Proof. We seek the test function in the form $v=\Phi\circ\log h$ for some convex, non-increasing, non-negative $C^{2}$-function $\Phi$. Compute

so using the properties of $h$ above,

To satisfy our conditions on $v$, it is enough to have

• •

  $\Phi(x)=0$ for $x\geq\log\delta$.

• •

  $|\Phi^{\prime}(x)|\lesssim 1$ for $2\log|t|\lesssim x\leq\log\delta$.

• •

  $-\frac{d}{dx}\log|\Phi^{\prime}|=\frac{\Phi^{\prime\prime}}{|\Phi^{\prime}|}\geq C_{4}^{\prime}\max\{h,h^{-1/n}|t|^{2/n}\}$ for $h=e^{x}\leq\delta,$ where $C_{4}^{\prime}>C_{2}^{\prime}/C_{1}^{\prime}$. Morever, for $x<\delta$, we need $\Phi^{\prime}<0$ so that $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$ has some strict positivity for $\delta/2<h<\delta$. Notice convexity of $\Phi$ is a consequence of these conditions.

To construct such $\Phi$, we can prescribe the behaviour near $x=\log\delta$ by $\Phi^{\prime}(x)=-e^{1/(x-\log\delta)}$ for $x<\log\delta$, and match this with a solution to

for some large enough $C_{4}^{\prime}$, such that $\Phi^{\prime}$ remains $C^{1}$ at the matching point. Integration shows that $|\Phi^{\prime}|$ remains uniformly bounded at $h\sim|t|^{2}$, or equivalently $x\sim 2\log|t|$. ∎

Proof. For any choice of $\theta_{i}$ the function $\phi(z_{1}e^{i\theta_{1}},\ldots z_{p}e^{i\theta_{n}},z_{p+1},\ldots,z_{n})$ is psh, since the $T^{p}$-action on $(\mathbb{C}^{*})^{p}$ is holomorphic. Thus the average function $\bar{\phi}$ is also psh as a function of $z_{1},\ldots z_{p}$. Any $T^{p}$-invariant psh function must be convex in the log coordinates, because for $x_{i}=\log|z_{i}|$,

∎

Proof. Let the global maximum of $u$ be achieved at $q_{0}\in E_{J}^{0}$, and denote the local potential of $u$ as $u_{\beta}$. Without loss of generality $u_{\beta}\leq 0$. We have $u_{\beta}(q_{0})\geq-C$ since $|u-u_{\beta}|\leq C$. Applying the mean value inequality around $q_{0}$, we find that the local average function $\bar{u}_{\beta}$ produced in Lemma 2.2 satisfies $\sup\bar{u}_{\beta}\geq-C$ for another uniform constant $C$. By the convexity of $\bar{u}_{\beta}$ its sup is almost achieved at the boundary of the chart, which is contained in a union of less deep strata $E_{J^{\prime}}^{0}$ with $J^{\prime}\subsetneq J$. Thus we can find a point $q^{\prime}$ with $u(q^{\prime})\geq-C$ that belongs to a less deep stratum; an induction shows that there is some $i\in I$, such that $\sup_{E_{i}^{0}}u\geq-C$.

For the $L^{1}$-bound we recall the following Harnack inequality argument. Suppose a coordinate ball $B(q,3R)$ is contained in a local chart in a small neighbourhood of $E_{i}^{0}$. Applying the mean value inequality to the local psh function associated to $u$, we see for $y\in B(q,R)$ that

hence the Harnack inequality

Applying this to a chain of balls connecting any two points in $E_{i}^{0}$ gives the $L^{1}$-bound $\int_{E_{i}^{0}}u\omega_{\mathcal{X}}|_{X_{t}}^{n}\geq-C^{\prime}$; the bound is uniform because the number of balls involved in the chain can be controlled independent of $t$. ∎

Proof. The $L^{1}$-estimate follows from the sup estimate as above, so the real problem is to transfer bounds between different $E_{i}^{0}$. This is nontrivial because the necks connecting $E_{i}^{0}$ with each other are highly degenerate.

Given one divisor $E_{0}$ such that $\int_{E_{0}^{0}}u\omega_{\mathcal{X}}|_{X_{t}}^{n}\geq-C,$ we produce a good test function $v$ by Lemma 2.1. Integrating by parts,

The LHS is the difference of $\int_{X_{t}}v(\omega_{t}+\sqrt{-1}\partial\bar{\partial}u)\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$ and $\int_{X_{t}}v\omega_{t}\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$, and since $-\log|t|\gtrsim v\geq 0$ both terms are bounded between $0$ and $C$. Thus

Now the form $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$ can only be negative on $\{h\sim|t|^{2}\}=E_{0}^{0}$, and is bounded below by $-C\omega_{\mathcal{X}}|_{X_{t}}^{n}$. Thus the positive part of the signed measure $u\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$ has total mass controlled by $\int_{E_{0}^{0}}|u|\omega_{\mathcal{X}}|_{X_{t}}^{n}\leq C$. Consequently, the negative part of the signed measure must also have total mass $\leq C$.

By construction, for any divisor $E_{j}$ intersecting $E_{0}$ there is a nontrivial amount of $\sqrt{-1}\partial\bar{\partial}v\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}$-measure inside $E_{j}^{0}$. This forces $\sup_{E_{j}^{0}}u\geq-C$. To summarize, we have transferred the sup bound from $E_{0}^{0}$ to any $E_{j}^{0}$ with $E_{j}\cap E_{0}\neq\emptyset$. Since the central fibre $X_{0}$ is connected, in at most $|I|$ steps this sup bound is transferred to all $E_{i}^{0}$ with $i\in I$. ∎

Proof. We induct on the depth of the strata. For $p=0$ this follows from Prop. 2.4. So let us assume the bound is achieved for depth $<p$. For a given chart, we consider the local psh function $u_{\beta}$ associated to $u$ and produce the convex average function $\bar{u}_{\beta}$ as in Lemma 2.2. Since a definite neighbourhood of the boundary of the chart lies inside less deep strata, we know that near the boundary $|\bar{u}_{\beta}|\leq C$ by the induction hypothesis and the convexity condition. Using convexity again in the interior of the chart we see $|\bar{u}_{\beta}|\leq C$ in the whole chart.

Within any log scale, by construction the local average $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}(u_{\beta}-\bar{u}_{\beta})=0.$ But $u_{\beta}\leq C$ by $u\leq 0$, hence

Using $|u-u_{\beta}|\leq C$ we conclude the local $L^{1}$-estimate on $u$. ∎

Proof. We choose the charts so that each point on $X_{t}$ is covered by $\leq C$ log scales. Summing over the local Skoda estimates from all log scales, $\int_{X_{t}}e^{-\alpha u}d\mu_{t}$ is bounded by

∎

Proof. First we observe that the choice of the Fubini-Study metric $\omega_{X}$ is immaterial. Given any two choices, the relative Kähler potential between them is bounded by $O(|\log|t||)$ for $0<|t|\ll 1$, because the pole order of a section near $t=0$ must be finite. Thus the relative Kähler potential between two choices of $\omega_{t}$ is bounded by $O(1)$ independent of $t$, which affects the Skoda constant $A$ but not its uniform nature.

We now pass to a finite base change and find a semistable reduction. The Calabi-Yau measure $d\mu_{t}$ on $X_{t}$ is independent of the parametrisation of the base, and is preserved under finite base change. Thus it is enough to prove it assuming $\omega_{X}$ agrees with a smooth Kähler metric on a semistable snc model $\mathcal{X}$; this is a special case of Theorem 2.9. ∎