Rank 2 stable vector bundles on Fano 3-folds of index 2
Let \((X,H)\) be a Fano 3-fold of the first kind with index 2, namely, the anticanonical \(-K_X\) is ample on \(X\) and \(-K_X=2H\) with \(\text{Pic} X =\mathbb{Z} H\) and \(H^3=d\). The authors prove at first that there exists a normalized rank 2 stable vector bundle \(E\) on \((X,H)\) with Chern classes \(c_1\), \(c_2\) if and only if \(c_1=0\) and \(c_2\geq 2\) or \(c_1=-1\) and \(c_2 \geq 2\) even. The main result of the paper states then that if \(E\) is a rank 2 reflexive sheaf on \((X,H)\) with Chern classes \(c_i\) then if \(c_1=-1\) and \(c_1(c_1+1)d- 3c_2 +6>0\) we have \(H^0 E\neq 0\). Finally, it is proved that if \(E\) is a normalized rank 2 stable reflexive sheaf on \((X,H)\) then the following hold: (i) if \(c_1=-1\), then \(c_3\leq c^2_2 -3c_2 +2\) (ii) if \(c_1=0\), then \(c_3\leq c_2- 4c_2+4\).
scientific article; zbMATH DE number 1076654