Lemma 13.13.51.9453F71CEE8F Assume that $\omega $ is a Kähler form on $X$ and let $f_1,f_2\in C^{1,\overline{1}}(X)$. Then the functions $f_1$, $f_2$ and $P_\omega (\min \{ f_1,f_2\} )$ coincide up to second order at almost every point of the contact set $\{ P_\omega (\min \{ f_1,f_2\} )=f_1=f_2\} $. In particular,
\[ \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1=f_2\right\} }\, \omega _{f_1}^ n=\mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1=f_2\right\} }\, \omega _{f_2}^ n. \]
We also have for $j\in \{ 1,2\} $,
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13.13.52
\begin{equation} \begin{aligned} \omega ^ n_{P_\omega \left(\min \{ f_1,f_2\} \right)} = & \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_1\right\} }\, \omega ^ n_{f_1} + \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_2\right\} }\, \omega ^ n_{f_2} \\ & - \mathds {1}_{\left\{ P_\omega (\min \{ f_1,f_2\} )=f_1=f_2\right\} }\, \omega ^ n_{f_ j}\\ \geq & \mathds {1}_{\left\{ P_\omega \left(\min \{ f_1,f_2\} \right)=f_ j\right\} }\, \omega ^ n_{f_ j}. \end{aligned} \end{equation}
Proof.
Set $\varphi \coloneqq P_\omega (\min \{ f_1,f_2\} )$. Since $\min \{ f_1,f_2\} \in C^0(X)$, we have $\varphi \leq \min \{ f_1,f_2\} $. By proposition 13.13.50, $\varphi \in C^{1,\overline{1}}(X)$. As $f_1,f_2,\varphi \in C^{1,\overline{1}}(X)$, the measures $\omega ^ n_{f_1}$, $\omega ^ n_{f_2}$ and $\omega ^ n_\varphi $ are all absolutely continuous.
Step 1: We show that at almost every contact point, $\varphi $ and $f_ i$ agree to order two.
For $i\in \{ 1,2,3\} $ set
\[ \Psi _1\coloneqq \varphi -f_1,\quad \Psi _2\coloneqq \varphi -f_2,\quad \Psi _3\coloneqq f_1-f_2. \]
Each $\Psi _ i$ lies in $C^{1,\overline{1}}(X)$.
We claim that for $i=1,2,3$, we have
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13.13.53
\begin{equation} \Psi _ i\textup{ vanishes to second order at almost every point of }\{ \Psi _ i=0\} . \end{equation}
To prove this, write $\{ \Psi _ i=0\} $ as the disjoint union $A_ i\sqcup B_ i\sqcup C_ i\sqcup D_ i$ of the following four sets:
$A_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i\neq 0\} $;
$B_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \textup{the classical real Hessian of }\Psi _ i\textup{ does not exist}\} $;
$C_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i=0\} \cap \{ \textup{the classical real Hessian exists and is non-zero}\} $;
$D_ i\coloneqq \{ \Psi _ i=0\} \cap \{ \mathrm{d}\Psi _ i=0\} \cap \{ \textup{the classical real Hessian exists and equals zero}\} $.
It suffices to prove $A_ i$, $B_ i$, $C_ i$ are Lebesgue null sets.
The set $A_ i$ is a $C^1$ real hypersurface in $X$, hence has Lebesgue measure zero.
Since $\Psi _ i\in W^{2,1}_{\mathrm{loc}}$, the classical pointwise real Hessian of $\Psi _ i$ exists almost everywhere, so $B_ i$ is Lebesgue null.
Finally $C_ i$ is Lebesgue null by Stampacchia's lemma. This proves (13.13.53).
Specialising (13.13.53) to $\Psi _1,\Psi _2$, the functions $\varphi $ and $f_ i$ agree to order two at almost every point of the contact set $\{ \varphi =f_ i\} $. Consequently $\omega +\mathrm{dd}^{\mathrm{c}}\varphi $ and $\omega +\mathrm{dd}^{\mathrm{c}}f_ i$ have the same matrix of coefficients almost everywhere on $\{ \varphi =f_ i\} $, and therefore so do their $n$-fold wedge powers:
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13.13.54
\begin{equation} \mathds {1}_{\{ \varphi =f_ i\} }\, \omega ^ n_\varphi = \mathds {1}_{\{ \varphi =f_ i\} }\, \omega ^ n_{f_ i},\qquad i\in \{ 1,2\} . \end{equation}
The analogous statement applied to $\Psi _3$ gives
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13.13.55
\begin{equation} \mathds {1}_{\{ f_1=f_2\} }\, \omega ^ n_{f_1} = \mathds {1}_{\{ f_1=f_2\} }\, \omega ^ n_{f_2}. \end{equation}
Step 2. The barrier $\min \{ f_1,f_2\} $ is continuous, hence quasi-continuous, so theorem 13.13.24 yields
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13.13.56
\begin{equation} \int _{X\setminus \left(\{ \varphi =f_1\} \cup \{ \varphi =f_2\} \right)}\omega ^ n_\varphi = \int _{\left\{ \varphi <\min \{ f_1,f_2\} \right\} }\omega ^ n_\varphi =0. \end{equation}
Set $V_1\coloneqq \{ f_1<f_2\} $ and $V_2\coloneqq \{ f_1>f_2\} $, both open. We may decompose
\[ \{ \varphi =f_1\} \cup \{ \varphi =f_2\} = \bigl (V_1\cap \{ \varphi =f_1\} \bigr ) \sqcup \bigl (V_2\cap \{ \varphi =f_2\} \bigr ) \sqcup \{ \varphi =f_1=f_2\} . \]
This combined with (13.13.56) and (13.13.54) yields
\[ \begin{aligned} \omega ^ n_\varphi ={}& \mathds {1}_{V_1\cap \{ \varphi =f_1\} }\, \omega ^ n_\varphi + \mathds {1}_{V_2\cap \{ \varphi =f_2\} }\, \omega ^ n_\varphi + \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_\varphi \\ ={}& \mathds {1}_{V_1\cap \{ \varphi =f_1\} }\, \omega ^ n_{f_1} + \mathds {1}_{V_2\cap \{ \varphi =f_2\} }\, \omega ^ n_{f_2} + \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_{f_ j}. \end{aligned} \]
On the other hand,
\[ \mathds {1}_{V_ j\cap \{ \varphi =f_ j\} }\, \omega ^ n_{f_ j} = \mathds {1}_{\{ \varphi =f_ j\} }\, \omega ^ n_{f_ j} - \mathds {1}_{\{ \varphi =f_1=f_2\} }\, \omega ^ n_{f_ j}. \]
Therefore,
\[ \omega ^ n_\varphi = \mathds {1}_{\{ \varphi =f_1\} }\, \omega ^ n_{f_1} + \mathds {1}_{\{ \varphi =f_2\} }\, \omega ^ n_{f_2} + \mathds {1}_{\{ \varphi =f_1=f_2\} } \, \bigl (\omega ^ n_{f_ j}-\omega ^ n_{f_1}-\omega ^ n_{f_2}\bigr ). \]
Together with (13.13.55) we obtain (13.13.52).
$\square$
Pome
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