2.1 Uniform Skoda inequality [00AZ]

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2.1 Uniform Skoda inequality

Given a Kähler manifold $(Y,\omega)$ , an upper semicontinuous $L^{1}_{loc}$ function $\phi\in PSH(Y,\omega)$ , if $\omega_{\phi}=\omega+dd^{c}\phi\geq 0$ . A Skoda type inequality captures the apriori regularity of such functions. The following uniform version is the main result in the author’s companion paper [33].

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.