3.3 Uniform L ∞ -estimate [008K]

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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.

3.3 Uniform L ∞ -estimate [008K]

3.3 Uniform L∞L^{\infty}-estimate

Theorem 3.1.

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