# Direct image of a vector bundle under birational morphism

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?

• you can take double dual $(f_∗E)^{**}$. – user21574 Jun 13 '16 at 9:47
• @Hassan Jolany Sorry I don't understand your comment. Can you elaborate? Even $(f_*E)^*$ will be reflexive right?. But I am interested to know when $f_*E$ itself is reflexive – gradstudent Jun 13 '16 at 9:49
• read this paper of Hartshorne gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002320002 – user21574 Jun 13 '16 at 9:57
• 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 $p$, which is not reflexive. – abx Jun 13 '16 at 10:30
• @ abx - I think your comment is so elegant you really ought to consider making it an answer so that people searching this will see it. – meh Jun 13 '16 at 14:36

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.