ScalingStacks

Theorem 2.1 . [009G]

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Theorem 2.1.

(Uniform Skoda estimate) Given a polarised algebraic degeneration family of Calabi-Yau manifolds π:X→S∖{0}\pi:X\to S\setminus\{0\} as in the Introduction. Let ωF​S\omega_{FS} be a fixed Fubini-Study metric on (X,c1​(L))(X,c_{1}(L)) induced by a projective embedding via the sections of a high power of LL, and use ωF​S,t=1|log⁡|t||​ωF​S|Xt\omega_{FS,t}=\frac{1}{|\log|t||}\omega_{FS}|_{X_{t}} to define a family of background metrics on XtX_{t} in the class 1|log⁡|t||​c1​(L)\frac{1}{|\log|t||}c_{1}(L). 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,ωF​S,t)​ with ​supXtu=0.\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.

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