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The Cauchy Riemann equation on singular spaces. (English) Zbl 1047.32500

The authors propose an intrinsic method for studying the \(\overline\partial\) problem on a complex variety \(X\) with singularities. They consider \(X\) locally as a branched cover over an open set in \(\mathbb C\), and they push the problem forward to \(\mathbb C\). While parts of the setup (such as existence) are established for a fairly general \(X\), the detailed estimates for the solution of the \(\overline\partial\) equation are proved for the specific variety \(X=\{(w_1,w_2,z)\in\mathbb C^3\colon w_1w_2=z^2\}\).
The paper is based loosely on ideas in the second author’s thesis. It is a nice piece of work, suggesting broad vistas for future research.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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References:

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