Log continuity of solutions of complex Monge-Amp\`ere equations

Hoang-Son Do; Duc-Viet Vu

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arXiv · 2312.04128

Log continuity of solutions of complex Monge-Amp\`ere equations

Hoang-Son Do·Duc-Viet Vu

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Abstract

Let $X$ be a compact K\"ahler manifold whose anticanonical cohomology class is semipositive. Let $L$ be a big and semi-ample line bundle on $X$ and $\alpha$ be the Chern class of $L$. We give a sufficient condition ensuring that the solution of the complex Monge-Amp\`ere equations in $\alpha$ with $L^p$ right-hand side ($p>1$) is $\log^M$-continuous for every constant $M>0$. As an application, we show that every singular Ricci-flat metric in a semi-ample integral class in a projective Calabi-Yau surface $X$ is globally $\log^M$-continuous with respect to a smooth metric on $X$.

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BibTeX RIS

Hoang-Son Do, Duc-Viet Vu. 2023-12-07. Log continuity of solutions of complex Monge-Amp\`ere equations. https://arxiv.org/abs/2312.04128

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