ScalingStacks

Theorem 2.1. (Uniform Skoda estimate) Given a polarised algebraic degeneration family of Calabi-Yau manifolds $\pi:X\to S\setminus\{0\}$ as in the Introduction. Let $\omega_{FS}$ be a fixed Fubini-Study metric on $(X,c_{1}(L))$ induced by a projective embedding via the sections of a high power of $L$, and use $\omega_{FS,t}=\frac{1}{|\log|t||}\omega_{FS}|_{X_{t}}$ to define a family of background metrics on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$. Then there are uniform positive constants $\alpha,A$ independent of $t$ for $0<|t|\ll 1$, such that for the normalised Calabi-Yau measures $d\mu_{t}$,

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