Families of homogeneous vector bundles on \({\mathbb{P}}^ 2\) (Q762571)

The trivial topological vector bundle of rank 3 on \({\mathbb{P}}^ 2({\mathbb{C}})\) has three distinct homogeneous structures, namely the trivial bundle, the bundle \(E_ 0=T_{{\mathbb{P}}^ 2}(-1)+{\mathcal O}_{{\mathbb{P}}^ 2}(-1)\) and \(E_ 0^{\vee}\). The bundles \(E_ 0\) and \(E_ 0^{\vee}\) are not rigid; the author constructs the versal deformation of \(E_ 0\) and also describes explicitly a family F of bundles with base \({\mathbb{P}}^ 1\) for which \(F_ 0\cong E_ 0,\quad F_{\infty}\cong E_ 0^{\vee}\) and \(F_ s\) is trivial for \(s\in {\mathbb{C}}^*\).