Proof.
First observe that by proposition 18.28.45, for any $k>0$, $N^ k_{v,K}$ and $N^ k_{v',K'}$ are both norms, hence the expressions inside the limit in (18.28.69) make sense. Moreover, (18.28.70) is just a special case of (18.28.69) for $K'=X$ and $v'= 0$.
To prove (18.28.69) it is enough to show that for any fixed $w \in C^\infty (X)$ we have
AC1B69834497
18.28.71
\begin{equation} \lim _{k\to \infty } \big ( \mathcal{L}_{k,\varphi }(N^ k_{v,K}) - \mathcal{L}_{k,\varphi }(N^ k_{w,X}) \big ) = \mathcal{E}_{\theta ,K}^{\varphi }(v) - \mathcal{E}_{\theta ,X}^{\varphi }(w)\, . \end{equation}
For $\epsilon \in (0,1)$ small enough, $\theta _{(1-\epsilon )\varphi }$ is still a Kähler current. Let us fix such $\epsilon $, along with an arbitrary $\epsilon ' \in (0,1)$.
Let $(v^-_{j})_ j$ and $(v^+_{j})_ j$ be the sequences of smooth functions constructed in 18.28.31 for the data $(K,v)$.
By proposition 5.20.17 there exists $k_0(\epsilon ,\epsilon ') \in \mathbb N$ such that
\[ \frac{1}{k} \log h^ k \otimes h_ T (s,s) \preceq (1-\epsilon )\varphi , \]
and $\frac{1}{k} \log h^ k \otimes h_ T (s,s) \in \mathrm{PSH}(X,\theta + \epsilon '\omega )$ for any $s \in \mathrm{H}^0(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi ))$, as long as $k \geq k_0(\epsilon ,\epsilon ')$.
In particular, 18.28.65 gives that
\[ \begin{aligned} N^ k_{P'_{\theta + \epsilon '\omega ,K}[(1-\epsilon )\varphi ](v),X}(s)=& N^ k_{v,K}(s),\\ N^ k_{P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^-),X}(s)=& N^ k_{v_{j}^-,X}(s),\\ N^ k_{P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^+),X}(s)=& N^ k_{v^+_{j},X}(s). \end{aligned} \]
As
\[ P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^-) \leq P'_{\theta + \epsilon '\omega ,K}[(1-\epsilon )\varphi ](v) \leq P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^+), \]
where the second inequality follows from 18.28.31 and the fact that $P'=P$ for the global continuous obstacles $v_ j^\pm $. By 18.28.42 we have
\[ N^ k_{v^+_{j},X}(s) \leq N^ k_{v,K}(s) \leq N^ k_{v_{j}^-,X}(s), \quad s \in \mathrm{H}^0(X,T \otimes L^ k \otimes \mathcal{I}(k\varphi )), k \geq k_0(\epsilon ,\epsilon '). \]
Composing with $\mathcal{L}_{k,\varphi }$ we arrive at
\[ \mathcal{L}_{k,\varphi }(N^ k_{v^-_{j},X})\leq \mathcal{L}_{k,\varphi }(N^ k_{v,K})\leq \mathcal{L}_{k,\varphi }(N^ k_{v^+_{j},X})\, , \quad k \geq k_0(\epsilon ,\epsilon '). \]
For any $j>0$, by corollary 18.28.64 we get
\[ \begin{aligned} \mathcal{E}_{\theta ,X}^{\varphi }(v^-_{j}) - \mathcal{E}_{\theta ,X}^{\varphi }(w)=& \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{v^-_{j},X})-\mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \varliminf _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \varlimsup _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{v^+_{j},X})-\mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ =& \mathcal{E}_{\theta ,X}^{\varphi }(v^+_{j}) - \mathcal{E}_{\theta ,X}^{\varphi }(w). \end{aligned} \]
Using 18.28.31, we can let $j \to \infty $ to arrive at
\[ \begin{split} \mathcal{E}_{\theta ,K}^{\varphi }(v) -\mathcal{E}_{\theta ,X}^{\varphi }(w) & \leq \varliminf _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ & \leq \varlimsup _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ & \leq \mathcal{E}_{\theta ,K}^{\varphi }(v) -\mathcal{E}_{\theta ,X}^{\varphi }(w). \end{split} \]
Hence, (18.28.71) follows.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.