Semistability criterion for parabolic vector bundles on curves (Q2902454)

For \(X\) an irreducible smooth projective curve over an algebraically closed field of characteristic zero, \(D\) a reduced effective divisor on \(X\) and \(r\) a positive integer, denote by \(\text{Vect}(X,D,r)\) the category of parabolic vector bundles on \(X\) with parabolic structure along \(D\) and parabolic weights integral multiples of \(1/r\). In this paper the following theorem is proved:NEWLINENEWLINEA parabolic vector bundle \(\mathcal{E}_\ast \in \text{Vect}(X,D,r)\) is semistable if and only if there is a parabolic vector bundle \(\mathcal{F}_\ast \in \text{Vect}(X,D,r)\) such that \(H^0(X, (\mathcal{E}_\ast\otimes \mathcal{F}_\ast)_0)=0=H^1(X,(\mathcal{E}_\ast\otimes \mathcal{F}_\ast)_0)\).NEWLINENEWLINEThe proof uses the theory of stacks, mainly the Grothendieck-Riemann-Roch Theorem for Deligne-Mumford stacks (cf. [\textit{B. Töen}, K-Theory 18, No. 1, 33--76 (1999; Zbl 0946.14004)]).NEWLINENEWLINEThe theorem generalizes the known semistability criterion of Faltings for vector bundles on curves and improves a former result of the first author (cf. the references given in the paper under review).