ScalingStacks

Theorem 4.7. (cf. [53, Theorem 2.6]) (Uniform $L^{1}$-stability) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$, satisfying the complex MA equations

$$\frac{\omega^{n}}{\text{Vol}(Y)}=d\mu,\quad\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}=d\nu$$

for probability measures $d\mu$ and $d\nu$. Assume

• •

  There is a Skoda type estimate

  $$\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.$$

• •

  The complement of $E_{0}=\{\phi>0\}$ has a mass lower bound

  $$\int_{E_{0}^{c}}d\mu\geq\lambda>0.$$

• •

  ($L^{1}$-stability assumption) The total variation $\int_{Y}|d\mu-d\nu|\leq s^{2n+3}<1$.

• •

  $\phi$ is smooth away from a (possibly empty) closed subset $S$ with $d\mu$-measure zero. Globally $\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}$.

Then for $0<s<s_{0}(\lambda,n,\alpha,A,A^{\prime})\ll 1$, there is a uniform estimate

$$\sup_{Y}\phi\leq C(\lambda,n,\alpha,A,A^{\prime})s.$$