Let $X$ be a smooth projective complex variety, and $D=\cup\_{j=1}^m D\_j$ a simple normal crossings divisor on $X$. Then we have an exact sequence
$$0\to \Omega\_X^1\to \Omega\_X^1(\log D)\to \oplus\_{j=1}^m i^\*\_j\mathcal{O}\_{D\_j}\to 0,$$
where $i\_j:D\_j\to X$ is the inclusion. Note that $H^0(i\_j^\*\mathcal{O}\_{D\_j})=\mathbb{C}$. So now I'm curious about which global residues $(r\_1,\dots,r\_m)\in \oplus\_{j=1}^m i^\*\_j\mathcal{O}\_{D\_j}$ can come from a global section $\omega\in H^0(\Omega\_X^1(\log D))$, so in other words, I want to know what the image of the map $H^0(\Omega\_X^1(\log D))\to \mathbb{C}^m$ is. 


This seems like a very natural question, but I cannot find any information about this. Hence some pointers to literature would be appreciated.



Admissible global residues on smooth variety with normal crossings divisor
Jason has given an essentially complete answer, which I'm just repeating it here, so that the question can be considered answered. (I think the sequence is OK, however. E.g. for $D=\{xy=0\}$ locally, $Res$ sends $fdx/x+gdy/y\to (f(0,y), g(x,0))$, and this clearly surjects.) The image of $H^0(\Omega\_X^1(\log D)\to \mathbb{C}^m$ is the kernel of the map $\mathbb{C}^m\to H^2(X,\mathbb{C})$ sending $(r\_i)\to \sum r\_i[D\_i]$


**Added** Regarding your comment, let me consider a more general situation $\omega\in H^0(\Omega^1(\log D))$ on a surface $X$. Suppose $D\_1$ and $D\_2$ meet at $p$. Let $\omega$ have residues $r\_i$ along $D\_i$. Let $\pi:Y\to X$ be the blow up at $p$, with exceptional divisor $E$. Then, according to my calculations, $\pi^\*\omega$ has residues $r\_i$ along the strict transforms $D\_i'$ of $D\_i$ and $r\_1+r\_2$ along $E$. So the cohomological condition still holds for $\pi^\*\omega$.


