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arXiv · 1804.01004

Estimates for the $\bar{\partial}$-equation on canonical surfaces

Mats Andersson·Richard Lärkäng·Jean Ruppenthal·Håkan Samuelsson Kalm·Elizabeth Wulcan
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Abstract

We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.

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Mats Andersson, Richard Lärkäng, Jean Ruppenthal, Håkan Samuelsson Kalm, Elizabeth Wulcan. 2018-04-03. Estimates for the $\bar{\partial}$-equation on canonical surfaces. https://doi.org/10.1007/s12220-019-00187-2

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