Global deformations of \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\)
Every holomorphic vector bundle over \(\mathbb{P}^1\) is a direct sum of line bundles. Hence the analytic classification of holomorphic \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) may be considered known. Not obvious is when a \(\mathbb{P}^2\)-bundle may be deformed to another \(\mathbb{P}^2\)-bundle. In this paper the author studies the complex analytic global deformation of \(\mathbb{P}^2\)-bundles on \(\mathbb{P}^1\). The author proves the following results. Theorem 1. The set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) with first Chern class divisible by 3 is closed and transitive under global deformations. Theorem 2. The set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) whose first Chern class is not divisible by 3 and which are not homeomorphic to \(\mathbb{P}^1\times \mathbb{P}^2\) is closed and transitive under global deformations. Theorem 3. The union of the set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) homeomorphic to \(\mathbb{P}^1\times\mathbb{P}^2\) and the set of all \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) homeomorphic to \(\mathbb{P}^1 \times\mathbb{P}^2\) is closed and transitive under global deformations. Furthermore, the author shows that there are infinitely many non-isomorphic \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) homeomorphic to \(\mathbb{P}^1\times \mathbb{P}^2\) which arise from topologically trivial unstable rank two vector bundles over \(\mathbb{P}^2\).
scientific article; zbMATH DE number 1295040