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

TL;DR

This paper proves that for semistable vector bundles on a smooth projective curve of genus at least 2, the original bundle can be uniquely reconstructed from its associated bundle on the symmetric power of the curve.

Contribution

It establishes a uniqueness result for reconstructing vector bundles on curves from their direct images on symmetric powers, a new insight in algebraic geometry.

Findings

Unique reconstruction of vector bundles from symmetric powers

Equality of associated bundles implies equality of original bundles

Applicable to semistable bundles on curves of genus ≥ 2

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