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\begin{document}

\title{Volume and regularity for big $(1,1)-$classes on Kähler manifolds }


\author{Robert Berman, Jean-Pierre Demailly}


\email{robertb@math.chalmers.se, Jean-Pierre.Demailly@ujf-grenoble.fr}


\curraddr{Institut Fourier, 100 rue des Maths, BP 74, 38402 St Martin d'Heres
(France)}

\maketitle
Let $(X,\omega)$ be a Kähler manifold. The aim in this preliminary
note is to prove conjecture $B$ below. But for the moment we can
only prove that it is equivalent to a Monge-Ampere estimate (conjecture
$A)$ which gives a {}``comparison principle'' for big classes with
a quantative controle on the influence of the singularities of the
functions.

\begin{conjecture*}
(A) Let $\alpha,\beta$ be smooth forms such that the classes $[\alpha],[\beta]$
in $H^{1,1}(X,\R)$ are ample and big respectively and let $\gamma_{\epsilon}=\beta-\epsilon\alpha,$
with $\epsilon\in]0,1].$ If $u_{\epsilon}$ is a $\gamma_{\epsilon}-$psh
function with minimal singularities the following estimate holds:\[
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{0})^{n}\leq\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{\epsilon})^{n}+o(\epsilon)\]

\end{conjecture*}
We will show that this conjecture is equivalent to the following main
conjecture in \cite{b-d-p-p}:

\begin{conjecture*}
(B) \label{thm:volume bound}Let $[\alpha_{0}]$ and $[\alpha_{1}]$
be two nef classes in $H^{1,1}(X,\R)$. Then \begin{equation}
\textrm{Vol\,}([\alpha_{0}]-[\alpha_{1}])\geq[\alpha_{0}]^{n}-n[\alpha_{0}]^{n-1}\wedge[\alpha_{1}].\label{eq:siu}\end{equation}
In particular $[\alpha_{0}]-[\alpha_{1}]$ is big if the right hand
side above is positive.
\end{conjecture*}
Since, conjecture $B$ does hold for integer classes (as a consequence
of the holomorphic Morse inequalities) the equivalence between the
conjectures then shows that conjecture $A$ too holds in this case.
Of course, one way to prove conjecture $B$ in general would hence
be to give an independent proof of conjecture $A$ valid for \emph{any}
big class in $H^{1,1}(X,\R).$


\section{Regularity of psh envelopes}

Let $[\alpha]$ be a pseudo-effective class in $H^{1,1}(X,\R)$ and
$\alpha$ a continuous representative. The corresponding \emph{$\alpha-$psh
envelope} $u$ is defined by \begin{equation}
u(x)=\sup_{\widetilde{u}\in\textrm{Psh$(X,\alpha)$}}\{\widetilde{u}(x):\,\widetilde{u}\leq0\}.\label{def:psh env}\end{equation}
Then $u$ is in $\textrm{Psh$(X,\alpha)$ }$\cite{g-z} and clearly
has \emph{minimal singularities.} Hence, if $[\alpha]$ is a big class
there is a Zariski-open set $\Omega$ (only depending on the class
$[\alpha])$ such that $u$ is locally bounded on $\Omega.$ It follows,
by the classical work of Bedford-Taylor, that the \emph{Monge-Ampere
measure} $(\alpha+dd^{c}u)^{n}$ is a well-defined positive measure
on $\Omega$ that we, extending by zero, identify with a measure on
$X.$ 

The following regularity theorem extends the corresponding result
in \cite{berm5} for integral classes.

\begin{thm}
\label{thm:regularity}Suppose that $[\alpha]$ is a big class in
with a smooth representative $\alpha.$ Let $u$ be the corresponding
$\alpha-$psh envelope. Then the restriction of $u$ to the Zariski-open
set $\Omega$ is in the class $\mathcal{C}^{1,1}(\Omega).$ In particular,
the second derivatives of $u$ exist almost everywhere on $\Omega$
and are locally bounded there. Moreover, \[
1_{\Omega}(\alpha+dd^{c}u)^{n}=1_{D}\alpha^{n}\]
 as measures on $X,$ where $D:=\{ u=0\}.$
\end{thm}
{[}add proof{]}


\section{Variation of volume }

Let $T$ be a big class in $H^{1,1}(X,\R).$ Following \cite{bo,b-d-p-p}
we define \begin{equation}
\textrm{Vol}\, T=\sup_{\alpha}\int_{X-\textrm{sing $(\alpha)$}}\alpha^{n}\label{def:vol as sup of anal}\end{equation}
where the supremum is taken over all positive currents $\alpha\in T$
with analytic singularities that we denote by $\textrm{sing $(\alpha)$}.$
If $T$ is merely pseudo-effective its volume may be defined as the
limit of $\textrm{Vol}\, T_{\epsilon},$ where $T_{\epsilon}=T+\epsilon[\omega],$
when $\epsilon>0$ tends to zero. As shown in \cite{bo} $\textrm{Vol}\, T>0$
precisely when $T$ is big.

Since, as observed in \cite{b-b}, the total mass of the Monge-Ampere
measure of any function in $\textrm{Psh$(X,\alpha)$ }$with minimal
singularities gives the volume of a big class $[\alpha]$ we have
the following consequence of the regularity Theorem \ref{thm:regularity}.

\begin{prop}
\label{pro:vol in terms of psh env}Let $[\alpha]$ be a big class
in $H^{1,1}(X,\R)$ and $\alpha$ a smooth representative. Then \[
\textrm{Vol\,}[\alpha]=\int_{\Omega}(\alpha+dd^{c}u)^{n}=\int_{D}\alpha^{n}\]
 where $u$ is the corresponding $\alpha-$psh envelope. 
\end{prop}
The proof in \cite{b-b} of the first equality in the previous proposition
just uses a simple comparison principle for the total mass of Monge-Ampere
measures. More generally, the following {}``moving'' or {}``positive''
intersection product is well-defined for $\theta_{1},...,\theta_{p}\in H^{1,1}(X,\R)$
big cohomology classes and $T_{i}\in\theta_{i}$ any positive current
with minimal singularities:\begin{equation}
\langle\theta_{1}\cdot...\cdot\theta_{p}\rangle:=1_{\Omega}T_{1}\wedge...\wedge T_{p}:\label{eq:pos prod}\end{equation}
 where $\Omega$ denotes any Zariski open set where the potentials
of $T_{i}$ are locally bounded (see the appendix in \cite{b-b}).
Will have use for the following continuity property: 

\begin{lem}
\label{lem:cont}Let $\alpha,\beta$ be smooth forms such that the
classes $[\alpha],[\beta]$ in $H^{1,1}(X,\R)$ are ample (with $\alpha>0)$
and big respectively and let $\gamma_{\epsilon}=\beta-\epsilon\alpha,$
with $\epsilon\in]0,1].$ Then if $u_{\epsilon}$ is the $\gamma_{\epsilon}-$psh
envelope \ref{def:psh env} we have that $u_{\epsilon}$ increases
to $u$ and $1_{\Omega_{1}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{p}$
converges weakly to $1_{\Omega_{1}}(\gamma_{0}+dd^{c}u_{0})^{p}$
as currents.
\end{lem}
\begin{proof}
First observe that $u_{\epsilon}(\leq0)$ increases when $\epsilon$
tends to zero and that the point-wise limit of $u_{\epsilon}$ equals
$u_{0}$ a.e. Accepting this for the moment the convergence of the
corresponding Monge-Ampere operators then follow from Theorem 11.6
in \cite{b-b}. Finally, to see that $u_{\epsilon}$ is increasing
observe that if $\epsilon'<\epsilon$ then $u_{\epsilon}$ is $\gamma_{\epsilon'}-$psh
(using that $\alpha>0).$ Since also $u_{\epsilon}\leq0$ it follows,
by the extremal definition of $u_{\epsilon'},$ that $u_{\epsilon}\leq u_{\epsilon'}.$
In particular, the point-wise limit $U$ of $u_{\epsilon}$ exists
and its u.s.c regularization $U^{*}$ is $\gamma_{0}-$psh (where
$U^{*}=U$ a.e.). To conclude the proof it is now enough to prove
that \begin{equation}
U\geq u_{0}\label{eq:pf of lemma cont}\end{equation}
 a.e. on $X.$ To this end fix $\delta>0$ and let $u_{0,\delta}:=u_{0}(1-\delta)+\delta u_{+}$
where $\gamma_{0}+dd^{c}u_{+}>\delta\alpha$ and $u_{+}\leq0,$ where
$u_{+}$ is locally bounded on a Zariski-open set. Comparing $u_{0,\delta}$
and $u_{\epsilon}$ as above then gives, $u_{0,\delta}\leq u_{\epsilon}$
for $\epsilon$ sufficently small. In particular, $u_{0,\delta}\leq U$
and letting $\delta$ tend to zero finally proves \ref{eq:pf of lemma cont}.
\end{proof}
If the following conjectured estimate holds then one obtains as a
consequence a proof of the main conjecture $B$ below. 

\begin{conjecture*}
(A) \label{con:dom}Let $\alpha,\beta$ be smooth forms such that
the classes $[\alpha],[\beta]$ in $H^{1,1}(X,\R)$ are ample (with
$\alpha>0)$ and big respectively and let $\gamma_{\epsilon}=\beta-\epsilon\alpha,$
with $\epsilon\in]0,1].$ If $u_{\epsilon}$ is a $\gamma_{\epsilon}-$psh
function with minimal singularities the following estimate holds:
\begin{equation}
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{0})^{n}\leq\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{\epsilon})^{n}+o(\epsilon)\label{eq:conj A a}\end{equation}
 In particular, if $u_{\epsilon}$ is the $\gamma_{\epsilon}-$psh
envelope \ref{def:psh env} one then obtains\begin{equation}
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}\gamma_{0}^{n}\leq n\epsilon(\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha+o(\epsilon)\label{eq:conj A b}\end{equation}
 
\end{conjecture*}
To see that the inequality \ref{eq:conj A a} implies the inequality
\ref{eq:conj A b} write $\gamma_{0}=\gamma_{\epsilon}+\epsilon\alpha.$
Expanding the $n$ th power in the r.h.s. of \ref{eq:conj A a} and
using that, for the $\gamma_{\epsilon}-$psh envelopoe $u_{\epsilon},$
the Monge-Ampere measure $(\gamma_{0}+dd^{c}u_{\epsilon})^{n}$ vanishes
on $\Omega_{\epsilon}\cap\{ u_{\epsilon}<0\}$ (which contains $\{ u_{\epsilon}<u_{0}\})$
then gives \[
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{0})^{n}\leq\epsilon\int_{\Omega_{1}\cap\{ u_{\epsilon}<0\}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha+O(\epsilon^{2})+o(\epsilon),\]
 which is precisely the inequality \ref{eq:conj A b}. 

The following main conjecture was stated in \cite{b-d-p-p}:

\begin{conjecture*}
(B) \label{thm:volume bound}Let $[\alpha_{0}]$ and $[\alpha_{1}]$
be two nef classes in $H^{1,1}(X,\R)$. Then \begin{equation}
\textrm{Vol\,}([\alpha_{0}]-[\alpha_{1}])\geq[\alpha_{0}]^{n}-n[\alpha_{0}]^{n-1}\wedge[\alpha_{1}].\label{eq:siu}\end{equation}
In particular $[\alpha_{0}]-[\alpha_{1}]$ is big if the right hand
side above is positive.
\end{conjecture*}

\subsection{Conjecture $A$ i\label{sub:Conjecture A impl}mplies conjecture
$B$}

\emph{Step 1: assume that $[\alpha_{0}],[\alpha_{1}]$ are ample and
$[\alpha_{0}]-[\alpha_{1}]$ is big.}

We may assume that the representatives $\alpha_{i}$ are smooth and
positive. Let $\beta_{t}=\alpha_{0}-t\alpha_{1},$ where $t\in[0,1].$
Then $[\beta_{t}]$ is big by assumption. By the log-concavity of
$\textrm{Vol}$ on the open big cone \cite{bo}, the directional derivative
\[
\frac{d}{dt}_{+}\textrm{Vol}[\beta_{t}]:=\lim_{0<\epsilon\rightarrow0}\frac{\textrm{Vol}[\beta_{t+\epsilon}]-\textrm{Vol}[\beta_{t}]}{\epsilon}\]
exists. To prove \ref{eq:siu} it is now enough to prove that 

\begin{equation}
\frac{d}{dt}_{+}\textrm{Vol}[\beta_{t}]\geq-n[\alpha_{0}]^{n-1}\wedge[\alpha_{1}],\label{eq:bd on dir der}\end{equation}
 since integrating over $t\in[0,1]$ then immediately gives the bound
\ref{eq:siu}. To this end fix $t$ and let $\gamma_{\epsilon}=\beta_{t+\epsilon}:=\beta_{t}-\epsilon\alpha_{1}$
(we will also write $\beta_{t}:=\gamma_{0}$ and $\alpha:=\alpha_{1}).$
\begin{equation}
\textrm{Claim:\,\,\,\,$\frac{d}{d\epsilon}_{+}$ Vol\,}[\gamma_{\epsilon}]\geq n\int_{\Omega}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha)\label{eq:claim for der}\end{equation}
 In the following we will simplify the notation by writing $\int_{\Omega}:=\int,$
where $\Omega$ denotes any Zariski open set. To prove the claim first
observe that by Proposition \ref{pro:vol in terms of psh env} we
have the identity \begin{equation}
\textrm{Vol}[\gamma_{\epsilon}]=\int(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n}=\int_{D_{\epsilon}}\gamma_{\epsilon}^{n},\label{eq:vol id}\end{equation}
where $D_{\epsilon}=\{ u_{\epsilon}=0\}.$ Hence, using the {}``exact
discrete Leibniz rule'' gives \[
\textrm{Vol}[\gamma_{\epsilon}]-\textrm{Vol}[\gamma]=\int_{_{D_{0}}}(\gamma_{\epsilon}^{n}-\gamma_{0}^{n})+\int_{_{D_{\epsilon}-D_{0}}}\gamma_{0}^{n}+\int_{_{D_{\epsilon}-D_{0}}}(\gamma_{\epsilon}^{n}-\gamma_{0}^{n}).\]
 Observing that (by lemma \ref{lem:cont}) $D_{\epsilon}$ is contained
in $D_{0}$ and using that $\gamma_{\epsilon}^{n}=\gamma^{n}+\epsilon n\gamma^{n-1}\wedge(-\alpha)+O(\epsilon^{2})$
we may then write the r.h.s above as \begin{equation}
\epsilon n\int_{D_{0}}\gamma_{0}^{n-1}\wedge(-\alpha)-\int_{D_{0}-D_{\epsilon}}\gamma_{0}^{n}+\epsilon n\int_{_{D_{0}-D_{\epsilon}}}\gamma_{0}^{n-1}\wedge\alpha+O(\epsilon^{2})\label{eq:difference of vol}\end{equation}
 By the inequality \ref{eq:conj A b} in conjecture $A$ the middle
term above may be estimated by \[
-\int_{D_{0}-D_{\epsilon}}\gamma_{0}^{n}\geq\int_{X-D_{0}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha+\int_{D_{0}-D_{\epsilon}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha,\]
 where we in the l.h.s have used that $D_{0}-D_{\epsilon}=\{ u=0,\, u_{\epsilon}<0\}\subset\{ u_{\epsilon}<u\}$
and in the r.h.s that $\{ u_{\epsilon}<u\}\subset\{ u_{\epsilon}<0\}=X-D_{0}\bigsqcup(D_{0}-D_{\epsilon})$
and that $dd^{c}u_{\epsilon}=0$ a.e. on $D_{0}$ (by Theorem \ref{thm:regularity}).
All in all we deduce that $(\textrm{Vol}[\gamma_{\epsilon}]-\textrm{Vol}[\gamma])/\epsilon n\geq$\[
\int_{D_{0}}\gamma_{0}^{n-1}\wedge(-\alpha)+\int_{X-D_{0}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge(-\alpha)\]
\[
+\int_{D_{0}-D_{\epsilon}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge(-\alpha)+\int_{_{D_{0}-D_{\epsilon}}}\gamma_{0}^{n-1}\wedge\alpha+o(\epsilon)/\epsilon,\]
 which may be written (again using that $dd^{c}u_{\epsilon}=0$ a.e.
on $D_{0}$) as \[
\int_{X}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge(-\alpha)+\int_{D_{0}}(\gamma_{0}^{n-1}-\gamma_{\epsilon}^{n-1})\wedge(-\alpha)\]
\[
0+\int_{_{D_{0}-D_{\epsilon}}}(\gamma_{0}^{n-1}-\gamma_{\epsilon}^{n-1})\wedge\alpha+o(\epsilon)/\epsilon.\]
But since $(\gamma_{0}^{n-1}-\gamma_{\epsilon}^{n-1})=O(\epsilon)$
on $D$ the previous expression becomes \[
\int(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge(-\alpha)+O(\epsilon)+o(\epsilon)/\epsilon.\]
Finally, applying the continuity property given by lemma \ref{lem:cont}
proves the claim \ref{eq:claim for der}.

Finally, by the same claim we have (in terms of the notation of positive
products in \ref{eq:pos prod}) that $\frac{d}{dt}_{+}\textrm{Vol}[\beta_{t}]\geq$
\[
-n\limsup_{\epsilon}\left\langle \alpha_{0}-t\alpha_{1}\right\rangle ^{n-1}\wedge(-\alpha_{1})\geq-n\int_{X}(\alpha_{0})^{n-1}\wedge(-\alpha_{1}),\]
(using that $\alpha_{0}-t\alpha_{1}\leq\alpha_{0}$ by positivity
and a simple monotonicity property of positive products \cite{b-b})
which finishes the proof of \ref{eq:bd on dir der}. 

\begin{proof}
\emph{Step 2: the general case.}

First observe that by a standard approximation argument we may assume
that $[\alpha_{1}]$ and $[\alpha_{2}]$ are ample. Now let $s$ be
the supremum over all $t\in[0,1]$ such that $[\beta_{t}](=[\alpha_{0}]-t[\alpha_{1}])$
is big. By step 1 we are done if we can prove that $s=1.$ To this
end first observe that for all $t$ as above \begin{equation}
\textrm{Vol}\,[\beta_{t}]\geq C>0.\label{pf of thm vol: lower bd vol}\end{equation}
 Indeed, since $\beta_{t}$ is big step 1 above gives the lower bound
\ref{eq:siu} (with $\alpha_{1}$ replaced by $t\alpha_{1})$ on $\textrm{Vol}\,[\beta_{t}],$
which in turn is bounded from below by $C=[\alpha_{0}]^{n}-n[\alpha_{0}]^{n-1}\wedge[\alpha_{1}],$
which is positive by assumption. This proves \ref{pf of thm vol: lower bd vol}.
But the continuity of the volume function (on all of the pseudo-effective
cone) \cite{bo} then implies that \ref{pf of thm vol: lower bd vol}
holds for the limit $\beta_{s}$ as well. In other words $\beta_{s}$
has positive volume and is hence big \cite{bo}. Since bigness is
an open condition we must have $s=1$ by the extremal definition of
$s.$ This finally concludes the proof.
\end{proof}

\section{Conjecture $B$ implies conjecture $A$}

Let us first observe that the following lemma gives a weak version
of conjecture $A$:

\begin{lem}
\label{lem:domination}Let $\alpha,\beta$ be smooth forms such that
the classes $[\alpha],[\beta]$ in $H^{1,1}(X,\R)$ are ample (with
$\alpha>0)$ and big respectively and let $\gamma_{\epsilon}=\beta-\epsilon\alpha,$
with $\epsilon\in]0,1].$ If $u_{\epsilon}$ is a $\gamma_{\epsilon}-$psh
function with minimal singularities the following estimate holds:
\[
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{0})^{n}\leq\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{\epsilon})^{n}-\delta(\epsilon)\]
 where \[
\lim_{0<\epsilon\rightarrow0}\delta(\epsilon)/\epsilon=\frac{d\textrm{Vol}[\gamma_{\epsilon}]}{d\epsilon}_{\epsilon=0}-n\int_{\Omega_{0}}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha).\]
In particular, if $u_{\epsilon}$ is the $\gamma_{\epsilon}-$psh
envelope \ref{def:psh env} one then obtains\[
\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{0}+dd^{c}u_{0})^{n}\leq n\epsilon(\int_{\Omega_{1}\cap\{ u_{\epsilon}<u_{0}\}}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha-\delta(\epsilon)\]

\end{lem}
\begin{proof}
First observe that the following general {}``comparison principle''
holds: let $\gamma$ represent a big class. If $v,v'$ are $\gamma-$psh
weights on $L$ that are locally bounded on a Zariski open subset
$\Omega$ then, if $v'$ has minimal singularities, \[
\int_{\{ v<v'\}}\MA(v')\leq\int_{\{ v<v'\}}\MA(v)+\int_{X}(\MA(v')-\MA(v))\]
where $\MA(v):=1_{\Omega}(\gamma+dd^{c}v)^{n}.$ Too see this we repeat
the proof of Proposition 2.6 in \cite{b-b}. Let $\e>0$. By Bedford-Taylor
(see \cite{d}) the following identity holds locally:\[
1_{\{ v<v'\}}(dd^{c}v')^{n}=1_{\{ u<v\}}(dd^{c}\max(v,v'))^{n}.\]
 Of course, this is trivial if $v,v'$ are continuous so that $\{ v<v'\}$
is open. In fact, in our case this will allways be the case. Now we
have \[
\int_{\{ v<v'-\e\}}\MA(v')=\int_{\{ v<v'-\e\}}\MA(v'-\e)=\int_{\{ v<v'-\e\}}\MA(\max(v,v'-\e))\]
 \[
\leq\int_{X}\MA(\max(v,v'-\e))-\int_{\{ v>v'-\e\}}\MA(\max(v,v'-\e))=\]
\[
=\int_{X}\MA(\max(v,v'-\e))-\int_{\{ v>v'-\e\}}\MA(v)=\]


\[
=\int_{X}\MA(\max(v,v'-\e))-(\int_{X}\MA(v)-\int_{\{ v\leq v'-\e\}}\MA(v)\leq\]
\[
(\int_{X}\MA(v')-\int_{X}\MA(v))+\int_{\{ v\leq v'-\e\}}\int_{\{ v<v'\}}\MA(v),\]
 where we have used that $\max(v,v'-\e)\geq v'-\e$ has minimal singularities,
since $v'$ does, so that their total Monge-Ampere masses coincide
(and equal $\textrm{Vol $[\gamma]$).}$ The result now follows by
letting $\e\rightarrow0$. 

Next, setting $v=u_{\epsilon}$ and $v'=u_{0}$ gives \[
(\int_{X}\MA(v')-\int_{X}\MA(v))=(\int_{\Omega}(\gamma_{0}+dd^{c}u_{0})^{n}-\int_{\Omega}(\gamma_{0}+dd^{c}u_{\epsilon})^{n})=\]
\[
=(\int_{\Omega}(\gamma_{0}+dd^{c}u_{0})^{n}-\int_{\Omega}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n})-\epsilon n\int_{\Omega}(\gamma_{\epsilon}+dd^{c}u_{\epsilon})^{n-1}\wedge\alpha+O(\epsilon^{2})\]
 where we have used that $\gamma_{0}=\gamma_{\epsilon}+\epsilon\alpha$
to expand the $n$ th power. Finally, applying the first equality
in Proposition \ref{pro:vol in terms of psh env} and the continuity
property in Lemma \ref{lem:cont} then concludes the proof of the
lemma. 
\end{proof}
\begin{rem}
The present description of the limit of $\delta(\epsilon)/\epsilon$
as the derivative of the volume appears unfortunately to be useless
for our purposes, i.e. for proving Conjecture $B.$ The problem is
that when using the previous arguments to prove the following following
crucial claim in section \ref{sub:Conjecture A impl}\[
\textrm{Claim:\,\,\,\,$\frac{d}{d\epsilon}_{+}$ Vol\,}[\gamma_{\epsilon}]\geq n\int_{X}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha_{1})\]
 one only gets, using the new error term $\delta(\epsilon):$\[
\textrm{$\frac{d}{d\epsilon}_{+}$ Vol\,}[\gamma_{\epsilon}]\geq n\int_{X}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha_{1})+\lim_{0<\epsilon\rightarrow0}\delta(\epsilon)/\epsilon,\]
 i.e. \[
\textrm{$\frac{d}{d\epsilon}_{+}$ Vol\,}[\gamma_{\epsilon}]\geq n\int_{X}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha_{1})+\]
\[
+\textrm{$\frac{d}{d\epsilon}_{+}$ Vol\,}[\gamma_{\epsilon}]-n\int_{\Omega_{0}}(\gamma_{0}+dd^{c}u_{0})^{n-1}\wedge(-\alpha).\]
which is equivalent to the fact that $0\geq0!$ This is indeed a true,
but not very useful fact...
\end{rem}
However, \emph{a posteriori} one should actually have $\delta(\epsilon)=o(\epsilon)$
in the lemma above, i.e. \begin{equation}
\frac{d\textrm{Vol}[\gamma_{\epsilon}]}{d\epsilon}_{\epsilon=0}=n\left\langle \gamma_{0}\right\rangle ^{n-1}\wedge(-\alpha)\label{eq:form for deriv as pos}\end{equation}
 in terms of the positive intersection product \ref{eq:pos prod}.
This means that the inequality in the claim above should actually
be an equality. In fact, this equality does holds in the case of integer
classes, as shown in \cite{b-f-j}. In general, the previous lemma
imples the converse relation between the two conjectures above. Indeed,
in \cite{b-f-j} it was shown how to deduce formula \ref{eq:form for deriv as pos}
from the bound in conjecture $B.$ Of course, the point in \cite{b-f-j}
was that the later conjecture does hold in the case of integer classes
(as a well-known consequence of the holomorphic Morse inequalities). 

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