On the separation of cohomology groups of increasing unions of \((1,1)\) convex-concave manifolds (Q818035)

It is known [\textit{A. Markoe}, Ann. Inst. Fourier 27, No. 3, 117--127 (1977; Zbl 0323.32014); \textit{A. Silva}, Ann. Inst. Fourier 28, No. 2, 187--200 (1978; Zbl 0365.32008)] that a union of an increasing sequence of complex Stein manifolds (of one and the same dimension) is not always Stein: it is Stein if and only if its first cohomology group in the sheaf of holomorphic functions is separated. A complex manifold is said to be \((1,1)\) convex-concave, if it admits a positive function that is strictly plurisubharmonic outside a compact subset and the preimage of any segment in \(\mathbb R_+\) is compact (thus, a piece of its boundary is convex, while the complementary piece is concave). The main result of the paper provides an example of an increasing sequence of \((1,1)\) convex-concave manifolds (of one and the same arbitrary dimension \(k\geq3\)) whose union (denoted by \(X\)) is not \((1,1)\)-convex-concave. It is known [\textit{J.-P. Ramis}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. (3) 27 (1973), 933--997 (1974; Zbl 0327.32001)] for any \((1,1)\) convex-concave manifold \(Y\) of dimension \(k\geq3\), any holomorphic vector bundle \(F\) on \(Y\) and any \(j\), \(1\leq j\leq k-1\), the \(j\)-th cohomology group of \(Y\) in the sheaf of holomorphic sections of \(F\) is separated. The manifold \(X\) is constructed in the paper together with a holomorphic line bundle on it for which the previous first cohomology group is not separated. (This together with the previous statement implies that \(X\) is not \((1,1)\) convex-concave.) The manifold \(X\) is the punctured space \(\mathbb C^k\setminus0\) blowed-up at a ``generic'' sequence of points converging to 0. It admits a natural holomorphic projection to \(\mathbb C^k\setminus0\). The \((1,1)\) convex-concave manifolds that exhaust \(X\) are the preimages of the complements of balls centered at 0; the latter balls decrease and contract to 0. The previous line bundle over \(X\) is the one associated to the blowing-up divisor.