A splitting criterion for rank 2 vector bundles on \(\mathbb{P}^ n\) (Q1902039)

This is an addendum to a paper by \textit{V. Ancona}, \textit{T. Peternell} and \textit{J. Wisńiewski} [Pac. J. Math. 163, No. 1, 17-42 (1994; Zbl 0808.14013)]. Here we prove (using heavily this paper) two criteria for the splitting of rank 2 algebraic vector bundles (one on \(\mathbb{P}^n\) and one on certain algebraic complete manifolds). -- More precisely, the aim here is to show why the proofs of theorem 10.5 and theorem 10.13 of the quoted paper give the following two theorems. Theorem 1. Let \(E\) be a rank 2 algebraic vector bundle on \(\mathbb{P}^n\) which satisfies the assumptions of theorem 10.5 of the quoted paper. Then \(E\) splits. Theorem 2. Let \(E\) be a rank 2 algebraic vector bundle on a projective manifold \(X\) with \((X, E)\) satisfying the assumption of theorem 10.13 of the quoted paper. Then \(E\) splits. The assumptions on \(X\) in theorem 2 are very restrictive (e.g. \(X\) is a Fano manifold with \(\text{Pic} (X) \cong \mathbb{Z})\). We only remark that the assumptions of theorem 1 are satisfied if there is a two dimensional projective family, \(S\), of lines in \(\mathbb{P}^n\) such that the splitting type of \(E |L\) is the same for all \(L \in S\).