On large families of bundles over algebraic surfaces (Q899234)

The authors prove that for any smooth projective varieties and any pair \((s,t)\) of positive real numbers with \(t>4s\) there exists a large family of order \((s,t)\), which is a sequence of stable vector bundles \(E_m\) such that their ranks and discriminant satisfy \(r_m = O(m^s)\) and \(\Delta_m = O(m^t)\). As a consequence they find counterexamples to the Strong Bogomolov Inequality \(\mathrm{SBI}_l\) for any polarized algebraic surface \((S,H)\) and any \(l>4\). Recall that for a positive real number \(l\), \((S, H)\) satisfies the \(\mathrm{SBI}_l\) if there is a positive constant \(\sigma\), depending only on the surface and the polarization, such that for any stable vector bundle \(E\) of rank \(r\) one has the inequality \(\Delta(E)\geq r^l\sigma\).