Proposition 4.3 (00A9) · ScalingStacks

Verified tagged author-source HTML · 2007.01384v1 · cited publication edition alignment unverified.

Proposition 4.3. For $0<|t|\ll 1$ , the CY potentials have a uniform bound $\left\lVert\phi_{CY,t}\right\rVert_{L^{\infty}}\leq C$ under the normalisation $\sup_{X_{t}}\phi_{CY,t}=0$ .

00AA

Proof. Combine the uniform $C^{0}$ -estimate Theorem 2.2 and the uniform Skoda estimate Theorem 2.1, we know the CY potential with respect to any fixed choice of Fubini-Study background metric is uniformly bounded for small $t$ . Here the implicit choice of $\epsilon$ in $\omega_{FS,t}$ does not matter because of Lemma 4.1. ∎