ScalingStacks

Theorem 8.5 . [01HC]

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Theorem 8.5.

Let LL be an ample line bundle on XX and ℒ∈Pic⁡(𝒳)\mathcal{L}\in\Pic(\mathcal{X}) an extension of LL to an SNC model 𝒳\mathcal{X}. Let θ∈𝒵1,1​(X)\theta\in\mathcal{Z}^{1,1}(X) be the curvature form of the corresponding model metric on LL. For m≫1m\gg 1 let 𝔞m⊂𝒪𝒳\mathfrak{a}_{m}\subset\mathcal{O}_{\mathcal{X}} be the (vertical) base-ideal of m​ℒm\mathcal{L} and set φm:=1m​log⁡|𝔞m|\varphi_{m}:=\tfrac{1}{m}\log|\mathfrak{a}_{m}|. Then φm\varphi_{m} is a θ\theta-psh model function and φm→Pθ​(0)\varphi_{m}\to P_{\theta}(0) uniformly on XX as m→∞m\to\infty.

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