Let $f:X\rightarrow Y$ be a birational morphism between smooth varieties and $F$ a torsion free sheaf. Is $f\_{\ast} F$ torsion free? If not, are there conditions on either $F$, $f$, $X$ or $Y$ that could ensure the torsion freeness of $F$? For example if $f$ is surj. and $F$ is the canonical bundle on $X$ a theorem of Kollar ensure the torsion freeness of $f\_\*(\omega\_X)$.



Torsion freeness and birational maps
Actually much more is true. In fact there is the following result, whose proof can be found in 


[Grothendieck - Dieudonné: EGA 1 (Elements de Géométrie Algebrique), Proposition 8.4.5 page 351].



> 
> **Proposition.** Let $X$, $Y$ be two integral schemes and $f \colon X \to Y$ be a dominant morphism. Then for any torsion-free $\mathcal{O}\_X$-module $\mathcal{F}$, the push-forward $f\_\* \mathcal{F}$ is a torsion-free $\mathcal{O}\_Y$-module. 
> 
> 
> 


Kollar's result is much deeper since he proves the torsion-freeness of the higher direct images $R^if\_\* \omega\_X$. 


