ScalingStacks

Theorem 1.3 . [007N]

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Theorem 1.3.

(Uniform Skoda estimate) Given a polarised algebraic Calabi-Yau degeneration family π:X→S∖{0}\pi:X\to S\setminus\{0\} as above. Then there are uniform positive constants α,A\alpha,A independent of tt for 0<|t|≪10<|t|\ll 1, such that for the normalised Calabi-Yau measures d​μtd\mu_{t},

∫Xte−α​u​d​μt≤A,∀u∈P​S​H​(Xt,ωt)​ with ​supXtu=0.\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.

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