Pome

Definition 16.11.5F80B4C1B3032

The same rooftop construction in the closed Kähler setting, with different notation. Both source versions retain the convention for an identically minus-infinity input.

Definition 16.11.5.F80B4C1B3032 For $u,v\in \mathrm{PSH}(X,\theta )$, the rooftop envelope of $u$ and $v$ is

\[ P_\theta (u,v)=P_\theta (\min \{ u,v\} ). \]

We also put $P_\theta (u,-\infty )=P_\theta (-\infty ,u)=-\infty $ for $u\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $.

Read the alternative statement and proof

Definition 16.11.5.F80B4C1B3032 Given $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$, we define their rooftop operator as follows:

6AE4825E4C6F 16.3.1
\begin{equation} \varphi \land \psi =\sup \left\{ \eta \in \mathrm{PSH}(X,\theta ):\eta \leq \varphi ,\eta \leq \psi \right\} . \end{equation}

For simplicity of notation, we extend the definition to the case where $\varphi $ or $\psi $ is constantly $-\infty $, in which case we simply set

\[ \varphi \land \psi =-\infty . \]
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