Stability of Kapranov bundles on quadrics
Let \(Q\) a smooth quadric hypersurface in \(\mathbb{P}^{n+1}\). In 1988 Kapranov introduced the vector bundles \(\psi_k\) on \(Q\) for \(0\leq k\leq n-1\) in order to construct a resolution of the diagonal in \(Q\times Q\). By using these bundles, the author gave a description of the derived category \(D^b(\text{Coh} (Q))\) and stated for quadrics a result analogous to Beilinson's theorem for projective spaces. For \(0\leq k\leq n-1\), the bundles \(\psi_k\) are simple and homogeneous but associated to a reducible representation. In this work we prove that they are conveniently suitably adapting to our case a criterion of stability for homogeneous vector bundles due to Rohmfeld. The proof consists of the analysis of the many possible homogeneous subvector bundles of the \(\psi_k\); without giving a complete description of them, we are able to bound their slope.
scientific article; zbMATH DE number 878427