Reconstructing vector bundles on curves from their direct image on symmetric powers

Abstract

Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector bundles ${\mathcal{F}}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is n$-thsymmetricpowerof$C$.Weprovethefollowing:Let$E_1$and$E_2$betwosemistablevectorbundleson$C$,with${\rm genus}(C)\, \geq\, 2$.If${\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)$forafixed$n$,then$E_1 \,=\, E_2$.