3 Application to Calabi-Yau degeneration [008H]

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3 Application to Calabi-Yau degeneration

We work in the setting of polarised algebraic degeneration of Calabi-Yau manifolds, as in the Introduction.

3.1 Calabi-Yau measure

The Calabi-Yau measure (2) is studied thoroughly in [3], but it is illustrative to recall it explicitly on a semistable snc model $\mathcal{X}$ . The discussion is local on the base, and we will follow the notations of section 2, e.g. the components of the central fibre are denoted as $E_{i}$ for $i\in I$ .

The canonical divisor $K_{\mathcal{X}}=\sum_{i}a_{i}E_{i}$ is supported on the central fibre, since $K_{X}$ is trivialised. Multiplying $\Omega$ by a power of $t$ , which does not change $d\mu_{t}$ , we may assume $\min a_{i}=0$ . In the local coordinates around $E_{J}$ away from the deeper strata,

\Omega=f_{J}\prod_{0}^{p}z_{i}^{a_{i}}dz_{i}\wedge\prod_{p+1}^{n}dz_{j}

for some nowhere vanishing local holomorphic function $f_{J}$ . Since $t=z_{0}\ldots z_{p}$ ,

\Omega_{t}=f_{J}z_{0}^{a_{0}}\ldots z_{p}^{a_{p}}\prod_{1}^{p}d\log z_{i}\wedge\prod_{p+1}^{n}dz_{j},

hence

\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|f_{J}|^{2}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{i}}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{j}\wedge d\overline{z}_{j}.

The total measure $\int_{X_{t}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is of the order $O(|\log|t||^{m})$ where

m=\max\{|J|-1:E_{J}\neq\emptyset,a_{i}=0\text{ for }i\in J\}.

Thus $d\mu_{t}$ satisfies a uniform upper bound of class $(a_{i})$ (cf. (4)).

3.2 Uniform Skoda estimate

We now prove the main theorem 1.3.

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. ∎

3.3 Uniform $L^{\infty}$ -estimate

We recall the following result proved using Kolodziej’s pluripotential theoretic methods (cf. [16, section 2.2] for an exposition based on [8][9]):

Theorem 3.1.

Let $(Y,\omega)$ be a compact Kähler manifold, and the Kähler potential $\phi$ solves the complex Monge-Ampère equation

\frac{(\omega+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\int_{Y}\omega^{n}}=d\mu,\quad\sup\phi=0.

Assume there are positive constants $\alpha,A$ , such that the Skoda type estimate (1) holds for $(Y,\omega,d\mu)$ :

\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.

Then $\left\lVert\phi\right\rVert_{C^{0}}\leq C(n,\alpha,A)$ .

The uniform $L^{\infty}$ -estimate for the Calabi-Yau potentials in Theorem 1.4 is an immediate consequence.