Is every $\omega\in H^0(U,\Omega_U^n)$ representable in $H^n(U,\mathbb{C})$ by an element from $H^0(X,\Omega_X^n(\log D))$?

Question

Let $U$ be a smooth variety, and $U\hookrightarrow X$ an smooth compactification with snc boundary $D=X\setminus U$. Suppose that $\omega\in H^0(U,\Omega^n_U)$ is global algebraic $n$-form on $U$. It defines a class in $H^n(U,\mathbb{C})=\mathbb{H}^n(X,\Omega_X^\bullet(\log D))$.

The form $\omega$ extends to a meromorphic form on $X$, denote it by $\tilde{\omega}$. This is not necessarily an element of $H^0(X,\Omega_X^n(\log D))$, since $\tilde{\omega}$ can have poles of higher order. Is there an element $\omega'\in H^0(X,\Omega_X^n(\log D))$ such that $\omega'|_U$ defines the same cohomology class as $\omega$ in $H^n(U,\mathbb{C})$?

Here are my thoughts: the Hodge spectral sequence degenerates, and so we have $$Gr_F^i(H^n(U,\mathbb{C}))=H^{n-i}(X,\Omega^i(\log D)),$$ and so non-canonically (I believe) $$H^n(U,\mathbb{C})=\bigoplus_i H^{n-i}(X,\Omega^i(\log D)).$$ Now it seems that the class defined by $\omega$ should be contributed by the summand $H^{0}(X,\Omega^n(\log D))$, which would imply the claim. However, to prove this I think one would need something analogous to what Peters and Steenbrink call a "Hodge decomposition in the strong sense" (page 45). However, I do not know if this kind of result exists for non-compact $U$?

Answer 1

This is not true. Take for $X$ an elliptic curve, for $D$ a point $p\in X$. The restriction map $H^1(X,\mathbb{C})\rightarrow H^1(U,\mathbb{C})$ is an isomorphism, and $H^0(X,\Omega ^1_X(\log D))=H^0(X,\Omega ^1_X)$. There is a form $\tilde{\omega } $ with a pole of order 2 at $p$; its restriction $\omega $ to $U$ is not cohomologous to the restriction of a holomorphic form on $X$.

Answer 2

This rarely happens. For example, when $U$ is an affine smooth variety, then by Grothendieck's algebraic de Rham theorem (or degeneration of Hodge spectral sequence at $E_2$),

$$H^n(U,\mathbb C)\cong\{\alpha\in H^0(U,\Omega^n)|d\alpha=0\}/\{d\beta|\beta\in H^0(U,\Omega^{n-1})\}.$$

In other words, $H^n(U,\mathbb C)$ are represented by closed algebraic $n$-forms on $U$. So if any closed algebraic $n$-form is cohomologous to some logarithmic form, then it would imply $$H^n(U,\mathbb C)=H^0(U,\Omega_U(\log D))=F^nH^n(U,\mathbb C),$$

which is a pretty strong condition.