Let $f:X\rightarrow Y$ be a birational map of smooth projective varieties over complex numbers. Let $E$ be a vector bundle on $X$. Will $f\_\*E$ be a reflexive sheaf. Is it possible to impose some additional conditions to ensure that the direct image is a reflexive sheaf?



Direct image of a vector bundle under birational morphism
As requested I put my comment into an answer: take for $f$ the simplest example of birational morphism, the blowing up of a point $p$ in a surface; let $E$ be the exceptional divisor. Then $f\_\*\mathcal{O}(-E)$ is the ideal sheaf of $p$, hence is not reflexive.


