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 Sn(C), where Sn(C) is the n-th symmetric power of C. We prove the following: Let E1 and E2 be two semistable vector bundles on C, with genus \({(C)\, \geq\, 2}\) . If \({{\mathcal F}_n(E_1)\,\simeq \, {\mathcal F}_n(E_2)}\) for a fixed n, then \({E_1 \,\simeq\, E_2}\) .