Corollary 2.12 (00PX) · ScalingStacks

00PX

Corollary 2.12. (Stability estimate) Let $(X,\omega)$ be a compact Kähler manifold, and $\phi,\psi\in PSH(X,\omega)\cap C^{0}$ , such that $\omega_{\phi}^{n}$ is absolutely continuous. Assume $\left\lVert\psi\right\rVert_{C^{0}}\leq A^{\prime}$ and the Skoda type estimate (2). Then there is a number $B(n,A,A^{\prime},\alpha)$ , such that if $\frac{\int_{\phi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{Vol(X)}<(2B)^{-2n}$ for some $t_{0}$ , then

\min(\phi-\psi)\geq-t_{0}-4B\left(\frac{\int_{\phi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{Vol(X)}\right)^{1/2n}.

00PY

Proof. If $\psi$ is smooth, this follows from Thm. 2.7 by changing $\omega$ into $\omega_{\psi}$ , and changing $\phi$ into $\phi-\psi$ , and checking the Skoda type estimate holds with modified constants. In general, one can approximate $\psi\in PSH(X,\omega)$ by a decreasing sequence of functions in $PSH(X,\omega)\cap C^{\infty}$ [1], and since $\psi\in C^{0}$ the convergence is uniform by Dini’s theorem. ∎