Constructing stable vector bundles from curves with torsion normal bundle (Q6539989)

Let \(X\) be a smooth complex projective surface and \(H\) a polarization. The aim is to prove the existence of rank \(2\) vector stable bundles \(E\) on \(X\) withprescribed Chern classes. The author point out that such \(c_2(E)\) generates the group \(CH_0(X)\) of \(0\)-cycles modulo rational equivalence. Moreover, the author also proves that all symmetric powers of the contructed bundle \(E\) are slope \(H\)-stable. Hence, it is not a surprise that the main theorem requires many assumptions: \(X\) has abelian torsion free goub, \(h^0(K_X) >h^0(\mathcal{O}_X)\), the first Chern classes is represented by a smooth curve \(C\subset X\) of even genus \(g\ge 2\) whose normal bundle is torsion in \(\mathrm{Pic}(X)\) of order at least \(4g-6\), no multiple of \(C\) is the support of a fibration of \(C\) and \(H^i(X\setminus C,\mathbb{Z})\), \(i=1,2\), if finitely generated (with these data \(c_2(E) =2g-2\)).