Let $X$ be a smooth projective variety over a field $k$ of characteristic zero and let $D$ be a simple normal crossing divisor on $X$, with irreducible components $D_i$.

Does there exist a nonzero global section $\omega \in H^0(X, \Omega^1_X(\log D))$ such that $\mathrm{Res}_{D_i}\omega \neq 0$ for every component $D_i$?

I think the case of curves should follow somehow from Riemann-Roch but I don't see how.

Thanks for your help