ScalingStacks

An upper semicontinuous $L^{1}$-function $\phi$ on a coordinate ball in $\mathbb{C}^{n}$ is called plurisubharmonic (psh) if it satisfies the sub mean value inequality when restricted to complex lines; this implies $dd^{c}\phi\geq 0$. The basic intuition is that regularity properties for psh functions in general dimensions are analogous to subharmonic functions on Riemann surfaces. This is captured by the local version of the Skoda inequality:

On a compact Kähler manifold $(Y,\omega)$, we say an upper semicontinuous $L^{1}$-function $\phi\in PSH(Y,\omega)$ if its sum with the local potential of $\omega$ is psh, so that $\omega_{\phi}=\omega+dd^{c}\phi\geq 0$. This is the generalised notion of Kähler potentials. The standard global analogue of the Skoda inequality is:

In our applications, we need to work with a polarized algebraic degeneration of Calabi-Yau manifolds $\pi:X\to S\setminus\{0\}$ near the large complex structure limit, as in section 3. 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 $\frac{1}{|\log|t||}c_{1}(L)$. The normalization factor $\frac{1}{|\log|t||}$ is to ensure two different choices of Fubini-Study reference metrics would differ by a potential with $C^{0}$ norm of order $O(1)$ independent of small $t$. Recall $d\mu_{t}$ is the normalized Calabi-Yau measure.

We adapted the Skoda inequality to a uniform version [51]:

The proof involves covering $X_{t}$ by plenty of small regions which look like standard balls in $\mathbb{C}^{n}$. An elementary but somewhat tricky construction of test function allows one to estimate $L^{1}$ norms of the local potentials. One then applies the local version of Skoda inequality to each small region, and sum over all regions.