Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems
The author constructs certain determinantal varieties associated to a rank two vector bundle on a projective variety \(S\) of dimension \(n\). These determinantal varieties have codimension \([n/2]\), the integral part of \(n/2\). By a careful analysis of these varieties, the author proves theorems about vector bundles on projective spaces. The analysis seems very interesting. Some of the applications though, we believe, are well known to the experts. At any rate some of them have never appeared before in print, to the best of our knowledge. For example, the fact that if \(E\) is a rank two vector bundle on \({\mathbb{P}}^n\) (\(n\geq 3\)) with \(\text{H}^1(\text{End} E)=0\), then it is split. It is easy to show that the condition implies that \(E\) is homogeneous and thus split if \(n\geq 4\). But the reviewer has not seen this fact written down before.
scientific article; zbMATH DE number 1347354