Geometric convexity properties of coverings of 1-convex surfaces (Q2659007)

The authors consider a \(1\)-convex complex surface \(X\), with a compact complex curve \(A\) as exceptional set. They study the geometric convexity properties of unramified coverings \(p: \tilde{X} \rightarrow X\). The main result of the paper asserts that \(\tilde{X}\) is \(p_5\)-convex if and only if \(\tilde{A}:= p^{-1}(A)\) does not contain an infinite Nori string of rational curves. For an arbitrary surface, it is shown that it is not \(p_5\)-convex if it contains an infinite Nori string of rational curves. An example of a covering \(\tilde{X}\) of a \(1\)-convex surface is constructed such that \(\tilde{X}\) is \(p_5\)-convex and \(p_3\)-convex but its cohomology group \(H^1(\tilde{X} ,\mathcal{O}_{\tilde{X}})\) is not separated. In this example, \(\tilde{X}\) contains an infinite Nori string of irrational curves.