Let $X$ be an noetherian, affine, isolated of dimension at least $2$ and $\pi:\widetilde{X} \to X$ a resolution of singularities. Let $\mathcal{E}$ be a locally-free sheaf on $\widetilde{X}$ such that $\pi\_\*\mathcal{E}$ is a free $\mathcal{O}\_X$-module. Is $\mathcal{E}$ necessarily trivial i.e., a direct sum of copies of $\mathcal{O}\_{\widetilde{X}}$?



Pushforward of locally free sheaves and resolution of singularities
No. If $X$ is the cone over a normal rational curve of degree $d > 1$ and $\tilde{X}$ is its blowup at the vertex with the exceptional curve $E$ then $\pi\_\*\mathcal{O}(nE) = \mathcal{O}$ for any $n \ge 0$.


