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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.

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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. A weaker version was proved in this one.

For examples when this fails see 6.3 of the older paper and 3.B, especially 3.2 of the newer one. For an example when an extension like this fails for symmetric tensors see 3.1.3 of the old paper.

For $1$-forms, as in your question there is a stronger result in this 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 paper mentioned showing that this stronger statement fails for $p>1$.

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  • $\begingroup$ Thanks Kovács, this answer is really helpful. By the way do you think log canonical singularity is the natural setting for extendability of 1-forms? Since I have also seen a result of Flenner (mentioned in your paper) that if codim(Sing X)>2 then the extension of 1-forms also exists. $\endgroup$
    – ormula
    Feb 20, 2015 at 6:19
  • $\begingroup$ For $p=\dim X$, the extendibility is almost the same as being lc. That suggests that lc is a natural class. Having said that I am not claiming that for $1$-forms it cannot hold in more generality. After all our paper with Graf is saying that for $1$-forms there is a stronger extension theorem than for forms with $p>1$. Then again, I would expect it to be relatively easy to find examples that are worse than log-canonical where it fails even for $1$-forms. The difficulty in going beyond the lc case is that I don't see a natural class of singularities where it would hold. $\endgroup$ Feb 20, 2015 at 19:26

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