Parabolic sheaves on higher dimensional varieties (Q1204223)

This paper is devoted to the study of parabolic sheaves on higher dimensional smooth projective varieties. First of all, the author defines the notions of \(\chi\)-semistability and \(\mu\)-semistability for parabolic sheaves. As in the theory of stable sheaves she gets \(\mu\text{-stable} \Rightarrow\chi\text{-stable}\Rightarrow\chi\text{- semistable}\Rightarrow\mu\text{-semistable}\). As main results she obtains the following: In a flat family of parabolic torsionfree sheaves, the \(\mu\)-semistable (resp. \(\mu\)-stable) parabolic sheaves form an open subset. Let \(E\) be a \(\mu\)-semistable (resp. \(\mu\)-stable) (with respect to \(H)\) torsionfree parabolic sheaf on a smooth projective variety of dimension \(\geq 2\). We assume that the flag on \(E\) is \[ {\mathcal F}:F_ 1(E)=E_ D\supset F_ 2(D)\supset\cdots\supset F_{r+1}(D)=0 \] with \(F_ 1(E)/F_ i(D)\) torsionfree for \(i=2,\ldots,r+1\). Then the restriction of \(E\) to a general complete intersection curve \(Y\) of high degree is \(\mu\)- semistable (resp. \(\mu\)-stable) (with respect to \(H_{| Y})\). The converse of this result also holds. Finally the author studies the relation between parabolic sheaves and representations of \(\pi_ 1(X- D)\), \(D\) being an irreducible and smooth divisor.