Let $(Y,\omega)$ be a compact Kähler manifold, and $d\mu$ be a measure on $Y$. We say $(Y,\omega,d\mu)$ satisfies the Skoda type inequality, if for any Kähler potential $u\in PSH(Y,\omega)$ normalised to $\sup u=0$,

$$\int_{Y}e^{-\alpha u}d\mu\leq A,$$

(1)

where $\alpha,A$ are independent of $u$. A prototype theorem is

We are interested in keeping track of these constants $\alpha,A$ as $(Y,\omega,d\mu)$ varies. The main theme of this paper is that oftentimes the Skoda constants can be chosen uniformly for quite flexible choices of probability measures $d\mu$, even when the complex structure degenerates severely. In the literature $\alpha$ is much studied (cf. [19][11]), and a very recent preprint [7] made aware to the author after the completion of this work contains a uniform estimate for both $\alpha,A$ in the related context of Kähler-Einstein manifolds.

Our main application is to algebraic degenerations of Calabi-Yau manifolds. We work over $\mathbb{C}$. Let $S$ be a smooth affine algebraic curve, with a point $0\in S$. An algebraic degeneration family is given by a submersive projective morphism $\pi:X\to S\setminus\{0\}$ with smooth connected $n$-dimensional fibres $X_{t}$ for $t\in S\setminus\{0\}$. A polarisation is given by an ample line bundle $L$ over $X$; the sections of a sufficiently high power of $L$ induces an embedding $X\to\mathbb{CP}^{N}$, hence a Fubini-Study metric $\omega_{X}$ on $(X,L)$. For $0<|t|\ll 1$, a fixed choice of $\omega_{X}$ induces rescaled background metrics $\omega_{t}=\frac{1}{|\log|t||}\omega_{X}|_{X_{t}}$ on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$.

A model of $X$ is a normal flat projective $S$-scheme $\mathcal{X}$ which agrees with $\pi:X\to S\setminus\{0\}$ over the punctured curve. It is called a semistable snc model if $\mathcal{X}$ is smooth, the central fibre over $0\in S$ is reduced and is a simple normal crossing divisor in $\mathcal{X}$. By the semistable reduction theorem [12, chapter 2], after finite base change to another smooth algebraic curve $S^{\prime}$, we can always find some semistable snc model for the degeneration family $X\times_{S}(S^{\prime}\setminus\{0\})$. Everything here is quasi-projective.

We say the degeneration family is Calabi-Yau if there is a trivialising section $\Omega$ of the canonical bundle $K_{X}$. Over a small disc $\mathbb{D}_{t}$ around $0\in S$, this induces holomorphic volume forms $\Omega_{t}$ on $X_{t}$ via $\Omega=dt\wedge\Omega_{t}$. The normalised Calabi-Yau measure on $X_{t}$ is the probability measure

$$d\mu_{t}=\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}.$$

(2)

Our main result is

This is proved by reducing to the semistable snc model case, and prove a general Skoda type estimate there (cf. Theorem 2.9). A major consequence, readily reaped using Kolodziej’s estimate (cf. Theorem 3.1), is