Extending holomorphic forms

Question

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.

Answer 1

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 $p>1$.