On the restriction of vector bundles to good families of curves
The author proves several results concerning vector bundles on projective manifolds by restricting them to suitable curves. In the first paragraph, he considers a threefold \(X\), the contraction of an extremal ray \(f : X \to Y\) and a vector bundle \(E\) on \(X\) and gives a criterion for \(E\) to be the pull-back of a vector bundle on \(Y\). In the rest of the paper, the author proves the semi-stability of some homogeneous vector bundles on \(\mathbb{P}^n\). Let us illustrate his method by his proof of the semi-stability of the kernel \(E\) of the evaluation map: \(H^0{\mathcal O}_\mathbb{P} (d)\otimes_k{\mathcal O}_\mathbb{P} \to{\mathcal O}_\mathbb{P} (d)\). According to some older results of the author and \textit{Ph. Ellia} [Math. Z. 196, 355-367 (1987; Zbl 0631.14029)] there exist non-special curves \(C\) in \(\mathbb{P}^n\) with \(H^0 \dot I_C (d) = 0\), \(H^1 \dot I_C (d) = 0\). It follows that \(H^0 (E |C) = 0\), \(H^1 (E |C) = 0\) and this easily implies that \(E |C\) is semistable, whence \(E\) is semistable.
scientific article; zbMATH DE number 711493
scientific article