Let $X$ be a normal variety over $\mathbb{C}$ and $\pi:\tilde{X}\rightarrow X$ a log resolution with (reduced) exceptional divisor $E$. Let $U$ be the smooth locus of $X$ and $\omega$ a holomorphic 1-form on $U$, when is it possible to extend $\omega$ to a holomorphic form on $\tilde{X}$ with at most logarithmic poles along $E$? In other words, when is $\pi\_\* \Omega^1\_\tilde{X}(\log E)$ reflexive? Also can someone share any examples where such extension fails? Thanks in advance.



Extending holomorphic forms
Here is a result which is sort of what you are asking:



> 
> **Theorem** (Greb-Kebekus-Kovács-Peternell)
>  Let $X$ be a complex quasi-projective variety of dimension $n$
>  and let $D$ be a $\mathbb Q$-divisor on $X$ such that the pair
>  $(X, D)$ is log canonical. Let $\pi:\widetilde X\to X$ be a log resolution
>  with $\pi$-exceptional set $E$ and $\widetilde D :=$ largest
>  reduced divisor contained in $\mathrm{supp}\ \pi^{−1}$(non-klt
>  locus), where the non-klt locus is the smallest closed subset
>  $W\subset X$ such that $(X\setminus W, D\setminus W)$ is klt. Then the
>  sheaves $\pi^\*\Omega\_ X ^p(\log \widetilde D)$ are reflexive, for
>  all $p\leq n$.
> 
> 
> 


This is proved in [this paper](http://link.springer.com/article/10.1007%2Fs10240-011-0036-0). A weaker version was proved in [this](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7089264&fileId=S0010437X09004321) one.


For examples when this fails see 6.3 of the [older](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7089264&fileId=S0010437X09004321) paper and 3.B, especially 3.2 of the [newer](http://link.springer.com/article/10.1007%2Fs10240-011-0036-0) one. For an example when an extension like this fails for symmetric tensors see 3.1.3 of the [old](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7089264&fileId=S0010437X09004321) paper.


For $1$-forms, as in your question there is a stronger result in [this](http://www.math.uiuc.edu/documenta/vol-19/27.html) paper:



> 
> **Theorem** (Graf-Kovács)
>  Let $(X, D)$ be a complex log canonical pair, and let $\pi\!: \widetilde X \to X$ be a log resolution of $(X, D)$. Then the sheaf
>  $
>  \pi\_\* \Omega\_{\widetilde X}^1(\log \widetilde D) 
>  $ 
>  is reflexive, where $\widetilde D$ is any reduced divisor such that
>  $$\mathrm{Exc}(\pi) \wedge \pi^{-1}(\lfloor D\rfloor)
>  \subseteq \mathrm{supp} \widetilde D \subseteq
>  \pi^{-1}(\lfloor D\rfloor). $$
> 
> 
> 


There is also an example in 7.4 of the [last](http://www.math.uiuc.edu/documenta/vol-19/27.html) paper mentioned showing that this stronger statement fails for $p>1$. 


