ScalingStacks

Proposition 1.3 . [032G]

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Proposition 1.3.

1) If ω1≤ω2\omega_{1}\leq\omega_{2} then P​S​H​(X,ω1)⊂P​S​H​(X,ω2)PSH(X,\omega_{1})\subset PSH(X,\omega_{2}).

2) ∀A∈ℝ+∗\forall A\in\mathbb{R}_{+}^{*}, P​S​H​(X,A​ω)=A⋅P​S​H​(X,ω)PSH(X,A\omega)=A\cdot PSH(X,\omega).

3) If ω′\omega^{\prime} is cohomologous to ω\omega, ω′=ω+d​dc​χ\omega^{\prime}=\omega+dd^{c}\chi, then

P​S​H​(X,ω′)=P​S​H​(X,ω)+χ.PSH(X,\omega^{\prime})=PSH(X,\omega)+\chi.

4) If φ,ψ∈P​S​H​(X,ω)\varphi,\psi\in PSH(X,\omega) then

max⁡(φ,ψ),φ+ψ2,log⁡[eφ+eψ]∈P​S​H​(X,ω)\max(\varphi,\psi)\;,\;\frac{\varphi+\psi}{2}\;,\;\log[e^{\varphi}+e^{\psi}]\in PSH(X,\omega)

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