Ample and spanned vector bundles with large \(c^ 2_ 1\) relative to \(c_ 2\) on surfaces (Q1261123)

Let \(E\) be an ample and spanned vector bundle of rank \(r\geq 2\) on a smooth projective surface \(X\) over the complex number field. The purpose of this paper is to show that \(c_ 1(E)^ 2\leq (c_ 2(E)+1)^ 2\) holds for every \((X,E)\), by describing the pair \((X,E)\) with \(c_ 1(E)^ 2\geq (c_ 2(E)+2)^ 2/2\) explicitly. Subsequently, we obtain a classification of \((X,E)\) with \(c_ 2(E)=3,4\) and \(c_ 1(E)^ 2> 4c_ 2(E)\).