ScalingStacks

Lemma 4.8 . [023R]

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Lemma 4.8.

For x2≪x1x_{2}\ll x_{1}, the Tian-Yau potential satisfies the bound

|ϕT​Y|=O⁡(x2n(n−1)),|d​ϕT​Y|=O⁡(x1−n−22​n​(n−1)​x2n2​(n−1)),|\phi_{TY}|=O(x_{2}^{\frac{n}{(n-1)}}),\quad|d\phi_{TY}|=O(x_{1}^{-\frac{n-2}{2n(n-1)}}x_{2}^{\frac{n}{2(n-1)}}),
‖d​dc​ϕT​Y‖k,α,l​o​c=O⁡(x1−n−2n⁡(n−1)).\left\lVert dd^{c}\phi_{TY}\right\rVert_{k,\alpha,loc}=O(x_{1}^{-\frac{n-2}{n(n-1)}}).

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