Sheaf of differential and its reflexive hull on a toric variety Let $X$ be a non-smooth toric variety, $\Omega_X$ be the sheaf of differentials, $\hat{\Omega}_X$ the double dual of $\Omega_X$. My questions are:


*

*Is there any chance that $\Omega_X=\hat{\Omega}_X$?

*If 1 is not true, is there any chance that we can relate the cohomology of the two sheaves?

*If 1 and 2 does not have an answer in general, hwo about weighted projective spaces? 

 A: 1) No, this usually does not happen. Take the singularity $x^2+y^2+z^2=0$. Then 
$$xdx+ ydy+zdz=0$$ and hence one obtains that 
$$\eta:=\frac{xdz-zdx}y=\frac{zdy-ydz}x.$$
Now this implies that $\eta$ is a differential form that is defined on the complement of the singular point and hence it is in the reflexive hull, but it cannot be extended into the singular point. So $\Omega_X$ is not reflexive.
2/3) There is of course the natural map $\Omega_X\to \hat\Omega_X$, so there is a map on cohomology, but
4) Why would you want to know the cohomology of $\Omega_X$? The interesting sheaf here is the reflexive hull of $\Omega_X$, that is, in your notation, $\hat\Omega_X$. As Karl points out that has the nice property that it is isomorphic to $\pi_*\Omega_{\widetilde X}$ if $\pi:\widetilde X \to X$ is a resolution of singularities. It also is the sheaf whose cohomology appears in the Hodge structure of these singularities. (Along with the reflexive hulls of $\Omega_X^p$ which are also isomorphic to $\pi_*\Omega_{\widetilde X}^p$ by the paper Karl quoted above).
