ScalingStacks

Proposition 2.10 . [038C]

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Proposition 2.10.

Let LL be an ample line bundle on XX, β„’{\mathscr{L}} an extension to a model 𝒳{{\mathscr{X}}} and ΞΈ=c1(L,βˆ₯βˆ₯β„’)βˆˆπ’΅1,1(X)\theta={c_{1}(L,{\|\ \|}_{\mathscr{L}})}\in\mathcal{Z}^{1,1}(X). For m>0m>0 let

(2.3) π”žm=Im​(H0​(𝒳,β„’βŠ—m)βŠ—Kβˆ˜β„’βŠ—βˆ’mβ†’π’ͺ𝒳){\mathfrak{a}}_{m}=\mbox{\rm Im}\,\bigl(H^{0}({{\mathscr{X}}},{\mathscr{L}}^{\otimes m})\otimes_{K^{\circ}}{\mathscr{L}}^{\otimes-m}\to{\mathcal{O}}_{{\mathscr{X}}}\bigr)

be the mm-th base ideal of β„’{\mathscr{L}} and Ο†m:=mβˆ’1​log⁑|π”žm|\varphi_{m}:=m^{-1}\log|{\mathfrak{a}}_{m}|. Then Ο†m∈PSHπ’Ÿβ€‹(X,ΞΈ)\varphi_{m}\in{\rm PSH}_{\mathscr{D}}(X,\theta) and

(2.4) limmβ†’βˆžΟ†m=supmβˆˆβ„•Ο†m=Pθ​(0)\lim_{m\to\infty}\varphi_{m}=\sup_{m\in\mathbb{N}}\varphi_{m}={P}_{\theta}(0)

pointwise on XanX^{\mathrm{an}}.

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