ScalingStacks

Proof. We seek the test function in the form $v=\Phi\circ\log h$ for some convex, non-increasing, non-negative $C^{2}$-function $\Phi$. Compute

$$\partial\bar{\partial}v=\Phi^{\prime\prime}\partial\log h\wedge\bar{\partial}\log h+\Phi^{\prime}(\partial\bar{\partial}\log\tilde{h}),$$

so using the properties of $h$ above,

$$\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq\begin{cases}\left(\Phi^{\prime\prime}C_{1}^{\prime}\min\{\frac{1}{h},h^{1/n}|t|^{-2/n}\}+\Phi^{\prime}C_{2}^{\prime}\right)\omega_{\mathcal{X}}|_{X_{t}}^{n},\quad&|t|^{2}\lesssim h\leq\delta,\\ C_{3}^{\prime}\Phi^{\prime}\omega_{\mathcal{X}}|_{X_{t}}^{n},\quad&h\lesssim|t|^{2}.\end{cases}$$

To satisfy our conditions on $v$, it is enough to have

• •

  $\Phi(x)=0$ for $x\geq\log\delta$.

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  $|\Phi^{\prime}(x)|\lesssim 1$ for $2\log|t|\lesssim x\leq\log\delta$.

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  $-\frac{d}{dx}\log|\Phi^{\prime}|=\frac{\Phi^{\prime\prime}}{|\Phi^{\prime}|}\geq C_{4}^{\prime}\max\{h,h^{-1/n}|t|^{2/n}\}$ for $h=e^{x}\leq\delta,$ where $C_{4}^{\prime}>C_{2}^{\prime}/C_{1}^{\prime}$. Morever, for $x<\delta$, we need $\Phi^{\prime}<0$ so that $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$ has some strict positivity for $\delta/2<h<\delta$. Notice convexity of $\Phi$ is a consequence of these conditions.

To construct such $\Phi$, we can prescribe the behaviour near $x=\log\delta$ by $\Phi^{\prime}(x)=-e^{1/(x-\log\delta)}$ for $x<\log\delta$, and match this with a solution to

$$-\frac{d}{dx}\log|\Phi^{\prime}|=C_{4}^{\prime}\max\{e^{x},e^{-x/n}|t|^{2/n}\},\quad x<\log\delta$$

for some large enough $C_{4}^{\prime}$, such that $\Phi^{\prime}$ remains $C^{1}$ at the matching point. Integration shows that $|\Phi^{\prime}|$ remains uniformly bounded at $h\sim|t|^{2}$, or equivalently $x\sim 2\log|t|$. ∎