ScalingStacks

Theorem 1.3. (Uniform Skoda estimate) Given a polarised algebraic Calabi-Yau degeneration family $\pi:X\to S\setminus\{0\}$ as above. Then there are uniform positive constants $\alpha,A$ independent of $t$ for $0<|t|\ll 1$, such that for the normalised Calabi-Yau measures $d\mu_{t}$,

$$\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{t})\text{ with }\sup_{X_{t}}u=0.$$