Non-vanishing theorems for rank two vector bundles on threefolds (Q2914228)

The authors consider vector bundles \(E\) on a smooth projective threefold \(X\), whose Picard group is \(\mathbb Z\) (or at least the group \(\mathrm{Num}(X)\) is \(\mathbb Z\)) and \(h^1O_X(n)=0\) for all \(n\). \(E\) is \textit{decomposable} when it is isomorphic to a sum of line bundles. The authors study cohomological conditions for the indecomposability of bundles \(E\). In particular, the authors extend previous results by C. Madonna and determine ranges for the integers \(n\) such that \(h^1(E(n))\neq 0\) for any indecomposable bundle of rank \(2\) on \(X\). These ranges (that can be empty) depend on the Chern classes and the level of stability of \(E\), as well as on the main invariants of the threefolds \(X\).