On the existence of simple reducible vector bundles on complex surfaces of algebraic dimension zero (Q1190721)

Let \(X\) be a minimal (compact complex connected) surface of algebraic dimension zero. The author determines the pairs \((c_ 1,c_ 2)\) for which there exists a simple reducible rank 2 holomorphic vector bundle \(E\) on \(X\) with these Chern classes. When \(c_ 1=0\), such an \(E\) exists precisely when \(c_ 2\geq 0\), with the following exceptions: \(-X\) is a torus, and \(c_ 2=0\) \(-X\) is in class VII and \(c_ 2=0\), unless \(b_ 2=0\), \(X\) has no divisors and \(c_ 1(X)\in NS(X)\) \(-X\) is a \(K3\) surface, and either it has curves and \(c_ 2\in\{0,1,2,3\}\), or it has no curve and \(0\leq c_ 2\leq\inf\{-\xi^ 2\mid\xi\in NS(X)\}\) (When \(X\) has no curve, this problem was already treated by \textit{V. Brinzǎnescu} and \textit{P. Flondor} [J. Reine Angew. Math. 363, 47-58 (1985; Zbl 0567.32009)]).