711}\wedge S\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D711}}\wedge S.\nonumber\end{eqnarray}$$
This follows by observing that
$$\begin{eqnarray}\displaystyle -\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p}\,dd^{c}\unicode[STIX]{x1D711}\wedge S & = & \displaystyle p\int _{X}(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})^{p-1}\,d(\max (0,\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711})\wedge d^{c}\unicode[STIX]{x1D711}\wedge S\nonumber\\ \displaystyle & = & \displaystyle -p\int _{\{\unicode[STIX]{x1D711}<0\}}(-\unicode[STIX]{x1D711})^{p-1}\,d\unicode[STIX]{x1D711}\wedge d^{c}\unicode[STIX]{x1D711}\wedge S\leqslant 0.\Box \nonumber\end{eqnarray}$$
3 Comparing distances
In this section, we show that
$I_{p}$
is equivalent to
$d_{p}$
(Theorem 3.6). Recall that
$$\begin{eqnarray}{\mathcal{H}}_{bd}:=\{\unicode[STIX]{x1D711}\in \operatorname{PSH}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X),\unicode[STIX]{x1D711}=P_{\unicode[STIX]{x1D714}}(f)~\text{for some}~f\in C^{0}(X)~\text{with}~dd^{c}f\leqslant C\unicode[STIX]{x1D714}_{X},C>0\}.\end{eqnarray}$$
In the following, we are going to use several times and in a crucial way the fact that Theorem 1.21 ensures
$$\begin{eqnarray}d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\frac{(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0})^{n}}{V}=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}\frac{(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{1})^{n}}{V},\quad \forall \unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}.\end{eqnarray}$$
3.1 Kiselman transform and geodesics
Let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
be the Mabuchi geodesic. For all
$x\in X$
,
$t\in [0,1]\mapsto \unicode[STIX]{x1D711}_{t}(x)\in \mathbb{R}$
is convex. It is natural to consider its Legendre transform,
$u_{s}(x):s\mapsto \sup _{t\in [0,1]}\{st-\unicode[STIX]{x1D711}_{t}(x)\}$
. This function is convex in
$s$
, but the dependence in
$x$
is
$-\unicode[STIX]{x1D714}$
-psh, so we rather consider
$-u_{s}$
. We finally change
$s$
in
$-s$
to obtain a more elegant formula,
$$\begin{eqnarray}\unicode[STIX]{x1D713}_{s}(x):=\inf _{0\leqslant t\leqslant 1}\{st+\unicode[STIX]{x1D711}_{t}(x)\}.\end{eqnarray}$$
Proposition 3.1. The functions
$x\mapsto \unicode[STIX]{x1D713}_{s}(x)$
are
$\unicode[STIX]{x1D714}$
-plurisubharmonic. In particular,
$x\mapsto \unicode[STIX]{x1D713}_{0}(x)=\inf _{0\leqslant t\leqslant 1}\unicode[STIX]{x1D711}_{t}(x)$
is
$\unicode[STIX]{x1D714}$
-psh.
This is the minimum principle of Kiselman [Reference KiselmanKis78]. For
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
, we let
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
denote the greatest
$\unicode[STIX]{x1D714}$
-psh function that lies below
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
. In the notation of Berman and Demailly [Reference Berman and DemaillyBD12]
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=P(\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})),\end{eqnarray}$$
while
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
is denoted
$P(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
in [Reference DarvasDar17c].
An important consequence of Kiselman’s minimum principle [Reference KiselmanKis78] is the following observation of Darvas and Rubinstein [Reference Darvas and RubinsteinDR16].
Proposition 3.2. The function
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
is a bounded
$\unicode[STIX]{x1D714}$
-psh, which has a locally bounded Laplacian on the ample locus of
$\unicode[STIX]{x1D6FC}=\{\unicode[STIX]{x1D714}\}$
, and its Monge–Ampère measure
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})$
is supported on the coincidence set
$$\begin{eqnarray}\{x\in X\mid \unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}(x)=\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})(x)\}.\end{eqnarray}$$
Moreover,
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})=\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{0}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{1}<\unicode[STIX]{x1D711}_{0}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{1})$
.
Let
$(\unicode[STIX]{x1D711}_{t})$
be the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
. Then, for all
$x\in X$
,
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}(x)=\inf _{t\in [0,1]}\unicode[STIX]{x1D711}_{t}(x).\end{eqnarray}$$
Proof. It follows from a classical balayage procedure that goes back to Bedford and Taylor [Reference Bedford and TaylorBT82] that
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})$
is supported on the coincidence set
$\{x\in X\mid \unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}(x)=\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})(x)\}$
. This holds true more generally for the Monge–Ampère measure of any envelope, namely
$$\begin{eqnarray}\unicode[STIX]{x1D7D9}_{\{P(h)<h\}}\operatorname{MA}(P(h))\equiv 0,\end{eqnarray}$$
where
$h$
is a bounded lower semi-continuous function.
We have observed in Proposition 3.1 that
$x\mapsto \inf _{t\in [0,1]}\unicode[STIX]{x1D711}_{t}(x)$
is a
$\unicode[STIX]{x1D714}$
-psh function. Since it lies both below
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
, we infer
$$\begin{eqnarray}\inf _{t\in [0,1]}\unicode[STIX]{x1D711}_{t}\leqslant \unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}.\end{eqnarray}$$
Conversely,
$(t,x)\mapsto \unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}(x)$
is a subgeodesic (independent of
$t$
); hence, for all
$t,x$
,
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}(x)\leqslant \unicode[STIX]{x1D711}_{t}(x)$
. Thus,
$\unicode[STIX]{x1D713}:=\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\inf _{t\in [0,1]}\unicode[STIX]{x1D711}_{t}$
; hence,
$\unicode[STIX]{x1D713}$
is bounded, thanks to Proposition 1.4.
By Proposition 3.1,
$\unicode[STIX]{x1D713}$
is
$\unicode[STIX]{x1D714}$
-psh, hence
$A\unicode[STIX]{x1D714}_{X}$
-psh for some Kähler form
$\unicode[STIX]{x1D714}_{X}$
and
$A>0$
. Thus,
$\sup _{X}\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D714}_{X}}\unicode[STIX]{x1D713}\geqslant -C$
for some
$C>0$
.
It follows from the work of Berman and Demailly [Reference Berman and DemaillyBD12] (see also [Reference BermanBer13, Theorem 1.2]) that for any compact subset
$K\subset \operatorname{Amp}(\unicode[STIX]{x1D6FC})$
, there exists
$C_{K}>0$
, such that, for all
$t\in [0,1]$
,
$$\begin{eqnarray}\sup _{K}\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D714}_{X}}\unicode[STIX]{x1D711}_{t}<C_{K}n.\end{eqnarray}$$
Thus
$(-\unicode[STIX]{x1D711}_{t})$
is a family of
$C_{K}\unicode[STIX]{x1D714}_{X}$
-psh functions in a neighborhood of
$K$
, which are uniformly bounded from above. Thus,
$$\begin{eqnarray}-\unicode[STIX]{x1D713}=\sup _{0\leqslant t\leqslant 1}(-\unicode[STIX]{x1D711}_{t})=-\inf _{0\leqslant t\leqslant 1}\unicode[STIX]{x1D711}_{t}\end{eqnarray}$$
is
$C_{K}\unicode[STIX]{x1D714}_{X}$
-psh near
$K$
, in particular
$\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D714}_{X}}\unicode[STIX]{x1D713}<C_{K}n$
. This means that
$\unicode[STIX]{x1D713}$
has a locally bounded Laplacian on
$\operatorname{Amp}(\unicode[STIX]{x1D6FC})$
.
It follows then from classical arguments that the measure
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})$
is absolutely continuous with respect to the Lebesgue measure. Since
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}$
(respectively
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1}$
) have locally bounded Laplacians in
$\operatorname{Amp}(\unicode[STIX]{x1D6FC})$
, it follows from [Reference Gilbarg and TrudingerGT83, Lemma 7.7] that their second partial derivatives agree on
$\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{0}\}$
(respectively on
$\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{1}\}$
), hence
$$\begin{eqnarray}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})=\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{0}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{1}<\unicode[STIX]{x1D711}_{0}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
We have used here the fact that none of the measures
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}),\operatorname{MA}(\unicode[STIX]{x1D711}_{0})$
,
$\operatorname{MA}(\unicode[STIX]{x1D711}_{1})$
charges the pluripolar set
$X\setminus \operatorname{Amp}(\unicode[STIX]{x1D6FC})$
.◻
A basic observation that we shall use on several occasions is the following.
Lemma 3.3. Assume
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
and let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
be the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
. Then
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant \Vert \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\Vert _{L^{\infty }(X)}.\end{eqnarray}$$
Moreover,
(i) if
$\unicode[STIX]{x1D711}_{0}(x)\leqslant \unicode[STIX]{x1D711}_{1}(x)$
for some
$x\in X$
, then
$\dot{\unicode[STIX]{x1D711}}_{1}(x)\geqslant 0$
;(ii) if
$\unicode[STIX]{x1D711}_{0}(x)\leqslant \unicode[STIX]{x1D711}_{1}(x)$
for all
$x\in X$
then
$\dot{\unicode[STIX]{x1D711}}_{t}(x)\geqslant 0$
for all
$x\in X$
and a.e.
$t\in [0,1]$
.
By symmetry, if
$\unicode[STIX]{x1D711}_{1}(x)\leqslant \unicode[STIX]{x1D711}_{0}(x)$
, it follows that
$\dot{\unicode[STIX]{x1D711}}_{0}(x)\leqslant 0$
. Moreover, if
$\unicode[STIX]{x1D711}_{1}(x)\leqslant \unicode[STIX]{x1D711}_{0}(x)$
for all
$x\in X$
then
$\dot{\unicode[STIX]{x1D711}}_{t}(x)\leqslant 0$
for a.e.
$x,t$
. Here, and in the following,
$\dot{\unicode[STIX]{x1D711}_{0}}$
,
$\dot{\unicode[STIX]{x1D711}_{1}}$
denote the right and left derivatives, respectively, while we recall that
$\dot{\unicode[STIX]{x1D711}}_{t}(x)$
is well defined for a.e.
$(x,t)$
.
Proof. From Theorem 1.21 we know that
$d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\,=\,\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})$
. Moreover, Proposition 1.4 ensures that
$|\dot{\unicode[STIX]{x1D711}}_{0}|\leqslant \Vert \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\Vert _{L^{\infty }(X)}$
. Hence, the first statement.
Assume
$\dot{\unicode[STIX]{x1D711}}_{1}(x)<0$
. Since
$t\mapsto \unicode[STIX]{x1D711}_{t}(x)$
is convex, we infer
$\dot{\unicode[STIX]{x1D711}_{t}}(x)\leqslant \dot{\unicode[STIX]{x1D711}_{1}}(x)<0$
. Thus,
$t\mapsto \unicode[STIX]{x1D711}_{t}(x)$
is decreasing, hence
$\unicode[STIX]{x1D711}_{1}(x)<\unicode[STIX]{x1D711}_{0}(x)$
, a contradiction. This proves part (i).
Assume now that
$\unicode[STIX]{x1D711}_{0}(x)\leqslant \unicode[STIX]{x1D711}_{1}(x)$
for all
$x\in X$
. Then
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{t}\leqslant \unicode[STIX]{x1D711}_{1}.\end{eqnarray}$$
The first of these inequalities follows from the fact that, by Proposition 1.4,
$$\begin{eqnarray}\unicode[STIX]{x1D711}=\sup \{u~u\in \text{PSH}(M,\unicode[STIX]{x1D714}):u\leqslant \unicode[STIX]{x1D711}_{0,1}~\text{on}~M\}\,,\end{eqnarray}$$
with
$\unicode[STIX]{x1D711}(x,t+is)=\unicode[STIX]{x1D711}_{t}(x)$
, and that
$\unicode[STIX]{x1D711}_{0}(x,t+is)=\unicode[STIX]{x1D711}_{0}(x)$
is a subsolution (i.e. a candidate in the envelope). The other inequality follows from the fact that
$\unicode[STIX]{x1D711}_{1}(x,t+is)=\unicode[STIX]{x1D711}_{1}(x)$
is a supersolution of (3) since
$(\unicode[STIX]{x1D714}+dd_{x,z}^{c}\unicode[STIX]{x1D711}_{1})^{n+1}=0$
and
$\unicode[STIX]{x1D711}_{1}\geqslant \unicode[STIX]{x1D711}_{0,1}$
. The same argument shows that
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{s}\leqslant \unicode[STIX]{x1D711}_{t}$
for all
$0<s<t$
and
$x\in X$
, hence
$\dot{\unicode[STIX]{x1D711}}_{t}(x)\geqslant 0$
for all
$x\in X$
and a.e.
$t\in [0,1]$
, since the derivative in time of
$\unicode[STIX]{x1D711}_{t}$
is well defined for a.e.
$t$
.◻
We now establish a very useful relation established by Darvas [Reference DarvasDar17c, Proposition 8.1] when
$\unicode[STIX]{x1D714}$
is Kähler (see also [Reference DarvasDar15, Corollary 4.14]).
Proposition 3.4. Assume
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
. Then, for all
$p\geqslant 1$
,
$$\begin{eqnarray}d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})+d_{p}^{p}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. We proceed by approximation, so as to reduce to the Kähler case. The identity is known to hold for
$d_{p,\unicode[STIX]{x1D700}}$
and
$\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}$
, where
$d_{p,\unicode[STIX]{x1D700}}$
denotes the distance associated with the Kähler form
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
and
$\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}$
is the greatest
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-psh function that lies below
$\min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
.
Using Theorem 1.21 and the triangle inequality, the proof boils down to checking that
$d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})\rightarrow 0$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
. The same arguments used in the proof of Proposition 1.16 yield
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})\leqslant d_{p,\unicode[STIX]{x1D700}^{\prime }}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}),\quad \unicode[STIX]{x1D700}<\unicode[STIX]{x1D700}^{\prime }.\end{eqnarray}$$
We claim that
$d_{p,\unicode[STIX]{x1D700}^{\prime }}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})$
goes to zero as
$\unicode[STIX]{x1D700}$
goes to zero, since
$\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}$
decreases to
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
. Indeed, observe that
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}^{\prime })\cap L^{\infty }(X)$
and that by Proposition 3.8 we know that
$$\begin{eqnarray}d_{p,\unicode[STIX]{x1D700}^{\prime }}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})\leqslant 2I_{p,\unicode[STIX]{x1D700}^{\prime }}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
The same arguments in the proof of Proposition 2.11 then show that
$I_{p,\unicode[STIX]{x1D700}^{\prime }}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{0}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{1})\rightarrow 0$
as
$\unicode[STIX]{x1D700}$
goes to zero. The conclusion then follows.◻
We note for later use the following consequence.
Corollary 3.5. If
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
, then
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
3.2 Comparing
$d_{p}$
and
$I_{p}$
The goal of this section is to establish that
$d_{p}$
and
$I_{p}$
are equivalent, extending [Reference DarvasDar15, Theorem 5.5].
Theorem 3.6. For all
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
,
$$\begin{eqnarray}2^{-1}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant 2^{4+(2n-1)/p}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
It follows from Definition 1.10 and Proposition 2.11 that
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=\lim _{\unicode[STIX]{x1D700}\rightarrow 0}d_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\quad \text{and}\quad I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=\lim _{\unicode[STIX]{x1D700}\rightarrow 0}I_{p,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
so it suffices to establish these inequalities when
$\unicode[STIX]{x1D714}$
is a Kähler form.
We nevertheless give a direct proof, valid when
$\unicode[STIX]{x1D714}$
is merely semi-positive, with several intermediate results of independent interest. Several of these results have been obtained by Darvas [Reference DarvasDar17b, Reference DarvasDar17c, Reference DarvasDar15] when
$\unicode[STIX]{x1D714}$
is Kähler.
Lemma 3.7. Assume that
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
satisfy
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
.
(i)
$d_{p}(\unicode[STIX]{x1D711}_{1},(\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2)\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
.(ii)
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant 2^{1+n/p}d_{p}(\unicode[STIX]{x1D711}_{0}/2,\unicode[STIX]{x1D711}_{1}/2)$
.(iii) If
$\unicode[STIX]{x1D711}_{1}=0$
then
$d_{p}(\unicode[STIX]{x1D711}_{0},0)\geqslant 2d_{p}(\unicode[STIX]{x1D711}_{0}/2,0)$
.(iv) If
$\unicode[STIX]{x1D713}\in {\mathcal{H}}_{bd}$
is such that
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D713}\leqslant \unicode[STIX]{x1D711}_{1}$
, then $$\begin{eqnarray}\max \{d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D713});d_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{1})\}\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. Let
$\unicode[STIX]{x1D711}_{t}$
(respectively
$\unicode[STIX]{x1D713}_{t}$
) denote the Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
(respectively
$(\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2$
) to
$\unicode[STIX]{x1D711}_{1}$
. Since
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
, it follows from Lemma 3.3(ii) that
$t\mapsto \unicode[STIX]{x1D711}_{t}$
,
$t\mapsto \unicode[STIX]{x1D713}_{t}$
are increasing and
$\unicode[STIX]{x1D711}_{t}\leqslant \unicode[STIX]{x1D713}_{t}$
, hence
$$\begin{eqnarray}\frac{\unicode[STIX]{x1D711}_{t}-\unicode[STIX]{x1D711}_{1}}{t-1}\geqslant \frac{\unicode[STIX]{x1D713}_{t}-\unicode[STIX]{x1D713}_{1}}{t-1}\,,\end{eqnarray}$$
since
$\unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D713}_{1}$
. Therefore,
$\dot{\unicode[STIX]{x1D711}}_{1}\geqslant \dot{\unicode[STIX]{x1D713}}_{1}\geqslant 0$
and we infer
$$\begin{eqnarray}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D713}_{1})=d_{p}\biggl(\unicode[STIX]{x1D711}_{1},\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)^{p}\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
This, together with Theorem 1.21, proves part (i).
Now let
$(\unicode[STIX]{x1D711}_{t})$
(respectively
$(\unicode[STIX]{x1D713}_{t})$
) denote the geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
(respectively
$\unicode[STIX]{x1D711}_{0}/2$
to
$\unicode[STIX]{x1D711}_{1}/2$
). Observe that
$t\mapsto \unicode[STIX]{x1D711}_{t},\unicode[STIX]{x1D713}_{t}$
are increasing, hence
$\dot{\unicode[STIX]{x1D711}}_{0}\geqslant 0$
. The family
$(\unicode[STIX]{x1D711}_{t}/2)$
is a subgeodesic joining
$\unicode[STIX]{x1D711}_{0}/2$
to
$\unicode[STIX]{x1D711}_{1}/2$
, hence
$\unicode[STIX]{x1D711}_{t}/2\leqslant \unicode[STIX]{x1D713}_{t}$
and
$$\begin{eqnarray}0\leqslant \frac{\dot{\unicode[STIX]{x1D711}}_{0}}{2}\leqslant \dot{\unicode[STIX]{x1D713}}_{0}\Longrightarrow |\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\leqslant 2^{p}|\dot{\unicode[STIX]{x1D713}}_{0}|^{p}.\end{eqnarray}$$
Moreover,
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant 2^{n}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}/2)$
, so we infer
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant 2^{n+p}d_{p}(\unicode[STIX]{x1D711}_{0}/2,\unicode[STIX]{x1D711}_{1}/2)^{p},\end{eqnarray}$$
which proves part (ii). A similar argument shows that
$$\begin{eqnarray}0\leqslant \dot{\unicode[STIX]{x1D713}}_{1}\leqslant \frac{\dot{\unicode[STIX]{x1D711}}_{1}}{2}\Longrightarrow |\dot{\unicode[STIX]{x1D713}}_{1}|^{p}\leqslant 2^{-p}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}.\end{eqnarray}$$
Now
$\operatorname{MA}(\unicode[STIX]{x1D711}_{1}/2)=\operatorname{MA}(\unicode[STIX]{x1D711}_{1})=\operatorname{MA}(0)$
when
$\unicode[STIX]{x1D711}_{1}=0$
, hence
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},0)^{p}=\int _{X}|\dot{\unicode[STIX]{x1D711}}_{1}|^{p}\operatorname{MA}(0)\geqslant 2^{p}d_{p}(\unicode[STIX]{x1D711}_{0}/2,0)^{p},\end{eqnarray}$$
which yields part (iii).
It remains to prove part (iv). Let
$(\unicode[STIX]{x1D711}_{t})_{0\leqslant t\leqslant 1}$
(respectively
$(\unicode[STIX]{x1D713}_{t})_{0\leqslant t\leqslant 1}$
) be the geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
(respectively
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D713}$
). Observe that
$\unicode[STIX]{x1D711}_{0}=\unicode[STIX]{x1D713}_{0}$
and
$\unicode[STIX]{x1D713}_{t}\leqslant \unicode[STIX]{x1D711}_{t}$
, hence
$\dot{\unicode[STIX]{x1D713}}_{0}\leqslant \dot{\unicode[STIX]{x1D711}}_{0}$
. Moreover,
$0\leqslant \dot{\unicode[STIX]{x1D713}}_{0}$
since
$t\mapsto \unicode[STIX]{x1D713}_{t}(x)$
is increasing. We infer
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D713})^{p}=\int _{X}|\dot{\unicode[STIX]{x1D713}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant \int _{X}|\dot{\unicode[STIX]{x1D711}}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})=d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}.\end{eqnarray}$$
The other inequality is proved similarly. ◻
Proposition 3.8. For all
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
,
$$\begin{eqnarray}0\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant 2I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Moreover, if
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
then
$I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant (\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}))^{1/p}$
and
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant \biggl(\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\biggr)^{1/p}\leqslant 2^{1+n/p}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. We first assume that
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
. The inequality
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant \biggl(\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\biggr)^{1/p}\end{eqnarray}$$
follows from Lemma 2.12. Let
$(\unicode[STIX]{x1D711}_{t})$
be the geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
. It follows from Lemma 3.3 that
$0\leqslant \dot{\unicode[STIX]{x1D711}}_{0}\leqslant \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\leqslant \dot{\unicode[STIX]{x1D711}}_{1}$
, hence
$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1})\leqslant \int _{X}(\dot{\unicode[STIX]{x1D711}}_{1})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1})=d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}\end{eqnarray}$$
and, similarly,
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}\leqslant \int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})$
.
We now show that
$\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant 2^{n+p}d(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}$
. Observe that
$(\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2\in {\mathcal{H}}_{bd}$
with
$\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant 2^{n}\,\operatorname{MA}((\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2)$
, hence
$$\begin{eqnarray}\displaystyle \int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}) & = & \displaystyle 2^{p}\int _{X}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}-\unicode[STIX]{x1D711}_{0}\biggr)^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{n+p}\int _{X}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}-\unicode[STIX]{x1D711}_{0}\biggr)^{p}\operatorname{MA}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{n+p}d_{p}\biggl(\unicode[STIX]{x1D711}_{0},\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)^{p},\nonumber\end{eqnarray}$$
as follows from the first step of the proof, since
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
. Lemma 3.7(iv) yields
$$\begin{eqnarray}d_{p}\biggl(\unicode[STIX]{x1D711}_{0},\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\leqslant d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
hence
$\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\leqslant 2^{n+p}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})^{p}$
.
We finally treat the first upper bound of the proposition, which does not require
$\unicode[STIX]{x1D711}_{0}$
to lie below
$\unicode[STIX]{x1D711}_{1}$
. It follows from the triangle inequality that
$$\begin{eqnarray}\displaystyle d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}) & {\leqslant} & \displaystyle d_{p}(\unicode[STIX]{x1D711}_{0},\max (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}))+d_{p}(\max (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\unicode[STIX]{x1D711}_{1})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \biggl(\int _{\{\unicode[STIX]{x1D711}_{0}<\unicode[STIX]{x1D711}_{1}\}}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\biggr)^{1/p}+\biggl(\int _{\{\unicode[STIX]{x1D711}_{0}>\unicode[STIX]{x1D711}_{1}\}}(\unicode[STIX]{x1D711}_{0}-\unicode[STIX]{x1D711}_{1})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1})\biggr)^{1/p}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{1-1/p}\biggl(\int _{X}|\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}|^{p}[\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\operatorname{MA}(\unicode[STIX]{x1D711}_{1})]\biggr)^{1/p}\nonumber\\ \displaystyle & = & \displaystyle 2\biggl(\int _{X}|\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}|^{p}\frac{[\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\operatorname{MA}(\unicode[STIX]{x1D711}_{1})]}{2}\biggr)^{1/p}\nonumber\end{eqnarray}$$
by using the elementary inequality
$a^{1/p}+b^{1/p}\leqslant 2^{1-1/p}(a+b)^{1/p}$
.◻
Remark 3.9. Working with
$\unicode[STIX]{x1D713}=t\unicode[STIX]{x1D711}_{0}+(1-t)\unicode[STIX]{x1D711}_{1}$
,
$0<t<1$
, instead of
$(\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2$
, one can improve this inequality and obtain
$$\begin{eqnarray}\biggl(\int _{X}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\biggr)^{1/p}\leqslant \frac{(n+p)^{1+n/p}}{p\,n^{n/p}}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
We now extend Lemma 3.7(i), following [Reference DarvasDar15, Lemma 5.3].
Lemma 3.10. For all
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
,
$$\begin{eqnarray}d_{p}\biggl(\unicode[STIX]{x1D711}_{0},\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\leqslant 2^{2+n/p}d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. When
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
, this follows from Lemma 3.7(i). Replacing
$\unicode[STIX]{x1D714}$
with
$\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}_{0}$
, we can assume, without loss of generality, that
$\unicode[STIX]{x1D711}_{0}=0$
. The triangle inequality yields
$$\begin{eqnarray}d_{p}\biggl(0,\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr)\leqslant d_{p}\biggl(0,0\vee \frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr)+d_{p}\biggl(0\vee \frac{\unicode[STIX]{x1D711}_{1}}{2},\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr).\end{eqnarray}$$
Observe that
$0\vee \unicode[STIX]{x1D711}_{1}\leqslant 0\vee \unicode[STIX]{x1D711}_{1}/2\leqslant \min (0,\unicode[STIX]{x1D711}_{1}/2)$
. It follows, therefore, from Lemma 3.7(iv) that
$$\begin{eqnarray}d_{p}\biggl(0,0\vee \frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr)+d_{p}\biggl(0\vee \frac{\unicode[STIX]{x1D711}_{1}}{2},\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr)\leqslant d_{p}(0,0\vee \unicode[STIX]{x1D711}_{1})+d_{p}\biggl(0\vee \unicode[STIX]{x1D711}_{1},\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr).\end{eqnarray}$$
Since
$0\vee \unicode[STIX]{x1D711}_{1}\leqslant 0$
and
$0\vee \unicode[STIX]{x1D711}_{1}\leqslant \unicode[STIX]{x1D711}_{1}/2$
, we can invoke Proposition 3.8 to obtain
$$\begin{eqnarray}\displaystyle d_{p}(0,0\vee \unicode[STIX]{x1D711}_{1})+d_{p}\biggl(0\vee \unicode[STIX]{x1D711}_{1},\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr) & {\leqslant} & \displaystyle \biggl(\int _{X}|0\vee \unicode[STIX]{x1D711}_{1}|^{p}\operatorname{MA}(0\vee \unicode[STIX]{x1D711}_{1})\biggr)^{1/p}\nonumber\\ \displaystyle & & \displaystyle +\,\biggl(\int _{X}\biggl|0\vee \unicode[STIX]{x1D711}_{1}-\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr|^{p}\operatorname{MA}(0\vee \unicode[STIX]{x1D711}_{1})\biggr)^{1/p}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2^{1-1/p}\biggl(\int _{X}\biggl[|0\vee \unicode[STIX]{x1D711}_{1}|^{p}+\biggl|0\vee \unicode[STIX]{x1D711}_{1}-\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr|^{p}\biggr]\operatorname{MA}(0\vee \unicode[STIX]{x1D711}_{1})\biggr)^{1/p}.\nonumber\end{eqnarray}$$
Recall now that the measure
$\operatorname{MA}(0\vee \unicode[STIX]{x1D711}_{1})$
is supported on the contact set
$S:=\{x\in X;0\vee \unicode[STIX]{x1D711}_{1}(x)=\min (0,\unicode[STIX]{x1D711}_{1})(x)\}$
. On this set, we have
$$\begin{eqnarray}|0\vee \unicode[STIX]{x1D711}_{1}|^{p}+\biggl|0\vee \unicode[STIX]{x1D711}_{1}-\frac{\unicode[STIX]{x1D711}_{1}}{2}\biggr|^{p}\leqslant 2|\unicode[STIX]{x1D711}_{1}|^{p}=2[|0\vee \unicode[STIX]{x1D711}_{1}|^{p}+|0\vee \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{1}|^{p}],\end{eqnarray}$$
while Proposition 3.8 yields
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{X}[|0\vee \unicode[STIX]{x1D711}_{1}|^{p}+|0\vee \unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{1}|^{p}]\operatorname{MA}(0\vee \unicode[STIX]{x1D711}_{1})\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant 2^{p+n}[d_{p}(0,0\vee \unicode[STIX]{x1D711}_{1})^{p}+d_{p}(0\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1})^{p}]=2^{p+n}d_{p}(0,\unicode[STIX]{x1D711}_{1})^{p},\nonumber\end{eqnarray}$$
where the last equality follows from Proposition 3.4. Altogether, this yields
$d_{p}(0,\unicode[STIX]{x1D711}_{1}/2)\leqslant 2^{2+n/p}d_{p}(0,\unicode[STIX]{x1D711}_{1})$
, as claimed.◻
We are now ready to prove Theorem 3.6.
Proof. We have already observed that
$d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\leqslant 2I_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
in Proposition 3.8, so we focus on the reverse control. Lemma 3.10 and Proposition 3.4 yield
$$\begin{eqnarray}\displaystyle 2^{2p+n}d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}) & {\geqslant} & \displaystyle d_{p}^{p}\biggl(\unicode[STIX]{x1D711}_{0},\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\nonumber\\ \displaystyle & = & \displaystyle d_{p}^{p}\biggl(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)+d_{p}^{p}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2},\unicode[STIX]{x1D711}_{0}\vee \frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr).\nonumber\end{eqnarray}$$
It follows from (8) together with the fact that
$2^{n}\operatorname{MA}((\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1})/2)\geqslant \operatorname{MA}(\unicode[STIX]{x1D711}_{0})$
that
$$\begin{eqnarray}d_{p}^{p}\biggl(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\geqslant \int _{X}\biggl(\unicode[STIX]{x1D711}_{0}-\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\vee \unicode[STIX]{x1D711}_{0}\biggr)^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\end{eqnarray}$$
and
$$\begin{eqnarray}d_{p}^{p}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2},\unicode[STIX]{x1D711}_{0}\vee \frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)\geqslant 2^{-n}\int _{X}\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}-\unicode[STIX]{x1D711}_{0}\vee \frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr)^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}).\end{eqnarray}$$
Hence
$$\begin{eqnarray}\displaystyle d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}) & {\geqslant} & \displaystyle 2^{-2(p+n)}\int _{X}\biggl[\biggl(\unicode[STIX]{x1D711}_{0}-\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\vee \unicode[STIX]{x1D711}_{0}\biggr)^{p}+\biggl(\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}-\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\vee \unicode[STIX]{x1D711}_{0}\biggr)^{p}\biggr]\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 2^{1-3p-2n}\int _{X}\biggl|\unicode[STIX]{x1D711}_{0}-\frac{\unicode[STIX]{x1D711}_{0}+\unicode[STIX]{x1D711}_{1}}{2}\biggr|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})\nonumber\\ \displaystyle & = & \displaystyle 2^{1-4p-2n}\int _{X}|\unicode[STIX]{x1D711}_{0}-\unicode[STIX]{x1D711}_{1}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}),\nonumber\end{eqnarray}$$
where in the last inequality we used the fact that
$|a-b|^{p}\leqslant 2^{p-1}(a^{p}+b^{p})$
, for any
$a,b\in \mathbb{R}^{+}$
.
Reversing the roles of
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
, we get
$$\begin{eqnarray}d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\geqslant 2^{1-4p-2n}\int _{X}|\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}|^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
from which it follows that
$d_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\geqslant 2^{1-4p-2n}I_{p}^{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$
.◻
3.3 Controlling the supremum
It follows from previous results that the supremum of a bounded potential with locally bounded Laplacian in
$\operatorname{Amp}(\unicode[STIX]{x1D6FC})$
is controlled by the distance to the base point.
Lemma 3.11. There exists
$C>0$
, such that for all
$\unicode[STIX]{x1D711}\in {\mathcal{H}}_{bd}$
,
$$\begin{eqnarray}-2^{4+2n}d_{1}(0,\unicode[STIX]{x1D711})\leqslant \sup _{X}\unicode[STIX]{x1D711}\leqslant 2^{4+2n}(n+1)d_{1}(0,\unicode[STIX]{x1D711})+C.\end{eqnarray}$$
Proof. If
$\sup _{X}\unicode[STIX]{x1D711}\leqslant 0$
, then
$\sup _{X}\unicode[STIX]{x1D711}\leqslant 0\leqslant (n+1)d_{1}(0,\unicode[STIX]{x1D711})+C$
, while
$$\begin{eqnarray}-d_{1}(0,\unicode[STIX]{x1D711})=E(\unicode[STIX]{x1D711})\leqslant \sup _{X}\unicode[STIX]{x1D711},\end{eqnarray}$$
as follows from Proposition 3.12. We therefore assume in what follows that
$\sup _{X}\unicode[STIX]{x1D711}\geqslant 0$
. If
$\unicode[STIX]{x1D711}\geqslant 0$
, Proposition 3.12 yields
$$\begin{eqnarray}\frac{1}{n+1}\int _{X}\unicode[STIX]{x1D711}\operatorname{MA}(0)\leqslant E(\unicode[STIX]{x1D711})=d_{1}(0,\unicode[STIX]{x1D711}).\end{eqnarray}$$
It is a classical consequence of the
$\unicode[STIX]{x1D714}$
-plurisubharmonicity [Reference Guedj and ZeriahiGZ05, Proposition 2.7] that there exists
$C>0$
such that, for all
$\unicode[STIX]{x1D711}\in \text{PSH}(X,\unicode[STIX]{x1D714})$
,
$$\begin{eqnarray}\sup _{X}\unicode[STIX]{x1D711}\leqslant \int _{X}\unicode[STIX]{x1D711}\operatorname{MA}(0)+C.\end{eqnarray}$$
Thus,
$\sup _{X}\unicode[STIX]{x1D711}\leqslant (n+1)d_{1}(0,\unicode[STIX]{x1D711})+C$
.
When
$\sup _{X}\unicode[STIX]{x1D711}\geqslant 0$
but
$\unicode[STIX]{x1D711}$
takes both positive and negative values, we set
$\unicode[STIX]{x1D713}=\max (0,\unicode[STIX]{x1D711})$
and observe that
$\sup _{X}\unicode[STIX]{x1D713}=\sup _{X}\unicode[STIX]{x1D711}$
. Using Propositions 2.13 and 3.8 and Theorem 3.6, we obtain
$$\begin{eqnarray}d_{1}(0,\max (0,\unicode[STIX]{x1D711}))\leqslant 2I_{1}(0,\max (0,\unicode[STIX]{x1D711}))\leqslant 2I_{1}(0,\unicode[STIX]{x1D711})\leqslant 2^{5-(2n-1)/p}d_{1}(0,\unicode[STIX]{x1D711}).\end{eqnarray}$$
The conclusion, therefore, follows from the previous case. ◻
Proposition 3.12. Assume
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{H}}_{bd}$
. Then
$$\begin{eqnarray}d_{1}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=E(\unicode[STIX]{x1D711})+E(\unicode[STIX]{x1D713})-2E(\unicode[STIX]{x1D711}\vee \unicode[STIX]{x1D713}).\end{eqnarray}$$
Proof. We proceed by approximation, so as to reduce to the Kähler case. By [Reference DarvasDar15, Corollary 4.14], we know that
$$\begin{eqnarray}d_{1,\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711})+E_{\unicode[STIX]{x1D714}_{e}}(\unicode[STIX]{x1D713})-2E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D713}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}_{X}$
,
$\unicode[STIX]{x1D711}\vee _{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D713}$
is the greatest
$\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}$
-psh function that lies below
$\min (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})$
, and
$E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
is as in (6). Since
$(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D711})^{n}$
converges weakly to
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}$
, we have that
$E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711})$
converges to
$E(\unicode[STIX]{x1D711})$
as
$\unicode[STIX]{x1D700}$
goes to
$0$
. The same holds for
$E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D713})$
. We then need to ensure that
$E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D711}\vee _{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}\unicode[STIX]{x1D713})$
converges to
$E(\unicode[STIX]{x1D711}\vee \unicode[STIX]{x1D713})$
. Denote
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D711}\vee _{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}\unicode[STIX]{x1D713}$
and
$\unicode[STIX]{x1D719}:=\unicode[STIX]{x1D711}\vee \unicode[STIX]{x1D713}$
. Fix
$\unicode[STIX]{x1D700}^{\prime }>\unicode[STIX]{x1D700}$
. Using Lemma 2.5 and the fact that
$\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}$
is decreasing to
$\unicode[STIX]{x1D719}$
, we get
$$\begin{eqnarray}\displaystyle 0\geqslant E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}})-E_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}(\unicode[STIX]{x1D719}) & = & \displaystyle \frac{1}{(n+1)V_{\unicode[STIX]{x1D700}}}\mathop{\sum }_{j=0}^{n}\int _{X}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}-\unicode[STIX]{x1D719})(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}})^{j}\wedge (\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D719})^{n-j}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \frac{1}{(n+1)V_{\unicode[STIX]{x1D700}}}\mathop{\sum }_{j=0}^{n}\int _{X}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}^{\prime }}-\unicode[STIX]{x1D719})(\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}})^{j}\wedge (\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}_{X}+dd^{c}\unicode[STIX]{x1D719})^{n-j}.\nonumber\end{eqnarray}$$
Letting first
$\unicode[STIX]{x1D700}$
to zero and the
$\unicode[STIX]{x1D700}^{\prime }$
, we get the result. The conclusion then follows from the previous arguments and Proposition 1.16.◻
4 The complete geodesic space
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
4.1 Metric completion
For
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
, we let
$\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{k}$
denote sequences of elements in
${\mathcal{H}}_{bd}$
decreasing to
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$
, respectively, and set
$$\begin{eqnarray}D_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}):=\liminf _{j,k\rightarrow +\infty }d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{k}).\end{eqnarray}$$
We list in the following proposition various properties of this extension.
Proposition 4.1.
(i)
$D_{p}$
is a distance on
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
, which coincides with
$d_{p}$
on
${\mathcal{H}}_{bd}$
.(ii) The definition of
$D_{p}$
is independent of the choice of the approximants.(iii)
$D_{p}$
is continuous along decreasing sequences in
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
.
Moreover all previous inequalities comparing
$d_{p}$
and
$I_{p}$
on
${\mathcal{H}}_{bd}$
extend to inequalities between
$D_{p}$
and
$I_{p}$
on
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
.
In the following, therefore, we will denote
$D_{p}$
by
$d_{p}$
.
Proof. It is a tedious exercise to verify that
$D_{p}$
defines a ‘semi-distance’, i.e. satisfies all properties of a distance but for the separation property. It follows from the definition of
$D_{p}$
and Proposition 2.11 that Theorem 3.6 extends in a natural way to potentials in
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
. If
$D_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=0$
, it therefore follows that
$I_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})=0$
, hence
$\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}$
by the domination principle.
One can check that
$D_{p}$
coincides with
$d_{p}$
on
${\mathcal{H}}_{bd}$
as follows: using part (ii), one can use the constant sequences
$\unicode[STIX]{x1D711}_{j}\equiv \unicode[STIX]{x1D711}$
and
$\unicode[STIX]{x1D713}_{k}\equiv \unicode[STIX]{x1D713}$
to obtain this equality.
We now prove part (ii). Let
$\unicode[STIX]{x1D711}_{j},u_{j}$
(respectively
$\unicode[STIX]{x1D713}_{k},v_{k}$
) denote two sequences of elements of
${\mathcal{H}}_{bd}$
decreasing to
$\unicode[STIX]{x1D711}$
(respectively
$\unicode[STIX]{x1D713}$
). We can assume without loss of generality that these sequences are intertwining, i.e. for all
$j,k\in \mathbb{N}$
, there exists
$\ell ,q\in \mathbb{N}$
, such that
$\unicode[STIX]{x1D711}_{j}\leqslant u_{\ell }$
and
$\unicode[STIX]{x1D713}_{k}\leqslant v_{q}$
, with similar reverse inequalities. It follows from Proposition 3.8 and the triangle inequality that
$$\begin{eqnarray}\displaystyle |d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{k})-d_{p}(u_{\ell },v_{q})| & {\leqslant} & \displaystyle d_{p}(\unicode[STIX]{x1D711}_{j},u_{\ell })+d_{p}(\unicode[STIX]{x1D713}_{k},v_{q})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 2I_{p}(\unicode[STIX]{x1D711}_{j},u_{\ell })+2I_{p}(\unicode[STIX]{x1D713}_{k},v_{q}).\nonumber\end{eqnarray}$$
Now, again by Proposition 3.8, we get
$$\begin{eqnarray}I_{p}(\unicode[STIX]{x1D711}_{j},u_{\ell })^{p}\leqslant \int _{X}(u_{\ell }-\unicode[STIX]{x1D711}_{j})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{j})\leqslant (p+1)^{n}\int _{X}(u_{\ell }-\unicode[STIX]{x1D711})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}),\end{eqnarray}$$
where the last inequality follows from [Reference Guedj and ZeriahiGZ07, Lemma 3.5]. The monotone convergence theorem, therefore, yields
$I_{p}(\unicode[STIX]{x1D711}_{j},u_{\ell })+I_{p}(\unicode[STIX]{x1D713}_{k},v_{q})\rightarrow 0$
as
$\ell ,q\rightarrow +\infty$
, proving part (ii).
One shows part (iii) with similar arguments. The extension of the inequalities comparing
$d_{p}$
and
$I_{p}$
follows from [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 2.17].◻
Proposition 4.2. The metric spaces
$({\mathcal{E}}_{\text{norm}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
and
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
are complete. The Mabuchi topology
$d_{p}$
dominates the topology induced by
$I$
: if a sequence converges for
$d_{p}$
, it converges in energy.
Proof. Let
$(\unicode[STIX]{x1D711}_{j})\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})^{\mathbb{N}}$
be a Cauchy sequence for
$d_{p}$
. We claim that there exists
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
, such that
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713})\rightarrow 0\quad \text{and}\quad I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{j})\rightarrow 0.\end{eqnarray}$$
Extracting and relabelling, we can assume that
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D711}_{j+1})\leqslant 2^{-j},\quad j\geqslant 1.\end{eqnarray}$$
Set
$\unicode[STIX]{x1D711}_{-1}\equiv 0$
and for
$k\geqslant j$
,
$\unicode[STIX]{x1D713}_{j,k}:=\unicode[STIX]{x1D711}_{j}\vee \unicode[STIX]{x1D711}_{j+1}\vee \cdots \vee \unicode[STIX]{x1D711}_{k}$
, and observe that
$\unicode[STIX]{x1D713}_{j,k}:=\unicode[STIX]{x1D711}_{j}\vee \unicode[STIX]{x1D713}_{j,k+1}$
. Hence, the Pythagorean formula gives
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j,k})\leqslant d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j+1,k})\leqslant 2^{-j}+d_{p}(\unicode[STIX]{x1D711}_{j+1},\unicode[STIX]{x1D713}_{j+1,k}).\end{eqnarray}$$
Repeating this argument, we get
$d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j,k})\leqslant 2^{-j+1}$
. We then have
$$\begin{eqnarray}\displaystyle d_{p}(0,\unicode[STIX]{x1D713}_{j,k}) & {\leqslant} & \displaystyle \mathop{\sum }_{\ell =-1}^{j-1}d_{p}(\unicode[STIX]{x1D711}_{\ell },\unicode[STIX]{x1D711}_{\ell +1})+d_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j,k})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{\ell =-1}^{j}d_{p}(\unicode[STIX]{x1D711}_{\ell },\unicode[STIX]{x1D711}_{\ell +1})+d_{p}(\unicode[STIX]{x1D711}_{j+1},\unicode[STIX]{x1D713}_{j+1,k})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle d_{p}(0,\unicode[STIX]{x1D711}_{1})+2+2^{-j+1}.\nonumber\end{eqnarray}$$
It follows from Theorem 3.6 that
$I_{p}(0,\unicode[STIX]{x1D713}_{j,k})$
is uniformly bounded, hence its decreasing limit
$\unicode[STIX]{x1D713}_{j}:=\lim _{k\rightarrow +\infty }\unicode[STIX]{x1D713}_{j,k}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
[Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Proposition 2.19]. From the above, we also have
$$\begin{eqnarray}d_{p}(0,\unicode[STIX]{x1D713}_{j})\leqslant d_{p}(0,\unicode[STIX]{x1D711}_{1})+2+2^{-j+1}.\end{eqnarray}$$
Lemma 3.11 then ensures that
$(\sup _{X}\unicode[STIX]{x1D713}_{j})_{j}$
is uniformly bounded, hence
$\unicode[STIX]{x1D713}_{j}$
increases a.e. towards
$\unicode[STIX]{x1D713}\in \operatorname{PSH}(X,\unicode[STIX]{x1D714})$
. Also,
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
thanks to [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Proposition 2.4]. Moreover, [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 2.17] yields
$$\begin{eqnarray}I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})+I_{p}(\unicode[STIX]{x1D713}_{j},\unicode[STIX]{x1D713})\longrightarrow 0.\end{eqnarray}$$
It follows, therefore, from Proposition 3.8 that
$d_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})\rightarrow 0$
and
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{j})\leqslant d_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})+d_{p}(\unicode[STIX]{x1D713}_{j},\unicode[STIX]{x1D711}_{j})\leqslant d_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})+2^{-j+1}\rightarrow 0.\end{eqnarray}$$
Recalling that
$\unicode[STIX]{x1D713}_{j}\leqslant \unicode[STIX]{x1D711}_{j}$
, it follows from the quasi-triangle inequality, Proposition 2.8, and Theorem 3.6 that
$$\begin{eqnarray}I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D711}_{j})\leqslant c_{n}\{I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})+I(\unicode[STIX]{x1D713}_{j},\unicode[STIX]{x1D711}_{j})\}\leqslant c_{n,p}\{I(\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}_{j})+d_{p}(\unicode[STIX]{x1D713}_{j},\unicode[STIX]{x1D711}_{j})\}\rightarrow 0.\Box\end{eqnarray}$$
Recall that the precompletion of a metric space
$(X,d)$
is the set of all Cauchy sequences
$C_{X}$
of
$X$
, together with the semi-distance
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\{x_{j}\},\{y_{j}\})=\lim _{j\rightarrow +\infty }d(x_{j},y_{j}).\end{eqnarray}$$
The metric completion
$(\overline{X},d)$
of
$(X,d)$
is the quotient space
$C_{X}/\sim$
, where
$$\begin{eqnarray}\{x_{j}\}\sim \{y_{j}\}\Longleftrightarrow \unicode[STIX]{x1D6FF}(\{x_{j}\},\{y_{j}\})=0,\end{eqnarray}$$
equipped with the induced distance, which we still denote
$d$
.
We are now taking advantage of the fact that
${\mathcal{H}}_{bd}$
lives inside the complete metric space
$({\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC}),d_{p})$
to conclude the following.
Theorem 4.3. The metric completion of
$({\mathcal{H}}_{bd},d_{p})$
is isometric to
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
.
Thanks to Theorem 3.6, an equivalent formulation of this statement is that the metric completion of
$({\mathcal{H}}_{bd},d_{p})$
is bi-Lipschitz equivalent to
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),I_{p})$
.
Proof. We work at the level of normalized potentials,
$$\begin{eqnarray}{\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714})=\{\unicode[STIX]{x1D711}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})\mid E(\unicode[STIX]{x1D711})=0\}\end{eqnarray}$$
and
${\mathcal{H}}_{0}:=\{\unicode[STIX]{x1D711}\in {\mathcal{H}}_{bd}\mid \unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711}\geqslant 0~\text{and}~E(\unicode[STIX]{x1D711})=0\}$
.
Since
$({\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
is a complete metric space that contains
${\mathcal{H}}_{0}$
, it suffices to show that the latter is dense in
${\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714})$
. Fix
$\unicode[STIX]{x1D711}\in {\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714})$
and let
$(\unicode[STIX]{x1D711}_{j})\in {\mathcal{H}}_{0}^{\mathbb{N}}$
be a sequence quasi-decreasing to
$\unicode[STIX]{x1D711}$
: the normalization condition
$E(\unicode[STIX]{x1D711}_{j})=0$
prevents us from getting a truly decreasing sequence; however,
$\unicode[STIX]{x1D711}_{j}+\unicode[STIX]{x1D700}_{j}$
is decreasing where
$\unicode[STIX]{x1D700}_{j}$
is a sequence of real numbers decreasing to zero. It follows from Proposition 3.8 that
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{j+\ell }+\unicode[STIX]{x1D700}_{j+l},\unicode[STIX]{x1D711}_{j}+\unicode[STIX]{x1D700}_{j})^{p}\leqslant \int _{X}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711}_{j+\ell })^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{j+\ell })+\unicode[STIX]{x1D700}_{j}.\end{eqnarray}$$
Now [Reference Guedj and ZeriahiGZ07, Lemma 3.5] shows that the latter is bounded from above by
$$\begin{eqnarray}(p+1)^{n}\int _{X}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711})^{p}\operatorname{MA}(\unicode[STIX]{x1D711})+\unicode[STIX]{x1D700}_{j},\end{eqnarray}$$
which converges to zero as
$j\rightarrow +\infty$
, as follows from the monotone convergence theorem. Therefore
$(\unicode[STIX]{x1D711}_{j})$
is a Cauchy sequence in
$({\mathcal{H}}_{0},d_{p})$
that converges to
$\unicode[STIX]{x1D711}$
, since
$$\begin{eqnarray}0\leqslant d_{p}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D711}_{j}+\unicode[STIX]{x1D700}_{j})\leqslant \liminf _{\ell \rightarrow +\infty }d_{p}(\unicode[STIX]{x1D711}_{j+\ell },\unicode[STIX]{x1D711}_{j})\leqslant 2(1+p)^{n/p}I_{p}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D711})+\unicode[STIX]{x1D700}_{j}^{1/p}\rightarrow 0,\end{eqnarray}$$
by Proposition 3.8 and [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 2.17].
We note the following alternative approach of independent interest. One first shows that
${\mathcal{H}}_{0}$
is dense in the set of all bounded
$\unicode[STIX]{x1D714}$
-psh functions. Given
$\unicode[STIX]{x1D711}\in {\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714})$
, one then considers its ‘canonical approximants’,
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}=\max (\unicode[STIX]{x1D711},-j)+\unicode[STIX]{x1D700}_{j}\in \text{PSH}_{0}(X,\unicode[STIX]{x1D714})\cap L^{\infty }(X)\,,\end{eqnarray}$$
which decrease towards
$\unicode[STIX]{x1D711}\in {\mathcal{E}}_{0}^{p}(X,\unicode[STIX]{x1D714})$
. It follows from Proposition 3.8 that
$$\begin{eqnarray}\displaystyle d_{p}(\unicode[STIX]{x1D711}_{j+\ell },\unicode[STIX]{x1D711}_{j})^{p} & {\leqslant} & \displaystyle o(1)+\int _{X}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711}_{j+\ell })^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{j+\ell })\nonumber\\ \displaystyle & = & \displaystyle o(1)+\int _{(\unicode[STIX]{x1D711}\leqslant -j-\ell )}\ell ^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{j+\ell })+\int _{(-j-\ell <\unicode[STIX]{x1D711}<-j)}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711}_{j+\ell })^{p}\operatorname{MA}(\unicode[STIX]{x1D711})\nonumber\\ \displaystyle & = & \displaystyle o(1)+\int _{(\unicode[STIX]{x1D711}\leqslant -j-\ell )}\ell ^{p}\operatorname{MA}(\unicode[STIX]{x1D711})+\int _{(-j-\ell <\unicode[STIX]{x1D711}<-j)}(\unicode[STIX]{x1D711}_{j}-\unicode[STIX]{x1D711}_{j+\ell })^{p}\operatorname{MA}(\unicode[STIX]{x1D711})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle o(1)+\int _{(\unicode[STIX]{x1D711}<-j)}\unicode[STIX]{x1D711}^{p}\operatorname{MA}(\unicode[STIX]{x1D711}),\nonumber\end{eqnarray}$$
where we have used the maximum principle, together with the fact that
$$\begin{eqnarray}\int _{(\unicode[STIX]{x1D711}\leqslant -k)}\operatorname{MA}(\unicode[STIX]{x1D711}_{k})=\int _{X}\operatorname{MA}(\unicode[STIX]{x1D711}_{k})-\int _{(\unicode[STIX]{x1D711}>-k)}\operatorname{MA}(\unicode[STIX]{x1D711}_{k})=\int _{(\unicode[STIX]{x1D711}\leqslant -k)}\operatorname{MA}(\unicode[STIX]{x1D711}),\end{eqnarray}$$
since
$\unicode[STIX]{x1D711}\in {\mathcal{E}}(X,\unicode[STIX]{x1D714})$
, as follows again from the maximum principle. We infer that
$(\unicode[STIX]{x1D711}_{j})$
is a Cauchy sequence, which converges to
$\unicode[STIX]{x1D711}$
.◻
We are now in a position to prove Theorem B of the introduction.
Corollary 4.4. Assume
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{Y}$
, where
$\unicode[STIX]{x1D714}_{Y}$
is a Hodge form. Then the metric completion
$(\overline{{\mathcal{H}}}_{\unicode[STIX]{x1D6FC}},d_{p})$
is isometric to
$({\mathcal{E}}^{p}(\unicode[STIX]{x1D6FC}),d_{p})$
. Similarly, the metric completion
$(\overline{{\mathcal{H}}}_{\unicode[STIX]{x1D714}},d_{p})$
is isometric to
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
.
Proof. Thanks to [Reference Coman, Guedj and ZeriahiCGZ13, Corollary C] we can ensure that the space
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
is dense in
${\mathcal{H}}_{bd}$
. The result then follows from Theorem 4.3.◻
4.2 Weak geodesics
4.2.1 Finite-energy geodesics
We now define finite-energy geodesics joining two finite-energy endpoints
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
. Fix
$j\in \mathbb{N}$
and consider
$\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{j}$
bounded
$\unicode[STIX]{x1D714}$
-psh functions decreasing to
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
. We let
$\unicode[STIX]{x1D711}_{t,j}$
denote the bounded geodesic joining
$\unicode[STIX]{x1D711}_{0}^{j}$
to
$\unicode[STIX]{x1D711}_{1}^{j}$
. It follows from the maximum principle that
$j\mapsto \unicode[STIX]{x1D711}_{t,j}$
is non-increasing. We can thus set
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{t}:=\lim _{j\rightarrow +\infty }\unicode[STIX]{x1D711}_{t,j}.\end{eqnarray}$$
Definition 4.5. The map
$(t,x)\mapsto \unicode[STIX]{x1D711}_{t}(x)$
is the (finite-energy) Mabuchi geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
.
The
$\unicode[STIX]{x1D711}_{t}$
indeed form a family of finite-energy functions: since
$t\mapsto E(\unicode[STIX]{x1D711}_{t,j})$
is affine (Lemma 2.5), we infer, for all
$j\in \mathbb{N}$
,
$$\begin{eqnarray}(1-t)E(\unicode[STIX]{x1D711}_{0})+tE(\unicode[STIX]{x1D711}_{1})\leqslant (1-t)E(\unicode[STIX]{x1D711}_{0}^{(j)})+tE(\unicode[STIX]{x1D711}_{1}^{(j)})=E(\unicode[STIX]{x1D711}_{t,j}),\end{eqnarray}$$
hence
$\unicode[STIX]{x1D711}_{t}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
with
$(1-t)E(\unicode[STIX]{x1D711}_{0})+tE(\unicode[STIX]{x1D711}_{1})=E(\unicode[STIX]{x1D711}_{t})$
.
It follows from the maximum principle that
$\unicode[STIX]{x1D711}_{t}$
is independent of the choice of the approximants
$\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{j}$
: if we set
$\unicode[STIX]{x1D711}(x,z):=\unicode[STIX]{x1D711}_{t}(x)$
,
$z=t+is$
, then
$\unicode[STIX]{x1D711}$
is a maximal
$\unicode[STIX]{x1D714}$
-psh function in
$X\times S$
, as a decreasing limit of maximal
$\unicode[STIX]{x1D714}$
-psh functions. It is thus the unique maximal
$\unicode[STIX]{x1D714}$
-psh function in
$X\times S$
with boundary values
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
.
When
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
belong to
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
, these weak geodesics are again metric geodesics in the complete metric space
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
.
Proposition 4.6. Given
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
, the Mabuchi geodesic
$\unicode[STIX]{x1D711}$
joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
lies in
${\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
and satisfies, for all
$t,s\in [0,1]$
,
$$\begin{eqnarray}d_{p}(\unicode[STIX]{x1D711}_{t},\unicode[STIX]{x1D711}_{s})=|t-s|\,d_{p}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Thus,
$({\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714}),d_{p})$
is a geodesic space.
Proof. We can assume, without loss of generality, that
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\leqslant 0$
. Fix
$j\in \mathbb{N}$
and consider
$\unicode[STIX]{x1D711}_{0}^{j},\unicode[STIX]{x1D711}_{1}^{j}$
bounded
$\unicode[STIX]{x1D714}$
-psh functions decreasing to
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$
. We let
$\unicode[STIX]{x1D711}_{t,j}$
denote the bounded geodesic joining
$\unicode[STIX]{x1D711}_{0}^{j}$
to
$\unicode[STIX]{x1D711}_{1}^{j}$
, which decreases towards
$\unicode[STIX]{x1D711}_{t}$
as
$j$
increases to
$+\infty$
. Observe that
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}\leqslant \unicode[STIX]{x1D711}_{0}^{j}\vee \unicode[STIX]{x1D711}_{1}^{j}\leqslant \unicode[STIX]{x1D711}_{t,j}.\end{eqnarray}$$
It therefore follows from [Reference Guedj and ZeriahiGZ07, Lemma 3.5] and Lemma 4.7 that
$$\begin{eqnarray}\int _{X}(-\unicode[STIX]{x1D711}_{t,j})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{t,j})\leqslant (p+1)^{n}\int _{X}(-\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})<+\infty \,,\end{eqnarray}$$
hence the monotone convergence theorem yields
$\int _{X}(-\unicode[STIX]{x1D711}_{t})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{t})<+\infty$
, for all
$t$
, i.e.
$\unicode[STIX]{x1D711}_{t}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
.
The remaining assertion is proved as in the case of bounded geodesics (Proposition 1.18). ◻
Lemma 4.7. Assume
$0\geqslant \unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
. Then
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{p}(X,\unicode[STIX]{x1D714})$
and
$$\begin{eqnarray}\int _{X}(-\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})\leqslant \int _{X}(-\unicode[STIX]{x1D711}_{0})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\int _{X}(-\unicode[STIX]{x1D711}_{1})^{p}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
Proof. It suffices to establish the claimed inequality when
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
and then proceed by approximation. It follows from Proposition 3.2 that
$$\begin{eqnarray}\operatorname{MA}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})\leqslant \unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{0}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{0})+\unicode[STIX]{x1D7D9}_{\{\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{1}\}}\operatorname{MA}(\unicode[STIX]{x1D711}_{1}).\end{eqnarray}$$
The inequality follows, since
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\leqslant 0$
.◻
4.2.2 (Non-)uniqueness of geodesics
Fix
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
. If the sets
$(\unicode[STIX]{x1D711}_{0}<\unicode[STIX]{x1D711}_{1})$
and
$(\unicode[STIX]{x1D711}_{0}>\unicode[STIX]{x1D711}_{1})$
are both non-empty, the function
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
differs from
$\unicode[STIX]{x1D711}_{0}$
and
$\unicode[STIX]{x1D711}_{1}$
and it follows from Proposition 3.4 that
$$\begin{eqnarray}d_{1}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})=d_{1}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1})+d_{1}(\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{1}),\end{eqnarray}$$
thus the concatenation of the geodesic joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
and of that joining
$\unicode[STIX]{x1D711}_{0}\vee \unicode[STIX]{x1D711}_{1}$
to
$\unicode[STIX]{x1D711}_{1}$
gives another minimizing path joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
.
When
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
, this argument no longer works, but there are nevertheless very many minimizing paths, as shown by the following result.
Lemma 4.8. Assume
$\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}\in {\mathcal{H}}_{bd}$
are such that
$\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}$
. Let
$(\unicode[STIX]{x1D713}_{t})_{0\leqslant t\leqslant 1}$
be a path joining
$\unicode[STIX]{x1D711}_{0}$
to
$\unicode[STIX]{x1D711}_{1}$
. Then
$$\begin{eqnarray}\ell _{1}(\unicode[STIX]{x1D713})=d_{1}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})\Longleftrightarrow \dot{\unicode[STIX]{x1D713}}_{t}(x)\geqslant 0\quad \text{for a.e.}~t,x.\end{eqnarray}$$
In particular,
$t\mapsto t\unicode[STIX]{x1D711}_{1}(x)+(1-t)\unicode[STIX]{x1D711}_{0}$
is a minimizing path for
$d_{1}$
, which is not a Mabuchi geodesic, unless
$\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}$
is constant.
Proof. Observe that
$$\begin{eqnarray}\displaystyle \ell _{1}(\unicode[STIX]{x1D713}) & = & \displaystyle \int _{0}^{1}\int _{X}|\dot{\unicode[STIX]{x1D713}}_{t}(x)|\operatorname{MA}(\unicode[STIX]{x1D713}_{t})\,dt\geqslant \biggl|\int _{0}^{1}\int _{X}\dot{\unicode[STIX]{x1D713}}_{t}(x)\operatorname{MA}(\unicode[STIX]{x1D713}_{t})\,dt\biggr|\nonumber\\ \displaystyle & = & \displaystyle \biggl|\int _{0}^{1}\frac{d}{dt}E(\unicode[STIX]{x1D713}_{t})\,dt\biggr|=|E(\unicode[STIX]{x1D711}_{1})-E(\unicode[STIX]{x1D711}_{0})|=d_{1}(\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}),\nonumber\end{eqnarray}$$
where the last identity follows from Proposition 3.12. There is equality if and only if
$|\dot{\unicode[STIX]{x1D713}}_{t}(x)|=\dot{\unicode[STIX]{x1D713}}_{t}(x)\geqslant 0$
for a.e.
$(t,x)$
(the sign has to be positive because
$\unicode[STIX]{x1D713}_{0}=\unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D713}_{1}$
).
In particular,
$t\mapsto \unicode[STIX]{x1D713}_{t}=t\unicode[STIX]{x1D711}_{1}(x)+(1-t)\unicode[STIX]{x1D711}_{0}$
has this property, since
$\dot{\unicode[STIX]{x1D713}}_{t}=\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}\geqslant 0$
. We recall that, since
$\unicode[STIX]{x1D713}_{t}$
is a smooth path, the geodesic equation can be written as
$$\begin{eqnarray}\ddot{\unicode[STIX]{x1D713}}_{t}\operatorname{MA}(\unicode[STIX]{x1D713}_{t})=\frac{n}{V}\,d\dot{\unicode[STIX]{x1D713}_{t}}\wedge d^{c}\dot{\unicode[STIX]{x1D713}_{t}}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{t}}^{n-1}\end{eqnarray}$$
(see § 1.1.1). Now
$\ddot{\unicode[STIX]{x1D713}}_{t}=0$
, hence
$t\mapsto \unicode[STIX]{x1D713}_{t}$
is not a Mabuchi geodesic, unless
$d(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})\wedge d^{c}(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0})\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{t}}^{n-1}=0$
for all
$t$
, i.e.
$\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{0}$
is constant.◻
Conversely, it follows from the work of Darvas [Reference DarvasDar17c, Lemma 6.12] (based on [Reference Calabi and ChenCC02, § 2.4]) that geodesics are unique in
${\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714})$
.
Theorem 4.9. Assume
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{Y}$
, where
$\unicode[STIX]{x1D714}_{Y}$
is a Hodge form. Then the space
$({\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714}),d_{2})$
is a
$\text{CAT}(0)$
space.
Complete CAT(0) spaces are also called Hadamard spaces. Recall that a CAT(0) space is a geodesic space that has non-positive curvature in the sense of Alexandrov. Hadamard spaces enjoy many interesting properties (uniqueness of geodesics, contractibility, convexity properties, etc., see [Reference Bridson and HaefligerBH99]).
Proof. By Corollary 4.4 we know that
$({\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714}),d_{2})$
is the completion of
$({\mathcal{H}}_{\unicode[STIX]{x1D714}},d_{2})$
and by Proposition 4.6 that it is a geodesic metric space. [Reference Bridson and HaefligerBH99, Exercise 1.9.1.c (p. 163)] ensures that
$({\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714}),d_{2})$
is a CAT(0) space if and only if the CN inequality of Bruhat and Tits [Reference Bruhat and TitsBT72] holds, i.e.
$\forall P,Q,R\in {\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714})$
and, for any
$M\in {\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714})$
such that
$d_{2}(Q,M)=d_{2}(R,M)=d_{2}(Q,R)/2$
(in other words
$M=\unicode[STIX]{x1D711}_{t}^{QR}|_{t=1/2}$
where
$\unicode[STIX]{x1D711}_{t}^{QR}$
is the geodesic joining
$Q,R$
), one has
$$\begin{eqnarray}d_{2}(P,M)^{2}\leqslant {\textstyle \frac{1}{2}}d_{2}(P,Q)^{2}+{\textstyle \frac{1}{2}}d_{2}(P,R)^{2}-{\textstyle \frac{1}{4}}d_{2}(Q,R)^{2}.\end{eqnarray}$$
Assume first that
$P,Q,R\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}\subset {\mathcal{H}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D700}}}$
. Then by [Reference Calabi and ChenCC02, § 2.4] (see also [Reference DarvasDar17c, Lemma 6.12]), we have that
$$\begin{eqnarray}d_{2,\unicode[STIX]{x1D700}}(P,M_{\unicode[STIX]{x1D700}})^{2}\leqslant {\textstyle \frac{1}{2}}d_{2,\unicode[STIX]{x1D700}}(P,Q)^{2}+{\textstyle \frac{1}{2}}d_{2,\unicode[STIX]{x1D700}}(P,R)^{2}-{\textstyle \frac{1}{4}}d_{2,\unicode[STIX]{x1D700}}(Q,R)^{2}\,,\end{eqnarray}$$
where
$M_{\unicode[STIX]{x1D700}}$
is the point of
$\unicode[STIX]{x1D700}$
-geodesic joining
$Q,R$
such that
$d_{2,\unicode[STIX]{x1D700}}(Q,M)=d_{2,\unicode[STIX]{x1D700}}(R,M)=d_{2,\unicode[STIX]{x1D700}}(Q,R)/2$
. Thanks to Theorem 1.13, the right-hand side in the inequality converges to the right-hand side of (9) as
$\unicode[STIX]{x1D700}$
goes to zero. We claim that
$d_{2,\unicode[STIX]{x1D700}}(P,M_{\unicode[STIX]{x1D700}})$
converges to
$d(P,M)$
. Observe first that
$M_{\unicode[STIX]{x1D700}}$
decreases to
$M$
, since
$\unicode[STIX]{x1D700}$
-geodesics decreases as
$\unicode[STIX]{x1D700}$
decrease to zero (Proposition 1.6). Moreover, the triangle inequality yields
$|d_{2,\unicode[STIX]{x1D700}}(P,M_{\unicode[STIX]{x1D700}})-d_{2,\unicode[STIX]{x1D700}}(P,M)|\leqslant d_{2,\unicode[STIX]{x1D700}}(M,M_{\unicode[STIX]{x1D700}})$
. Since
$M,M_{\unicode[STIX]{x1D700}}$
are both bounded, it follows from Theorem 3.6 and Proposition 2.11 that
$d_{2,\unicode[STIX]{x1D700}^{\prime }}(M,M_{\unicode[STIX]{x1D700}})\rightarrow 0$
as
$\unicode[STIX]{x1D700}\rightarrow 0$
. This proves the claim.
If
$P,Q,R\in {\mathcal{E}}^{2}(X,\unicode[STIX]{x1D714})$
, we choose smooth approximants
$P_{k},Q_{k},R_{k}\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}$
decreasing to
$P,Q,R$
. The above arguments ensure that
$$\begin{eqnarray}d_{2}(P_{k},M_{k})^{2}\leqslant {\textstyle \frac{1}{2}}d_{2}(P_{k},Q_{k})^{2}+{\textstyle \frac{1}{2}}d_{2}(P_{k},R_{k})^{2}-{\textstyle \frac{1}{4}}d_{2}(Q_{k},R_{k})^{2}.\end{eqnarray}$$
The comparison principle implies that
$M_{k}$
decreases to
$M$
as
$k$
goes to
$+\infty$
. It then follows from Propositions 3.8 and 4.1 that
$d_{2}(M,M_{k})\rightarrow 0$
as
$k$
goes to
$+\infty$
. This, together with Proposition 4.1, gives (9) when letting
$k\rightarrow +\infty$
.◻
5 Singular Kähler–Einstein metrics of positive curvature
The existence of singular Kähler–Einstein metrics of non-positive curvature has been established in [Reference Eyssidieux, Guedj and ZeriahiEGZ09], generalizing the fundamental work of Aubin [Reference AubinAub78] and Yau [Reference YauYau78]. They always exist, provided the underlying variety has mild singularities and the first Chern class is non-positive.
Singular Kähler–Einstein metrics of positive curvature are more difficult to construct. It is already so in the smooth case [Reference Chen, Donaldson and SunCDS15]. Their first properties have been obtained in [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ]. In § 5.3, pushing further these works, we provide a necessary and sufficient analytic condition for their existence, generalizing a result of Tian [Reference TianTia97] and Phong et al. [Reference Phong, Song, Sturm and WeinkovePSSW08].
5.1 Log terminal singularities
A pair
$(Y,D)$
is the data of a connected normal compact complex variety
$Y$
and an effective
$\mathbb{Q}$
-divisor
$D$
, such that
$K_{Y}+D$
is
$\mathbb{Q}$
-Cartier. We write
$$\begin{eqnarray}Y_{0}:=Y_{\text{reg}}\setminus \text{Supp}\,D.\end{eqnarray}$$
Given a log resolution
$\unicode[STIX]{x1D70B}:X\rightarrow Y$
of
$(Y,D)$
(which may be chosen to be an isomorphism over
$Y_{0}$
), there exists a unique
$\mathbb{Q}$
-divisor
$\sum _{i}a_{i}E_{i}$
, whose push-forward to
$Y$
is
$-D$
, such that
$$\begin{eqnarray}K_{X}=\unicode[STIX]{x1D70B}^{\ast }(K_{Y}+D)+\mathop{\sum }_{i}a_{i}E_{i}.\end{eqnarray}$$
Definition 5.1. The pair
$(Y,D)$
is klt if
$a_{j}>-1$
for all
$j$
.
The same condition will then hold for all log resolutions of
$Y$
. When
$D=0$
, one says that
$Y$
is log terminal when the pair
$(Y,0)$
is klt. We have the following analytic interpretation. Fix
$r\in \mathbb{N}^{\ast }$
such that
$r(K_{Y}+D)$
is Cartier. If
$\unicode[STIX]{x1D70E}$
is a nowhere-vanishing section of the corresponding line bundle over a small open set
$U$
of
$Y$
, then
$$\begin{eqnarray}(i^{rn^{2}}\unicode[STIX]{x1D70E}\wedge \bar{\unicode[STIX]{x1D70E}})^{1/r}\end{eqnarray}$$
defines a smooth, positive volume form on
$U_{0}:=U\cap Y_{0}$
. If
$f_{j}$
is a local equation of
$E_{j}$
around a point of
$\unicode[STIX]{x1D70B}^{-1}(U)$
, then
$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }(i^{rn^{2}}\unicode[STIX]{x1D70E}\wedge \bar{\unicode[STIX]{x1D70E}})^{1/r}=\mathop{\prod }_{i}|f_{i}|^{2a_{i}}\,dV\end{eqnarray}$$
locally on
$\unicode[STIX]{x1D70B}^{-1}(U)$
for some local volume form
$dV$
. Since
$\sum _{i}E_{i}$
has normal crossings, this shows that
$(Y,D)$
is klt if and only if each volume form of the form of (11) has locally finite mass near singular points of
$Y$
.
The previous construction globalizes as follows.
Definition 5.2. Let
$(Y,D)$
be a pair and let
$\unicode[STIX]{x1D719}$
be a smooth Hermitian metric on the
$\mathbb{Q}$
-line bundle
$-(K_{Y}+D)$
. The corresponding adapted measure
$\text{mes}_{\unicode[STIX]{x1D719}}$
on
$Y_{\text{reg}}$
is locally defined by choosing a nowhere-zero section
$\unicode[STIX]{x1D70E}$
of
$r(K_{Y}+D)$
over a small open set
$U$
and setting
$$\begin{eqnarray}\text{mes}_{\unicode[STIX]{x1D719}}:=(i^{rn^{2}}\unicode[STIX]{x1D70E}\wedge \overline{\unicode[STIX]{x1D70E}})^{1/r}/|\unicode[STIX]{x1D70E}|_{r\unicode[STIX]{x1D719}}^{2/r}.\end{eqnarray}$$
The point is that the measure
$\text{mes}_{\unicode[STIX]{x1D719}}$
does not depend on the choice of
$\unicode[STIX]{x1D70E}$
; hence, it is globally defined. This discussion shows that
$$\begin{eqnarray}(Y,D)~\text{is klt}~\Longleftrightarrow ~\text{mes}_{\unicode[STIX]{x1D719}}~\text{has finite total mass on}~Y,\end{eqnarray}$$
in which case we view it as a Radon measure on the whole of
$Y$
.
5.2 Kähler–Einstein metrics on log Fano pairs
Definition 5.3. A log Fano pair is a klt pair
$(Y,D)$
such that
$Y$
is projective and
$-(K_{Y}+D)$
is ample.
Let
$(Y,D)$
be a log Fano pair. Fix a reference smooth strictly psh metric
$\unicode[STIX]{x1D719}_{0}$
on
$-(K_{Y}+D)$
, with curvature
$\unicode[STIX]{x1D714}_{0}$
and adapted measure
$\unicode[STIX]{x1D707}_{0}=\text{mes}_{\unicode[STIX]{x1D719}_{0}}$
. We normalize
$\unicode[STIX]{x1D719}_{0}$
so that
$\unicode[STIX]{x1D707}_{0}$
is a probability measure. The volume of
$(Y,D)$
is
$$\begin{eqnarray}V:=c_{1}(Y,D)^{n}=\int _{X}\unicode[STIX]{x1D714}_{0}^{n}.\end{eqnarray}$$
Definition 5.4. A Kähler–Einstein metric
$T$
for the log Fano pair
$(Y,D)$
is a finite-energy current
$T\in c_{1}(Y,D)$
. such that
$T^{n}=V\cdot \unicode[STIX]{x1D707}_{T}$
.
We now list some important properties of these objects, established in [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Reference BerndtssonBer15, Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ].
(i) A Kähler–Einstein metric
$\unicode[STIX]{x1D714}$
is automatically smooth on
$Y_{0}$
, with continuous potentials on
$Y$
, and it satisfies $$\begin{eqnarray}\operatorname{Ric}(\unicode[STIX]{x1D714}_{\mathit{KE}})=\unicode[STIX]{x1D714}_{\mathit{KE}}+[D]\quad \text{on}~Y_{\text{reg}}.\end{eqnarray}$$
(ii) The definition of a log Fano pair requires the singularities to be klt. This condition is, in fact, necessary to obtain K–E metrics on
$Y_{\text{reg}}$
.(iii) The Kähler–Einstein equation reads
$(\unicode[STIX]{x1D714}_{0}+dd^{c}\unicode[STIX]{x1D719})^{n}=e^{-\unicode[STIX]{x1D719}+c}\unicode[STIX]{x1D707}_{0}$
for some constant
$c\in \mathbb{R}$
. If we choose a log resolution, the equation becomes
$(\unicode[STIX]{x1D714}+dd^{c}\unicode[STIX]{x1D711})^{n}=e^{-\unicode[STIX]{x1D711}+c}\widetilde{\unicode[STIX]{x1D707}}_{0}$
, where
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D714}_{0}$
is semi-positive and big and
$\widetilde{\unicode[STIX]{x1D707}}_{0}=\prod _{i}|f_{i}|^{2a_{i}}\,dV$
.(iv) The potential
$\unicode[STIX]{x1D711}$
belongs to
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
and maximizes the functional Conversely, any maximizer of$$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D711}):=E(\unicode[STIX]{x1D711})+\log \biggl[\int _{\tilde{X}}e^{-\unicode[STIX]{x1D711}}d\widetilde{\unicode[STIX]{x1D707}}_{0}\biggr].\end{eqnarray}$$
${\mathcal{F}}$
is a Kähler–Einstein metric.(v) Two Kähler–Einstein metrics are connected by the flow of a holomorphic vector field that leaves
$D$
invariant.(vi) If the functional
${\mathcal{F}}$
is proper (i.e. if
$E(\unicode[STIX]{x1D711}_{j})\rightarrow -\infty \Rightarrow {\mathcal{F}}(\unicode[STIX]{x1D711}_{j})\rightarrow -\infty$
), then there exists a unique Kähler–Einstein metric.
Here,
$[D]$
is the integration current on
$D|_{Y_{\text{reg}}}$
. Writing
$\operatorname{Ric}(\unicode[STIX]{x1D714}_{\mathit{KE}})$
on
$Y_{\text{reg}}$
implicitly means that the positive measure
$\unicode[STIX]{x1D714}_{\mathit{KE}}^{n}|_{Y_{\text{reg}}}$
corresponds to a singular metric on
$-K_{Y_{\text{reg}}}$
, whose curvature is then
$\operatorname{Ric}(\unicode[STIX]{x1D714}_{KE})$
by definition.
5.3 The analytic criterion
Following an idea of Darvas and Rubinstein [Reference Darvas and RubinsteinDR17], we now extend [Reference TianTia97, Theorem 1.6] and [Reference Phong, Song, Sturm and WeinkovePSSW08] by proving the following.
Theorem 5.5. Let
$(Y,D)$
be a log Fano pair. It admits a unique Kähler–Einstein metric if and only if there exists
$\unicode[STIX]{x1D700},M>0$
such that, for all
$\unicode[STIX]{x1D711}\in {\mathcal{H}}_{\text{norm}}$
,
$$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D711})\leqslant -\unicode[STIX]{x1D700}d_{1}(0,\unicode[STIX]{x1D711})+M.\end{eqnarray}$$
This is Theorem D of the introduction.
Proof. We are going to use Theorem B. Note that
$\unicode[STIX]{x1D714}_{Y}\in c_{1}(-K_{X}-D)$
is a Hodge form. One implication is given by [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorems 4.8 and 5.4]: if
$$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D711})\leqslant -\unicode[STIX]{x1D700}d_{1}(0,\unicode[STIX]{x1D711})+M,\end{eqnarray}$$
then
${\mathcal{F}}$
is proper, hence there exists a unique Kähler–Einstein metric.
So we assume now that there exists
$\unicode[STIX]{x1D714}$
, a unique Kähler–Einstein metric, which we take as our base point of
${\mathcal{H}}_{\unicode[STIX]{x1D714}}$
. It is the unique maximizer of
${\mathcal{F}}$
on
${\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
,
$$\begin{eqnarray}{\mathcal{F}}(0)=\sup _{\unicode[STIX]{x1D711}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})}{\mathcal{F}}(\unicode[STIX]{x1D711}),\end{eqnarray}$$
as follows from [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Theorem 6.6], [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorems 4.8 and 5.3].
Note that
${\mathcal{F}}$
is invariant by translations, so we actually consider the restriction of
${\mathcal{F}}$
on
${\mathcal{E}}_{\text{norm}}^{1}(X,\unicode[STIX]{x1D714})=\{\unicode[STIX]{x1D711}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714}),\sup _{X}\unicode[STIX]{x1D711}=0\}$
. Assume for contradiction that there is no
$\unicode[STIX]{x1D700}>0$
such that
${\mathcal{F}}(\unicode[STIX]{x1D711})\leqslant -\unicode[STIX]{x1D700}d_{1}(0,\unicode[STIX]{x1D711})+M$
for all
$\unicode[STIX]{x1D711}\in {\mathcal{H}}_{\text{norm}}$
, where we set
$M:={\mathcal{F}}(0)+1$
. Then we can find a sequence
$(\unicode[STIX]{x1D711}_{j})\in {\mathcal{H}}_{\unicode[STIX]{x1D714}}^{\mathbb{N}}$
such that
$\sup _{X}\unicode[STIX]{x1D711}_{j}=0$
and
$$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D711}_{j})>-\frac{d_{1}(0,\unicode[STIX]{x1D711}_{j})}{j+1}+{\mathcal{F}}(0)+1.\end{eqnarray}$$
If
$E(\unicode[STIX]{x1D711}_{j})$
does not blow up to
$-\infty$
, we reach a contradiction: up to extracting and relabelling, we can assume that
$E(\unicode[STIX]{x1D711}_{j})$
is bounded and
$\unicode[STIX]{x1D711}_{j}$
converges to some
$\unicode[STIX]{x1D713}\in {\mathcal{E}}^{1}(X,\unicode[STIX]{x1D714})$
. Since
${\mathcal{F}}$
is upper semi-continuous, we infer
${\mathcal{F}}(\unicode[STIX]{x1D713})\geqslant {\mathcal{F}}(0)+1$
, a contradiction.
So we assume now that
$E(\unicode[STIX]{x1D711}_{j})\rightarrow -\infty$
. It follows from Lemma 3.12 that
$d_{j}:=d_{1}(0,\unicode[STIX]{x1D711}_{j})=-E(\unicode[STIX]{x1D711}_{j})\rightarrow +\infty$
. We let
$(\unicode[STIX]{x1D711}_{t,j})_{0\leqslant t\leqslant d_{j}}$
denote the Mabuchi geodesic with unit speed joining
$0$
to
$\unicode[STIX]{x1D711}_{j}$
and set
$\unicode[STIX]{x1D713}_{j}:=\unicode[STIX]{x1D711}_{1,j}$
. Note that the arguments in Lemma 3.3 show that
$t\mapsto \unicode[STIX]{x1D711}_{t,j}$
is decreasing, hence
$\unicode[STIX]{x1D711}_{j}\leqslant \unicode[STIX]{x1D713}_{j}\leqslant 0$
. In particular,
$\sup _{X}\unicode[STIX]{x1D713}_{j}=0$
, while, by definition,
$d_{1}(0,\unicode[STIX]{x1D713}_{j})=1=-E(\unicode[STIX]{x1D713}_{j})$
.
It now follows from Berndtsson’s convexity result [Reference BerndtssonBer15, § 6.2] and its generalization to the singular context [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorem 11.1] that the map
$t\mapsto {\mathcal{F}}(\unicode[STIX]{x1D711}_{t,j})$
is concave. We infer
$$\begin{eqnarray}0\geqslant {\mathcal{F}}(\unicode[STIX]{x1D711}_{1,j})-{\mathcal{F}}(\unicode[STIX]{x1D711}_{0,j})\geqslant \frac{{\mathcal{F}}(\unicode[STIX]{x1D711}_{d_{j},j})-{\mathcal{F}}(\unicode[STIX]{x1D711}_{0,j})}{d_{j}}>-\frac{1}{j+1}+\frac{1}{d_{j}},\end{eqnarray}$$
thus
${\mathcal{F}}(\unicode[STIX]{x1D713}_{j})\rightarrow {\mathcal{F}}(0)$
. This shows that
$(\unicode[STIX]{x1D713}_{j})$
is a maximizing sequence for
${\mathcal{F}}$
, which therefore strongly converges to
$0$
, by [Reference Berman, Boucksom, Eyssidieux, Guedj and ZeriahiBBEGZ, Theorem 5.3.3]. This yields a contradiction, since
$d_{1}(0,\unicode[STIX]{x1D713}_{j})=1$
.◻
Acknowledgements
We thank Darvas, Lu and Zeriahi for useful conversations.
E.D.N. was supported by a Marie Sklodowska Curie individual fellowship 660940KRFCY (MSCAIF). V.G. was partially supported by the ANR project GRACK.
This paper is based on work supported also by the NSF Grant DMS-1440140, while E.D.N. was in residence at the MSRI, during the Spring 2016 semester. It is partially based on lecture notes of V.G. [Reference GuedjGue14], after a series of lectures he gave at KIAS in April 2013. The authors thank J.-M. Hwang and M. Paun for their invitation and the staff of KIAS and MSRI for providing excellent conditions of work.
