\(q\)-completeness and \(q\)-concavity of the union of open subspaces (Q1910195)

It is proved that the union of two open subspaces \(D_1\) and \(D_2\) of a complex space \(X\) is \((q_1 + q_2)\)-complete if \(D_i\) is \(q_i\)-complete, \(i = 1,2\). This has been previously proved by M. Peternell by a much more complicated method. Our proof is rather elementary and is based on some special convex functions of one real variable. Our method enables us to treat also the \(q\)-concave situation. In fact if \(D_i\) are \(q_i\)-concave, \(i = 1,2\), then their union (resp. intersection) is \((q_1 + q_2 - 1)\)-concave (resp. \((q_1 + q_2)\)-concave). An example of two 1-concave open sets in \(P^2\) whose intersection is not 1-concave is shown. It is also proved that if a complex space \(X\) is an increasing union of \(q\)-complete open subspaces then \(X\) is \(2q\)-complete. Note the particular case \(q = 1\).